Subgroups — #15686094

Order 2944.

348 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093150	C₂	1	1	yes	maximal, characteristic
2	#13093143	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092772	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092770	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092762	C₂	1	1	yes	maximal, characteristic
6	#13092717	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092716	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092701	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092639	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092637	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092615	C₂	1	1	yes	maximal, characteristic
12	C₈×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092509	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092502	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
20	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
21	C₂×#67275	C₂²	1	1	yes	characteristic
22	C₂×#67274	C₂²	1	1	yes	characteristic
23	#10625111	C₂²	1	1	yes	characteristic
24	#10625107	C₂²	1	1	yes	characteristic
25	#10625100	—	2	1	no	—
26	C₄×C₂²⋊D₂₃	—	2	1	no	—
27	#10625064	C₂²	1	1	yes	characteristic
28	#10625061	—	2	1	no	—
29	#10625061	C₂²	1	1	yes	characteristic
30	#10625059	—	2	1	no	—
31	#10625059	C₂²	1	1	yes	characteristic
32	#10625058	C₂²	1	1	yes	characteristic
33	#10625056	—	2	1	no	—
34	#10625054	—	2	1	no	—
35	#10625046	C₂²	1	1	yes	characteristic
36	#10625041	—	2	1	no	—
37	#10625041	C₂²	1	1	yes	characteristic
38	C₄×Dic₄₆	—	2	1	no	—
39	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
40	C₂₃×#106	C₂²	1	1	yes	characteristic
41	C₂₃×#103	C₂²	1	1	yes	characteristic
42	C₂₃×#102	C₂²	1	1	yes	characteristic
43	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
44	#10625014	C₂²	1	1	yes	characteristic
45	#10625014	C₂²	1	1	yes	characteristic
46	#10625003	C₂²	1	1	yes	characteristic
47	#10625003	C₂²	1	1	yes	characteristic
48	#10624998	C₂²	1	1	yes	characteristic
49	#10624998	C₂²	1	1	yes	characteristic
50	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
53	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
54	#10624992	C₂²	1	1	yes	characteristic
55	#10624991	C₂²	1	1	yes	characteristic
56	#10624990	C₂²	1	1	yes	characteristic
57	#10624989	C₂²	1	1	yes	characteristic
58	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂×C₂²⋊D₂₃	—	2	1	no	—
63	C₂×C₂²⋊D₂₃	—	2	1	no	—
64	C₂×D₉₂	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₂₃	—	2	1	no	—
66	C₂×Dic₄₆	—	2	1	no	—
67	C₂×Dic₄₆	D₄	1	1	yes	characteristic
68	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
69	SD₁₆×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₂₃	—	2	1	no	—
76	C₄²×C₂₃	—	2	1	no	—
77	C₄²×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₂₃	—	2	1	no	—
79	C₂²⋊Dic₂₃	—	2	1	no	—
80	#67275	—	2	2	no	—
81	#67274	—	2	2	no	—
82	(C₂×D₂₃)⋊C₄	—	2	1	no	—
83	(C₂×D₂₃)⋊C₄	—	2	1	no	—
84	(C₂×D₂₃)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₂₃	—	2	1	no	—
86	C₄⋊Dic₂₃	—	2	1	no	—
87	C₄⋊Dic₂₃	—	2	1	no	—
88	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
89	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
91	Dic₂₃⋊C₄	—	2	1	no	—
92	Dic₂₃⋊C₄	—	2	1	no	—
93	Dic₂₃⋊C₄	—	2	1	no	—
94	Dic₂₃⋊C₄	—	2	1	no	—
95	Dic₂₃⋊C₄	—	2	1	no	—
96	Dic₂₃⋊C₄	D₄	1	1	yes	characteristic
97	Dic₂₃⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₂₃	—	2	1	no	—
99	C₄×Dic₂₃	—	2	1	no	—
100	C₄×Dic₂₃	—	2	1	no	—
101	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
102	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂³×C₂₃	—	2	1	no	—
106	C₂²×D₂₃	—	2	1	no	—
107	Q₈×C₂₃	—	2	1	no	—
108	Q₈×C₂₃	#41	1	2	yes	—
109	D₄×C₂₃	—	2	1	no	—
110	D₄×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	—	8	1	no	—
115	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₂₃	—	4	2	no	—
119	C₂²⋊D₂₃	—	8	1	no	—
120	C₂×Dic₂₃	—	2	1	no	—
121	C₂×Dic₂₃	—	2	1	no	—
122	C₂×Dic₂₃	—	2	1	no	—
123	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
127	D₉₂	—	2	1	no	—
128	D₉₂	—	2	1	no	—
129	C₄×D₂₃	—	8	1	no	—
130	Dic₄₆	—	2	2	no	—
131	Dic₄₆	—	2	1	no	—
132	Dic₄₆	—	2	1	no	—
133	Dic₄₆	—	4	2	no	—
134	C₈×C₂₃	—	2	1	no	—
135	C₈×C₂₃	—	2	1	no	—
136	C₂₃⋊C₈	—	2	1	no	—
137	C₂₃⋊C₈	—	2	1	no	—
138	#3365	—	23	1	no	maximal, Sylow
139	C₂²×C₂₃	—	4	1	no	—
140	C₂²×C₂₃	—	4	1	no	—
141	C₂²×C₂₃	—	4	1	no	—
142	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₂₃	—	4	1	no	—
144	C₂×D₂₃	—	4	1	no	—
145	C₂×D₂₃	—	4	1	no	—
146	C₄×C₂₃	—	2	2	no	—
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	C₂×#41	1	1	yes	characteristic
152	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₂₃	—	2	1	no	—
154	Dic₂₃	—	2	1	no	—
155	Dic₂₃	—	2	1	no	—
156	Dic₂₃	—	4	1	no	—
157	Dic₂₃	—	4	1	no	—
158	Dic₂₃	—	4	1	no	—
159	Dic₂₃	#135	1	2	yes	—
160	#559	—	23	1	no	—
161	#557	—	23	1	no	—
162	#488	—	23	1	no	—
163	#488	—	23	1	no	—
164	#486	—	23	1	no	—
165	#478	—	23	1	no	—
166	#478	—	23	1	no	—
167	#475	—	23	1	no	—
168	#464	—	23	1	no	—
169	#464	—	23	1	no	—
170	#460	—	23	1	no	—
171	C₈×Q₈	—	23	1	no	—
172	C₄×Q₁₆	—	23	1	no	—
173	C₄×SD₁₆	—	23	1	no	—
174	C₄×SD₁₆	D₂₃	1	1	yes	characteristic, complemented
175	C₂×C₂₃	—	8	1	no	—
176	C₂×C₂₃	#548	1	1	yes	characteristic
177	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
178	C₂×C₂₃	#577	1	1	yes	characteristic
179	D₂₃	—	8	1	no	—
180	C₂×Q₁₆	—	23	1	no	—
181	C₂×SD₁₆	—	23	1	no	—
182	C₂×SD₁₆	C₂×D₂₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	23	1	no	—
184	C₄⋊Q₈	—	46	1	no	—
185	#127	—	23	1	no	—
186	#125	—	23	1	no	—
187	#125	—	23	1	no	—
188	#125	—	46	1	no	—
189	#125	—	46	1	no	—
190	#124	—	23	1	no	—
191	#124	—	23	1	no	—
192	#124	—	46	1	no	—
193	#123	—	46	1	no	—
194	#123	—	46	1	no	—
195	C₄×Q₈	—	23	1	no	—
196	C₄×Q₈	—	23	1	no	—
197	C₄×Q₈	—	23	1	no	—
198	C₄×Q₈	—	23	1	no	—
199	C₄×Q₈	—	46	1	no	—
200	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
201	C₄×D₄	—	23	1	no	—
202	C₄×D₄	—	46	1	no	—
203	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
204	#107	—	23	1	no	—
205	#106	—	23	1	no	—
206	#106	C₂×D₂₃	1	1	yes	characteristic
207	C₄⋊C₈	—	23	1	no	—
208	C₄⋊C₈	—	23	1	no	—
209	C₄⋊C₈	—	23	1	no	—
210	#103	—	23	1	no	—
211	#103	—	23	1	no	—
212	#103	—	23	1	no	—
213	#103	C₂×D₂₃	1	1	yes	characteristic
214	#102	—	23	1	no	—
215	#102	—	23	1	no	—
216	#102	—	23	1	no	—
217	#102	—	23	1	no	—
218	#102	—	23	1	no	—
219	#102	C₂×D₂₃	1	1	yes	characteristic
220	C₄×C₈	—	23	1	no	—
221	C₄×C₈	—	23	1	no	—
222	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
223	C₂₃	#3365	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₈	—	23	1	no	—
225	C₂×Q₈	—	23	1	no	—
226	C₂×Q₈	—	46	1	no	—
227	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
228	C₂×D₄	—	23	1	no	—
229	C₂×D₄	—	46	1	no	—
230	C₂×D₄	—	46	1	no	—
231	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	46	1	no	—
234	Q₁₆	—	46	2	no	—
235	SD₁₆	—	2	2	no	—
236	SD₁₆	—	46	2	no	—
237	C₂×C₈	—	23	1	no	—
238	C₂×C₈	—	23	1	no	—
239	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	23	1	no	—
243	C₄⋊C₄	—	23	1	no	—
244	C₄⋊C₄	—	23	1	no	—
245	C₄⋊C₄	—	23	1	no	—
246	C₄⋊C₄	—	23	1	no	—
247	C₄⋊C₄	—	46	1	no	—
248	C₄⋊C₄	—	46	1	no	—
249	C₄⋊C₄	—	46	1	no	—
250	C₄⋊C₄	—	46	1	no	—
251	C₄⋊C₄	—	46	1	no	—
252	C₄⋊C₄	—	46	1	no	—
253	C₄⋊C₄	—	46	1	no	—
254	C₄⋊C₄	—	46	1	no	—
255	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	46	1	no	—
259	C₂²⋊C₄	—	46	1	no	—
260	C₂²⋊C₄	—	46	1	no	—
261	C₂²⋊C₄	—	46	1	no	—
262	C₂²⋊C₄	—	46	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	23	1	no	—
265	C₄²	—	23	1	no	—
266	C₄²	—	46	1	no	—
267	C₄²	—	46	1	no	—
268	C₄²	—	46	1	no	—
269	C₄²	C₂²×D₂₃	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	46	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	46	2	no	—
274	Q₈	—	46	1	no	—
275	Q₈	—	46	1	no	—
276	Q₈	—	92	2	no	—
277	Q₈	#67288	1	2	yes	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	46	1	no	—
281	D₄	—	46	1	no	—
282	D₄	—	92	2	no	—
283	D₄	—	184	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	23	1	no	—
289	C₂×C₄	—	23	1	no	—
290	C₂×C₄	—	23	1	no	—
291	C₂×C₄	—	23	1	no	—
292	C₂×C₄	—	46	1	no	—
293	C₂×C₄	—	46	1	no	—
294	C₂×C₄	—	46	1	no	—
295	C₂×C₄	—	184	1	no	—
296	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
297	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	46	1	no	—
302	C₈	—	46	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	92	1	no	—
307	C₂²	—	92	1	no	—
308	C₂²	—	92	1	no	—
309	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	23	2	no	—
316	C₄	—	46	1	no	—
317	C₄	—	46	1	no	—
318	C₄	—	46	1	no	—
319	C₄	—	92	1	no	—
320	C₄	—	92	1	no	—
321	C₄	—	92	1	no	—
322	C₄	C₂×#67288	1	1	yes	characteristic
323	C₄	#10625161	1	1	yes	characteristic
324	C₂	—	8	1	no	—
325	C₂	—	184	1	no	—
326	C₂	#13093076	1	1	yes	characteristic
327	C₂	#135×D₂₃	1	1	yes	characteristic
328	C₂	#13093223	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₄×SD₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	#10625111	C₂²
hypercenter	C₄×SD₁₆	D₂₃
hypocenter	C₂₃	#3365
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. #13093143	C₂	semidirect
3. #13092772	C₂	semidirect
4. #13092770	C₂	semidirect
6. #13092717	C₂	semidirect
7. #13092716	C₂	semidirect
8. #13092701	C₂	semidirect
9. #13092639	C₂	semidirect
10. #13092637	C₂	semidirect
12. C₈×Dic₄₆	C₂	semidirect
13. #13092509	C₂	semidirect
14. #13092502	C₂	semidirect
15. C₄×SD₁₆×C₂₃	C₂	semidirect
174. C₄×SD₁₆	D₂₃	semidirect
223. C₂₃	#3365	semidirect

Maximal subgroups

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

subgroup	index
#13093150	2
#13093143	2
#13092772	2
#13092770	2
#13092762	2
#13092717	2
#13092716	2
#13092701	2
#13092639	2
#13092637	2
#13092615	2
C₈×Dic₄₆	2
#13092509	2
#13092502	2
C₄×SD₁₆×C₂₃	2
#3365	23