Subgroups — C₂₃×#19127 #19166753

Order 5888.

642 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3106	C₂	1	1	yes	maximal, characteristic
4	C₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1684	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1683	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1682	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1681	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1672	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1671	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1666	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1665	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1544	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1543	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#533	C₂²	1	1	yes	characteristic
17	C₂₃×#532	C₂²	1	1	yes	characteristic
18	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#498	C₂²	1	1	yes	characteristic
20	C₂₃×#492	C₂²	1	1	yes	characteristic
21	C₂₃×#487	C₂²	1	1	yes	characteristic
22	C₂₃×#486	C₂²	1	1	yes	characteristic
23	C₂₃×#485	—	2	1	no	—
24	C₂₃×#482	—	2	1	no	—
25	C₂₃×#477	C₂²	1	1	yes	characteristic
26	C₂₃×#476	C₂²	1	1	yes	characteristic
27	C₂₃×#475	C₂²	1	1	yes	characteristic
28	C₂₃×#474	C₂²	1	1	yes	characteristic
29	C₂₃×#465	—	2	1	no	—
30	C₂₃×#465	—	2	1	no	—
31	C₂₃×#462	C₂²	1	1	yes	characteristic
32	C₂₃×#461	C₂²	1	1	yes	characteristic
33	C₂₃×#460	C₂²	1	1	yes	characteristic
34	C₂₃×#459	C₂²	1	1	yes	characteristic
35	C₂₃×#452	—	2	1	no	—
36	C₂₃×#449	—	2	1	no	—
37	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₂₃	—	2	1	no	—
40	C₄×SD₁₆×C₂₃	—	2	1	no	—
41	C₄×SD₁₆×C₂₃	—	2	1	no	—
42	C₄×SD₁₆×C₂₃	—	2	1	no	—
43	C₄×SD₁₆×C₂₃	—	2	1	no	—
44	C₄×D₈×C₂₃	—	2	1	no	—
45	C₈×D₄×C₂₃	—	2	1	no	—
46	C₈×D₄×C₂₃	—	2	1	no	—
47	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
48	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
49	C₂₃×#431	—	2	1	no	—
50	C₂₃×#431	C₂²	1	1	yes	characteristic
51	C₂₃×#431	C₂²	1	1	yes	characteristic
52	C₂₃×#431	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₂₃	—	2	1	no	—
54	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
55	C₂₃×#335	C₂²	1	1	yes	characteristic
56	C₂₃×#335	C₂²	1	1	yes	characteristic
57	C₂₃×#334	C₂²	1	1	yes	characteristic
58	C₂₃×#334	C₂²	1	1	yes	characteristic
59	C₂₃×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#333	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	C₂²	1	1	yes	characteristic
63	C₂₃×#332	C₂²	1	1	yes	characteristic
64	C₂₃×#332	C₂²	1	1	yes	characteristic
65	C₂₃×#331	C₂²	1	1	yes	characteristic
66	C₂₃×#331	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	—	2	1	no	—
68	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
69	C₂×Q₁₆×C₂₃	—	2	1	no	—
70	C₂×SD₁₆×C₂₃	—	2	1	no	—
71	C₂×SD₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	—	2	1	no	—
74	C₂×D₈×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	—	2	1	no	—
76	C₂²×C₈×C₂₃	—	2	1	no	—
77	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
79	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₂₃×#127	C₂³	1	1	yes	characteristic
83	C₂₃×#125	—	2	1	no	—
84	C₂₃×#124	—	2	1	no	—
85	C₂₃×C₂²⋊Q₈	—	2	1	no	—
86	C₂₃×C₂²⋊Q₈	—	2	1	no	—
87	C₂₃×#121	—	2	1	no	—
88	C₂₃×#121	—	2	1	no	—
89	C₄×Q₈×C₂₃	—	2	1	no	—
90	C₄×Q₈×C₂₃	—	2	1	no	—
91	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
92	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
93	C₄×D₄×C₂₃	—	2	1	no	—
94	C₄×D₄×C₂₃	—	2	1	no	—
95	C₄×D₄×C₂₃	—	2	1	no	—
96	C₄×D₄×C₂₃	—	2	1	no	—
97	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
98	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂₃×#117	—	2	1	no	—
100	C₂₃×#117	—	2	1	no	—
101	C₂₃×#117	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	—	2	1	no	—
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂₃×#107	—	2	1	no	—
105	C₂₃×#107	—	2	1	no	—
106	C₂₃×#106	—	2	1	no	—
107	C₂₃×#106	—	2	1	no	—
108	C₂₃×#106	—	2	1	no	—
109	C₂₃×#106	—	2	1	no	—
110	C₂₃×C₄⋊C₈	—	2	1	no	—
111	C₂₃×C₄⋊C₈	—	2	1	no	—
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×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₂₃×#103	—	2	1	no	—
119	C₂₃×#103	—	2	1	no	—
120	C₂₃×#103	—	2	1	no	—
121	C₂₃×#103	—	2	1	no	—
122	C₂₃×#103	—	2	1	no	—
123	C₂₃×#103	—	2	1	no	—
124	C₂₃×#102	—	2	1	no	—
125	C₂₃×#102	—	2	1	no	—
126	C₂₃×#102	—	2	1	no	—
127	C₂₃×#102	—	2	1	no	—
128	C₂₃×#102	—	2	1	no	—
129	C₂₃×#102	—	2	1	no	—
130	C₂₃×C₂²⋊C₈	—	2	1	no	—
131	C₂₃×C₂²⋊C₈	—	2	1	no	—
132	C₂₃×C₂²⋊C₈	—	2	1	no	—
133	C₂₃×C₂²⋊C₈	—	2	1	no	—
134	C₂₃×C₂²⋊C₈	—	2	1	no	—
135	C₄×C₈×C₂₃	—	2	1	no	—
136	C₄×C₈×C₂₃	—	2	1	no	—
137	C₄×C₈×C₂₃	—	2	1	no	—
138	C₄×C₈×C₂₃	—	2	1	no	—
139	C₄×C₈×C₂₃	—	2	1	no	—
140	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
141	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	#41×C₂₃	—	2	2	no	—
145	#41×C₂₃	—	2	2	no	—
146	#41×C₂₃	—	8	2	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂×D₄×C₂₃	—	2	1	no	—
152	C₂×D₄×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	—	2	1	no	—
154	C₂×D₄×C₂₃	—	2	1	no	—
155	C₂×D₄×C₂₃	—	4	1	no	—
156	C₂×D₄×C₂₃	C₂×D₄	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	—	4	1	no	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
164	Q₁₆×C₂₃	—	4	2	no	—
165	SD₁₆×C₂₃	—	4	2	no	—
166	SD₁₆×C₂₃	—	4	2	no	—
167	SD₁₆×C₂₃	—	4	2	no	—
168	SD₁₆×C₂₃	—	4	2	no	—
169	D₈×C₂₃	—	4	2	no	—
170	C₂×C₈×C₂₃	—	2	2	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	4	1	no	—
178	C₂×C₈×C₂₃	—	4	2	no	—
179	C₂×C₈×C₂₃	—	4	2	no	—
180	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	4	1	no	—
193	C₄⋊C₄×C₂₃	—	4	1	no	—
194	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂²⋊C₄×C₂₃	—	2	1	no	—
197	C₂²⋊C₄×C₂₃	—	2	1	no	—
198	C₂²⋊C₄×C₂₃	—	2	1	no	—
199	C₂²⋊C₄×C₂₃	—	4	1	no	—
200	C₂²⋊C₄×C₂₃	—	4	1	no	—
201	C₂²⋊C₄×C₂₃	—	4	1	no	—
202	C₄²×C₂₃	—	2	1	no	—
203	C₄²×C₂₃	—	2	1	no	—
204	C₄²×C₂₃	—	2	1	no	—
205	C₄²×C₂₃	—	4	1	no	—
206	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₂₃	#41	1	1	yes	characteristic
209	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
210	#19127	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂³×C₂₃	—	2	1	no	—
212	C₂³×C₂₃	—	4	1	no	—
213	C₂³×C₂₃	#121	1	1	yes	characteristic
214	Q₈×C₂₃	—	2	1	no	—
215	Q₈×C₂₃	—	4	1	no	—
216	Q₈×C₂₃	—	4	1	no	—
217	Q₈×C₂₃	—	4	1	no	—
218	Q₈×C₂₃	—	4	1	no	—
219	Q₈×C₂₃	—	8	1	no	—
220	Q₈×C₂₃	#135	1	2	yes	—
221	D₄×C₂₃	—	2	4	no	—
222	D₄×C₂₃	—	2	1	no	—
223	D₄×C₂₃	—	4	1	no	—
224	D₄×C₂₃	—	4	1	no	—
225	D₄×C₂₃	—	4	1	no	—
226	D₄×C₂₃	—	4	1	no	—
227	D₄×C₂₃	—	8	1	no	—
228	D₄×C₂₃	—	8	2	no	—
229	D₄×C₂₃	#135	1	2	yes	—
230	C₂×C₄×C₂₃	—	2	4	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	2	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	8	2	no	—
246	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
247	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
248	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
252	C₈×C₂₃	—	2	1	no	—
253	C₈×C₂₃	—	2	1	no	—
254	C₈×C₂₃	—	2	1	no	—
255	C₈×C₂₃	—	4	1	no	—
256	C₈×C₂₃	—	4	1	no	—
257	C₈×C₂₃	—	4	1	no	—
258	C₈×C₂₃	—	4	1	no	—
259	C₈×C₂₃	—	4	1	no	—
260	C₈×C₂₃	—	4	1	no	—
261	C₈×C₂₃	#135	1	2	yes	—
262	#3167	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#3107	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#3106	C₂×C₂₃	1	1	yes	characteristic
265	C₈×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1684	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1683	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1682	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1681	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1672	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1671	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1666	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1665	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#1544	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#1543	C₂×C₂₃	1	1	yes	characteristic, complemented
277	C₂²×C₂₃	—	2	1	no	—
278	C₂²×C₂₃	—	2	1	no	—
279	C₂²×C₂₃	—	2	2	no	—
280	C₂²×C₂₃	—	2	1	no	—
281	C₂²×C₂₃	—	4	1	no	—
282	C₂²×C₂₃	—	4	1	no	—
283	C₂²×C₂₃	—	8	1	no	—
284	C₂²×C₂₃	—	8	1	no	—
285	C₂²×C₂₃	—	8	1	no	—
286	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
287	C₄×C₂₃	—	2	1	no	—
288	C₄×C₂₃	—	2	1	no	—
289	C₄×C₂₃	—	2	2	no	—
290	C₄×C₂₃	—	2	1	no	—
291	C₄×C₂₃	—	4	1	no	—
292	C₄×C₂₃	—	4	1	no	—
293	C₄×C₂₃	—	8	1	no	—
294	C₄×C₂₃	—	8	1	no	—
295	C₄×C₂₃	—	8	1	no	—
296	C₄×C₂₃	#471	1	2	yes	—
297	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
298	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
299	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
300	C₄×C₂₃	#575	1	1	yes	characteristic
301	#533	C₂²×C₂₃	1	1	yes	characteristic
302	#532	C₂²×C₂₃	1	1	yes	characteristic
303	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
304	#498	C₂²×C₂₃	1	1	yes	characteristic
305	#492	C₂²×C₂₃	1	1	yes	characteristic
306	#487	C₂²×C₂₃	1	1	yes	characteristic
307	#486	C₂²×C₂₃	1	1	yes	characteristic
308	#485	—	2	1	no	—
309	#482	—	2	1	no	—
310	#477	C₂²×C₂₃	1	1	yes	characteristic
311	#476	C₂²×C₂₃	1	1	yes	characteristic
312	#475	C₂²×C₂₃	1	1	yes	characteristic
313	#474	C₂²×C₂₃	1	1	yes	characteristic
314	#465	—	2	1	no	—
315	#465	—	2	1	no	—
316	#462	C₂²×C₂₃	1	1	yes	characteristic
317	#461	C₂²×C₂₃	1	1	yes	characteristic
318	#460	C₂²×C₂₃	1	1	yes	characteristic
319	#459	C₂²×C₂₃	1	1	yes	characteristic
320	#452	—	2	1	no	—
321	#449	—	2	1	no	—
322	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
323	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
324	C₄×Q₁₆	—	2	1	no	—
325	C₄×SD₁₆	—	2	1	no	—
326	C₄×SD₁₆	—	2	1	no	—
327	C₄×SD₁₆	—	2	1	no	—
328	C₄×SD₁₆	—	2	1	no	—
329	C₄×D₈	—	2	1	no	—
330	C₈×D₄	—	2	1	no	—
331	C₈×D₄	—	2	1	no	—
332	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
333	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
334	#431	—	2	1	no	—
335	#431	C₂²×C₂₃	1	1	yes	characteristic
336	#431	C₂²×C₂₃	1	1	yes	characteristic
337	#431	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×C₄×C₈	—	2	1	no	—
339	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
340	#335	C₂²×C₂₃	1	1	yes	characteristic
341	#335	C₂²×C₂₃	1	1	yes	characteristic
342	#334	C₂²×C₂₃	1	1	yes	characteristic
343	#334	C₂²×C₂₃	1	1	yes	characteristic
344	#333	C₂²×C₂₃	1	1	yes	characteristic
345	#333	C₂²×C₂₃	1	1	yes	characteristic
346	#332	C₂²×C₂₃	1	1	yes	characteristic
347	#332	C₂²×C₂₃	1	1	yes	characteristic
348	#332	C₂²×C₂₃	1	1	yes	characteristic
349	#332	C₂²×C₂₃	1	1	yes	characteristic
350	#331	C₂²×C₂₃	1	1	yes	characteristic
351	#331	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×C₂₃	—	4	1	no	—
353	C₂×C₂₃	—	4	2	no	—
354	C₂×C₂₃	—	16	1	no	—
355	C₂×C₂₃	#3035	1	1	yes	characteristic
356	C₂×C₂₃	#3050	1	1	yes	characteristic
357	C₂×C₂₃	C₂×#471	1	1	yes	characteristic
358	C₂×#41	—	2	1	no	—
359	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
360	C₂×Q₁₆	—	2	1	no	—
361	C₂×SD₁₆	—	2	1	no	—
362	C₂×SD₁₆	—	2	1	no	—
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×SD₁₆	—	2	1	no	—
365	C₂×D₈	—	2	1	no	—
366	C₂²×C₈	—	2	1	no	—
367	C₂²×C₈	—	2	1	no	—
368	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
369	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
370	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
371	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
372	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
373	#127	C₂³×C₂₃	1	1	yes	characteristic
374	#125	—	2	1	no	—
375	#124	—	2	1	no	—
376	C₂²⋊Q₈	—	2	1	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	#121	—	2	1	no	—
379	#121	—	2	1	no	—
380	C₄×Q₈	—	2	1	no	—
381	C₄×Q₈	—	2	1	no	—
382	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
383	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	—	2	1	no	—
387	C₄×D₄	—	2	1	no	—
388	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
389	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
390	#117	—	2	1	no	—
391	#117	—	2	1	no	—
392	#117	D₄×C₂₃	1	1	yes	characteristic
393	C₂×C₄²	—	2	1	no	—
394	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
395	#107	—	2	1	no	—
396	#107	—	2	1	no	—
397	#106	—	2	1	no	—
398	#106	—	2	1	no	—
399	#106	—	2	1	no	—
400	#106	—	2	1	no	—
401	C₄⋊C₈	—	2	1	no	—
402	C₄⋊C₈	—	2	1	no	—
403	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
404	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
407	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
408	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
409	#103	—	2	1	no	—
410	#103	—	2	1	no	—
411	#103	—	2	1	no	—
412	#103	—	2	1	no	—
413	#103	—	2	1	no	—
414	#103	—	2	1	no	—
415	#102	—	2	1	no	—
416	#102	—	2	1	no	—
417	#102	—	2	1	no	—
418	#102	—	2	1	no	—
419	#102	—	2	1	no	—
420	#102	—	2	1	no	—
421	C₂²⋊C₈	—	2	1	no	—
422	C₂²⋊C₈	—	2	1	no	—
423	C₂²⋊C₈	—	2	1	no	—
424	C₂²⋊C₈	—	2	1	no	—
425	C₂²⋊C₈	—	2	1	no	—
426	C₄×C₈	—	2	1	no	—
427	C₄×C₈	—	2	1	no	—
428	C₄×C₈	—	2	1	no	—
429	C₄×C₈	—	2	1	no	—
430	C₄×C₈	—	2	1	no	—
431	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
432	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
433	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
434	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
435	C₂₃	#19127	1	1	yes	characteristic, direct product factor, Sylow
436	#41	—	2	2	no	—
437	#41	—	2	2	no	—
438	#41	—	8	2	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	4	1	no	—
448	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	4	1	no	—
453	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
456	Q₁₆	—	4	2	no	—
457	SD₁₆	—	4	2	no	—
458	SD₁₆	—	4	2	no	—
459	SD₁₆	—	4	2	no	—
460	SD₁₆	—	4	2	no	—
461	D₈	—	4	2	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	4	1	no	—
470	C₂×C₈	—	4	2	no	—
471	C₂×C₈	—	4	2	no	—
472	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
473	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
474	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
475	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	4	1	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
487	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	1	no	—
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₂²⋊C₄	—	4	1	no	—
493	C₂²⋊C₄	—	4	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	4	1	no	—
498	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
499	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
500	C₄²	#41×C₂₃	1	1	yes	characteristic
501	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
502	C₂³	—	2	1	no	—
503	C₂³	—	4	1	no	—
504	C₂³	C₂₃×#121	1	1	yes	characteristic
505	Q₈	—	2	1	no	—
506	Q₈	—	4	1	no	—
507	Q₈	—	4	1	no	—
508	Q₈	—	4	1	no	—
509	Q₈	—	4	1	no	—
510	Q₈	—	8	1	no	—
511	Q₈	C₂₃×#135	1	2	yes	—
512	D₄	—	2	4	no	—
513	D₄	—	2	1	no	—
514	D₄	—	4	1	no	—
515	D₄	—	4	1	no	—
516	D₄	—	4	1	no	—
517	D₄	—	4	1	no	—
518	D₄	—	8	1	no	—
519	D₄	—	8	2	no	—
520	D₄	C₂₃×#135	1	2	yes	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	8	2	no	—
537	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
538	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
539	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
540	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
542	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
543	C₈	—	2	1	no	—
544	C₈	—	2	1	no	—
545	C₈	—	2	1	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	1	no	—
548	C₈	—	4	1	no	—
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	1	no	—
552	C₈	C₂₃×#135	1	2	yes	—
553	C₂²	—	2	1	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	1	no	—
559	C₂²	—	8	1	no	—
560	C₂²	—	8	1	no	—
561	C₂²	—	8	1	no	—
562	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
563	C₄	—	2	1	no	—
564	C₄	—	2	1	no	—
565	C₄	—	2	2	no	—
566	C₄	—	2	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	8	1	no	—
570	C₄	—	8	1	no	—
571	C₄	—	8	1	no	—
572	C₄	C₂₃×#471	1	2	yes	—
573	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
574	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
575	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
576	C₄	C₂₃×#575	1	1	yes	characteristic
577	C₂	—	4	1	no	—
578	C₂	—	4	2	no	—
579	C₂	—	16	1	no	—
580	C₂	C₂₃×#3035	1	1	yes	characteristic
581	C₂	C₂₃×#3050	1	1	yes	characteristic
582	C₂	C₂×C₂₃×#471	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#121
prime radical	C₂₃×#3106	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3167	C₂	semidirect
2. C₂₃×#3107	C₂	semidirect
4. C₈×#41×C₂₃	C₂	semidirect
5. C₂₃×#1684	C₂	semidirect
6. C₂₃×#1683	C₂	semidirect
7. C₂₃×#1682	C₂	semidirect
8. C₂₃×#1681	C₂	semidirect
9. C₂₃×#1672	C₂	semidirect
10. C₂₃×#1671	C₂	semidirect
11. C₂₃×#1666	C₂	semidirect
12. C₂₃×#1665	C₂	semidirect
13. C₂₃×#1564	C₂	semidirect
14. C₂₃×#1544	C₂	semidirect
15. C₂₃×#1543	C₂	semidirect
210. #19127	C₂₃	direct (N × H)
262. #3167	C₂×C₂₃	semidirect
263. #3107	C₂×C₂₃	semidirect
265. C₈×#41	C₂×C₂₃	semidirect
266. #1684	C₂×C₂₃	semidirect
267. #1683	C₂×C₂₃	semidirect
268. #1682	C₂×C₂₃	semidirect
269. #1681	C₂×C₂₃	semidirect
270. #1672	C₂×C₂₃	semidirect
271. #1671	C₂×C₂₃	semidirect
272. #1666	C₂×C₂₃	semidirect
273. #1665	C₂×C₂₃	semidirect
274. #1564	C₂×C₂₃	semidirect
275. #1544	C₂×C₂₃	semidirect
276. #1543	C₂×C₂₃	semidirect
435. C₂₃	#19127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3167	2
C₂₃×#3107	2
C₂₃×#3106	2
C₈×#41×C₂₃	2
C₂₃×#1684	2
C₂₃×#1683	2
C₂₃×#1682	2
C₂₃×#1681	2
C₂₃×#1672	2
C₂₃×#1671	2
C₂₃×#1666	2
C₂₃×#1665	2
C₂₃×#1564	2
C₂₃×#1544	2
C₂₃×#1543	2
#19127	23

Subgroups — C23×#19127 #19166753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19127 #19166753