Subgroups — C₂₃×#8694 #19156320

Order 5888.

384 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2966	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1509	C₂	1	2	yes	maximal
5	C₂₃×#1285	C₂	1	2	yes	maximal, complemented
6	C₂₃×#529	C₂²	1	1	yes	characteristic
7	C₂₃×#446	C₄	1	1	yes	characteristic, complemented
8	C₈×Q₈×C₂₃	C₄	1	1	yes	characteristic, complemented
9	C₂₃×#436	C₄	1	1	yes	characteristic, complemented
10	C₂₃×#435	—	2	1	no	—
11	C₈×D₄×C₂₃	—	2	1	no	—
12	C₈×D₄×C₂₃	C₄	1	1	yes	characteristic, complemented
13	C₂₃×#433	—	2	1	no	—
14	C₂₃×#432	C₂²	1	2	yes	—
15	C₂₃×#432	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#106	—	2	1	no	—
17	C₂₃×#405	—	2	1	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂₃×#340	—	2	1	no	—
21	C₂₃×#335	—	2	1	no	—
22	C₂₃×#334	—	2	1	no	—
23	C₂₃×#328	—	2	2	no	—
24	C₂₃×#327	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	—	2	1	no	—
27	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₂₃×#126	—	2	1	no	—
30	C₂₃×#125	Q₈	1	1	yes	characteristic
31	C₂₃×#124	Q₈	1	1	yes	characteristic
32	C₂₃×#123	—	2	1	no	—
33	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
34	C₂₃×#121	D₄	1	1	yes	characteristic
35	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₂₃	—	2	1	no	—
37	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂₃×#117	—	2	1	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄²×C₂₃	—	2	2	no	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×#106	—	2	4	no	—
43	C₂₃×C₄⋊C₈	—	2	1	no	—
44	C₂₃×C₄⋊C₈	—	2	1	no	—
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂₃×C₂²⋊C₈	—	2	1	no	—
49	C₂₃×C₂²⋊C₈	—	2	1	no	—
50	C₂₃×C₂²⋊C₈	—	2	1	no	—
51	C₂₃×C₂²⋊C₈	—	2	2	no	—
52	C₂₃×#97	—	2	1	no	—
53	C₂₃×#97	—	2	1	no	—
54	C₂₃×#97	—	2	2	no	—
55	C₄×C₈×C₂₃	—	2	1	no	—
56	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
60	C₂₃×#95	—	2	2	no	—
61	C₂×Q₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₄×C₂₃	—	2	1	no	—
63	C₂×D₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₄×C₂₃	—	4	2	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
69	M₁₆×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	1	no	—
72	C₂×C₈×C₂₃	—	2	1	no	—
73	C₂×C₈×C₂₃	—	4	1	no	—
74	C₂×C₈×C₂₃	—	4	2	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₄⋊C₄×C₂₃	—	2	4	no	—
85	C₄⋊C₄×C₂₃	—	4	2	no	—
86	C₄⋊C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₂₃	—	2	1	no	—
89	C₂²⋊C₄×C₂₃	—	2	1	no	—
90	C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₂²⋊C₄×C₂₃	—	2	1	no	—
93	C₄²×C₂₃	—	2	1	no	—
94	C₄²×C₂₃	—	8	2	no	—
95	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₂₃	#41	1	1	yes	characteristic
99	#8694	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	—	2	1	no	—
101	C₂³×C₂₃	#121	1	1	yes	characteristic
102	Q₈×C₂₃	—	4	1	no	—
103	D₄×C₂₃	—	4	1	no	—
104	D₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	—	4	1	no	—
113	C₂×C₄×C₂₃	—	4	1	no	—
114	C₂×C₄×C₂₃	—	4	4	no	—
115	C₂×C₄×C₂₃	—	8	2	no	—
116	C₂×C₄×C₂₃	—	8	2	no	—
117	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₂₃	—	2	1	no	—
124	C₈×C₂₃	—	2	1	no	—
125	C₈×C₂₃	—	2	1	no	—
126	C₈×C₂₃	—	4	1	no	—
127	C₈×C₂₃	—	4	1	no	—
128	C₈×C₂₃	—	8	2	no	—
129	C₈×C₂₃	#104	1	2	yes	—
130	#2966	C₂×C₂₃	1	1	yes	characteristic
131	C₄×#106	C₂×C₂₃	1	1	yes	characteristic, complemented
132	#1568	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#1509	C₂×C₂₃	1	2	yes	—
134	#1285	C₂×C₂₃	1	2	yes	complemented
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	—	4	1	no	—
138	C₂²×C₂₃	—	4	1	no	—
139	C₂²×C₂₃	—	4	1	no	—
140	C₂²×C₂₃	#380	1	1	yes	characteristic
141	C₂²×C₂₃	#471	1	2	yes	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	2	1	no	—
146	C₄×C₂₃	—	2	1	no	—
147	C₄×C₂₃	—	4	1	no	—
148	C₄×C₂₃	—	4	1	no	—
149	C₄×C₂₃	—	4	1	no	—
150	C₄×C₂₃	—	8	4	no	—
151	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
152	C₄×C₂₃	#421	1	1	yes	characteristic
153	#529	C₂²×C₂₃	1	1	yes	characteristic
154	#446	C₄×C₂₃	1	1	yes	characteristic, complemented
155	C₈×Q₈	C₄×C₂₃	1	1	yes	characteristic, complemented
156	#436	C₄×C₂₃	1	1	yes	characteristic, complemented
157	#435	—	2	1	no	—
158	C₈×D₄	—	2	1	no	—
159	C₈×D₄	C₄×C₂₃	1	1	yes	characteristic, complemented
160	#433	—	2	1	no	—
161	#432	C₂²×C₂₃	1	2	yes	—
162	#432	C₂²×C₂₃	1	1	yes	characteristic
163	C₂×#106	—	2	1	no	—
164	#405	—	2	1	no	—
165	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
166	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
167	#340	—	2	1	no	—
168	#335	—	2	1	no	—
169	#334	—	2	1	no	—
170	#328	—	2	2	no	—
171	#327	—	2	2	no	—
172	C₂×C₂₃	—	2	2	no	—
173	C₂×C₂₃	—	8	1	no	—
174	C₂×C₂₃	#1806	1	1	yes	characteristic
175	C₂×C₂₃	#1812	1	1	yes	characteristic
176	C₂×C₂₃	#1815	1	1	yes	characteristic
177	C₂×M₁₆	—	2	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
181	#126	—	2	1	no	—
182	#125	Q₈×C₂₃	1	1	yes	characteristic
183	#124	Q₈×C₂₃	1	1	yes	characteristic
184	#123	—	2	1	no	—
185	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
186	#121	D₄×C₂₃	1	1	yes	characteristic
187	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
190	#117	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
194	#106	—	2	4	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
199	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
200	C₂²⋊C₈	—	2	1	no	—
201	C₂²⋊C₈	—	2	1	no	—
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	#97	—	2	1	no	—
205	#97	—	2	1	no	—
206	#97	—	2	2	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
209	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
210	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
211	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
212	#95	—	2	2	no	—
213	C₂₃	#8694	1	1	yes	characteristic, direct product factor, Sylow
214	C₂×Q₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
240	C₄⋊C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	2	1	no	—
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	8	2	no	—
248	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
250	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
251	C₄²	#41×C₂₃	1	1	yes	characteristic
252	C₂³	—	2	1	no	—
253	C₂³	C₂₃×#121	1	1	yes	characteristic
254	Q₈	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
272	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
273	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
274	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	8	2	no	—
281	C₈	C₂₃×#104	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	C₂₃×#380	1	1	yes	characteristic
288	C₂²	C₂₃×#471	1	2	yes	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	8	4	no	—
298	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
299	C₄	C₂₃×#421	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	8	1	no	—
302	C₂	C₂₃×#1806	1	1	yes	characteristic
303	C₂	C₂₃×#1812	1	1	yes	characteristic
304	C₂	C₂₃×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#380
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂₃×#121	D₄
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
2. C₄×C₂₃×#106	C₂	semidirect
3. C₂₃×#1568	C₂	semidirect
5. C₂₃×#1285	C₂	semidirect
7. C₂₃×#446	C₄	semidirect
8. C₈×Q₈×C₂₃	C₄	semidirect
9. C₂₃×#436	C₄	semidirect
12. C₈×D₄×C₂₃	C₄	semidirect
99. #8694	C₂₃	direct (N × H)
131. C₄×#106	C₂×C₂₃	semidirect
132. #1568	C₂×C₂₃	semidirect
134. #1285	C₂×C₂₃	semidirect
154. #446	C₄×C₂₃	semidirect
155. C₈×Q₈	C₄×C₂₃	semidirect
156. #436	C₄×C₂₃	semidirect
159. C₈×D₄	C₄×C₂₃	semidirect
213. C₂₃	#8694	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×#2966	2
C₄×C₂₃×#106	2
C₂₃×#1568	2
C₂₃×#1509	2
C₂₃×#1285	2
#8694	23

Subgroups — C23×#8694 #19156320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8694 #19156320