Subgroups — C₂₃×#18052 #19165678

Order 5888.

674 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2991	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2974	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1594	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1589	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1588	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1586	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1583	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1581	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1569	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1488	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1486	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×AGL(1,ℤ/8)	C₄	1	1	yes	characteristic
17	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
20	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
21	C₂₃×#442	C₄	1	1	yes	characteristic
22	C₂₃×#442	C₄	1	1	yes	characteristic
23	C₂₃×#440	C₄	1	1	yes	characteristic
24	C₂₃×#440	C₄	1	1	yes	characteristic
25	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
26	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#436	—	2	1	no	—
30	C₂₃×#436	—	2	1	no	—
31	C₂₃×#436	C₂²	1	1	yes	characteristic
32	C₂₃×#436	C₂²	1	1	yes	characteristic
33	C₂₃×#436	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×#436	C₂²	1	1	yes	characteristic, complemented
35	C₂₃×#435	—	2	1	no	—
36	C₂₃×#435	—	2	1	no	—
37	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#432	C₂²	1	1	yes	characteristic
40	C₂₃×#431	—	2	1	no	—
41	C₂₃×#431	C₂²	1	1	yes	characteristic
42	C₂₃×#431	C₂²	1	1	yes	characteristic
43	C₂₃×#428	C₄	1	1	yes	characteristic
44	C₂₃×#423	—	2	1	no	—
45	C₂₃×#423	C₂²	1	1	yes	characteristic
46	C₂₃×#418	C₄	1	1	yes	characteristic
47	C₂₃×#417	C₄	1	1	yes	characteristic
48	C₂₃×#407	—	2	1	no	—
49	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
50	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
80	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
81	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₈×C₂₃	—	2	1	no	—
83	C₂²×C₈×C₂₃	—	2	1	no	—
84	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₄×D₄×C₂₃	—	2	1	no	—
87	C₄×D₄×C₂₃	—	2	1	no	—
88	C₄×D₄×C₂₃	—	2	1	no	—
89	C₄×D₄×C₂₃	—	2	1	no	—
90	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
91	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
92	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
93	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂₃×#117	—	2	1	no	—
97	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄²×C₂₃	—	2	1	no	—
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
103	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
104	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
105	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
106	C₂₃×C₄⋊C₈	—	2	1	no	—
107	C₂₃×C₄⋊C₈	—	2	1	no	—
108	C₂₃×C₄⋊C₈	—	2	1	no	—
109	C₂₃×C₄⋊C₈	—	2	1	no	—
110	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
111	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
112	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
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×#103	C₂×C₄	1	1	yes	characteristic
117	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
118	C₂₃×#102	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×C₂²⋊C₈	—	2	1	no	—
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×C₂²⋊C₈	D₄	1	1	yes	characteristic
133	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
134	C₂₃×#97	—	2	1	no	—
135	C₂₃×#97	C₂×C₄	1	1	yes	characteristic
136	C₂₃×#97	C₂×C₄	1	1	yes	characteristic
137	C₄×C₈×C₂₃	—	2	1	no	—
138	C₄×C₈×C₂₃	—	2	1	no	—
139	C₄×C₈×C₂₃	C₂³	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	C₂⁴×C₂₃	—	2	1	no	—
146	#41×C₂₃	—	2	2	no	—
147	#41×C₂₃	—	4	1	no	—
148	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
149	C₂×D₄×C₂₃	—	2	2	no	—
150	C₂×D₄×C₂₃	—	2	1	no	—
151	C₂×D₄×C₂₃	—	2	1	no	—
152	C₂×D₄×C₂₃	—	4	1	no	—
153	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
154	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
155	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	—	2	1	no	—
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	—	8	1	no	—
165	C₂²×C₄×C₂₃	—	8	1	no	—
166	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
169	SD₁₆×C₂₃	—	2	2	no	—
170	SD₁₆×C₂₃	M₁₆	1	4	yes	complemented
171	D₈×C₂₃	—	2	2	no	—
172	D₈×C₂₃	—	2	2	no	—
173	M₁₆×C₂₃	—	2	2	no	—
174	M₁₆×C₂₃	—	4	1	no	—
175	M₁₆×C₂₃	—	8	1	no	—
176	M₁₆×C₂₃	—	8	1	no	—
177	M₁₆×C₂₃	—	8	1	no	—
178	M₁₆×C₂₃	—	8	1	no	—
179	M₁₆×C₂₃	—	8	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	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	—	8	1	no	—
187	C₂×C₈×C₂₃	—	8	1	no	—
188	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
189	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
190	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
191	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
192	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
194	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₄⋊C₄×C₂₃	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	—	2	1	no	—
199	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
200	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
201	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
202	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
203	C₂²⋊C₄×C₂₃	—	2	1	no	—
204	C₂²⋊C₄×C₂₃	—	2	1	no	—
205	C₂²⋊C₄×C₂₃	—	2	1	no	—
206	C₂²⋊C₄×C₂₃	—	2	1	no	—
207	C₂²⋊C₄×C₂₃	—	2	1	no	—
208	C₂²⋊C₄×C₂₃	—	2	1	no	—
209	C₂²⋊C₄×C₂₃	—	2	1	no	—
210	C₄²×C₂₃	—	2	1	no	—
211	C₄²×C₂₃	—	2	1	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
214	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₂₃	#41	1	1	yes	characteristic
216	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
217	#18052	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂³×C₂₃	—	2	1	no	—
219	C₂³×C₂₃	—	2	1	no	—
220	C₂³×C₂₃	—	2	1	no	—
221	C₂³×C₂₃	—	4	2	no	—
222	C₂³×C₂₃	—	4	2	no	—
223	C₂³×C₂₃	—	8	1	no	—
224	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
225	Q₈×C₂₃	—	2	1	no	—
226	Q₈×C₂₃	C₂×M₁₆	1	2	yes	—
227	D₄×C₂₃	—	2	1	no	—
228	D₄×C₂₃	—	2	1	no	—
229	D₄×C₂₃	—	2	1	no	—
230	D₄×C₂₃	—	2	1	no	—
231	D₄×C₂₃	—	2	1	no	—
232	D₄×C₂₃	—	4	2	no	—
233	D₄×C₂₃	—	4	2	no	—
234	D₄×C₂₃	C₂×M₁₆	1	2	yes	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
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	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	2	no	—
246	C₂×C₄×C₂₃	—	4	2	no	—
247	C₂×C₄×C₂₃	—	4	1	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	4	1	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	8	1	no	—
254	C₂×C₄×C₂₃	—	8	1	no	—
255	C₂×C₄×C₂₃	—	8	1	no	—
256	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
257	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
258	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
259	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
260	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
262	C₈×C₂₃	—	2	1	no	—
263	C₈×C₂₃	—	2	1	no	—
264	C₈×C₂₃	—	2	1	no	—
265	C₈×C₂₃	—	2	1	no	—
266	C₈×C₂₃	—	2	1	no	—
267	C₈×C₂₃	—	2	1	no	—
268	C₈×C₂₃	—	2	1	no	—
269	C₈×C₂₃	—	4	1	no	—
270	C₈×C₂₃	—	4	1	no	—
271	C₈×C₂₃	—	4	1	no	—
272	C₈×C₂₃	—	4	1	no	—
273	C₈×C₂₃	—	4	1	no	—
274	C₈×C₂₃	—	4	1	no	—
275	C₈×C₂₃	C₂×M₁₆	1	2	yes	—
276	#2991	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#2974	C₂×C₂₃	1	1	yes	characteristic, complemented
278	C₄×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic
279	#1594	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#1589	C₂×C₂₃	1	1	yes	characteristic, complemented
281	#1588	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#1586	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#1583	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#1581	C₂×C₂₃	1	1	yes	characteristic, complemented
285	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
286	C₈×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#1569	C₂×C₂₃	1	1	yes	characteristic, complemented
288	#1488	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#1486	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#1453	C₂×C₂₃	1	1	yes	characteristic, complemented
291	C₂²×C₂₃	—	2	1	no	—
292	C₂²×C₂₃	—	2	2	no	—
293	C₂²×C₂₃	—	2	1	no	—
294	C₂²×C₂₃	—	2	1	no	—
295	C₂²×C₂₃	—	4	1	no	—
296	C₂²×C₂₃	—	4	1	no	—
297	C₂²×C₂₃	—	4	1	no	—
298	C₂²×C₂₃	—	4	1	no	—
299	C₂²×C₂₃	—	4	1	no	—
300	C₂²×C₂₃	—	4	1	no	—
301	C₂²×C₂₃	—	4	1	no	—
302	C₂²×C₂₃	—	4	1	no	—
303	C₂²×C₂₃	—	8	4	no	—
304	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
305	C₄×C₂₃	—	2	1	no	—
306	C₄×C₂₃	—	2	2	no	—
307	C₄×C₂₃	—	2	1	no	—
308	C₄×C₂₃	—	2	1	no	—
309	C₄×C₂₃	—	2	1	no	—
310	C₄×C₂₃	—	4	1	no	—
311	C₄×C₂₃	—	4	1	no	—
312	C₄×C₂₃	—	4	1	no	—
313	C₄×C₂₃	—	4	1	no	—
314	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
315	C₄×C₂₃	#568	1	1	yes	characteristic
316	C₄×C₂₃	#576	1	1	yes	characteristic
317	C₄×C₂₃	#577	1	1	yes	characteristic
318	C₂×AGL(1,ℤ/8)	C₄×C₂₃	1	1	yes	characteristic
319	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
321	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
322	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
323	#442	C₄×C₂₃	1	1	yes	characteristic
324	#442	C₄×C₂₃	1	1	yes	characteristic
325	#440	C₄×C₂₃	1	1	yes	characteristic
326	#440	C₄×C₂₃	1	1	yes	characteristic
327	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
328	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
329	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
330	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
331	#436	—	2	1	no	—
332	#436	—	2	1	no	—
333	#436	C₂²×C₂₃	1	1	yes	characteristic
334	#436	C₂²×C₂₃	1	1	yes	characteristic
335	#436	C₂²×C₂₃	1	1	yes	characteristic, complemented
336	#436	C₂²×C₂₃	1	1	yes	characteristic, complemented
337	#435	—	2	1	no	—
338	#435	—	2	1	no	—
339	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
340	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
341	#432	C₂²×C₂₃	1	1	yes	characteristic
342	#431	—	2	1	no	—
343	#431	C₂²×C₂₃	1	1	yes	characteristic
344	#431	C₂²×C₂₃	1	1	yes	characteristic
345	#428	C₄×C₂₃	1	1	yes	characteristic
346	#423	—	2	1	no	—
347	#423	C₂²×C₂₃	1	1	yes	characteristic
348	#418	C₄×C₂₃	1	1	yes	characteristic
349	#417	C₄×C₂₃	1	1	yes	characteristic
350	#407	—	2	1	no	—
351	C₂×C₂²⋊C₈	—	2	1	no	—
352	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
353	#335	C₂²×C₂₃	1	1	yes	characteristic, complemented
354	#335	C₂²×C₂₃	1	1	yes	characteristic, complemented
355	#334	C₂²×C₂₃	1	1	yes	characteristic, complemented
356	#334	C₂²×C₂₃	1	1	yes	characteristic, complemented
357	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
358	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
359	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
360	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
361	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
362	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
363	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
364	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
365	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
366	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
367	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
368	C₈²	C₂²×C₂₃	1	1	yes	characteristic, complemented
369	C₂×C₂₃	—	4	1	no	—
370	C₂×C₂₃	—	4	2	no	—
371	C₂×C₂₃	—	8	1	no	—
372	C₂×C₂₃	—	8	1	no	—
373	C₂×C₂₃	D₄×M₁₆	1	1	yes	characteristic
374	C₂×C₂₃	#2946	1	1	yes	characteristic
375	C₂×C₂₃	#2955	1	1	yes	characteristic
376	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
377	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
378	C₂³×C₄	—	2	1	no	—
379	AGL(1,ℤ/8)	—	2	4	no	—
380	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
381	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
382	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
383	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
384	C₂×M₁₆	—	2	1	no	—
385	C₂×M₁₆	—	2	1	no	—
386	C₂×M₁₆	—	2	1	no	—
387	C₂×M₁₆	—	2	1	no	—
388	C₂×M₁₆	—	2	1	no	—
389	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
390	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
391	C₂²×C₈	—	2	1	no	—
392	C₂²×C₈	—	2	1	no	—
393	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
394	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
395	C₄×D₄	—	2	1	no	—
396	C₄×D₄	—	2	1	no	—
397	C₄×D₄	—	2	1	no	—
398	C₄×D₄	—	2	1	no	—
399	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
400	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
401	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
402	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
403	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
404	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
405	#117	—	2	1	no	—
406	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
408	C₂×C₂²⋊C₄	—	2	1	no	—
409	C₂×C₄²	—	2	1	no	—
410	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
411	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
412	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
413	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
414	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
415	C₄⋊C₈	—	2	1	no	—
416	C₄⋊C₈	—	2	1	no	—
417	C₄⋊C₈	—	2	1	no	—
418	C₄⋊C₈	—	2	1	no	—
419	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
420	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
421	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
422	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
423	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
424	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
425	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
426	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
427	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
428	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
429	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
430	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
431	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
432	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
433	C₂²⋊C₈	—	2	1	no	—
434	C₂²⋊C₈	—	2	1	no	—
435	C₂²⋊C₈	—	2	1	no	—
436	C₂²⋊C₈	—	2	1	no	—
437	C₂²⋊C₈	—	2	1	no	—
438	C₂²⋊C₈	—	2	1	no	—
439	C₂²⋊C₈	—	2	1	no	—
440	C₂²⋊C₈	—	2	1	no	—
441	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
442	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
443	#97	—	2	1	no	—
444	#97	C₂×C₄×C₂₃	1	1	yes	characteristic
445	#97	C₂×C₄×C₂₃	1	1	yes	characteristic
446	C₄×C₈	—	2	1	no	—
447	C₄×C₈	—	2	1	no	—
448	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
449	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
450	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
451	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
452	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
453	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
454	C₂₃	#18052	1	1	yes	characteristic, direct product factor, Sylow
455	C₂⁴	—	2	1	no	—
456	#41	—	2	2	no	—
457	#41	—	4	1	no	—
458	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	4	1	no	—
463	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
464	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
465	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	8	1	no	—
475	C₂²×C₄	—	8	1	no	—
476	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
479	SD₁₆	—	2	2	no	—
480	SD₁₆	M₁₆×C₂₃	1	4	yes	complemented
481	D₈	—	2	2	no	—
482	D₈	—	2	2	no	—
483	M₁₆	—	2	2	no	—
484	M₁₆	—	4	1	no	—
485	M₁₆	—	8	1	no	—
486	M₁₆	—	8	1	no	—
487	M₁₆	—	8	1	no	—
488	M₁₆	—	8	1	no	—
489	M₁₆	—	8	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	8	1	no	—
497	C₂×C₈	—	8	1	no	—
498	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
499	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
500	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
501	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
502	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
503	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
504	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
510	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
511	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
512	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
513	C₂²⋊C₄	—	2	1	no	—
514	C₂²⋊C₄	—	2	1	no	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	2	1	no	—
518	C₂²⋊C₄	—	2	1	no	—
519	C₂²⋊C₄	—	2	1	no	—
520	C₄²	—	2	1	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	2	1	no	—
523	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
524	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
525	C₄²	#41×C₂₃	1	1	yes	characteristic
526	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
527	C₂³	—	2	1	no	—
528	C₂³	—	2	1	no	—
529	C₂³	—	2	1	no	—
530	C₂³	—	4	2	no	—
531	C₂³	—	4	2	no	—
532	C₂³	—	8	1	no	—
533	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
534	Q₈	—	2	1	no	—
535	Q₈	C₂×M₁₆×C₂₃	1	2	yes	—
536	D₄	—	2	1	no	—
537	D₄	—	2	1	no	—
538	D₄	—	2	1	no	—
539	D₄	—	2	1	no	—
540	D₄	—	2	1	no	—
541	D₄	—	4	2	no	—
542	D₄	—	4	2	no	—
543	D₄	C₂×M₁₆×C₂₃	1	2	yes	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	8	1	no	—
563	C₂×C₄	—	8	1	no	—
564	C₂×C₄	—	8	1	no	—
565	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
566	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
567	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
568	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
569	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
570	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
571	C₈	—	2	1	no	—
572	C₈	—	2	1	no	—
573	C₈	—	2	1	no	—
574	C₈	—	2	1	no	—
575	C₈	—	2	1	no	—
576	C₈	—	2	1	no	—
577	C₈	—	2	1	no	—
578	C₈	—	4	1	no	—
579	C₈	—	4	1	no	—
580	C₈	—	4	1	no	—
581	C₈	—	4	1	no	—
582	C₈	—	4	1	no	—
583	C₈	—	4	1	no	—
584	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	1	no	—
588	C₂²	—	2	1	no	—
589	C₂²	—	4	1	no	—
590	C₂²	—	4	1	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	1	no	—
593	C₂²	—	4	1	no	—
594	C₂²	—	4	1	no	—
595	C₂²	—	4	1	no	—
596	C₂²	—	4	1	no	—
597	C₂²	—	8	4	no	—
598	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
599	C₄	—	2	1	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	1	no	—
602	C₄	—	2	1	no	—
603	C₄	—	2	1	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	1	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	1	no	—
608	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
609	C₄	C₂₃×#568	1	1	yes	characteristic
610	C₄	C₂₃×#576	1	1	yes	characteristic
611	C₄	C₂₃×#577	1	1	yes	characteristic
612	C₂	—	4	1	no	—
613	C₂	—	4	2	no	—
614	C₂	—	8	1	no	—
615	C₂	—	8	1	no	—
616	C₂	D₄×M₁₆×C₂₃	1	1	yes	characteristic
617	C₂	C₂₃×#2946	1	1	yes	characteristic
618	C₂	C₂₃×#2955	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 ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2991	C₂	semidirect
2. C₂₃×#2974	C₂	semidirect
4. C₂₃×#1594	C₂	semidirect
5. C₂₃×#1589	C₂	semidirect
6. C₂₃×#1588	C₂	semidirect
7. C₂₃×#1586	C₂	semidirect
8. C₂₃×#1583	C₂	semidirect
9. C₂₃×#1581	C₂	semidirect
10. C₂₃×SD₁₆⋊C₈	C₂	semidirect
11. C₈×SD₁₆×C₂₃	C₂	semidirect
12. C₂₃×#1569	C₂	semidirect
13. C₂₃×#1488	C₂	semidirect
14. C₂₃×#1486	C₂	semidirect
15. C₂₃×#1453	C₂	semidirect
19. C₈×Q₈×C₂₃	C₂²	semidirect
20. C₈×Q₈×C₂₃	C₂²	semidirect
33. C₂₃×#436	C₂²	semidirect
34. C₂₃×#436	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
170. SD₁₆×C₂₃	M₁₆	semidirect
217. #18052	C₂₃	direct (N × H)
276. #2991	C₂×C₂₃	semidirect
277. #2974	C₂×C₂₃	semidirect
279. #1594	C₂×C₂₃	semidirect
280. #1589	C₂×C₂₃	semidirect
281. #1588	C₂×C₂₃	semidirect
282. #1586	C₂×C₂₃	semidirect
283. #1583	C₂×C₂₃	semidirect
284. #1581	C₂×C₂₃	semidirect
285. SD₁₆⋊C₈	C₂×C₂₃	semidirect
286. C₈×SD₁₆	C₂×C₂₃	semidirect
287. #1569	C₂×C₂₃	semidirect
288. #1488	C₂×C₂₃	semidirect
289. #1486	C₂×C₂₃	semidirect
290. #1453	C₂×C₂₃	semidirect
321. C₈×Q₈	C₂²×C₂₃	semidirect
322. C₈×Q₈	C₂²×C₂₃	semidirect
335. #436	C₂²×C₂₃	semidirect
336. #436	C₂²×C₂₃	semidirect
353. #335	C₂²×C₂₃	semidirect
354. #335	C₂²×C₂₃	semidirect
355. #334	C₂²×C₂₃	semidirect
356. #334	C₂²×C₂₃	semidirect
357. Q₈⋊C₈	C₂²×C₂₃	semidirect
358. Q₈⋊C₈	C₂²×C₂₃	semidirect
363. D₄⋊C₈	C₂²×C₂₃	semidirect
364. D₄⋊C₈	C₂²×C₂₃	semidirect
365. #322	C₂²×C₂₃	semidirect
366. #322	C₂²×C₂₃	semidirect
367. #322	C₂²×C₂₃	semidirect
368. C₈²	C₂²×C₂₃	semidirect
454. C₂₃	#18052	direct (N × H)
480. SD₁₆	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2991	2
C₂₃×#2974	2
C₄×C₂₃×AGL(1,ℤ/8)	2
C₂₃×#1594	2
C₂₃×#1589	2
C₂₃×#1588	2
C₂₃×#1586	2
C₂₃×#1583	2
C₂₃×#1581	2
C₂₃×SD₁₆⋊C₈	2
C₈×SD₁₆×C₂₃	2
C₂₃×#1569	2
C₂₃×#1488	2
C₂₃×#1486	2
C₂₃×#1453	2
#18052	23

Subgroups — C23×#18052 #19165678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18052 #19165678