Subgroups — #15682056

Order 2944.

402 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625043	C₂	1	1	yes	maximal, characteristic
2	#13092479	C₂	1	2	yes	maximal, complemented
3	#13092477	C₂	1	2	yes	maximal, complemented
4	#13092471	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092470	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092467	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092442	C₂	1	2	yes	maximal, complemented
8	#13092379	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
10	#13092374	C₂	1	2	yes	maximal, complemented
11	C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×(C₂×D₂₃)⋊C₄	C₄	1	1	yes	characteristic
15	C₂×(C₂×D₂₃)⋊C₄	C₄	1	1	yes	characteristic
16	C₂×Dic₂₃⋊C₄	C₄	1	1	yes	characteristic
17	C₂×Dic₂₃⋊C₄	C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
19	C₂×#67262	C₂²	1	2	yes	—
20	C₂×C₈×D₂₃	C₂²	1	2	yes	—
21	#10625043	C₄	1	1	yes	characteristic
22	#10625043	C₄	1	1	yes	characteristic
23	#10625043	C₄	1	1	yes	characteristic
24	#10625043	C₄	1	1	yes	characteristic
25	#10625043	C₂²	1	1	yes	characteristic
26	#10625043	C₂²	1	1	yes	characteristic
27	#10625043	C₂²	1	1	yes	characteristic
28	#10625043	C₂²	1	1	yes	characteristic
29	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
30	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂₃×#97	C₂²	1	2	yes	complemented
33	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
34	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
35	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
36	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
37	#10624997	C₂²	1	2	yes	complemented
38	Dic₂₃⋊C₈	C₂²	1	2	yes	complemented
39	Dic₂₃⋊C₈	C₂²	1	2	yes	complemented
40	C₈×Dic₂₃	C₂²	1	2	yes	complemented
41	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
42	C₄×C₂₃⋊C₈	C₂²	1	2	yes	complemented
43	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₂₃	—	2	4	no	—
51	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂×C₈×C₂₃	C₂³	1	2	yes	—
57	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₂₃	C₂³	1	1	yes	characteristic
60	C₄²×C₂₃	C₂³	1	1	yes	characteristic
61	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
62	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
63	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
64	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
65	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
66	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
67	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
68	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
69	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
70	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
71	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
72	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
73	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
74	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
75	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
76	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
80	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
82	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
83	#67262	—	2	4	no	—
84	C₈×D₂₃	—	2	4	no	—
85	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₂₃	—	2	2	no	—
87	C₂²×D₂₃	—	2	2	no	—
88	C₂²×D₂₃	—	4	1	no	—
89	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	4	1	no	—
94	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₂₃	—	2	2	no	—
101	C₂×Dic₂₃	—	2	2	no	—
102	C₂×Dic₂₃	—	4	1	no	—
103	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₂₃	—	2	2	no	—
110	C₄×D₂₃	—	2	2	no	—
111	C₄×D₂₃	—	2	2	no	—
112	C₄×D₂₃	#41	1	4	yes	—
113	C₈×C₂₃	—	2	2	no	—
114	C₈×C₂₃	—	2	2	no	—
115	C₂₃⋊C₈	—	2	2	no	—
116	C₂₃⋊C₈	—	2	2	no	—
117	#2924	—	23	1	no	maximal, Sylow
118	C₂²×C₂₃	—	2	1	no	—
119	C₂²×C₂₃	—	2	1	no	—
120	C₂²×C₂₃	#131	1	2	yes	—
121	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×D₂₃	—	2	1	no	—
123	C₂×D₂₃	—	2	1	no	—
124	C₂×D₂₃	—	2	1	no	—
125	C₂×D₂₃	—	2	1	no	—
126	C₂×D₂₃	—	2	1	no	—
127	C₂×D₂₃	—	4	4	no	—
128	C₂×D₂₃	#117	1	2	yes	—
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	C₂×#41	1	2	yes	—
135	Dic₂₃	—	2	1	no	—
136	Dic₂₃	—	2	1	no	—
137	Dic₂₃	—	2	1	no	—
138	Dic₂₃	—	2	1	no	—
139	Dic₂₃	—	2	1	no	—
140	Dic₂₃	#117	1	2	yes	—
141	C₂×#117	—	23	1	no	—
142	#433	—	23	2	no	—
143	#433	—	23	2	no	—
144	#432	—	23	1	no	—
145	#432	D₂₃	1	1	yes	characteristic, complemented
146	#431	—	23	1	no	—
147	#431	—	23	1	no	—
148	#424	—	23	2	no	—
149	#407	—	23	1	no	—
150	C₂×C₂²⋊C₈	—	23	1	no	—
151	#405	—	23	2	no	—
152	C₂×C₂₃	—	2	2	no	—
153	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
154	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
155	C₂×C₂₃	#568	1	1	yes	characteristic
156	D₂₃	—	2	2	no	—
157	D₂₃	—	4	1	no	—
158	C₂³×C₄	—	23	1	no	—
159	C₂×M₁₆	—	23	2	no	—
160	C₂²×C₈	—	23	2	no	—
161	#117	—	23	1	no	—
162	#117	—	23	1	no	—
163	#117	—	23	1	no	—
164	#117	—	23	1	no	—
165	#117	—	23	1	no	—
166	#117	—	23	1	no	—
167	#117	—	23	1	no	—
168	#117	—	23	1	no	—
169	C₂×C₄⋊C₄	—	23	1	no	—
170	C₂×C₄⋊C₄	—	23	1	no	—
171	C₂×C₂²⋊C₄	—	23	1	no	—
172	C₂×C₂²⋊C₄	—	23	1	no	—
173	C₂×C₄²	—	23	1	no	—
174	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
175	C₄⋊C₈	—	23	2	no	—
176	C₄⋊C₈	—	23	2	no	—
177	C₄⋊C₈	—	23	2	no	—
178	C₄⋊C₈	C₂×D₂₃	1	2	yes	complemented
179	C₂²⋊C₈	—	23	1	no	—
180	C₂²⋊C₈	—	23	1	no	—
181	C₂²⋊C₈	—	23	2	no	—
182	C₂²⋊C₈	—	23	2	no	—
183	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
184	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
185	#97	—	23	2	no	—
186	#97	C₂×D₂₃	1	2	yes	complemented
187	C₄×C₈	—	23	2	no	—
188	C₄×C₈	—	23	2	no	—
189	C₂₃	#2924	1	1	yes	characteristic, complemented, Sylow
190	C₂⁴	—	23	1	no	—
191	C₂²×C₄	—	23	1	no	—
192	C₂²×C₄	—	23	1	no	—
193	C₂²×C₄	—	23	1	no	—
194	C₂²×C₄	—	23	1	no	—
195	C₂²×C₄	—	23	1	no	—
196	C₂²×C₄	—	23	1	no	—
197	C₂²×C₄	—	23	1	no	—
198	C₂²×C₄	—	46	4	no	—
199	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
202	M₁₆	—	46	4	no	—
203	C₂×C₈	—	23	2	no	—
204	C₂×C₈	—	23	2	no	—
205	C₂×C₈	—	46	4	no	—
206	C₂×C₈	C₂²×D₂₃	1	2	yes	—
207	C₂×C₈	C₂²×D₂₃	1	2	yes	—
208	C₄⋊C₄	—	23	1	no	—
209	C₄⋊C₄	—	23	1	no	—
210	C₄⋊C₄	—	23	1	no	—
211	C₄⋊C₄	—	23	1	no	—
212	C₄⋊C₄	—	23	1	no	—
213	C₄⋊C₄	—	23	1	no	—
214	C₄⋊C₄	—	23	1	no	—
215	C₄⋊C₄	—	23	1	no	—
216	C₂²⋊C₄	—	23	1	no	—
217	C₂²⋊C₄	—	23	1	no	—
218	C₂²⋊C₄	—	23	1	no	—
219	C₂²⋊C₄	—	23	1	no	—
220	C₂²⋊C₄	—	23	1	no	—
221	C₂²⋊C₄	—	23	1	no	—
222	C₂²⋊C₄	—	23	1	no	—
223	C₂²⋊C₄	—	23	1	no	—
224	C₄²	—	23	1	no	—
225	C₄²	—	23	1	no	—
226	C₄²	—	23	1	no	—
227	C₄²	—	23	1	no	—
228	C₄²	C₄×D₂₃	1	1	yes	characteristic
229	C₄²	C₄×D₂₃	1	1	yes	characteristic
230	C₄²	C₂²×D₂₃	1	1	yes	characteristic
231	C₄²	C₂²×D₂₃	1	1	yes	characteristic
232	C₂³	—	23	1	no	—
233	C₂³	—	23	1	no	—
234	C₂³	—	46	2	no	—
235	C₂³	—	46	2	no	—
236	C₂³	—	92	1	no	—
237	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	23	1	no	—
242	C₂×C₄	—	23	4	no	—
243	C₂×C₄	—	23	1	no	—
244	C₂×C₄	—	23	1	no	—
245	C₂×C₄	—	23	1	no	—
246	C₂×C₄	—	23	1	no	—
247	C₂×C₄	—	23	1	no	—
248	C₂×C₄	—	46	2	no	—
249	C₂×C₄	—	46	2	no	—
250	C₂×C₄	—	46	2	no	—
251	C₂×C₄	—	46	2	no	—
252	C₂×C₄	—	46	2	no	—
253	C₂×C₄	—	92	1	no	—
254	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	46	2	no	—
263	C₈	—	46	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	23	2	no	—
267	C₂²	—	46	1	no	—
268	C₂²	—	46	1	no	—
269	C₂²	—	46	1	no	—
270	C₂²	—	46	1	no	—
271	C₂²	—	46	1	no	—
272	C₂²	—	92	4	no	—
273	C₂²	#10625078	1	2	yes	—
274	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	23	2	no	—
281	C₄	—	46	1	no	—
282	C₄	—	46	1	no	—
283	C₄	—	46	1	no	—
284	C₄	—	46	1	no	—
285	C₄	—	46	1	no	—
286	C₄	#41×D₂₃	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	46	2	no	—
289	C₂	—	92	1	no	—
290	C₂	#117×D₂₃	1	1	yes	characteristic
291	C₂	C₂×#10625078	1	1	yes	characteristic
292	C₂	#13093183	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#432	C₂
prime radical	C₂³×D₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#432	D₂₃
hypocenter	C₂₃	#2924
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #13092479	C₂	semidirect
3. #13092477	C₂	semidirect
4. #13092471	C₂	semidirect
5. #13092470	C₂	semidirect
6. #13092467	C₂	semidirect
7. #13092442	C₂	semidirect
10. #13092374	C₂	semidirect
11. C₂₃×#432	C₂	semidirect
29. C₂₃×C₄⋊C₈	C₂²	semidirect
32. C₂₃×#97	C₂²	semidirect
37. #10624997	C₂²	semidirect
38. Dic₂₃⋊C₈	C₂²	semidirect
39. Dic₂₃⋊C₈	C₂²	semidirect
40. C₈×Dic₂₃	C₂²	semidirect
41. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
42. C₄×C₂₃⋊C₈	C₂²	semidirect
145. #432	D₂₃	semidirect
178. C₄⋊C₈	C₂×D₂₃	semidirect
186. #97	C₂×D₂₃	semidirect
189. C₂₃	#2924	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625043	2
#13092479	2
#13092477	2
#13092471	2
#13092470	2
#13092467	2
#13092442	2
#13092379	2
C₂²⋊C₈×D₂₃	2
#13092374	2
C₂₃×#432	2
#2924	23