Subgroups — C₂₃×#19623 #19167249

Order 5888.

848 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3208	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3206	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3151	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1715	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1714	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1708	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1706	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
13	C₂₃×#533	C₂²	1	1	yes	characteristic
14	C₂₃×#532	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
17	C₂₃×#501	—	2	2	no	—
18	C₂₃×#500	C₂²	1	2	yes	complemented
19	C₂₃×#498	C₂²	1	2	yes	complemented
20	C₂₃×#498	C₂²	1	1	yes	characteristic, complemented
21	C₂₃×#496	—	2	2	no	—
22	C₂₃×#493	C₂²	1	2	yes	—
23	C₂₃×#492	C₂²	1	2	yes	—
24	C₂₃×#492	C₂²	1	2	yes	—
25	C₂₃×#492	C₂²	1	1	yes	characteristic, complemented
26	C₂₃×#492	C₂²	1	2	yes	complemented
27	C₂₃×#487	C₂²	1	1	yes	characteristic, complemented
28	C₂₃×#487	C₂²	1	2	yes	complemented
29	C₂₃×#486	—	2	2	no	—
30	C₂₃×#486	C₂²	1	2	yes	—
31	C₂₃×#486	C₂²	1	2	yes	—
32	C₂₃×#486	C₂²	1	2	yes	—
33	C₂₃×#486	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×#465	—	2	2	no	—
35	C₂₃×#461	—	2	2	no	—
36	C₂₃×#460	—	2	2	no	—
37	C₂₃×#450	—	2	2	no	—
38	C₂₃×#448	—	2	2	no	—
39	C₄×SD₁₆×C₂₃	—	2	2	no	—
40	C₂₃×#431	C₂²	1	1	yes	characteristic
41	C₂₃×#428	—	2	2	no	—
42	C₂×C₂₃×#102	—	2	2	no	—
43	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
45	C₂₃×#322	C₂²	1	2	yes	complemented
46	C₈²×C₂₃	C₂²	1	2	yes	complemented
47	C₂×#41×C₂₃	—	2	1	no	—
48	C₂²×Q₈×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	—	2	1	no	—
51	C₂²×D₄×C₂₃	—	2	1	no	—
52	C₂₃×AGL(1,ℤ/8)	—	4	8	no	—
53	C₂×SD₁₆×C₂₃	—	2	4	no	—
54	C₂×SD₁₆×C₂₃	—	2	2	no	—
55	C₂×SD₁₆×C₂₃	—	2	4	no	—
56	C₂×SD₁₆×C₂₃	—	2	4	no	—
57	C₂×D₈×C₂₃	—	2	4	no	—
58	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
59	C₂²×C₈×C₂₃	D₄	1	2	yes	—
60	C₂₃×C₄⋊Q₈	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	—	2	1	no	—
62	C₂₃×C₄⋊Q₈	—	2	1	no	—
63	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₂₃×#127	—	2	1	no	—
68	C₂₃×#127	—	2	1	no	—
69	C₂₃×#127	—	2	1	no	—
70	C₂₃×#127	C₂³	1	2	yes	—
71	C₂₃×#127	C₂³	1	1	yes	characteristic
72	C₂₃×#125	—	2	1	no	—
73	C₂₃×#124	—	2	1	no	—
74	C₂₃×C₂²⋊Q₈	—	2	1	no	—
75	C₂₃×C₂²⋊Q₈	—	2	1	no	—
76	C₂₃×#121	—	2	1	no	—
77	C₂₃×#121	—	2	1	no	—
78	C₄×Q₈×C₂₃	—	2	1	no	—
79	C₄×Q₈×C₂₃	—	2	1	no	—
80	C₄×D₄×C₂₃	—	2	1	no	—
81	C₄×D₄×C₂₃	—	2	1	no	—
82	C₂₃×#117	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
86	C₂₃×#107	—	2	4	no	—
87	C₂₃×#106	—	2	4	no	—
88	C₂₃×#106	—	2	2	no	—
89	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
90	C₂₃×#103	—	2	2	no	—
91	C₂₃×#103	—	2	4	no	—
92	C₂₃×#102	—	2	4	no	—
93	C₂₃×#102	—	2	4	no	—
94	C₂₃×#102	—	2	2	no	—
95	C₂₃×#102	—	2	4	no	—
96	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
97	C₂₃×#97	D₄	1	2	yes	—
98	C₄×C₈×C₂₃	D₄	1	2	yes	—
99	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₂₃	C₂³	1	2	yes	—
101	C₄×C₈×C₂₃	C₂³	1	2	yes	—
102	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
103	C₂⁴×C₂₃	—	4	1	no	—
104	#41×C₂₃	—	4	4	no	—
105	C₂×Q₈×C₂₃	—	2	1	no	—
106	C₂×Q₈×C₂₃	—	2	2	no	—
107	C₂×Q₈×C₂₃	—	2	1	no	—
108	C₂×Q₈×C₂₃	—	2	1	no	—
109	C₂×Q₈×C₂₃	—	4	1	no	—
110	C₂×Q₈×C₂₃	—	8	2	no	—
111	C₂×D₄×C₂₃	—	2	2	no	—
112	C₂×D₄×C₂₃	—	2	1	no	—
113	C₂×D₄×C₂₃	—	2	2	no	—
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	—	4	4	no	—
118	C₂×D₄×C₂₃	—	4	1	no	—
119	C₂×D₄×C₂₃	—	4	1	no	—
120	C₂×D₄×C₂₃	—	4	1	no	—
121	C₂×D₄×C₂₃	—	4	1	no	—
122	C₂×D₄×C₂₃	—	4	4	no	—
123	C₂×D₄×C₂₃	—	8	2	no	—
124	C₂²×C₄×C₂₃	—	4	1	no	—
125	C₂²×C₄×C₂₃	—	4	1	no	—
126	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
129	SD₁₆×C₂₃	—	4	4	no	—
130	SD₁₆×C₂₃	—	4	8	no	—
131	SD₁₆×C₂₃	—	8	4	no	—
132	SD₁₆×C₂₃	—	8	4	no	—
133	D₈×C₂₃	—	4	8	no	—
134	M₁₆×C₂₃	SD₁₆	1	8	yes	complemented
135	C₂×C₈×C₂₃	—	4	2	no	—
136	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
138	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
139	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
140	C₄⋊C₄×C₂₃	—	2	2	no	—
141	C₄⋊C₄×C₂₃	—	2	1	no	—
142	C₄⋊C₄×C₂₃	—	2	1	no	—
143	C₄⋊C₄×C₂₃	—	2	1	no	—
144	C₄⋊C₄×C₂₃	—	2	1	no	—
145	C₄⋊C₄×C₂₃	—	2	1	no	—
146	C₄⋊C₄×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₄⋊C₄×C₂₃	—	4	1	no	—
149	C₄⋊C₄×C₂₃	—	4	1	no	—
150	C₄⋊C₄×C₂₃	—	4	1	no	—
151	C₄⋊C₄×C₂₃	—	4	1	no	—
152	C₂²⋊C₄×C₂₃	—	4	1	no	—
153	C₂²⋊C₄×C₂₃	—	4	1	no	—
154	C₂²⋊C₄×C₂₃	—	4	1	no	—
155	C₄²×C₂₃	—	4	1	no	—
156	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
160	#19623	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₂₃	—	4	1	no	—
162	C₂³×C₂₃	—	4	2	no	—
163	C₂³×C₂₃	—	8	2	no	—
164	C₂³×C₂₃	—	8	2	no	—
165	C₂³×C₂₃	—	8	2	no	—
166	C₂³×C₂₃	#127	1	1	yes	characteristic
167	Q₈×C₂₃	—	4	2	no	—
168	Q₈×C₂₃	—	4	2	no	—
169	Q₈×C₂₃	—	4	2	no	—
170	Q₈×C₂₃	—	4	4	no	—
171	Q₈×C₂₃	—	8	2	no	—
172	D₄×C₂₃	—	4	2	no	—
173	D₄×C₂₃	—	4	4	no	—
174	D₄×C₂₃	—	4	2	no	—
175	D₄×C₂₃	—	4	4	no	—
176	D₄×C₂₃	—	4	2	no	—
177	D₄×C₂₃	—	4	4	no	—
178	D₄×C₂₃	—	8	2	no	—
179	D₄×C₂₃	—	8	2	no	—
180	D₄×C₂₃	—	8	2	no	—
181	D₄×C₂₃	—	8	2	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	2	2	no	—
184	C₂×C₄×C₂₃	—	4	1	no	—
185	C₂×C₄×C₂₃	—	4	1	no	—
186	C₂×C₄×C₂₃	—	4	1	no	—
187	C₂×C₄×C₂₃	—	4	2	no	—
188	C₂×C₄×C₂₃	—	8	2	no	—
189	C₂×C₄×C₂₃	—	8	2	no	—
190	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
191	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
192	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
193	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
194	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
197	C₈×C₂₃	—	2	2	no	—
198	C₈×C₂₃	—	2	2	no	—
199	C₈×C₂₃	—	2	4	no	—
200	C₈×C₂₃	C₂×SD₁₆	1	8	yes	—
201	#3208	C₂×C₂₃	1	1	yes	characteristic, complemented
202	#3206	C₂×C₂₃	1	1	yes	characteristic, complemented
203	#3167	C₂×C₂₃	1	1	yes	characteristic, complemented
204	#3165	C₂×C₂₃	1	1	yes	characteristic, complemented
205	#3151	C₂×C₂₃	1	1	yes	characteristic
206	C₂×#486	C₂×C₂₃	1	1	yes	characteristic, complemented
207	#1715	C₂×C₂₃	1	2	yes	complemented
208	#1714	C₂×C₂₃	1	2	yes	complemented
209	#1708	C₂×C₂₃	1	2	yes	complemented
210	#1706	C₂×C₂₃	1	2	yes	complemented
211	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
212	C₂²×C₂₃	—	2	1	no	—
213	C₂²×C₂₃	—	2	1	no	—
214	C₂²×C₂₃	—	8	1	no	—
215	C₂²×C₂₃	—	8	2	no	—
216	C₂²×C₂₃	—	8	2	no	—
217	C₂²×C₂₃	—	8	1	no	—
218	C₂²×C₂₃	—	8	2	no	—
219	C₂²×C₂₃	—	8	1	no	—
220	C₂²×C₂₃	—	16	4	no	—
221	C₂²×C₂₃	#492	1	2	yes	—
222	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
223	C₄×C₂₃	—	2	1	no	—
224	C₄×C₂₃	—	2	1	no	—
225	C₄×C₂₃	—	8	1	no	—
226	C₄×C₂₃	—	8	1	no	—
227	C₄×C₂₃	—	8	2	no	—
228	C₄×C₂₃	—	8	1	no	—
229	C₄×C₂₃	#492	1	2	yes	—
230	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
231	C₄×C₂₃	#575	1	2	yes	—
232	C₄×C₂₃	#576	1	2	yes	—
233	C₂×AGL(1,ℤ/8)	—	2	2	no	—
234	#533	C₂²×C₂₃	1	1	yes	characteristic
235	#532	C₂²×C₂₃	1	1	yes	characteristic
236	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
237	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
238	#501	—	2	2	no	—
239	#500	C₂²×C₂₃	1	2	yes	complemented
240	#498	C₂²×C₂₃	1	2	yes	complemented
241	#498	C₂²×C₂₃	1	1	yes	characteristic, complemented
242	#496	—	2	2	no	—
243	#493	C₂²×C₂₃	1	2	yes	—
244	#492	C₂²×C₂₃	1	2	yes	—
245	#492	C₂²×C₂₃	1	2	yes	—
246	#492	C₂²×C₂₃	1	1	yes	characteristic, complemented
247	#492	C₂²×C₂₃	1	2	yes	complemented
248	#487	C₂²×C₂₃	1	1	yes	characteristic, complemented
249	#487	C₂²×C₂₃	1	2	yes	complemented
250	#486	—	2	2	no	—
251	#486	C₂²×C₂₃	1	2	yes	—
252	#486	C₂²×C₂₃	1	2	yes	—
253	#486	C₂²×C₂₃	1	2	yes	—
254	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
255	#465	—	2	2	no	—
256	#461	—	2	2	no	—
257	#460	—	2	2	no	—
258	#450	—	2	2	no	—
259	#448	—	2	2	no	—
260	C₄×SD₁₆	—	2	2	no	—
261	#431	C₂²×C₂₃	1	1	yes	characteristic
262	#428	—	2	2	no	—
263	C₂×#102	—	2	2	no	—
264	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
266	#322	C₂²×C₂₃	1	2	yes	complemented
267	C₈²	C₂²×C₂₃	1	2	yes	complemented
268	C₂×C₂₃	—	2	2	no	—
269	C₂×C₂₃	—	16	1	no	—
270	C₂×C₂₃	—	16	2	no	—
271	C₂×C₂₃	C₂×#492	1	1	yes	characteristic
272	C₂×C₂₃	#3153	1	1	yes	characteristic
273	C₂×C₂₃	#3155	1	1	yes	characteristic
274	C₂×#41	—	2	1	no	—
275	C₂²×Q₈	—	2	1	no	—
276	C₂²×D₄	—	2	1	no	—
277	C₂²×D₄	—	2	1	no	—
278	C₂²×D₄	—	2	1	no	—
279	AGL(1,ℤ/8)	—	4	8	no	—
280	C₂×SD₁₆	—	2	4	no	—
281	C₂×SD₁₆	—	2	2	no	—
282	C₂×SD₁₆	—	2	4	no	—
283	C₂×SD₁₆	—	2	4	no	—
284	C₂×D₈	—	2	4	no	—
285	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
286	C₂²×C₈	D₄×C₂₃	1	2	yes	—
287	C₄⋊Q₈	—	2	1	no	—
288	C₄⋊Q₈	—	2	1	no	—
289	C₄⋊Q₈	—	2	1	no	—
290	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
291	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
292	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
293	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
294	#127	—	2	1	no	—
295	#127	—	2	1	no	—
296	#127	—	2	1	no	—
297	#127	C₂³×C₂₃	1	2	yes	—
298	#127	C₂³×C₂₃	1	1	yes	characteristic
299	#125	—	2	1	no	—
300	#124	—	2	1	no	—
301	C₂²⋊Q₈	—	2	1	no	—
302	C₂²⋊Q₈	—	2	1	no	—
303	#121	—	2	1	no	—
304	#121	—	2	1	no	—
305	C₄×Q₈	—	2	1	no	—
306	C₄×Q₈	—	2	1	no	—
307	C₄×D₄	—	2	1	no	—
308	C₄×D₄	—	2	1	no	—
309	#117	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
313	#107	—	2	4	no	—
314	#106	—	2	4	no	—
315	#106	—	2	2	no	—
316	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
317	#103	—	2	2	no	—
318	#103	—	2	4	no	—
319	#102	—	2	4	no	—
320	#102	—	2	4	no	—
321	#102	—	2	2	no	—
322	#102	—	2	4	no	—
323	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
324	#97	D₄×C₂₃	1	2	yes	—
325	C₄×C₈	D₄×C₂₃	1	2	yes	—
326	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
327	C₄×C₈	C₂³×C₂₃	1	2	yes	—
328	C₄×C₈	C₂³×C₂₃	1	2	yes	—
329	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
330	C₂₃	#19623	1	1	yes	characteristic, direct product factor, Sylow
331	C₂⁴	—	4	1	no	—
332	#41	—	4	4	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	—	4	1	no	—
338	C₂×Q₈	—	8	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	4	4	no	—
346	C₂×D₄	—	4	1	no	—
347	C₂×D₄	—	4	1	no	—
348	C₂×D₄	—	4	1	no	—
349	C₂×D₄	—	4	1	no	—
350	C₂×D₄	—	4	4	no	—
351	C₂×D₄	—	8	2	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	—	4	1	no	—
354	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
357	SD₁₆	—	4	4	no	—
358	SD₁₆	—	4	8	no	—
359	SD₁₆	—	8	4	no	—
360	SD₁₆	—	8	4	no	—
361	D₈	—	4	8	no	—
362	M₁₆	SD₁₆×C₂₃	1	8	yes	complemented
363	C₂×C₈	—	4	2	no	—
364	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
365	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
366	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
367	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	4	1	no	—
377	C₄⋊C₄	—	4	1	no	—
378	C₄⋊C₄	—	4	1	no	—
379	C₄⋊C₄	—	4	1	no	—
380	C₂²⋊C₄	—	4	1	no	—
381	C₂²⋊C₄	—	4	1	no	—
382	C₂²⋊C₄	—	4	1	no	—
383	C₄²	—	4	1	no	—
384	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
385	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
386	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
387	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
388	C₂³	—	4	1	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	8	2	no	—
391	C₂³	—	8	2	no	—
392	C₂³	—	8	2	no	—
393	C₂³	C₂₃×#127	1	1	yes	characteristic
394	Q₈	—	4	2	no	—
395	Q₈	—	4	2	no	—
396	Q₈	—	4	2	no	—
397	Q₈	—	4	4	no	—
398	Q₈	—	8	2	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	4	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	4	no	—
403	D₄	—	4	2	no	—
404	D₄	—	4	4	no	—
405	D₄	—	8	2	no	—
406	D₄	—	8	2	no	—
407	D₄	—	8	2	no	—
408	D₄	—	8	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	8	2	no	—
416	C₂×C₄	—	8	2	no	—
417	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
418	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
419	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
420	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
421	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
423	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
424	C₈	—	2	2	no	—
425	C₈	—	2	2	no	—
426	C₈	—	2	4	no	—
427	C₈	C₂×SD₁₆×C₂₃	1	8	yes	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	8	1	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	1	no	—
434	C₂²	—	8	2	no	—
435	C₂²	—	8	1	no	—
436	C₂²	—	16	4	no	—
437	C₂²	C₂₃×#492	1	2	yes	—
438	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
439	C₄	—	2	1	no	—
440	C₄	—	2	1	no	—
441	C₄	—	8	1	no	—
442	C₄	—	8	1	no	—
443	C₄	—	8	2	no	—
444	C₄	—	8	1	no	—
445	C₄	C₂₃×#492	1	2	yes	—
446	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
447	C₄	C₂₃×#575	1	2	yes	—
448	C₄	C₂₃×#576	1	2	yes	—
449	C₂	—	2	2	no	—
450	C₂	—	16	2	no	—
451	C₂	—	16	1	no	—
452	C₂	C₂×C₂₃×#492	1	1	yes	characteristic
453	C₂	C₂₃×#3153	1	1	yes	characteristic
454	C₂	C₂₃×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3151	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3208	C₂	semidirect
2. C₂₃×#3206	C₂	semidirect
3. C₂₃×#3167	C₂	semidirect
4. C₂₃×#3165	C₂	semidirect
6. C₂×C₂₃×#486	C₂	semidirect
7. C₂₃×#1715	C₂	semidirect
8. C₂₃×#1714	C₂	semidirect
9. C₂₃×#1708	C₂	semidirect
10. C₂₃×#1706	C₂	semidirect
11. C₈×M₁₆×C₂₃	C₂	semidirect
18. C₂₃×#500	C₂²	semidirect
19. C₂₃×#498	C₂²	semidirect
20. C₂₃×#498	C₂²	semidirect
25. C₂₃×#492	C₂²	semidirect
26. C₂₃×#492	C₂²	semidirect
27. C₂₃×#487	C₂²	semidirect
28. C₂₃×#487	C₂²	semidirect
33. C₂₃×#486	C₂²	semidirect
45. C₂₃×#322	C₂²	semidirect
46. C₈²×C₂₃	C₂²	semidirect
134. M₁₆×C₂₃	SD₁₆	semidirect
160. #19623	C₂₃	direct (N × H)
201. #3208	C₂×C₂₃	semidirect
202. #3206	C₂×C₂₃	semidirect
203. #3167	C₂×C₂₃	semidirect
204. #3165	C₂×C₂₃	semidirect
206. C₂×#486	C₂×C₂₃	semidirect
207. #1715	C₂×C₂₃	semidirect
208. #1714	C₂×C₂₃	semidirect
209. #1708	C₂×C₂₃	semidirect
210. #1706	C₂×C₂₃	semidirect
211. C₈×M₁₆	C₂×C₂₃	semidirect
239. #500	C₂²×C₂₃	semidirect
240. #498	C₂²×C₂₃	semidirect
241. #498	C₂²×C₂₃	semidirect
246. #492	C₂²×C₂₃	semidirect
247. #492	C₂²×C₂₃	semidirect
248. #487	C₂²×C₂₃	semidirect
249. #487	C₂²×C₂₃	semidirect
254. #486	C₂²×C₂₃	semidirect
266. #322	C₂²×C₂₃	semidirect
267. C₈²	C₂²×C₂₃	semidirect
330. C₂₃	#19623	direct (N × H)
362. M₁₆	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3208	2
C₂₃×#3206	2
C₂₃×#3167	2
C₂₃×#3165	2
C₂₃×#3151	2
C₂×C₂₃×#486	2
C₂₃×#1715	2
C₂₃×#1714	2
C₂₃×#1708	2
C₂₃×#1706	2
C₈×M₁₆×C₂₃	2
#19623	23

Subgroups — C23×#19623 #19167249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19623 #19167249