Subgroups — #14736138

Order 2816.

606 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	#13078651	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₂₂	C₂	1	1	yes	maximal, characteristic
3	#13068017	C₂	1	2	yes	maximal, complemented
4	#13068007	C₂	1	2	yes	maximal, complemented
5	#13067995	C₂	1	4	yes	maximal, complemented
6	#13067364	C₂	1	1	yes	maximal, characteristic
7	#13067273	C₂	1	2	yes	maximal
8	C₁₁×#3160	C₂	1	1	yes	maximal, characteristic
9	C₁₁×#533	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₂	C₂²	1	1	yes	characteristic
11	C₁₁×#500	C₂²	1	1	yes	characteristic
12	C₁₁×#500	C₂²	1	1	yes	characteristic
13	C₁₁×#499	—	2	1	no	—
14	C₁₁×#499	—	2	1	no	—
15	C₁₁×#498	C₂²	1	2	yes	—
16	C₁₁×#427	—	2	1	no	—
17	C₁₁×#427	—	2	1	no	—
18	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₁⋊C₈	—	2	1	no	—
20	C₂×C₈×Dic₁₁	—	2	1	no	—
21	#10622704	C₂²	1	4	yes	—
22	#10622703	C₂²	1	4	yes	—
23	#10622663	—	2	2	no	—
24	#10622657	—	2	2	no	—
25	C₄×#4326	—	2	4	no	—
26	#10622538	C₂²	1	2	yes	—
27	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
28	C₈×Dic₂₂	—	2	1	no	—
29	C₈×Dic₂₂	—	2	1	no	—
30	C₈×Dic₂₂	C₂²	1	2	yes	—
31	C₈×Dic₂₂	C₂²	1	2	yes	—
32	#10622185	C₂²	1	4	yes	—
33	#10622184	C₂²	1	4	yes	—
34	#10622096	C₂²	1	2	yes	—
35	#10622095	C₂²	1	2	yes	—
36	C₂²×Dic₂₂	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
39	C₁₁×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₁₁×C₄⋊Q₈	C₂³	1	2	yes	—
41	C₁₁×#125	—	2	2	no	—
42	C₁₁×C₂²⋊Q₈	—	2	2	no	—
43	C₁₁×C₂²⋊Q₈	—	2	2	no	—
44	C₄×Q₈×C₁₁	—	2	2	no	—
45	C₄×Q₈×C₁₁	—	2	2	no	—
46	C₁₁×#117	—	2	2	no	—
47	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
48	C₂×#4326	—	2	4	no	—
49	C₂×C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
50	C₂×Dic₁₁⋊C₄	—	2	1	no	—
51	C₂×C₄×Dic₁₁	—	2	1	no	—
52	C₂²×C₁₁⋊C₈	—	2	1	no	—
53	C₄×Dic₂₂	—	2	1	no	—
54	C₄×Dic₂₂	—	2	1	no	—
55	C₄×Dic₂₂	D₄	1	2	yes	—
56	C₄×Dic₂₂	C₂³	1	2	yes	—
57	C₁₁×#107	—	2	1	no	—
58	C₁₁×#107	—	2	1	no	—
59	C₁₁×#107	—	2	1	no	—
60	C₁₁×#107	—	2	1	no	—
61	C₁₁×#106	—	2	2	no	—
62	C₁₁×#106	—	2	2	no	—
63	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
67	#66899	—	2	4	no	—
68	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
69	Dic₁₁⋊C₈	—	2	2	no	—
70	Dic₁₁⋊C₈	—	2	2	no	—
71	C₈×Dic₁₁	—	2	2	no	—
72	C₈×Dic₁₁	—	2	2	no	—
73	#66874	—	2	4	no	—
74	#66871	—	2	4	no	—
75	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
76	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
77	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
78	C₄×C₁₁⋊C₈	—	2	2	no	—
79	#19118	—	11	1	no	maximal, Sylow
80	C₂×Q₈×C₁₁	—	2	2	no	—
81	C₂×Q₈×C₁₁	—	2	2	no	—
82	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
85	C₂²×Dic₁₁	—	2	1	no	—
86	C₂²×Dic₁₁	—	2	1	no	—
87	C₂×Dic₂₂	—	2	4	no	—
88	C₂×Dic₂₂	—	2	1	no	—
89	C₂×Dic₂₂	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₁	—	2	2	no	—
91	C₂×C₈×C₁₁	Q₁₆	1	4	yes	complemented
92	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
96	C₁₁×C₄⋊C₄	—	2	2	no	—
97	C₁₁×C₄⋊C₄	—	2	2	no	—
98	C₁₁×C₄⋊C₄	—	2	2	no	—
99	C₁₁×C₄⋊C₄	—	2	2	no	—
100	C₁₁×C₄⋊C₄	—	4	2	no	—
101	C₁₁×C₄⋊C₄	—	4	2	no	—
102	C₁₁×C₂²⋊C₄	—	4	2	no	—
103	C₄²×C₁₁	—	4	2	no	—
104	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₁	#41	1	1	yes	characteristic
107	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
108	#4326	—	4	8	no	—
109	C₄⋊Dic₁₁	—	2	1	no	—
110	C₄⋊Dic₁₁	C₂×D₄	1	2	yes	—
111	Dic₁₁⋊C₄	—	2	2	no	—
112	Dic₁₁⋊C₄	—	2	2	no	—
113	C₄×Dic₁₁	—	2	2	no	—
114	C₄×Dic₁₁	—	2	2	no	—
115	C₂×C₁₁⋊C₈	—	2	4	no	—
116	C₂×C₁₁⋊C₈	—	2	2	no	—
117	C₂×C₁₁⋊C₈	—	4	2	no	—
118	#3160	D₁₁	1	1	yes	characteristic
119	#3104	—	11	2	no	—
120	C₂×C₈×Q₈	—	11	1	no	—
121	#1676	—	11	2	no	—
122	#1674	—	11	2	no	—
123	#1672	—	11	4	no	—
124	#1564	—	11	1	no	—
125	#1542	—	11	2	no	—
126	C₂³×C₁₁	#121	1	1	yes	characteristic
127	Q₈×C₁₁	—	4	4	no	—
128	Q₈×C₁₁	—	8	2	no	—
129	C₂×C₄×C₁₁	—	2	2	no	—
130	C₂×C₄×C₁₁	—	4	1	no	—
131	C₂×C₄×C₁₁	—	4	2	no	—
132	C₂×C₄×C₁₁	—	4	2	no	—
133	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
137	C₂×C₄×C₁₁	#137	1	2	yes	—
138	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₁₁	—	2	4	no	—
142	C₂×Dic₁₁	—	2	2	no	—
143	C₂×Dic₁₁	—	2	2	no	—
144	C₂×Dic₁₁	—	4	2	no	—
145	Dic₂₂	—	2	4	no	—
146	Dic₂₂	—	2	2	no	—
147	Dic₂₂	#135	1	4	yes	—
148	C₈×C₁₁	—	2	1	no	—
149	C₈×C₁₁	—	2	1	no	—
150	C₈×C₁₁	C₂×Q₁₆	1	2	yes	—
151	C₈×C₁₁	C₂×Q₁₆	1	2	yes	—
152	C₁₁⋊C₈	—	4	4	no	—
153	C₁₁⋊C₈	—	4	2	no	—
154	#533	C₂×D₁₁	1	2	yes	—
155	C₂×C₄×Q₈	—	11	1	no	—
156	#500	C₂×D₁₁	1	1	yes	characteristic
157	#500	C₂×D₁₁	1	1	yes	characteristic
158	#499	—	2	1	no	—
159	#499	—	2	1	no	—
160	#498	C₂×D₁₁	1	2	yes	—
161	#484	—	22	2	no	—
162	#477	—	11	4	no	—
163	#462	—	11	4	no	—
164	#451	—	22	2	no	—
165	C₈×Q₈	—	11	2	no	—
166	C₈×Q₈	—	11	2	no	—
167	C₈×Q₈	—	22	1	no	—
168	C₈×Q₈	—	22	1	no	—
169	C₄×Q₁₆	—	22	4	no	—
170	#431	—	11	2	no	—
171	#427	—	2	1	no	—
172	#427	—	2	1	no	—
173	C₂×C₄⋊C₈	—	11	1	no	—
174	C₂×C₄⋊C₈	—	22	1	no	—
175	C₂×C₄×C₈	—	22	1	no	—
176	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
177	#335	—	11	2	no	—
178	#334	—	11	2	no	—
179	#333	—	11	4	no	—
180	#332	—	11	4	no	—
181	C₂²×C₁₁	—	2	1	no	—
182	C₂²×C₁₁	—	2	1	no	—
183	C₂²×C₁₁	#462	1	2	yes	—
184	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
185	C₄×C₁₁	—	2	1	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	8	2	no	—
189	C₄×C₁₁	—	8	2	no	—
190	C₄×C₁₁	#462	1	2	yes	—
191	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
192	C₄×C₁₁	C₂×#135	1	2	yes	—
193	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
194	Dic₁₁	—	2	4	no	—
195	Dic₁₁	—	4	2	no	—
196	C₂²×Q₈	—	11	1	no	—
197	C₂×Q₁₆	—	22	4	no	—
198	C₂²×C₈	—	22	1	no	—
199	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
200	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
201	C₄⋊Q₈	C₂²×D₁₁	1	2	yes	—
202	C₄⋊Q₈	C₂²×D₁₁	1	2	yes	—
203	#125	—	2	2	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	—	11	2	no	—
209	C₄×Q₈	—	11	2	no	—
210	C₄×Q₈	—	22	1	no	—
211	C₄×Q₈	—	22	1	no	—
212	#117	—	2	2	no	—
213	C₂×C₄⋊C₄	—	11	1	no	—
214	C₂×C₄⋊C₄	—	22	1	no	—
215	C₂×C₄²	—	22	1	no	—
216	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
217	#107	—	2	1	no	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	22	4	no	—
222	#106	—	2	2	no	—
223	#106	—	2	2	no	—
224	C₄⋊C₈	—	11	2	no	—
225	C₄⋊C₈	—	11	4	no	—
226	C₄⋊C₈	—	22	2	no	—
227	C₄⋊C₈	—	22	2	no	—
228	C₄⋊C₈	—	22	1	no	—
229	#103	—	22	4	no	—
230	#103	—	22	4	no	—
231	C₂²⋊C₈	—	22	2	no	—
232	C₄×C₈	—	22	2	no	—
233	C₄×C₈	—	22	2	no	—
234	C₄×C₈	—	22	2	no	—
235	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
236	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
239	C₂×C₁₁	—	2	2	no	—
240	C₂×C₁₁	C₂×#462	1	1	yes	characteristic
241	C₂×C₁₁	#3050	1	1	yes	characteristic
242	C₂×C₁₁	#3067	1	1	yes	characteristic
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	11	2	no	—
246	C₂×Q₈	—	22	4	no	—
247	C₂×Q₈	—	22	1	no	—
248	C₂²×C₄	—	22	1	no	—
249	C₂²×C₄	—	22	1	no	—
250	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
251	C₂²×C₄	#4341	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
253	Q₁₆	—	44	8	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	22	4	no	—
256	C₂×C₈	—	22	2	no	—
257	C₂×C₈	—	44	2	no	—
258	C₂×C₈	#4326	1	4	yes	complemented
259	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	11	2	no	—
270	C₄⋊C₄	—	22	1	no	—
271	C₄⋊C₄	—	22	2	no	—
272	C₄⋊C₄	—	22	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	—	22	2	no	—
276	C₄²	—	22	2	no	—
277	C₄²	D₄×D₁₁	1	1	yes	characteristic
278	C₄²	#4341	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
280	C₄²	C₂³×D₁₁	1	1	yes	characteristic
281	C₁₁	#19118	1	1	yes	characteristic, complemented, Sylow
282	C₂³	#66991	1	1	yes	characteristic
283	Q₈	—	4	4	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	11	4	no	—
286	Q₈	—	22	4	no	—
287	Q₈	—	22	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	22	4	no	—
293	C₂×C₄	—	22	2	no	—
294	C₂×C₄	—	22	2	no	—
295	C₂×C₄	—	44	2	no	—
296	C₂×C₄	#66991	1	1	yes	characteristic
297	C₂×C₄	#66991	1	1	yes	characteristic
298	C₂×C₄	#66991	1	1	yes	characteristic
299	C₂×C₄	#66995	1	2	yes	—
300	C₂×C₄	C₂×#4326	1	2	yes	—
301	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
302	C₂×C₄	C₂×#4341	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	44	4	no	—
307	C₈	—	44	2	no	—
308	C₈	C₂×#4326	1	2	yes	—
309	C₈	C₂×#4326	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#10622703	1	2	yes	—
313	C₂²	C₂×#66991	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	22	4	no	—
320	C₄	—	44	2	no	—
321	C₄	#10622703	1	2	yes	—
322	C₄	D₁₁×#135	1	2	yes	—
323	C₄	C₂×#66995	1	1	yes	characteristic
324	C₄	C₂²×#4326	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	C₂×#10622703	1	1	yes	characteristic
327	C₂	#13078193	1	1	yes	characteristic
328	C₂	#13078332	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#66991
commutator, cosocle	C₄²×C₁₁	C₂⁴
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#3160	C₂
socle	C₂²×C₁₁	C₂×#121
prime radical	C₂³×C₁₁	#121
hypercenter	#3160	D₁₁
hypocenter	C₁₁	#19118
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #13068017	C₂	semidirect
4. #13068007	C₂	semidirect
5. #13067995	C₂	semidirect
91. C₂×C₈×C₁₁	Q₁₆	semidirect
258. C₂×C₈	#4326	semidirect
281. C₁₁	#19118	semidirect

Maximal subgroups

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

subgroup	index
#13078651	2
C₂×C₈×Dic₂₂	2
#13068017	2
#13068007	2
#13067995	2
#13067364	2
#13067273	2
C₁₁×#3160	2
#19118	11