Subgroups — #14907158

Order 2816.

674 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622532	C₂	1	1	yes	maximal, characteristic
2	#13077491	C₂	1	2	yes	maximal, complemented
3	#13070483	C₂	1	2	yes	maximal, complemented
4	#13070478	C₂	1	2	yes	maximal, complemented
5	Q₃₂×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₁₇₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070462	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₁×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	#13070327	C₂	1	2	yes	maximal, complemented
10	#13070211	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₁₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#66972	C₂²	1	2	yes	—
13	#10623105	—	2	2	no	—
14	C₂×C₁₁×Q₃₂	C₂²	1	1	yes	characteristic
15	C₄×C₁₁×Q₁₆	C₂²	1	2	yes	—
16	Q₁₆×Dic₁₁	C₂²	1	2	yes	—
17	C₂×#66892	C₂²	1	2	yes	—
18	#10622639	—	2	2	no	—
19	#10622532	C₄	1	8	yes	complemented
20	C₂×D₁₇₆	C₂²	1	1	yes	characteristic
21	C₂×#66862	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₁₁	C₂²	1	1	yes	characteristic
23	D₁₁×#107	C₂²	1	2	yes	—
24	#10622485	C₂²	1	2	yes	—
25	#10622430	—	2	2	no	—
26	#10622416	—	2	2	no	—
27	C₄×D₈₈	C₂²	1	2	yes	—
28	#10622302	C₂²	1	1	yes	characteristic
29	C₁₁×#366	C₂²	1	1	yes	characteristic
30	C₁₁×#358	C₂²	1	2	yes	—
31	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
32	#10622202	C₂²	1	2	yes	—
33	#10622147	C₂²	1	2	yes	—
34	#10622142	C₂²	1	1	yes	characteristic
35	#10622140	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₁	C₂²	1	1	yes	characteristic
37	#10622129	C₂²	1	2	yes	—
38	#10622127	C₂²	1	2	yes	—
39	#10622099	C₂²	1	1	yes	characteristic
40	C₂×#4343	—	2	2	no	—
41	C₂×C₁₁×Q₁₆	C₂³	1	2	yes	—
42	C₄×Q₈×C₁₁	—	2	2	no	—
43	Q₈×Dic₁₁	—	2	2	no	—
44	C₂×#4325	—	2	2	no	—
45	#66972	—	2	4	no	—
46	#66972	C₂×C₄	1	8	yes	complemented
47	C₂×D₈₈	C₂³	1	2	yes	—
48	C₂×#4313	D₄	1	1	yes	characteristic
49	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
50	#66946	—	2	2	no	—
51	#66946	—	4	2	no	—
52	C₄⋊C₄×D₁₁	—	2	2	no	—
53	C₄⋊C₄×D₁₁	—	4	2	no	—
54	C₄×D₄₄	—	2	2	no	—
55	C₄×D₄₄	—	4	2	no	—
56	#66925	—	4	2	no	—
57	#66925	D₄	1	1	yes	characteristic
58	C₁₁×Q₃₂	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
61	C₁₁×#107	C₂³	1	2	yes	—
62	C₁₁×#103	—	2	2	no	—
63	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
64	#66899	—	2	2	no	—
65	#66892	C₂×C₄	1	8	yes	complemented
66	#66885	—	2	2	no	—
67	#66882	C₂³	1	2	yes	—
68	#66879	D₄	1	1	yes	characteristic
69	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
70	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
72	#66873	—	2	2	no	—
73	#66871	—	2	2	no	—
74	#66867	D₄	1	1	yes	characteristic
75	D₁₇₆	C₂×C₄	1	4	yes	complemented
76	#66862	—	2	2	no	—
77	C₁₆×D₁₁	C₂×C₄	1	4	yes	complemented
78	#33459	—	11	1	no	maximal, Sylow
79	C₂×Q₈×C₁₁	—	2	2	no	—
80	#4343	—	2	8	no	—
81	#4343	—	4	4	no	—
82	C₂×D₄₄	—	2	2	no	—
83	C₂×D₄₄	—	4	2	no	—
84	C₂×C₄×D₁₁	—	4	2	no	—
85	C₂×C₄×D₁₁	—	4	2	no	—
86	C₂×C₄×D₁₁	—	4	2	no	—
87	C₂×C₄×D₁₁	—	4	2	no	—
88	C₂×C₄×D₁₁	C₂×D₄	1	1	yes	characteristic
89	C₁₁×Q₁₆	—	2	2	no	—
90	C₁₁×Q₁₆	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
93	C₁₁×C₄⋊C₄	—	2	2	no	—
94	C₁₁×C₄⋊C₄	—	4	2	no	—
95	C₄²×C₁₁	—	4	2	no	—
96	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
97	#4325	—	2	8	no	—
98	(C₂×D₁₁)⋊C₄	—	4	2	no	—
99	(C₂×D₁₁)⋊C₄	—	4	2	no	—
100	(C₂×D₁₁)⋊C₄	D₈	1	1	yes	characteristic
101	(C₂×D₁₁)⋊C₄	D₈	1	1	yes	characteristic
102	C₄⋊Dic₁₁	—	2	2	no	—
103	C₄⋊Dic₁₁	—	4	2	no	—
104	Dic₁₁⋊C₄	—	4	2	no	—
105	Dic₁₁⋊C₄	D₈	1	1	yes	characteristic
106	Dic₁₁⋊C₄	D₈	1	1	yes	characteristic
107	C₄×Dic₁₁	—	4	2	no	—
108	C₄×Dic₁₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
111	D₈₈	—	2	2	no	—
112	D₈₈	C₂²×C₄	1	4	yes	—
113	#4313	—	2	2	no	—
114	C₈×D₁₁	C₂²×C₄	1	2	yes	—
115	C₈×D₁₁	#41	1	2	yes	—
116	C₁₁×C₁₆	—	2	1	no	—
117	C₁₁×C₁₆	C₂²×C₄	1	2	yes	—
118	C₁₁⋊C₁₆	—	2	1	no	—
119	C₁₁⋊C₁₆	C₂²×C₄	1	2	yes	—
120	C₂×#508	—	11	1	no	—
121	#2951	—	11	2	no	—
122	#2175	—	11	2	no	—
123	#2175	—	11	2	no	—
124	C₄×Q₃₂	—	11	1	no	—
125	C₄×Q₃₂	D₁₁	1	1	yes	characteristic, complemented
126	C₄×D₁₆	—	11	1	no	—
127	C₄×D₁₆	—	11	1	no	—
128	C₂×#366	—	11	1	no	—
129	#2139	—	11	2	no	—
130	#2108	—	11	1	no	—
131	C₂²×D₁₁	—	4	2	no	—
132	C₂²×D₁₁	C₂×D₈	1	1	yes	characteristic
133	Q₈×C₁₁	—	2	4	no	—
134	Q₈×C₁₁	—	4	2	no	—
135	C₂×C₄×C₁₁	—	4	2	no	—
136	C₂×C₄×C₁₁	—	4	2	no	—
137	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₁₁	—	4	2	no	—
141	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
143	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
144	D₄₄	—	2	4	no	—
145	D₄₄	—	4	8	no	—
146	D₄₄	—	4	2	no	—
147	C₄×D₁₁	—	4	8	no	—
148	C₄×D₁₁	—	16	2	no	—
149	C₄×D₁₁	—	16	2	no	—
150	C₄×D₁₁	—	16	2	no	—
151	C₄×D₁₁	C₄×D₄	1	2	yes	—
152	C₄×D₁₁	#135	1	2	yes	—
153	C₈×C₁₁	—	2	1	no	—
154	C₈×C₁₁	C₂³×C₄	1	1	yes	characteristic
155	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
156	C₁₁⋊C₈	—	2	1	no	—
157	C₁₁⋊C₈	C₄×D₄	1	2	yes	—
158	C₂×#135	—	11	2	no	—
159	#520	—	22	2	no	—
160	#508	—	11	8	no	—
161	C₂×Q₃₂	C₂×D₁₁	1	1	yes	characteristic
162	C₂×SD₃₂	—	11	2	no	—
163	C₂×D₁₆	—	11	1	no	—
164	C₂×M₃₂	—	11	1	no	—
165	C₂²×C₁₆	—	11	1	no	—
166	#440	—	22	2	no	—
167	#440	—	22	2	no	—
168	C₄×Q₁₆	—	11	2	no	—
169	C₄×Q₁₆	C₂×D₁₁	1	2	yes	—
170	C₄×D₈	—	11	2	no	—
171	C₄×D₈	—	11	2	no	—
172	C₂×#107	—	11	2	no	—
173	#417	—	22	2	no	—
174	#405	—	11	1	no	—
175	#366	—	11	1	no	—
176	#366	—	11	2	no	—
177	#366	C₂×D₁₁	1	1	yes	characteristic
178	#358	—	11	2	no	—
179	#358	C₂×D₁₁	1	2	yes	—
180	#357	—	11	2	no	—
181	#357	—	11	2	no	—
182	#346	—	11	1	no	—
183	#346	—	11	1	no	—
184	C₄×C₁₆	—	11	1	no	—
185	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
186	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
187	C₂×D₁₁	—	2	1	no	—
188	C₂×D₁₁	—	2	1	no	—
189	C₂×D₁₁	—	4	4	no	—
190	C₂×D₁₁	—	8	2	no	—
191	C₂×D₁₁	—	8	2	no	—
192	C₂×D₁₁	C₄×D₈	1	2	yes	—
193	C₄×C₁₁	—	2	1	no	—
194	C₄×C₁₁	—	2	1	no	—
195	C₄×C₁₁	—	4	4	no	—
196	C₄×C₁₁	—	8	2	no	—
197	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
198	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
199	Dic₁₁	—	2	1	no	—
200	Dic₁₁	—	2	1	no	—
201	Dic₁₁	—	8	2	no	—
202	Dic₁₁	C₄×D₈	1	2	yes	—
203	C₂×#41	—	22	2	no	—
204	#135	—	11	8	no	—
205	#135	—	22	4	no	—
206	C₂×Q₁₆	C₂²×D₁₁	1	2	yes	—
207	C₂×SD₁₆	—	22	2	no	—
208	C₂×D₈	—	11	2	no	—
209	C₂×M₁₆	—	11	1	no	—
210	C₂²×C₈	—	11	1	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	22	2	no	—
213	C₄×D₄	—	22	2	no	—
214	C₄×D₄	—	22	2	no	—
215	C₄×D₄	—	44	2	no	—
216	C₄×D₄	—	44	2	no	—
217	#117	—	11	1	no	—
218	#117	—	44	2	no	—
219	C₂×C₄⋊C₄	—	22	2	no	—
220	C₂×C₄⋊C₄	—	44	2	no	—
221	Q₃₂	C₄×D₁₁	1	4	yes	complemented
222	SD₃₂	—	11	8	no	—
223	D₁₆	—	11	4	no	—
224	M₃₂	—	22	2	no	—
225	C₂×C₁₆	—	11	1	no	—
226	C₂×C₁₆	—	11	4	no	—
227	C₂×C₁₆	—	11	1	no	—
228	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
230	#107	—	11	2	no	—
231	#107	—	22	2	no	—
232	#107	C₂²×D₁₁	1	2	yes	—
233	#103	—	2	2	no	—
234	#103	—	22	2	no	—
235	#102	—	22	2	no	—
236	#102	—	22	2	no	—
237	#97	—	11	1	no	—
238	#97	—	11	1	no	—
239	C₄×C₈	—	11	1	no	—
240	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
241	C₂×C₁₁	C₂×C₄×D₈	1	1	yes	characteristic
242	C₂×C₁₁	#3415	1	1	yes	characteristic
243	C₂×C₁₁	#3417	1	1	yes	characteristic
244	D₁₁	—	2	2	no	—
245	D₁₁	—	8	4	no	—
246	#41	—	22	8	no	—
247	#41	—	44	4	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×D₄	—	22	2	no	—
250	C₂×D₄	—	44	2	no	—
251	C₂²×C₄	—	11	1	no	—
252	C₂²×C₄	—	44	2	no	—
253	C₂²×C₄	—	44	2	no	—
254	C₂²×C₄	—	44	2	no	—
255	C₂²×C₄	—	44	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	C₂×C₄×D₁₁	1	4	yes	—
258	SD₁₆	—	22	8	no	—
259	D₈	—	11	4	no	—
260	D₈	—	22	2	no	—
261	M₁₆	—	22	2	no	—
262	C₂×C₈	—	11	1	no	—
263	C₂×C₈	—	11	2	no	—
264	C₂×C₈	—	11	2	no	—
265	C₂×C₈	—	11	1	no	—
266	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
267	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	11	1	no	—
271	C₄⋊C₄	—	11	1	no	—
272	C₄⋊C₄	—	22	2	no	—
273	C₄⋊C₄	—	44	2	no	—
274	C₄⋊C₄	—	44	2	no	—
275	C₂²⋊C₄	—	11	1	no	—
276	C₂²⋊C₄	—	11	1	no	—
277	C₂²⋊C₄	—	44	2	no	—
278	C₂²⋊C₄	—	44	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	11	1	no	—
281	C₄²	—	44	2	no	—
282	C₄²	D₄×D₁₁	1	1	yes	characteristic
283	C₁₆	—	2	1	no	—
284	C₁₆	—	11	2	no	—
285	C₁₆	—	22	1	no	—
286	C₁₆	C₂×C₄×D₁₁	1	2	yes	—
287	C₁₁	#33459	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	11	1	no	—
289	C₂³	—	44	2	no	—
290	Q₈	—	2	4	no	—
291	Q₈	—	4	2	no	—
292	D₄	—	22	4	no	—
293	D₄	—	44	8	no	—
294	D₄	—	44	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	11	1	no	—
298	C₂×C₄	—	11	2	no	—
299	C₂×C₄	—	11	2	no	—
300	C₂×C₄	—	11	1	no	—
301	C₂×C₄	—	11	1	no	—
302	C₂×C₄	—	44	8	no	—
303	C₂×C₄	—	44	2	no	—
304	C₂×C₄	—	176	2	no	—
305	C₂×C₄	—	176	2	no	—
306	C₂×C₄	—	176	2	no	—
307	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
308	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	11	2	no	—
312	C₈	—	22	1	no	—
313	C₈	C₂²×C₄×D₁₁	1	1	yes	characteristic
314	C₈	#41×D₁₁	1	1	yes	characteristic
315	C₂²	—	11	2	no	—
316	C₂²	—	22	1	no	—
317	C₂²	—	22	1	no	—
318	C₂²	—	44	4	no	—
319	C₂²	—	88	2	no	—
320	C₂²	—	88	2	no	—
321	C₂²	C₂×D₈×D₁₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	8	2	no	—
326	C₄	—	11	2	no	—
327	C₄	—	22	1	no	—
328	C₄	—	22	1	no	—
329	C₄	—	88	2	no	—
330	C₄	C₄×D₄×D₁₁	1	1	yes	characteristic
331	C₄	D₁₁×#135	1	1	yes	characteristic
332	C₂	—	22	2	no	—
333	C₂	—	88	4	no	—
334	C₂	C₄×D₈×D₁₁	1	1	yes	characteristic
335	C₂	#13081754	1	1	yes	characteristic
336	C₂	#13081765	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₁
commutator	C₈×C₁₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₄×C₁₁×Q₃₂	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	C₂×#10622532	C₂
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₄×Q₃₂	D₁₁
hypocenter	C₁₁	#33459
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #13077491	C₂	semidirect
3. #13070483	C₂	semidirect
4. #13070478	C₂	semidirect
5. Q₃₂×Dic₁₁	C₂	semidirect
6. C₄×D₁₇₆	C₂	semidirect
7. #13070462	C₂	semidirect
8. D₁₁×#366	C₂	semidirect
9. #13070327	C₂	semidirect
10. #13070211	C₂	semidirect
11. C₄×C₁₁×Q₃₂	C₂	semidirect
19. #10622532	C₄	semidirect
46. #66972	C₂×C₄	semidirect
58. C₁₁×Q₃₂	C₂×C₄	semidirect
65. #66892	C₂×C₄	semidirect
75. D₁₇₆	C₂×C₄	semidirect
77. C₁₆×D₁₁	C₂×C₄	semidirect
125. C₄×Q₃₂	D₁₁	semidirect
221. Q₃₂	C₄×D₁₁	semidirect
287. C₁₁	#33459	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622532	2
#13077491	2
#13070483	2
#13070478	2
Q₃₂×Dic₁₁	2
C₄×D₁₇₆	2
#13070462	2
D₁₁×#366	2
#13070327	2
#13070211	2
C₄×C₁₁×Q₃₂	2
#33459	11