Subgroups — #15683029

Order 2944.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625122	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#67267	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625095	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092673	C₂	1	2	yes	maximal, complemented
5	#13092661	C₂	1	2	yes	maximal, complemented
6	#13092656	C₂	1	2	yes	maximal, complemented
7	#13092644	C₂	1	2	yes	maximal, complemented
8	#13092460	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092425	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092418	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#525	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
15	C₂₃×#123	—	2	1	no	—
16	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂₃×#121	C₂²	1	1	yes	characteristic
19	C₂₃×#121	C₂²	1	1	yes	characteristic
20	C₂₃×#120	—	2	1	no	—
21	C₄×D₄×C₂₃	—	2	1	no	—
22	C₄×D₄×C₂₃	—	2	1	no	—
23	C₂₃×#117	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
25	#10625122	—	2	4	no	—
26	C₂×#67275	C₂²	1	2	yes	complemented
27	C₂×#67273	C₂²	1	2	yes	complemented
28	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	#10625095	—	2	1	no	—
30	#10625095	—	2	1	no	—
31	#10625095	C₂²	1	2	yes	—
32	#10625095	C₂²	1	2	yes	—
33	C₂×Dic₂₃⋊C₄	—	2	1	no	—
34	C₂×#67267	—	2	4	no	—
35	C₂×#67267	C₂²	1	2	yes	—
36	C₂×#67267	C₂²	1	1	yes	characteristic
37	C₂×#67267	C₂²	1	1	yes	characteristic
38	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
39	#10625017	C₂²	1	2	yes	complemented
40	#10625014	C₂²	1	2	yes	complemented
41	#10624992	C₂²	1	2	yes	complemented
42	#10624992	C₂²	1	2	yes	complemented
43	#10624990	C₂²	1	2	yes	complemented
44	#10624989	C₂²	1	2	yes	complemented
45	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
46	#41×C₂₃	—	2	2	no	—
47	#41×C₂₃	—	2	2	no	—
48	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	—	2	1	no	—
51	C₂×D₄×C₂₃	—	2	1	no	—
52	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Dic₂₃	—	2	1	no	—
63	C₂²×Dic₂₃	—	2	1	no	—
64	C₂×Dic₄₆	—	2	2	no	—
65	C₂×Dic₄₆	—	2	2	no	—
66	C₂×Dic₄₆	—	2	1	no	—
67	C₂×Dic₄₆	C₂³	1	2	yes	—
68	C₄⋊C₄×C₂₃	—	2	1	no	—
69	C₄⋊C₄×C₂₃	—	2	1	no	—
70	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₂²⋊C₄×C₂₃	—	2	1	no	—
75	C₂²⋊C₄×C₂₃	—	2	1	no	—
76	C₂²⋊C₄×C₂₃	—	2	1	no	—
77	C₄²×C₂₃	—	2	1	no	—
78	C₂²⋊Dic₂₃	—	2	2	no	—
79	C₂²⋊Dic₂₃	—	2	2	no	—
80	#67275	—	2	4	no	—
81	#67273	—	2	4	no	—
82	C₄⋊Dic₂₃	—	2	1	no	—
83	C₄⋊Dic₂₃	C₂³	1	2	yes	—
84	Dic₂₃⋊C₄	—	2	2	no	—
85	Dic₂₃⋊C₄	—	2	2	no	—
86	#67267	—	2	2	no	—
87	#67267	—	2	4	no	—
88	#67267	D₄	1	4	yes	complemented
89	C₂×C₂₃⋊C₈	—	2	4	no	—
90	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
91	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	—	2	1	no	—
95	C₂³×C₂₃	—	4	1	no	—
96	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
99	Q₈×C₂₃	—	2	1	no	—
100	Q₈×C₂₃	—	2	1	no	—
101	D₄×C₂₃	—	2	1	no	—
102	D₄×C₂₃	—	2	1	no	—
103	D₄×C₂₃	—	4	2	no	—
104	D₄×C₂₃	—	4	2	no	—
105	D₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	#41	1	2	yes	—
122	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₂₃	—	2	2	no	—
124	C₂×Dic₂₃	—	2	2	no	—
125	C₂×Dic₂₃	—	4	2	no	—
126	C₂×Dic₂₃	—	4	2	no	—
127	Dic₄₆	—	2	2	no	—
128	Dic₄₆	—	2	2	no	—
129	Dic₄₆	—	4	2	no	—
130	C₂₃⋊C₈	—	2	2	no	—
131	C₂₃⋊C₈	C₂×D₄	1	4	yes	—
132	#3056	—	23	1	no	maximal, Sylow
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	—	2	1	no	—
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	—	2	1	no	—
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	—	4	4	no	—
141	C₂²×C₂₃	—	4	1	no	—
142	C₂²×C₂₃	—	4	1	no	—
143	C₂²×C₂₃	—	4	1	no	—
144	C₂²×C₂₃	#121	1	2	yes	—
145	C₂²×C₂₃	#137	1	2	yes	complemented
146	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₂₃	—	2	1	no	—
148	C₄×C₂₃	—	2	1	no	—
149	C₄×C₂₃	—	4	1	no	—
150	C₄×C₂₃	—	4	1	no	—
151	C₄×C₂₃	—	4	1	no	—
152	C₄×C₂₃	#121	1	2	yes	—
153	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
154	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
155	Dic₂₃	—	4	2	no	—
156	Dic₂₃	—	4	2	no	—
157	C₂×#137	—	23	1	no	—
158	C₂²×M₁₆	—	23	1	no	—
159	#525	D₂₃	1	1	yes	characteristic
160	C₂×C₂²⋊Q₈	—	23	1	no	—
161	#470	—	23	2	no	—
162	#468	—	23	2	no	—
163	#467	—	23	2	no	—
164	#465	—	23	2	no	—
165	#428	—	23	1	no	—
166	#419	—	23	1	no	—
167	#417	—	23	1	no	—
168	C₂×C₂₃	—	2	2	no	—
169	C₂×C₂₃	—	2	2	no	—
170	C₂×C₂₃	—	4	1	no	—
171	C₂×C₂₃	—	8	1	no	—
172	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
173	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
174	C₂×C₂₃	#575	1	1	yes	characteristic
175	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
176	C₂²×Q₈	—	23	1	no	—
177	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
178	#137	—	46	4	no	—
179	C₂×Q₁₆	—	23	2	no	—
180	C₂×SD₁₆	—	23	2	no	—
181	C₂×M₁₆	—	23	1	no	—
182	C₂×M₁₆	—	23	1	no	—
183	C₂×M₁₆	—	23	2	no	—
184	C₂×M₁₆	—	46	4	no	—
185	C₂²×C₈	—	23	2	no	—
186	#123	—	2	1	no	—
187	C₂²⋊Q₈	—	23	2	no	—
188	C₂²⋊Q₈	—	23	2	no	—
189	C₂²⋊Q₈	—	46	1	no	—
190	C₂²⋊Q₈	—	46	1	no	—
191	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
192	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
193	#121	C₂×D₂₃	1	1	yes	characteristic
194	#121	C₂×D₂₃	1	1	yes	characteristic
195	#120	—	2	1	no	—
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	—	2	1	no	—
198	#117	C₂×D₂₃	1	1	yes	characteristic
199	C₂×C₄⋊C₄	—	23	1	no	—
200	C₂×C₄⋊C₄	—	46	1	no	—
201	C₂×C₂²⋊C₄	—	46	1	no	—
202	#107	—	23	2	no	—
203	#106	—	23	2	no	—
204	#103	—	23	2	no	—
205	#103	—	23	2	no	—
206	#103	—	23	2	no	—
207	#102	—	23	2	no	—
208	C₂₃	#3056	1	1	yes	characteristic, complemented, Sylow
209	C₂⁴	C₂²⋊D₂₃	1	1	yes	characteristic
210	#41	—	2	2	no	—
211	#41	—	2	2	no	—
212	C₂×Q₈	—	23	2	no	—
213	C₂×Q₈	—	46	1	no	—
214	C₂×Q₈	—	46	2	no	—
215	C₂×Q₈	—	46	2	no	—
216	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	46	1	no	—
225	C₂²×C₄	—	46	1	no	—
226	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
232	Q₁₆	—	46	4	no	—
233	SD₁₆	—	46	4	no	—
234	M₁₆	—	23	4	no	—
235	M₁₆	—	46	4	no	—
236	M₁₆	—	46	2	no	—
237	C₂×C₈	—	23	2	no	—
238	C₂×C₈	—	23	2	no	—
239	C₂×C₈	—	46	4	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	23	2	no	—
243	C₄⋊C₄	—	46	1	no	—
244	C₄⋊C₄	—	46	2	no	—
245	C₄⋊C₄	—	46	2	no	—
246	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
247	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	46	2	no	—
254	C₂²⋊C₄	—	46	2	no	—
255	C₄²	—	2	1	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
261	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
262	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
263	Q₈	—	2	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	46	2	no	—
266	Q₈	—	46	2	no	—
267	Q₈	—	92	2	no	—
268	D₄	—	2	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	46	2	no	—
285	C₂×C₄	—	46	2	no	—
286	C₂×C₄	—	92	2	no	—
287	C₂×C₄	—	92	2	no	—
288	C₂×C₄	D₄×D₂₃	1	2	yes	—
289	C₂×C₄	#67290	1	2	yes	—
290	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
294	C₈	—	23	4	no	—
295	C₈	—	46	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	#10625110	1	2	yes	—
308	C₂²	#10625122	1	2	yes	complemented
309	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	92	2	no	—
316	C₄	—	92	2	no	—
317	C₄	#10625110	1	2	yes	—
318	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
319	C₄	C₂×#67290	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	8	1	no	—
324	C₂	C₂×#10625110	1	1	yes	characteristic
325	C₂	C₂×#10625122	1	1	yes	characteristic
326	C₂	#13093215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting, prime radical	C₂₃×#525	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	#525	D₂₃
hypocenter	C₂₃	#3056
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#10625122	C₂	semidirect
2. C₂²×#67267	C₂	semidirect
3. C₂×#10625095	C₂	semidirect
4. #13092673	C₂	semidirect
5. #13092661	C₂	semidirect
6. #13092656	C₂	semidirect
7. #13092644	C₂	semidirect
8. #13092460	C₂	semidirect
9. #13092425	C₂	semidirect
10. #13092418	C₂	semidirect
26. C₂×#67275	C₂²	semidirect
27. C₂×#67273	C₂²	semidirect
39. #10625017	C₂²	semidirect
40. #10625014	C₂²	semidirect
41. #10624992	C₂²	semidirect
42. #10624992	C₂²	semidirect
43. #10624990	C₂²	semidirect
44. #10624989	C₂²	semidirect
88. #67267	D₄	semidirect
145. C₂²×C₂₃	#137	semidirect
208. C₂₃	#3056	semidirect
308. C₂²	#10625122	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625122	2
C₂²×#67267	2
C₂×#10625095	2
#13092673	2
#13092661	2
#13092656	2
#13092644	2
#13092460	2
#13092425	2
#13092418	2
C₂₃×#525	2
#3056	23