Subgroups — C₂₃×#18046 #19165672

Order 5888.

720 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2970	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
3	D₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1590	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1586	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1582	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1581	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1580	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1569	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1490	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1489	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×AGL(1,ℤ/8)	C₄	1	1	yes	characteristic
17	C₂²×M₁₆×C₂₃	—	2	1	no	—
18	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#446	C₂²	1	1	yes	characteristic, complemented
21	C₂₃×#446	C₂²	1	1	yes	characteristic, complemented
22	C₂₃×#442	C₄	1	1	yes	characteristic
23	C₂₃×#442	C₄	1	1	yes	characteristic
24	C₂₃×#440	C₄	1	1	yes	characteristic
25	C₂₃×#440	C₄	1	1	yes	characteristic
26	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#436	C₂²	1	1	yes	characteristic
31	C₂₃×#436	C₂²	1	1	yes	characteristic
32	C₂₃×#435	—	2	1	no	—
33	C₂₃×#435	—	2	1	no	—
34	C₂₃×#435	—	2	1	no	—
35	C₂₃×#435	—	2	1	no	—
36	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
37	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
39	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
40	C₂₃×#432	—	2	1	no	—
41	C₂₃×#432	C₂²	1	1	yes	characteristic
42	C₂₃×#428	C₄	1	1	yes	characteristic
43	C₂₃×#423	C₂²	1	1	yes	characteristic
44	C₂₃×#418	C₄	1	1	yes	characteristic
45	C₂₃×#417	C₄	1	1	yes	characteristic
46	C₂₃×#407	—	2	1	no	—
47	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
49	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×#97	—	2	1	no	—
51	C₂₃×#335	C₂²	1	1	yes	characteristic, complemented
52	C₂₃×#335	C₂²	1	1	yes	characteristic, complemented
53	C₂₃×#334	C₂²	1	1	yes	characteristic, complemented
54	C₂₃×#334	C₂²	1	1	yes	characteristic, complemented
55	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
56	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
59	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
60	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
61	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
62	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₂₃×#322	C₂²	1	1	yes	characteristic, complemented
64	C₂₃×#322	C₂²	1	1	yes	characteristic, complemented
65	C₂₃×#322	C₂²	1	1	yes	characteristic, complemented
66	C₈²×C₂₃	C₂²	1	1	yes	characteristic, complemented
67	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
69	C₂³×C₄×C₂₃	—	2	1	no	—
70	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
71	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
72	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
73	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
74	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
75	C₂×M₁₆×C₂₃	—	2	1	no	—
76	C₂×M₁₆×C₂₃	—	2	1	no	—
77	C₂×M₁₆×C₂₃	—	2	1	no	—
78	C₂×M₁₆×C₂₃	—	2	1	no	—
79	C₂×M₁₆×C₂₃	—	4	4	no	—
80	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
81	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
82	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
83	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
84	C₂²×C₈×C₂₃	—	2	1	no	—
85	C₂²×C₈×C₂₃	—	2	1	no	—
86	C₂²×C₈×C₂₃	—	2	1	no	—
87	C₂²×C₈×C₂₃	—	2	1	no	—
88	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
89	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
90	C₄×D₄×C₂₃	—	2	1	no	—
91	C₄×D₄×C₂₃	—	2	1	no	—
92	C₄×D₄×C₂₃	—	2	1	no	—
93	C₄×D₄×C₂₃	—	2	1	no	—
94	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
95	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
96	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
97	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂₃×#117	—	2	1	no	—
101	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄²×C₂₃	—	2	1	no	—
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
107	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
108	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
109	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
110	C₂₃×C₄⋊C₈	—	2	1	no	—
111	C₂₃×C₄⋊C₈	—	2	1	no	—
112	C₂₃×C₄⋊C₈	—	2	1	no	—
113	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
118	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
119	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
120	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
121	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
122	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
123	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
124	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
125	C₂₃×C₂²⋊C₈	—	2	1	no	—
126	C₂₃×C₂²⋊C₈	—	2	1	no	—
127	C₂₃×C₂²⋊C₈	—	2	1	no	—
128	C₂₃×C₂²⋊C₈	—	2	1	no	—
129	C₂₃×C₂²⋊C₈	—	2	1	no	—
130	C₂₃×C₂²⋊C₈	—	2	1	no	—
131	C₂₃×C₂²⋊C₈	—	2	1	no	—
132	C₂₃×#97	—	2	1	no	—
133	C₂₃×#97	—	2	1	no	—
134	C₂₃×#97	—	2	1	no	—
135	C₂₃×#97	—	2	1	no	—
136	C₂₃×#97	C₂×C₄	1	1	yes	characteristic
137	C₂₃×#97	C₂×C₄	1	1	yes	characteristic
138	C₂₃×#97	D₄	1	1	yes	characteristic
139	C₂₃×#97	D₄	1	1	yes	characteristic
140	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
146	C₂⁴×C₂₃	—	2	1	no	—
147	#41×C₂₃	—	2	2	no	—
148	#41×C₂₃	—	4	1	no	—
149	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
150	C₂×D₄×C₂₃	—	2	2	no	—
151	C₂×D₄×C₂₃	—	2	1	no	—
152	C₂×D₄×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	—	4	1	no	—
154	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
155	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
156	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	4	2	no	—
166	C₂²×C₄×C₂₃	—	4	2	no	—
167	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
170	SD₁₆×C₂₃	—	2	2	no	—
171	SD₁₆×C₂₃	—	2	2	no	—
172	D₈×C₂₃	—	2	2	no	—
173	D₈×C₂₃	M₁₆	1	4	yes	complemented
174	M₁₆×C₂₃	—	2	2	no	—
175	M₁₆×C₂₃	—	2	2	no	—
176	M₁₆×C₂₃	—	4	1	no	—
177	M₁₆×C₂₃	—	4	1	no	—
178	M₁₆×C₂₃	—	4	2	no	—
179	M₁₆×C₂₃	—	4	2	no	—
180	M₁₆×C₂₃	—	4	2	no	—
181	M₁₆×C₂₃	—	4	2	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	—	2	1	no	—
184	C₂×C₈×C₂₃	—	2	1	no	—
185	C₂×C₈×C₂₃	—	2	1	no	—
186	C₂×C₈×C₂₃	—	2	1	no	—
187	C₂×C₈×C₂₃	—	2	1	no	—
188	C₂×C₈×C₂₃	—	4	2	no	—
189	C₂×C₈×C₂₃	—	4	2	no	—
190	C₂×C₈×C₂₃	—	4	2	no	—
191	C₂×C₈×C₂₃	—	4	2	no	—
192	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
193	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
194	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
195	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
196	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₂₃	—	2	1	no	—
201	C₄⋊C₄×C₂₃	—	2	1	no	—
202	C₄⋊C₄×C₂₃	—	2	1	no	—
203	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
204	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
205	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
206	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
207	C₂²⋊C₄×C₂₃	—	2	1	no	—
208	C₂²⋊C₄×C₂₃	—	2	1	no	—
209	C₂²⋊C₄×C₂₃	—	2	1	no	—
210	C₂²⋊C₄×C₂₃	—	2	1	no	—
211	C₂²⋊C₄×C₂₃	—	2	1	no	—
212	C₂²⋊C₄×C₂₃	—	2	1	no	—
213	C₂²⋊C₄×C₂₃	—	2	1	no	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	2	1	no	—
216	C₄²×C₂₃	—	2	1	no	—
217	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
218	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₂₃	#41	1	1	yes	characteristic
220	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
221	#18046	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂³×C₂₃	—	2	1	no	—
223	C₂³×C₂₃	—	2	1	no	—
224	C₂³×C₂₃	—	2	1	no	—
225	C₂³×C₂₃	—	4	2	no	—
226	C₂³×C₂₃	—	4	2	no	—
227	C₂³×C₂₃	—	8	1	no	—
228	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
229	Q₈×C₂₃	—	2	1	no	—
230	Q₈×C₂₃	—	2	1	no	—
231	D₄×C₂₃	—	2	1	no	—
232	D₄×C₂₃	—	2	1	no	—
233	D₄×C₂₃	—	2	1	no	—
234	D₄×C₂₃	—	2	1	no	—
235	D₄×C₂₃	—	4	2	no	—
236	D₄×C₂₃	—	4	2	no	—
237	D₄×C₂₃	C₂×M₁₆	1	2	yes	—
238	D₄×C₂₃	C₂×M₁₆	1	2	yes	complemented
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	1	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	2	no	—
250	C₂×C₄×C₂₃	—	4	2	no	—
251	C₂×C₄×C₂₃	—	4	2	no	—
252	C₂×C₄×C₂₃	—	4	2	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	1	no	—
259	C₂×C₄×C₂₃	—	8	1	no	—
260	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
261	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
262	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
263	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
264	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
266	C₈×C₂₃	—	2	2	no	—
267	C₈×C₂₃	—	2	1	no	—
268	C₈×C₂₃	—	2	1	no	—
269	C₈×C₂₃	—	2	1	no	—
270	C₈×C₂₃	—	2	1	no	—
271	C₈×C₂₃	—	2	1	no	—
272	C₈×C₂₃	—	2	1	no	—
273	C₈×C₂₃	—	4	1	no	—
274	C₈×C₂₃	—	4	1	no	—
275	C₈×C₂₃	—	4	1	no	—
276	C₈×C₂₃	—	4	1	no	—
277	C₈×C₂₃	—	4	1	no	—
278	C₈×C₂₃	C₂×M₁₆	1	2	yes	complemented
279	C₈×C₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
280	#2970	C₂×C₂₃	1	1	yes	characteristic, complemented
281	C₄×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic
282	D₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#1592	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#1590	C₂×C₂₃	1	1	yes	characteristic, complemented
285	#1586	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#1585	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#1582	C₂×C₂₃	1	1	yes	characteristic, complemented
288	#1581	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#1580	C₂×C₂₃	1	1	yes	characteristic, complemented
290	C₈×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#1569	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#1490	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#1489	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#1455	C₂×C₂₃	1	1	yes	characteristic, complemented
295	C₂²×C₂₃	—	2	1	no	—
296	C₂²×C₂₃	—	2	2	no	—
297	C₂²×C₂₃	—	2	2	no	—
298	C₂²×C₂₃	—	2	1	no	—
299	C₂²×C₂₃	—	2	1	no	—
300	C₂²×C₂₃	—	4	1	no	—
301	C₂²×C₂₃	—	4	1	no	—
302	C₂²×C₂₃	—	4	1	no	—
303	C₂²×C₂₃	—	4	1	no	—
304	C₂²×C₂₃	—	4	1	no	—
305	C₂²×C₂₃	—	4	1	no	—
306	C₂²×C₂₃	—	4	1	no	—
307	C₂²×C₂₃	—	8	4	no	—
308	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
309	C₄×C₂₃	—	2	1	no	—
310	C₄×C₂₃	—	2	1	no	—
311	C₄×C₂₃	—	2	1	no	—
312	C₄×C₂₃	—	2	1	no	—
313	C₄×C₂₃	—	4	1	no	—
314	C₄×C₂₃	—	4	1	no	—
315	C₄×C₂₃	—	4	1	no	—
316	C₄×C₂₃	—	4	1	no	—
317	C₄×C₂₃	—	4	1	no	—
318	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
319	C₄×C₂₃	#568	1	1	yes	characteristic
320	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₄×C₂₃	#575	1	1	yes	characteristic
322	C₂×AGL(1,ℤ/8)	C₄×C₂₃	1	1	yes	characteristic
323	C₂²×M₁₆	—	2	1	no	—
324	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
325	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
326	#446	C₂²×C₂₃	1	1	yes	characteristic, complemented
327	#446	C₂²×C₂₃	1	1	yes	characteristic, complemented
328	#442	C₄×C₂₃	1	1	yes	characteristic
329	#442	C₄×C₂₃	1	1	yes	characteristic
330	#440	C₄×C₂₃	1	1	yes	characteristic
331	#440	C₄×C₂₃	1	1	yes	characteristic
332	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
333	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
334	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
335	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
336	#436	C₂²×C₂₃	1	1	yes	characteristic
337	#436	C₂²×C₂₃	1	1	yes	characteristic
338	#435	—	2	1	no	—
339	#435	—	2	1	no	—
340	#435	—	2	1	no	—
341	#435	—	2	1	no	—
342	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
343	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
344	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
345	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
346	#432	—	2	1	no	—
347	#432	C₂²×C₂₃	1	1	yes	characteristic
348	#428	C₄×C₂₃	1	1	yes	characteristic
349	#423	C₂²×C₂₃	1	1	yes	characteristic
350	#418	C₄×C₂₃	1	1	yes	characteristic
351	#417	C₄×C₂₃	1	1	yes	characteristic
352	#407	—	2	1	no	—
353	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
354	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
355	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×#97	—	2	1	no	—
357	#335	C₂²×C₂₃	1	1	yes	characteristic, complemented
358	#335	C₂²×C₂₃	1	1	yes	characteristic, complemented
359	#334	C₂²×C₂₃	1	1	yes	characteristic, complemented
360	#334	C₂²×C₂₃	1	1	yes	characteristic, complemented
361	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
362	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
363	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
364	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
365	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
366	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
367	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
368	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
369	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
370	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
371	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
372	C₈²	C₂²×C₂₃	1	1	yes	characteristic, complemented
373	C₂×C₂₃	—	4	1	no	—
374	C₂×C₂₃	—	4	2	no	—
375	C₂×C₂₃	—	4	2	no	—
376	C₂×C₂₃	—	8	1	no	—
377	C₂×C₂₃	D₄×M₁₆	1	1	yes	characteristic
378	C₂×C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
379	C₂×C₂₃	#2956	1	1	yes	characteristic
380	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
381	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
382	C₂³×C₄	—	2	1	no	—
383	AGL(1,ℤ/8)	—	2	4	no	—
384	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
385	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
386	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
387	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
388	C₂×M₁₆	—	2	1	no	—
389	C₂×M₁₆	—	2	1	no	—
390	C₂×M₁₆	—	2	1	no	—
391	C₂×M₁₆	—	2	1	no	—
392	C₂×M₁₆	—	4	4	no	—
393	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
394	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
395	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
396	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
397	C₂²×C₈	—	2	1	no	—
398	C₂²×C₈	—	2	1	no	—
399	C₂²×C₈	—	2	1	no	—
400	C₂²×C₈	—	2	1	no	—
401	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
402	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
408	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
409	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
410	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
411	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
412	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
413	#117	—	2	1	no	—
414	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
416	C₂×C₂²⋊C₄	—	2	1	no	—
417	C₂×C₄²	—	2	1	no	—
418	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
419	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
420	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
421	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
422	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
423	C₄⋊C₈	—	2	1	no	—
424	C₄⋊C₈	—	2	1	no	—
425	C₄⋊C₈	—	2	1	no	—
426	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
427	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
428	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
429	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
430	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
431	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
432	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
433	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
434	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
435	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
436	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
437	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
438	C₂²⋊C₈	—	2	1	no	—
439	C₂²⋊C₈	—	2	1	no	—
440	C₂²⋊C₈	—	2	1	no	—
441	C₂²⋊C₈	—	2	1	no	—
442	C₂²⋊C₈	—	2	1	no	—
443	C₂²⋊C₈	—	2	1	no	—
444	C₂²⋊C₈	—	2	1	no	—
445	#97	—	2	1	no	—
446	#97	—	2	1	no	—
447	#97	—	2	1	no	—
448	#97	—	2	1	no	—
449	#97	C₂×C₄×C₂₃	1	1	yes	characteristic
450	#97	C₂×C₄×C₂₃	1	1	yes	characteristic
451	#97	D₄×C₂₃	1	1	yes	characteristic
452	#97	D₄×C₂₃	1	1	yes	characteristic
453	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
454	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
455	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
456	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
457	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
458	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
459	C₂₃	#18046	1	1	yes	characteristic, direct product factor, Sylow
460	C₂⁴	—	2	1	no	—
461	#41	—	2	2	no	—
462	#41	—	4	1	no	—
463	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	4	1	no	—
468	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
469	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
470	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	4	2	no	—
480	C₂²×C₄	—	4	2	no	—
481	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
483	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
484	SD₁₆	—	2	2	no	—
485	SD₁₆	—	2	2	no	—
486	D₈	—	2	2	no	—
487	D₈	M₁₆×C₂₃	1	4	yes	complemented
488	M₁₆	—	2	2	no	—
489	M₁₆	—	2	2	no	—
490	M₁₆	—	4	1	no	—
491	M₁₆	—	4	1	no	—
492	M₁₆	—	4	2	no	—
493	M₁₆	—	4	2	no	—
494	M₁₆	—	4	2	no	—
495	M₁₆	—	4	2	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	4	2	no	—
503	C₂×C₈	—	4	2	no	—
504	C₂×C₈	—	4	2	no	—
505	C₂×C₈	—	4	2	no	—
506	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
507	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
508	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
509	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
510	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
511	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
512	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
513	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
518	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
519	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
520	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
521	C₂²⋊C₄	—	2	1	no	—
522	C₂²⋊C₄	—	2	1	no	—
523	C₂²⋊C₄	—	2	1	no	—
524	C₂²⋊C₄	—	2	1	no	—
525	C₂²⋊C₄	—	2	1	no	—
526	C₂²⋊C₄	—	2	1	no	—
527	C₂²⋊C₄	—	2	1	no	—
528	C₄²	—	2	1	no	—
529	C₄²	—	2	1	no	—
530	C₄²	—	2	1	no	—
531	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
532	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
533	C₄²	#41×C₂₃	1	1	yes	characteristic
534	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
535	C₂³	—	2	1	no	—
536	C₂³	—	2	1	no	—
537	C₂³	—	2	1	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	4	2	no	—
540	C₂³	—	8	1	no	—
541	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
542	Q₈	—	2	1	no	—
543	Q₈	—	2	1	no	—
544	D₄	—	2	1	no	—
545	D₄	—	2	1	no	—
546	D₄	—	2	1	no	—
547	D₄	—	2	1	no	—
548	D₄	—	4	2	no	—
549	D₄	—	4	2	no	—
550	D₄	C₂×M₁₆×C₂₃	1	2	yes	—
551	D₄	C₂×M₁₆×C₂₃	1	2	yes	complemented
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	8	1	no	—
573	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
574	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
575	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
576	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
577	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
578	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
579	C₈	—	2	2	no	—
580	C₈	—	2	1	no	—
581	C₈	—	2	1	no	—
582	C₈	—	2	1	no	—
583	C₈	—	2	1	no	—
584	C₈	—	2	1	no	—
585	C₈	—	2	1	no	—
586	C₈	—	4	1	no	—
587	C₈	—	4	1	no	—
588	C₈	—	4	1	no	—
589	C₈	—	4	1	no	—
590	C₈	—	4	1	no	—
591	C₈	C₂×M₁₆×C₂₃	1	2	yes	complemented
592	C₈	C₂₃×AGL(1,ℤ/8)	1	2	yes	complemented
593	C₂²	—	2	1	no	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	4	1	no	—
599	C₂²	—	4	1	no	—
600	C₂²	—	4	1	no	—
601	C₂²	—	4	1	no	—
602	C₂²	—	4	1	no	—
603	C₂²	—	4	1	no	—
604	C₂²	—	4	1	no	—
605	C₂²	—	8	4	no	—
606	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	1	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	1	no	—
613	C₄	—	4	1	no	—
614	C₄	—	4	1	no	—
615	C₄	—	4	1	no	—
616	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
617	C₄	C₂₃×#568	1	1	yes	characteristic
618	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
619	C₄	C₂₃×#575	1	1	yes	characteristic
620	C₂	—	4	1	no	—
621	C₂	—	4	2	no	—
622	C₂	—	4	2	no	—
623	C₂	—	8	1	no	—
624	C₂	D₄×M₁₆×C₂₃	1	1	yes	characteristic
625	C₂	C₄×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
626	C₂	C₂₃×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂×C₂₃×AGL(1,ℤ/8)	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×M₁₆×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#2970	C₂	semidirect
3. D₄×M₁₆×C₂₃	C₂	semidirect
4. C₂₃×#1592	C₂	semidirect
5. C₂₃×#1590	C₂	semidirect
6. C₂₃×#1586	C₂	semidirect
7. C₂₃×#1585	C₂	semidirect
8. C₂₃×#1582	C₂	semidirect
9. C₂₃×#1581	C₂	semidirect
10. C₂₃×#1580	C₂	semidirect
11. C₈×D₈×C₂₃	C₂	semidirect
12. C₂₃×#1569	C₂	semidirect
13. C₂₃×#1490	C₂	semidirect
14. C₂₃×#1489	C₂	semidirect
15. C₂₃×#1455	C₂	semidirect
20. C₂₃×#446	C₂²	semidirect
21. C₂₃×#446	C₂²	semidirect
38. C₈×D₄×C₂₃	C₂²	semidirect
39. C₈×D₄×C₂₃	C₂²	semidirect
51. C₂₃×#335	C₂²	semidirect
52. C₂₃×#335	C₂²	semidirect
53. C₂₃×#334	C₂²	semidirect
54. C₂₃×#334	C₂²	semidirect
55. C₂₃×Q₈⋊C₈	C₂²	semidirect
56. C₂₃×Q₈⋊C₈	C₂²	semidirect
61. C₂₃×D₄⋊C₈	C₂²	semidirect
62. C₂₃×D₄⋊C₈	C₂²	semidirect
63. C₂₃×#322	C₂²	semidirect
64. C₂₃×#322	C₂²	semidirect
65. C₂₃×#322	C₂²	semidirect
66. C₈²×C₂₃	C₂²	semidirect
173. D₈×C₂₃	M₁₆	semidirect
221. #18046	C₂₃	direct (N × H)
238. D₄×C₂₃	C₂×M₁₆	semidirect
278. C₈×C₂₃	C₂×M₁₆	semidirect
279. C₈×C₂₃	AGL(1,ℤ/8)	semidirect
280. #2970	C₂×C₂₃	semidirect
282. D₄×M₁₆	C₂×C₂₃	semidirect
283. #1592	C₂×C₂₃	semidirect
284. #1590	C₂×C₂₃	semidirect
285. #1586	C₂×C₂₃	semidirect
286. #1585	C₂×C₂₃	semidirect
287. #1582	C₂×C₂₃	semidirect
288. #1581	C₂×C₂₃	semidirect
289. #1580	C₂×C₂₃	semidirect
290. C₈×D₈	C₂×C₂₃	semidirect
291. #1569	C₂×C₂₃	semidirect
292. #1490	C₂×C₂₃	semidirect
293. #1489	C₂×C₂₃	semidirect
294. #1455	C₂×C₂₃	semidirect
326. #446	C₂²×C₂₃	semidirect
327. #446	C₂²×C₂₃	semidirect
344. C₈×D₄	C₂²×C₂₃	semidirect
345. C₈×D₄	C₂²×C₂₃	semidirect
357. #335	C₂²×C₂₃	semidirect
358. #335	C₂²×C₂₃	semidirect
359. #334	C₂²×C₂₃	semidirect
360. #334	C₂²×C₂₃	semidirect
361. Q₈⋊C₈	C₂²×C₂₃	semidirect
362. Q₈⋊C₈	C₂²×C₂₃	semidirect
367. D₄⋊C₈	C₂²×C₂₃	semidirect
368. D₄⋊C₈	C₂²×C₂₃	semidirect
369. #322	C₂²×C₂₃	semidirect
370. #322	C₂²×C₂₃	semidirect
371. #322	C₂²×C₂₃	semidirect
372. C₈²	C₂²×C₂₃	semidirect
459. C₂₃	#18046	direct (N × H)
487. D₈	M₁₆×C₂₃	semidirect
551. D₄	C₂×M₁₆×C₂₃	semidirect
591. C₈	C₂×M₁₆×C₂₃	semidirect
592. C₈	C₂₃×AGL(1,ℤ/8)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#2970	2
C₄×C₂₃×AGL(1,ℤ/8)	2
D₄×M₁₆×C₂₃	2
C₂₃×#1592	2
C₂₃×#1590	2
C₂₃×#1586	2
C₂₃×#1585	2
C₂₃×#1582	2
C₂₃×#1581	2
C₂₃×#1580	2
C₈×D₈×C₂₃	2
C₂₃×#1569	2
C₂₃×#1490	2
C₂₃×#1489	2
C₂₃×#1455	2
#18046	23

Subgroups — C23×#18046 #19165672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18046 #19165672