Subgroups — #14737620

Order 2816.

602 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13078417	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₂₂	C₂	1	1	yes	maximal, characteristic
3	#13068071	C₂	1	4	yes	maximal, complemented
4	#13068017	C₂	1	2	yes	maximal, complemented
5	#13067983	C₂	1	2	yes	maximal, complemented
6	#13067375	C₂	1	1	yes	maximal, characteristic
7	#13067306	C₂	1	2	yes	maximal
8	C₁₁×#3170	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×#494	C₂²	1	1	yes	characteristic
13	C₁₁×#487	C₂²	1	2	yes	—
14	C₁₁×#467	—	2	1	no	—
15	C₁₁×#467	—	2	1	no	—
16	C₄×C₁₁×Q₁₆	—	2	1	no	—
17	C₄×C₁₁×Q₁₆	—	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	#10622707	C₂²	1	4	yes	—
22	#10622704	C₂²	1	4	yes	—
23	#10622676	—	2	4	no	—
24	#10622617	—	2	2	no	—
25	#10622608	—	2	2	no	—
26	#10622537	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	#10622096	C₂²	1	2	yes	—
36	C₂²×Dic₂₂	D₄	1	1	yes	characteristic
37	C₂×C₁₁×Q₁₆	—	2	1	no	—
38	C₂×C₁₁×Q₁₆	—	2	1	no	—
39	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
41	C₁₁×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₁₁×C₄⋊Q₈	C₂³	1	2	yes	—
43	C₁₁×#125	—	2	2	no	—
44	C₁₁×C₂²⋊Q₈	—	2	2	no	—
45	C₁₁×C₂²⋊Q₈	—	2	2	no	—
46	C₄×Q₈×C₁₁	—	2	2	no	—
47	C₄×Q₈×C₁₁	—	2	2	no	—
48	C₁₁×#117	—	2	2	no	—
49	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
51	C₂×Dic₁₁⋊C₄	—	2	1	no	—
52	C₂×C₄×Dic₁₁	—	2	1	no	—
53	C₂×#4318	—	2	2	no	—
54	C₂²×C₁₁⋊C₈	—	2	1	no	—
55	C₄×Dic₂₂	—	2	1	no	—
56	C₄×Dic₂₂	—	2	1	no	—
57	C₄×Dic₂₂	D₄	1	2	yes	—
58	C₄×Dic₂₂	C₂³	1	2	yes	—
59	C₁₁×#107	—	2	1	no	—
60	C₁₁×#107	—	2	1	no	—
61	C₁₁×#103	—	2	2	no	—
62	C₁₁×#103	—	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	Dic₁₁⋊C₈	—	2	2	no	—
68	Dic₁₁⋊C₈	—	2	2	no	—
69	C₈×Dic₁₁	—	2	2	no	—
70	C₈×Dic₁₁	—	2	2	no	—
71	#66874	—	2	4	no	—
72	#66874	—	2	4	no	—
73	#66872	—	2	4	no	—
74	#66871	—	2	4	no	—
75	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
76	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
77	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
78	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
79	#19262	—	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	1	no	—
88	C₂×Dic₂₂	—	2	2	no	—
89	C₂×Dic₂₂	Q₁₆	1	4	yes	complemented
90	C₂×Dic₂₂	C₂×D₄	1	2	yes	—
91	C₁₁×Q₁₆	—	4	2	no	—
92	C₁₁×Q₁₆	—	4	2	no	—
93	C₂×C₈×C₁₁	—	2	2	no	—
94	C₂×C₈×C₁₁	—	4	1	no	—
95	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
99	C₁₁×C₄⋊C₄	—	2	2	no	—
100	C₁₁×C₄⋊C₄	—	2	2	no	—
101	C₁₁×C₄⋊C₄	—	2	2	no	—
102	C₁₁×C₄⋊C₄	—	2	2	no	—
103	C₁₁×C₄⋊C₄	—	4	2	no	—
104	C₁₁×C₄⋊C₄	—	4	2	no	—
105	C₁₁×C₂²⋊C₄	—	4	2	no	—
106	C₄²×C₁₁	—	4	2	no	—
107	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₁	#41	1	1	yes	characteristic
110	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
111	C₄⋊Dic₁₁	—	2	1	no	—
112	C₄⋊Dic₁₁	C₂×D₄	1	2	yes	—
113	Dic₁₁⋊C₄	—	2	2	no	—
114	Dic₁₁⋊C₄	—	2	2	no	—
115	C₄×Dic₁₁	—	2	2	no	—
116	C₄×Dic₁₁	—	2	2	no	—
117	#4318	—	4	4	no	—
118	C₂×C₁₁⋊C₈	—	2	4	no	—
119	C₂×C₁₁⋊C₈	—	2	2	no	—
120	C₂×C₁₁⋊C₈	—	4	2	no	—
121	#3170	D₁₁	1	1	yes	characteristic
122	#3078	—	11	2	no	—
123	C₂×C₈×Q₈	—	11	1	no	—
124	#1684	—	11	4	no	—
125	#1676	—	11	2	no	—
126	#1670	—	11	2	no	—
127	#1567	—	11	1	no	—
128	#1550	—	11	2	no	—
129	C₂³×C₁₁	#121	1	1	yes	characteristic
130	Q₈×C₁₁	—	4	2	no	—
131	Q₈×C₁₁	—	4	2	no	—
132	Q₈×C₁₁	—	8	2	no	—
133	C₂×C₄×C₁₁	—	2	1	no	—
134	C₂×C₄×C₁₁	—	2	1	no	—
135	C₂×C₄×C₁₁	—	4	1	no	—
136	C₂×C₄×C₁₁	—	4	2	no	—
137	C₂×C₄×C₁₁	—	4	2	no	—
138	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
142	C₂×C₄×C₁₁	#135	1	2	yes	—
143	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₁₁	—	2	2	no	—
147	C₂×Dic₁₁	—	2	4	no	—
148	C₂×Dic₁₁	—	2	2	no	—
149	C₂×Dic₁₁	—	4	2	no	—
150	Dic₂₂	—	2	2	no	—
151	Dic₂₂	—	2	4	no	—
152	Dic₂₂	C₂×Q₁₆	1	4	yes	—
153	C₈×C₁₁	—	2	1	no	—
154	C₈×C₁₁	—	2	1	no	—
155	C₈×C₁₁	—	2	1	no	—
156	C₈×C₁₁	#137	1	2	yes	—
157	C₁₁⋊C₈	—	4	4	no	—
158	C₁₁⋊C₈	—	4	2	no	—
159	#533	C₂×D₁₁	1	2	yes	—
160	C₂×C₄×Q₈	—	11	1	no	—
161	#500	C₂×D₁₁	1	1	yes	characteristic
162	#494	C₂×D₁₁	1	1	yes	characteristic
163	#487	C₂×D₁₁	1	2	yes	—
164	#479	—	22	4	no	—
165	#477	—	11	4	no	—
166	#475	—	11	4	no	—
167	#467	—	2	1	no	—
168	#467	—	2	1	no	—
169	C₈×Q₈	—	11	2	no	—
170	C₈×Q₈	—	11	2	no	—
171	C₈×Q₈	—	22	1	no	—
172	C₈×Q₈	—	22	1	no	—
173	C₄×Q₁₆	—	2	1	no	—
174	C₄×Q₁₆	—	2	1	no	—
175	#428	—	22	2	no	—
176	#423	—	11	2	no	—
177	C₂×C₄⋊C₈	—	11	1	no	—
178	C₂×C₄⋊C₈	—	22	1	no	—
179	#419	—	22	2	no	—
180	C₂×C₄×C₈	—	22	1	no	—
181	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
182	#335	—	11	2	no	—
183	#335	—	11	2	no	—
184	#333	—	11	4	no	—
185	#332	—	11	4	no	—
186	C₂²×C₁₁	—	2	1	no	—
187	C₂²×C₁₁	—	2	1	no	—
188	C₂²×C₁₁	#467	1	2	yes	—
189	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
190	C₄×C₁₁	—	2	1	no	—
191	C₄×C₁₁	—	2	1	no	—
192	C₄×C₁₁	—	2	1	no	—
193	C₄×C₁₁	—	8	2	no	—
194	C₄×C₁₁	—	8	2	no	—
195	C₄×C₁₁	#467	1	2	yes	—
196	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
197	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
198	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
199	C₄×C₁₁	#575	1	1	yes	characteristic
200	Dic₁₁	—	2	4	no	—
201	Dic₁₁	—	4	2	no	—
202	C₂²×Q₈	—	11	1	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×M₁₆	—	22	2	no	—
206	C₂²×C₈	—	22	1	no	—
207	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
208	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
209	C₄⋊Q₈	C₂²×D₁₁	1	2	yes	—
210	C₄⋊Q₈	C₂²×D₁₁	1	2	yes	—
211	#125	—	2	2	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	11	2	no	—
217	C₄×Q₈	—	11	2	no	—
218	C₄×Q₈	—	22	1	no	—
219	C₄×Q₈	—	22	1	no	—
220	#117	—	2	2	no	—
221	C₂×C₄⋊C₄	—	11	1	no	—
222	C₂×C₄⋊C₄	—	22	1	no	—
223	C₂×C₄²	—	22	1	no	—
224	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
225	#107	—	2	1	no	—
226	#107	—	2	1	no	—
227	#107	—	22	4	no	—
228	#106	—	22	4	no	—
229	C₄⋊C₈	—	11	2	no	—
230	C₄⋊C₈	—	11	4	no	—
231	C₄⋊C₈	—	22	2	no	—
232	C₄⋊C₈	—	22	2	no	—
233	C₄⋊C₈	—	22	2	no	—
234	C₄⋊C₈	—	22	1	no	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#103	—	22	4	no	—
238	#103	—	22	4	no	—
239	C₄×C₈	—	22	2	no	—
240	C₄×C₈	—	22	2	no	—
241	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
242	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
245	C₂×C₁₁	—	2	2	no	—
246	C₂×C₁₁	#3037	1	1	yes	characteristic
247	C₂×C₁₁	C₂×#467	1	1	yes	characteristic
248	C₂×C₁₁	#3067	1	1	yes	characteristic
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	11	4	no	—
252	C₂×Q₈	—	11	2	no	—
253	C₂×Q₈	—	22	2	no	—
254	C₂×Q₈	—	22	1	no	—
255	C₂²×C₄	—	22	1	no	—
256	C₂²×C₄	—	22	1	no	—
257	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
258	C₂²×C₄	#4341	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
260	Q₁₆	—	4	2	no	—
261	Q₁₆	—	4	2	no	—
262	M₁₆	—	44	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	22	4	no	—
266	C₂×C₈	—	22	2	no	—
267	C₂×C₈	—	44	2	no	—
268	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	11	2	no	—
279	C₄⋊C₄	—	22	1	no	—
280	C₄⋊C₄	—	22	2	no	—
281	C₄⋊C₄	—	22	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	22	2	no	—
285	C₄²	—	22	2	no	—
286	C₄²	D₄×D₁₁	1	1	yes	characteristic
287	C₄²	#4341	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
289	C₄²	C₂³×D₁₁	1	1	yes	characteristic
290	C₁₁	#19262	1	1	yes	characteristic, complemented, Sylow
291	C₂³	#66991	1	1	yes	characteristic
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	8	2	no	—
295	Q₈	—	11	4	no	—
296	Q₈	—	22	2	no	—
297	Q₈	—	22	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	22	4	no	—
304	C₂×C₄	—	22	2	no	—
305	C₂×C₄	—	22	2	no	—
306	C₂×C₄	—	44	2	no	—
307	C₂×C₄	Q₁₆×D₁₁	1	2	yes	—
308	C₂×C₄	#66972	1	2	yes	—
309	C₂×C₄	#66991	1	1	yes	characteristic
310	C₂×C₄	#66991	1	1	yes	characteristic
311	C₂×C₄	#66991	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
313	C₂×C₄	C₂×#4341	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	44	4	no	—
319	C₈	—	44	2	no	—
320	C₈	#67004	1	2	yes	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	#10622801	1	2	yes	—
324	C₂²	C₂×#66991	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	—	22	4	no	—
331	C₄	—	44	2	no	—
332	C₄	#10622801	1	2	yes	—
333	C₄	C₂×Q₁₆×D₁₁	1	1	yes	characteristic
334	C₄	C₂×#66972	1	1	yes	characteristic
335	C₄	#10623353	1	1	yes	characteristic
336	C₄	C₂×#67004	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	#13078084	1	1	yes	characteristic
339	C₂	C₂×#10622801	1	1	yes	characteristic
340	C₂	#13078332	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#66991
commutator, cosocle	C₄²×C₁₁	C₂⁴
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#3170	C₂
socle	C₂²×C₁₁	C₂×#121
prime radical	C₂³×C₁₁	#121
hypercenter	#3170	D₁₁
hypocenter	C₁₁	#19262
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #13068071	C₂	semidirect
4. #13068017	C₂	semidirect
5. #13067983	C₂	semidirect
89. C₂×Dic₂₂	Q₁₆	semidirect
290. C₁₁	#19262	semidirect

Maximal subgroups

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

subgroup	index
#13078417	2
C₂×C₈×Dic₂₂	2
#13068071	2
#13068017	2
#13067983	2
#13067375	2
#13067306	2
C₁₁×#3170	2
#19262	11