Subgroups — #14908823

Order 2816.

542 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₄₄	C₂	1	1	yes	maximal, characteristic
2	#13070811	C₂	1	2	yes	maximal
3	#13070755	C₂	1	2	yes	maximal, complemented
4	#13070663	C₂	1	2	yes	maximal, complemented
5	#13070549	C₂	1	2	yes	maximal
6	#13070485	C₂	1	4	yes	maximal, complemented
7	#13070419	C₂	1	1	yes	maximal, characteristic
8	C₁₁×#3170	C₂	1	1	yes	maximal, characteristic
9	C₁₁×#533	—	2	1	no	—
10	C₂²×Dic₄₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₂	—	2	1	no	—
12	C₁₁×#500	C₂²	1	1	yes	characteristic
13	C₁₁×#494	C₂²	1	1	yes	characteristic
14	C₁₁×#487	—	2	1	no	—
15	C₁₁×#467	—	2	1	no	—
16	C₁₁×#467	C₂²	1	2	yes	—
17	C₄×C₁₁×Q₁₆	—	2	1	no	—
18	C₄×C₁₁×Q₁₆	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
20	C₂×#66893	C₂²	1	4	yes	—
21	C₂×#66882	C₂²	1	1	yes	characteristic
22	C₂×#66880	—	2	1	no	—
23	C₄×Dic₄₄	—	2	1	no	—
24	C₄×Dic₄₄	—	2	1	no	—
25	C₄×Dic₄₄	C₂²	1	2	yes	—
26	C₄×Dic₄₄	C₂²	1	2	yes	—
27	#10622202	C₂²	1	4	yes	—
28	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
29	#10622130	C₂²	1	4	yes	—
30	#10622128	C₂²	1	4	yes	—
31	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
32	#10622099	C₂²	1	2	yes	—
33	C₂²×Dic₂₂	—	2	1	no	—
34	C₂×C₁₁×Q₁₆	—	2	1	no	—
35	C₂×C₁₁×Q₁₆	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
38	C₁₁×C₄⋊Q₈	—	2	1	no	—
39	C₁₁×C₄⋊Q₈	—	2	1	no	—
40	C₁₁×#125	—	4	1	no	—
41	C₁₁×C₂²⋊Q₈	—	2	2	no	—
42	C₁₁×C₂²⋊Q₈	—	4	1	no	—
43	C₄×Q₈×C₁₁	—	2	2	no	—
44	C₄×Q₈×C₁₁	—	4	1	no	—
45	C₁₁×#117	—	4	1	no	—
46	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₁₁	—	2	1	no	—
48	C₂×Dic₁₁⋊C₄	—	4	1	no	—
49	C₂×C₄×Dic₁₁	—	4	1	no	—
50	C₂×Dic₄₄	—	2	1	no	—
51	C₂×Dic₄₄	—	2	4	no	—
52	C₂×Dic₄₄	C₂³	1	2	yes	—
53	C₄×Dic₂₂	—	2	2	no	—
54	C₄×Dic₂₂	—	2	2	no	—
55	C₄×Dic₂₂	—	4	1	no	—
56	C₄×Dic₂₂	—	4	1	no	—
57	C₁₁×#107	—	2	1	no	—
58	C₁₁×#107	C₂³	1	2	yes	—
59	C₁₁×#103	—	2	2	no	—
60	C₁₁×#103	—	2	2	no	—
61	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
65	#66893	—	2	8	no	—
66	#66882	—	2	1	no	—
67	#66882	C₂³	1	2	yes	—
68	#66880	—	2	2	no	—
69	#66880	—	2	2	no	—
70	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
71	#33673	—	11	1	no	maximal, Sylow
72	C₂×Q₈×C₁₁	—	2	2	no	—
73	C₂×Q₈×C₁₁	—	4	1	no	—
74	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₁₁	—	4	1	no	—
78	C₂²×Dic₁₁	—	4	1	no	—
79	C₂×Dic₂₂	—	2	4	no	—
80	C₂×Dic₂₂	—	2	2	no	—
81	C₂×Dic₂₂	—	4	2	no	—
82	C₂×Dic₂₂	—	4	1	no	—
83	C₁₁×Q₁₆	—	2	4	no	—
84	C₁₁×Q₁₆	—	4	2	no	—
85	C₂×C₈×C₁₁	—	2	2	no	—
86	C₂×C₈×C₁₁	—	4	1	no	—
87	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
91	C₁₁×C₄⋊C₄	—	2	2	no	—
92	C₁₁×C₄⋊C₄	—	4	1	no	—
93	C₁₁×C₄⋊C₄	—	4	1	no	—
94	C₁₁×C₄⋊C₄	—	4	2	no	—
95	C₁₁×C₄⋊C₄	—	4	2	no	—
96	C₁₁×C₄⋊C₄	—	4	1	no	—
97	C₁₁×C₂²⋊C₄	—	4	2	no	—
98	C₄²×C₁₁	—	4	2	no	—
99	C₄²×C₁₁	D₈	1	1	yes	characteristic
100	C₄²×C₁₁	D₈	1	1	yes	characteristic
101	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
103	C₄⋊Dic₁₁	—	2	2	no	—
104	C₄⋊Dic₁₁	—	4	1	no	—
105	Dic₁₁⋊C₄	—	4	2	no	—
106	Dic₁₁⋊C₄	—	4	2	no	—
107	C₄×Dic₁₁	—	4	2	no	—
108	C₄×Dic₁₁	—	4	2	no	—
109	Dic₄₄	—	2	4	no	—
110	Dic₄₄	—	2	2	no	—
111	Dic₄₄	#41	1	4	yes	—
112	C₁₁⋊C₁₆	—	2	4	no	—
113	#3170	D₁₁	1	1	yes	characteristic
114	C₂×C₄×Q₁₆	—	11	1	no	—
115	#2235	—	11	2	no	—
116	#2225	—	11	2	no	—
117	#2209	—	11	2	no	—
118	#2188	—	11	2	no	—
119	#2176	—	11	4	no	—
120	#2163	—	11	1	no	—
121	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
122	Q₈×C₁₁	—	4	2	no	—
123	Q₈×C₁₁	—	4	2	no	—
124	Q₈×C₁₁	—	8	2	no	—
125	C₂×C₄×C₁₁	—	2	2	no	—
126	C₂×C₄×C₁₁	—	2	1	no	—
127	C₂×C₄×C₁₁	—	2	1	no	—
128	C₂×C₄×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	—	4	2	no	—
130	C₂×C₄×C₁₁	—	4	2	no	—
131	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₁₁	—	4	4	no	—
138	C₂×Dic₁₁	—	4	2	no	—
139	C₂×Dic₁₁	—	4	2	no	—
140	C₂×Dic₁₁	—	8	2	no	—
141	Dic₂₂	—	2	4	no	—
142	Dic₂₂	—	4	4	no	—
143	Dic₂₂	—	4	2	no	—
144	C₈×C₁₁	—	2	1	no	—
145	C₈×C₁₁	—	2	1	no	—
146	C₈×C₁₁	—	2	1	no	—
147	C₈×C₁₁	C₂×#41	1	2	yes	—
148	C₂²×Q₁₆	—	11	1	no	—
149	#533	—	2	1	no	—
150	C₂×C₄×Q₈	—	22	1	no	—
151	C₂×Q₃₂	—	11	4	no	—
152	#500	C₂×D₁₁	1	1	yes	characteristic
153	#494	C₂×D₁₁	1	1	yes	characteristic
154	#487	—	2	1	no	—
155	#467	—	2	1	no	—
156	#467	C₂×D₁₁	1	2	yes	—
157	C₄×Q₁₆	—	2	1	no	—
158	C₄×Q₁₆	—	11	2	no	—
159	C₄×Q₁₆	—	11	2	no	—
160	C₄×Q₁₆	—	22	1	no	—
161	C₄×Q₁₆	—	22	1	no	—
162	C₄×Q₁₆	C₂×D₁₁	1	2	yes	—
163	C₂×#107	—	11	1	no	—
164	C₂×#103	—	22	1	no	—
165	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
166	#367	—	11	4	no	—
167	C₄⋊C₁₆	—	11	2	no	—
168	#358	—	11	4	no	—
169	#358	—	11	4	no	—
170	C₂²⋊C₁₆	—	11	2	no	—
171	#346	—	11	2	no	—
172	C₂²×C₁₁	—	2	1	no	—
173	C₂²×C₁₁	—	2	1	no	—
174	C₂²×C₁₁	#510	1	2	yes	—
175	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
176	C₄×C₁₁	—	2	1	no	—
177	C₄×C₁₁	—	2	1	no	—
178	C₄×C₁₁	—	2	1	no	—
179	C₄×C₁₁	—	2	1	no	—
180	C₄×C₁₁	—	8	2	no	—
181	C₄×C₁₁	—	8	2	no	—
182	C₄×C₁₁	#510	1	2	yes	—
183	C₄×C₁₁	#525	1	1	yes	characteristic
184	C₄×C₁₁	#575	1	1	yes	characteristic
185	Dic₁₁	—	4	4	no	—
186	Dic₁₁	—	8	2	no	—
187	C₂²×Q₈	—	22	1	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	11	2	no	—
190	C₂×Q₁₆	—	22	4	no	—
191	C₂×Q₁₆	—	22	1	no	—
192	C₂×Q₁₆	C₂²×D₁₁	1	2	yes	—
193	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	1	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	#125	—	4	1	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	C₂²⋊Q₈	—	4	1	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×Q₈	—	4	1	no	—
202	C₄×Q₈	—	22	2	no	—
203	C₄×Q₈	—	22	2	no	—
204	C₄×Q₈	—	44	1	no	—
205	C₄×Q₈	—	44	1	no	—
206	#117	—	4	1	no	—
207	C₂×C₄⋊C₄	—	22	1	no	—
208	C₂×C₄⋊C₄	—	44	1	no	—
209	C₂×C₄²	—	44	1	no	—
210	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
211	Q₃₂	—	22	8	no	—
212	C₂×C₁₆	—	11	4	no	—
213	#107	—	2	1	no	—
214	#107	—	11	2	no	—
215	#107	—	22	1	no	—
216	#107	C₂²×D₁₁	1	2	yes	—
217	#103	—	2	2	no	—
218	#103	—	2	2	no	—
219	#103	—	22	2	no	—
220	#103	—	22	2	no	—
221	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
222	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
225	C₂×C₁₁	—	2	2	no	—
226	C₂×C₁₁	#3100	1	1	yes	characteristic
227	C₂×C₁₁	C₂×#510	1	1	yes	characteristic
228	C₂×C₁₁	C₂×#510	1	1	yes	characteristic
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	22	4	no	—
232	C₂×Q₈	—	22	2	no	—
233	C₂×Q₈	—	44	2	no	—
234	C₂×Q₈	—	44	1	no	—
235	C₂²×C₄	—	44	1	no	—
236	C₂²×C₄	—	44	1	no	—
237	C₂²×C₄	#4323	1	1	yes	characteristic
238	C₂²×C₄	#4323	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
240	Q₁₆	—	2	4	no	—
241	Q₁₆	—	4	2	no	—
242	Q₁₆	—	11	4	no	—
243	Q₁₆	—	22	4	no	—
244	Q₁₆	—	22	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	22	2	no	—
258	C₄⋊C₄	—	44	1	no	—
259	C₄⋊C₄	—	44	2	no	—
260	C₄⋊C₄	—	44	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	44	2	no	—
264	C₄²	—	44	2	no	—
265	C₄²	#4323	1	1	yes	characteristic
266	C₄²	#4323	1	1	yes	characteristic
267	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
269	C₁₆	—	22	4	no	—
270	C₁₁	#33673	1	1	yes	characteristic, complemented, Sylow
271	C₂³	C₂×#4323	1	1	yes	characteristic
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	8	2	no	—
275	Q₈	—	22	4	no	—
276	Q₈	—	44	4	no	—
277	Q₈	—	44	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	44	4	no	—
285	C₂×C₄	—	44	2	no	—
286	C₂×C₄	—	44	2	no	—
287	C₂×C₄	—	88	2	no	—
288	C₂×C₄	C₂×#4323	1	1	yes	characteristic
289	C₂×C₄	C₂×#4323	1	1	yes	characteristic
290	C₂×C₄	C₂×#4323	1	1	yes	characteristic
291	C₂×C₄	C₂×#4323	1	1	yes	characteristic
292	C₂×C₄	C₂×#4323	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	#41×D₁₁	1	2	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	#10622792	1	2	yes	—
301	C₂²	C₂²×#4323	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	44	4	no	—
309	C₄	—	88	2	no	—
310	C₄	#10622808	1	2	yes	—
311	C₄	#10623353	1	1	yes	characteristic
312	C₄	#10623359	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	#13078612	1	1	yes	characteristic
315	C₂	C₂×#10622808	1	1	yes	characteristic
316	C₂	C₂×#10622792	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#4323
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#3170	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	C₂³×C₁₁	C₂×D₈
hypercenter	#3170	D₁₁
hypocenter	C₁₁	#33673
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #13070755	C₂	semidirect
4. #13070663	C₂	semidirect
6. #13070485	C₂	semidirect
270. C₁₁	#33673	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₄₄	2
#13070811	2
#13070755	2
#13070663	2
#13070549	2
#13070485	2
#13070419	2
C₁₁×#3170	2
#33673	11