Subgroups — C₂₃×#17891 #19165517

Order 5888.

724 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2965	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1594	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1592	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1590	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1589	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1477	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×AGL(1,ℤ/8)	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₂₃	—	2	1	no	—
14	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#446	C₂²	1	1	yes	characteristic, complemented
17	C₂₃×#446	C₂²	1	1	yes	characteristic, complemented
18	C₂₃×#442	C₄	1	2	yes	—
19	C₂₃×#440	C₄	1	2	yes	—
20	C₄×SD₁₆×C₂₃	C₂²	1	2	yes	—
21	C₄×D₈×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#436	—	2	1	no	—
23	C₂₃×#436	—	2	1	no	—
24	C₂₃×#436	C₂²	1	2	yes	—
25	C₂₃×#436	C₂²	1	2	yes	—
26	C₂₃×#436	C₂²	1	1	yes	characteristic, complemented
27	C₂₃×#436	C₂²	1	1	yes	characteristic, complemented
28	C₂₃×#435	—	2	1	no	—
29	C₂₃×#435	—	2	1	no	—
30	C₂₃×#431	—	2	1	no	—
31	C₂₃×#431	C₂²	1	1	yes	characteristic
32	C₂₃×#431	C₂²	1	1	yes	characteristic
33	C₂₃×#428	C₄	1	1	yes	characteristic
34	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂₃×#418	C₄	1	1	yes	characteristic
36	C₂₃×#417	C₄	1	1	yes	characteristic
37	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
38	C₄×M₁₆×C₂₃	—	2	1	no	—
39	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#335	C₂²	1	2	yes	complemented
42	C₂₃×#334	C₂²	1	2	yes	complemented
43	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
44	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
45	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
46	C₂₃×D₄⋊C₈	C₂²	1	2	yes	complemented
47	C₂₃×#322	C₂²	1	2	yes	complemented
48	C₈²×C₂₃	C₂²	1	2	yes	complemented
49	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₂₃	—	2	1	no	—
52	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
53	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
54	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
55	C₂×M₁₆×C₂₃	—	2	2	no	—
56	C₂×M₁₆×C₂₃	—	2	2	no	—
57	C₂×M₁₆×C₂₃	—	2	1	no	—
58	C₂×M₁₆×C₂₃	—	2	1	no	—
59	C₂×M₁₆×C₂₃	—	4	4	no	—
60	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₂₃	—	2	1	no	—
63	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
65	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₂₃	—	2	1	no	—
68	C₄×D₄×C₂₃	—	2	1	no	—
69	C₄×D₄×C₂₃	—	2	1	no	—
70	C₄×D₄×C₂₃	—	2	1	no	—
71	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
72	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
73	C₄×D₄×C₂₃	C₂³	1	2	yes	—
74	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₂₃×#117	—	2	1	no	—
76	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄²×C₂₃	—	2	1	no	—
80	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₂₃×#107	C₂×C₄	1	2	yes	—
82	C₂₃×#106	C₂×C₄	1	2	yes	—
83	C₂₃×C₄⋊C₈	—	2	1	no	—
84	C₂₃×C₄⋊C₈	—	2	1	no	—
85	C₂₃×C₄⋊C₈	—	2	1	no	—
86	C₂₃×C₄⋊C₈	—	2	1	no	—
87	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
89	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
90	C₂₃×#103	C₂×C₄	1	2	yes	—
91	C₂₃×#102	C₂×C₄	1	2	yes	—
92	C₂₃×#102	C₂×C₄	1	2	yes	—
93	C₂₃×#102	C₂×C₄	1	2	yes	—
94	C₂₃×C₂²⋊C₈	—	2	1	no	—
95	C₂₃×C₂²⋊C₈	—	2	1	no	—
96	C₂₃×C₂²⋊C₈	—	2	2	no	—
97	C₂₃×C₂²⋊C₈	—	2	2	no	—
98	C₂₃×C₂²⋊C₈	—	2	1	no	—
99	C₂₃×C₂²⋊C₈	—	2	1	no	—
100	C₂₃×#97	—	2	1	no	—
101	C₂₃×#97	—	2	1	no	—
102	C₂₃×#97	C₂×C₄	1	2	yes	—
103	C₄×C₈×C₂₃	—	2	1	no	—
104	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
105	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
106	C₄×C₈×C₂₃	C₂³	1	2	yes	—
107	C₄×C₈×C₂₃	C₂³	1	2	yes	—
108	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
110	C₂⁴×C₂₃	—	2	1	no	—
111	#41×C₂₃	—	2	2	no	—
112	#41×C₂₃	—	2	2	no	—
113	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄×C₂₃	—	2	1	no	—
115	C₂×D₄×C₂₃	—	2	1	no	—
116	C₂×D₄×C₂₃	—	2	2	no	—
117	C₂×D₄×C₂₃	—	2	2	no	—
118	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
120	C₂²×C₄×C₂₃	—	2	1	no	—
121	C₂²×C₄×C₂₃	—	2	1	no	—
122	C₂²×C₄×C₂₃	—	2	1	no	—
123	C₂²×C₄×C₂₃	—	2	2	no	—
124	C₂²×C₄×C₂₃	—	2	1	no	—
125	C₂²×C₄×C₂₃	—	2	2	no	—
126	C₂²×C₄×C₂₃	—	4	2	no	—
127	C₂²×C₄×C₂₃	—	4	2	no	—
128	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
131	SD₁₆×C₂₃	—	2	4	no	—
132	D₈×C₂₃	—	2	4	no	—
133	M₁₆×C₂₃	—	2	2	no	—
134	M₁₆×C₂₃	—	4	2	no	—
135	M₁₆×C₂₃	—	4	4	no	—
136	M₁₆×C₂₃	—	4	2	no	—
137	M₁₆×C₂₃	—	4	4	no	—
138	M₁₆×C₂₃	M₁₆	1	4	yes	complemented
139	C₂×C₈×C₂₃	—	2	2	no	—
140	C₂×C₈×C₂₃	—	2	2	no	—
141	C₂×C₈×C₂₃	—	2	1	no	—
142	C₂×C₈×C₂₃	—	2	1	no	—
143	C₂×C₈×C₂₃	—	2	2	no	—
144	C₂×C₈×C₂₃	—	4	1	no	—
145	C₂×C₈×C₂₃	—	4	2	no	—
146	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
147	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
148	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₄⋊C₄×C₂₃	—	2	1	no	—
153	C₄⋊C₄×C₂₃	—	2	1	no	—
154	C₄⋊C₄×C₂₃	—	2	1	no	—
155	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
156	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
157	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
158	C₂²⋊C₄×C₂₃	—	2	1	no	—
159	C₂²⋊C₄×C₂₃	—	2	2	no	—
160	C₂²⋊C₄×C₂₃	—	2	1	no	—
161	C₂²⋊C₄×C₂₃	—	2	2	no	—
162	C₂²⋊C₄×C₂₃	—	2	1	no	—
163	C₄²×C₂₃	—	2	1	no	—
164	C₄²×C₂₃	—	2	1	no	—
165	C₄²×C₂₃	—	2	1	no	—
166	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₂₃	#41	1	1	yes	characteristic
169	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
170	#17891	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₂₃	—	2	1	no	—
172	C₂³×C₂₃	—	2	2	no	—
173	C₂³×C₂₃	—	4	2	no	—
174	C₂³×C₂₃	—	4	2	no	—
175	C₂³×C₂₃	—	4	2	no	—
176	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
177	Q₈×C₂₃	—	2	1	no	—
178	Q₈×C₂₃	—	2	1	no	—
179	D₄×C₂₃	—	2	1	no	—
180	D₄×C₂₃	—	2	2	no	—
181	D₄×C₂₃	—	2	2	no	—
182	D₄×C₂₃	—	2	1	no	—
183	D₄×C₂₃	—	4	2	no	—
184	D₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	2	1	no	—
186	C₂×C₄×C₂₃	—	2	1	no	—
187	C₂×C₄×C₂₃	—	2	1	no	—
188	C₂×C₄×C₂₃	—	2	1	no	—
189	C₂×C₄×C₂₃	—	2	1	no	—
190	C₂×C₄×C₂₃	—	2	2	no	—
191	C₂×C₄×C₂₃	—	4	1	no	—
192	C₂×C₄×C₂₃	—	4	2	no	—
193	C₂×C₄×C₂₃	—	4	4	no	—
194	C₂×C₄×C₂₃	—	4	2	no	—
195	C₂×C₄×C₂₃	—	4	2	no	—
196	C₂×C₄×C₂₃	—	4	1	no	—
197	C₂×C₄×C₂₃	—	4	1	no	—
198	C₂×C₄×C₂₃	—	4	2	no	—
199	C₂×C₄×C₂₃	—	4	2	no	—
200	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
203	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
204	C₂×C₄×C₂₃	#131	1	2	yes	—
205	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
206	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
208	C₈×C₂₃	—	2	2	no	—
209	C₈×C₂₃	—	2	1	no	—
210	C₈×C₂₃	—	2	1	no	—
211	C₈×C₂₃	—	2	1	no	—
212	C₈×C₂₃	—	4	2	no	—
213	C₈×C₂₃	—	4	2	no	—
214	C₈×C₂₃	—	4	1	no	—
215	C₈×C₂₃	—	4	1	no	—
216	C₈×C₂₃	C₂×M₁₆	1	4	yes	complemented
217	C₈×C₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
218	#2965	C₂×C₂₃	1	1	yes	characteristic, complemented
219	C₄×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic
220	C₂×#436	C₂×C₂₃	1	1	yes	characteristic, complemented
221	#1594	C₂×C₂₃	1	2	yes	complemented
222	#1592	C₂×C₂₃	1	2	yes	complemented
223	#1590	C₂×C₂₃	1	2	yes	complemented
224	#1589	C₂×C₂₃	1	2	yes	complemented
225	#1565	C₂×C₂₃	1	1	yes	characteristic, complemented
226	#1478	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#1477	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₂²×C₂₃	—	2	1	no	—
230	C₂²×C₂₃	—	2	1	no	—
231	C₂²×C₂₃	—	4	1	no	—
232	C₂²×C₂₃	—	4	2	no	—
233	C₂²×C₂₃	—	4	2	no	—
234	C₂²×C₂₃	—	4	1	no	—
235	C₂²×C₂₃	—	4	1	no	—
236	C₂²×C₂₃	—	4	2	no	—
237	C₂²×C₂₃	—	8	4	no	—
238	C₂²×C₂₃	#436	1	2	yes	—
239	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
240	C₄×C₂₃	—	2	1	no	—
241	C₄×C₂₃	—	2	1	no	—
242	C₄×C₂₃	—	2	1	no	—
243	C₄×C₂₃	—	4	1	no	—
244	C₄×C₂₃	—	4	1	no	—
245	C₄×C₂₃	—	4	2	no	—
246	C₄×C₂₃	—	4	1	no	—
247	C₄×C₂₃	#436	1	2	yes	—
248	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
249	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
250	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₄×C₂₃	#575	1	1	yes	characteristic
252	C₂×AGL(1,ℤ/8)	C₄×C₂₃	1	1	yes	characteristic
253	C₂²×M₁₆	—	2	1	no	—
254	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
256	#446	C₂²×C₂₃	1	1	yes	characteristic, complemented
257	#446	C₂²×C₂₃	1	1	yes	characteristic, complemented
258	#442	C₄×C₂₃	1	2	yes	—
259	#440	C₄×C₂₃	1	2	yes	—
260	C₄×SD₁₆	C₂²×C₂₃	1	2	yes	—
261	C₄×D₈	C₂²×C₂₃	1	2	yes	—
262	#436	—	2	1	no	—
263	#436	—	2	1	no	—
264	#436	C₂²×C₂₃	1	2	yes	—
265	#436	C₂²×C₂₃	1	2	yes	—
266	#436	C₂²×C₂₃	1	1	yes	characteristic, complemented
267	#436	C₂²×C₂₃	1	1	yes	characteristic, complemented
268	#435	—	2	1	no	—
269	#435	—	2	1	no	—
270	#431	—	2	1	no	—
271	#431	C₂²×C₂₃	1	1	yes	characteristic
272	#431	C₂²×C₂₃	1	1	yes	characteristic
273	#428	C₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
275	#418	C₄×C₂₃	1	1	yes	characteristic
276	#417	C₄×C₂₃	1	1	yes	characteristic
277	C₂×C₂²⋊C₈	—	2	1	no	—
278	C₄×M₁₆	—	2	1	no	—
279	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
280	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
281	#335	C₂²×C₂₃	1	2	yes	complemented
282	#334	C₂²×C₂₃	1	2	yes	complemented
283	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
284	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
285	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
286	D₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
287	#322	C₂²×C₂₃	1	2	yes	complemented
288	C₈²	C₂²×C₂₃	1	2	yes	complemented
289	C₂×C₂₃	—	2	2	no	—
290	C₂×C₂₃	—	8	1	no	—
291	C₂×C₂₃	—	8	2	no	—
292	C₂×C₂₃	C₂×#436	1	1	yes	characteristic
293	C₂×C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
294	C₂×C₂₃	#2955	1	1	yes	characteristic
295	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
296	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
297	C₂³×C₄	—	2	1	no	—
298	AGL(1,ℤ/8)	—	2	4	no	—
299	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
300	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
301	C₂×M₁₆	—	2	2	no	—
302	C₂×M₁₆	—	2	2	no	—
303	C₂×M₁₆	—	2	1	no	—
304	C₂×M₁₆	—	2	1	no	—
305	C₂×M₁₆	—	4	4	no	—
306	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
307	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
308	C₂²×C₈	—	2	1	no	—
309	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
310	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
311	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
312	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
313	C₄×D₄	—	2	1	no	—
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
318	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
319	C₄×D₄	C₂³×C₂₃	1	2	yes	—
320	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
321	#117	—	2	1	no	—
322	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
324	C₂×C₂²⋊C₄	—	2	1	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
327	#107	C₂×C₄×C₂₃	1	2	yes	—
328	#106	C₂×C₄×C₂₃	1	2	yes	—
329	C₄⋊C₈	—	2	1	no	—
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	—	2	1	no	—
333	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
334	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
335	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
336	#103	C₂×C₄×C₂₃	1	2	yes	—
337	#102	C₂×C₄×C₂₃	1	2	yes	—
338	#102	C₂×C₄×C₂₃	1	2	yes	—
339	#102	C₂×C₄×C₂₃	1	2	yes	—
340	C₂²⋊C₈	—	2	1	no	—
341	C₂²⋊C₈	—	2	1	no	—
342	C₂²⋊C₈	—	2	2	no	—
343	C₂²⋊C₈	—	2	2	no	—
344	C₂²⋊C₈	—	2	1	no	—
345	C₂²⋊C₈	—	2	1	no	—
346	#97	—	2	1	no	—
347	#97	—	2	1	no	—
348	#97	C₂×C₄×C₂₃	1	2	yes	—
349	C₄×C₈	—	2	1	no	—
350	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
351	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
352	C₄×C₈	C₂³×C₂₃	1	2	yes	—
353	C₄×C₈	C₂³×C₂₃	1	2	yes	—
354	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
355	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
356	C₂₃	#17891	1	1	yes	characteristic, direct product factor, Sylow
357	C₂⁴	—	2	1	no	—
358	#41	—	2	2	no	—
359	#41	—	2	2	no	—
360	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
366	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
377	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
378	SD₁₆	—	2	4	no	—
379	D₈	—	2	4	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	4	4	no	—
382	M₁₆	—	4	2	no	—
383	M₁₆	—	4	4	no	—
384	M₁₆	—	4	2	no	—
385	M₁₆	M₁₆×C₂₃	1	4	yes	complemented
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	4	1	no	—
392	C₂×C₈	—	4	2	no	—
393	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
394	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
395	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
397	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
398	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
403	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
404	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
405	C₂²⋊C₄	—	2	1	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	1	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	1	no	—
413	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
414	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
415	C₄²	#41×C₂₃	1	1	yes	characteristic
416	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
417	C₂³	—	2	1	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
423	Q₈	—	2	1	no	—
424	Q₈	—	2	1	no	—
425	D₄	—	2	1	no	—
426	D₄	—	2	2	no	—
427	D₄	—	2	2	no	—
428	D₄	—	2	1	no	—
429	D₄	—	4	2	no	—
430	D₄	—	4	2	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
447	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
448	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
449	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
450	C₂×C₄	C₂₃×#131	1	2	yes	—
451	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
452	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	2	1	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	2	no	—
460	C₈	—	4	1	no	—
461	C₈	—	4	1	no	—
462	C₈	C₂×M₁₆×C₂₃	1	4	yes	complemented
463	C₈	C₂₃×AGL(1,ℤ/8)	1	2	yes	complemented
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	4	1	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	1	no	—
470	C₂²	—	4	1	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	8	4	no	—
473	C₂²	C₂₃×#436	1	2	yes	—
474	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	4	1	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	1	no	—
481	C₄	—	4	1	no	—
482	C₄	C₂₃×#436	1	2	yes	—
483	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
484	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
485	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
486	C₄	C₂₃×#575	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	8	2	no	—
489	C₂	—	8	1	no	—
490	C₂	C₂×C₂₃×#436	1	1	yes	characteristic
491	C₂	C₄×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
492	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 ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2965	C₂	semidirect
3. C₂×C₂₃×#436	C₂	semidirect
4. C₂₃×#1594	C₂	semidirect
5. C₂₃×#1592	C₂	semidirect
6. C₂₃×#1590	C₂	semidirect
7. C₂₃×#1589	C₂	semidirect
8. C₂₃×#1565	C₂	semidirect
9. C₂₃×#1478	C₂	semidirect
10. C₂₃×#1477	C₂	semidirect
11. C₈×M₁₆×C₂₃	C₂	semidirect
16. C₂₃×#446	C₂²	semidirect
17. C₂₃×#446	C₂²	semidirect
26. C₂₃×#436	C₂²	semidirect
27. C₂₃×#436	C₂²	semidirect
41. C₂₃×#335	C₂²	semidirect
42. C₂₃×#334	C₂²	semidirect
43. C₂₃×Q₈⋊C₈	C₂²	semidirect
46. C₂₃×D₄⋊C₈	C₂²	semidirect
47. C₂₃×#322	C₂²	semidirect
48. C₈²×C₂₃	C₂²	semidirect
138. M₁₆×C₂₃	M₁₆	semidirect
170. #17891	C₂₃	direct (N × H)
216. C₈×C₂₃	C₂×M₁₆	semidirect
217. C₈×C₂₃	AGL(1,ℤ/8)	semidirect
218. #2965	C₂×C₂₃	semidirect
220. C₂×#436	C₂×C₂₃	semidirect
221. #1594	C₂×C₂₃	semidirect
222. #1592	C₂×C₂₃	semidirect
223. #1590	C₂×C₂₃	semidirect
224. #1589	C₂×C₂₃	semidirect
225. #1565	C₂×C₂₃	semidirect
226. #1478	C₂×C₂₃	semidirect
227. #1477	C₂×C₂₃	semidirect
228. C₈×M₁₆	C₂×C₂₃	semidirect
256. #446	C₂²×C₂₃	semidirect
257. #446	C₂²×C₂₃	semidirect
266. #436	C₂²×C₂₃	semidirect
267. #436	C₂²×C₂₃	semidirect
281. #335	C₂²×C₂₃	semidirect
282. #334	C₂²×C₂₃	semidirect
283. Q₈⋊C₈	C₂²×C₂₃	semidirect
286. D₄⋊C₈	C₂²×C₂₃	semidirect
287. #322	C₂²×C₂₃	semidirect
288. C₈²	C₂²×C₂₃	semidirect
356. C₂₃	#17891	direct (N × H)
385. M₁₆	M₁₆×C₂₃	semidirect
462. C₈	C₂×M₁₆×C₂₃	semidirect
463. 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×#2965	2
C₄×C₂₃×AGL(1,ℤ/8)	2
C₂×C₂₃×#436	2
C₂₃×#1594	2
C₂₃×#1592	2
C₂₃×#1590	2
C₂₃×#1589	2
C₂₃×#1565	2
C₂₃×#1478	2
C₂₃×#1477	2
C₈×M₁₆×C₂₃	2
#17891	23

Subgroups — C23×#17891 #19165517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17891 #19165517