Subgroups — #14906953

Order 2816.

682 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622514	C₂	1	1	yes	maximal, characteristic, complemented
2	#13077310	C₂	1	1	yes	maximal, characteristic
3	#13070446	C₂	1	4	yes	maximal, complemented
4	#13070439	C₂	1	4	yes	maximal, complemented
5	#13070374	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070229	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070227	C₂	1	1	yes	maximal, characteristic, complemented
8	#13070212	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	#10623322	C₄	1	1	yes	characteristic
12	C₂×#66954	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×M₃₂	—	2	4	no	—
14	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
15	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
16	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
18	#10622820	C₄	1	1	yes	characteristic
19	#10622724	C₄	1	4	yes	—
20	#10622723	C₄	1	1	yes	characteristic
21	C₈×C₂²⋊D₁₁	C₂²	1	4	yes	—
22	#10622715	C₄	1	1	yes	characteristic
23	#10622714	C₂²	1	1	yes	characteristic
24	C₂×#66876	C₂²	1	1	yes	characteristic
25	C₂×#66876	C₂²	1	1	yes	characteristic
26	#10622514	—	2	4	no	—
27	C₂×#66862	C₂²	1	2	yes	complemented
28	C₂×C₁₆×D₁₁	C₂²	1	2	yes	complemented
29	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
30	(C₂×D₁₁)⋊C₁₆	C₂²	1	4	yes	complemented
31	#10622140	C₂²	1	2	yes	complemented
32	Dic₁₁⋊C₁₆	C₂²	1	4	yes	complemented
33	C₁₆×Dic₁₁	C₂²	1	2	yes	complemented
34	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₂×#4339	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	—	2	2	no	—
37	C₂²×C₈×C₁₁	—	2	2	no	—
38	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
42	#67006	C₈	1	2	yes	—
43	#66983	C₈	1	2	yes	—
44	#66982	C₈	1	2	yes	—
45	C₄×C₂²⋊D₁₁	C₂×C₄	1	4	yes	—
46	#66979	C₂×C₄	1	1	yes	characteristic
47	#66977	C₈	1	2	yes	—
48	C₂×#4318	C₂×C₄	1	1	yes	characteristic
49	C₂×#4318	C₂×C₄	1	1	yes	characteristic
50	#66954	—	2	2	no	—
51	#66954	—	2	2	no	—
52	C₂×#4313	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₁₁	C₂³	1	2	yes	—
54	C₁₁×M₃₂	—	2	4	no	—
55	C₁₁×M₃₂	—	2	4	no	—
56	C₂×C₁₁×C₁₆	—	2	2	no	—
57	C₂×C₁₁×C₁₆	D₄	1	4	yes	complemented
58	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
59	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
61	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
62	(C₂×D₁₁)⋊C₈	C₂×C₄	1	4	yes	—
63	#66879	C₂×C₄	1	2	yes	—
64	Dic₁₁⋊C₈	C₂×C₄	1	4	yes	—
65	C₈×Dic₁₁	C₂³	1	2	yes	—
66	#66876	—	2	2	no	—
67	#66876	—	2	2	no	—
68	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
69	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
70	#66862	—	2	4	no	—
71	C₁₆×D₁₁	—	2	4	no	—
72	#33433	—	11	1	no	maximal, Sylow
73	C₂⁴×C₁₁	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	—	2	2	no	—
75	C₂²×C₄×C₁₁	—	2	2	no	—
76	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₁₁	C₂×C₈	1	2	yes	—
81	C₂×C₂²⋊D₁₁	C₂×C₈	1	4	yes	—
82	#4339	—	2	2	no	—
83	#4339	—	2	2	no	—
84	C₂×D₄₄	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
86	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
87	C₂×C₈×C₁₁	—	2	4	no	—
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	2	2	no	—
90	C₂×C₈×C₁₁	—	2	2	no	—
91	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₁	#41	1	2	yes	—
95	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
96	C₂²⋊Dic₁₁	C₂×C₈	1	4	yes	—
97	C₂²⋊Dic₁₁	C₂×C₈	1	2	yes	—
98	(C₂×D₁₁)⋊C₄	C₂×C₈	1	4	yes	—
99	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
100	Dic₁₁⋊C₄	C₂×C₈	1	4	yes	—
101	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
102	#4318	—	2	2	no	—
103	#4318	—	2	2	no	—
104	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
106	#4313	—	2	4	no	—
107	C₈×D₁₁	—	2	2	no	—
108	C₈×D₁₁	—	2	2	no	—
109	C₁₁×C₁₆	—	2	2	no	—
110	C₁₁×C₁₆	C₂×D₄	1	2	yes	—
111	C₁₁×C₁₆	C₂×D₄	1	2	yes	—
112	C₁₁⋊C₁₆	—	2	2	no	—
113	C₁₁⋊C₁₆	—	2	2	no	—
114	C₂×#504	—	11	1	no	—
115	C₂²×M₃₂	D₁₁	1	1	yes	characteristic, complemented
116	#2929	—	11	1	no	—
117	#2169	—	11	4	no	—
118	#2168	—	11	4	no	—
119	#2151	—	11	1	no	—
120	#2113	—	11	1	no	—
121	#2112	—	11	1	no	—
122	#2108	—	11	1	no	—
123	C₂³×C₁₁	—	2	2	no	—
124	C₂³×C₁₁	—	4	2	no	—
125	C₂³×C₁₁	C₂²×C₈	1	2	yes	—
126	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
127	C₂²×D₁₁	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	—	2	2	no	—
130	C₂×C₄×C₁₁	—	2	2	no	—
131	C₂×C₄×C₁₁	—	2	4	no	—
132	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₁₁	#131	1	2	yes	—
136	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
137	C₂²⋊D₁₁	—	2	4	no	—
138	C₂²⋊D₁₁	—	4	4	no	—
139	C₂×Dic₁₁	C₂²×C₈	1	4	yes	—
140	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
141	D₄₄	—	2	2	no	—
142	C₄×D₁₁	—	2	2	no	—
143	C₄×D₁₁	—	2	2	no	—
144	Dic₂₂	—	2	2	no	—
145	C₈×C₁₁	—	2	2	no	—
146	C₈×C₁₁	C₄×D₄	1	2	yes	—
147	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
149	C₁₁⋊C₈	—	2	2	no	—
150	C₁₁⋊C₈	—	2	2	no	—
151	C₂×#131	—	11	1	no	—
152	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
153	#525	—	11	1	no	—
154	#504	—	22	4	no	—
155	C₂×M₃₂	—	2	4	no	—
156	C₂×M₃₂	—	11	1	no	—
157	C₂×M₃₂	—	11	1	no	—
158	C₂×M₃₂	—	11	2	no	—
159	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
160	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
161	C₂²×C₁₆	—	11	2	no	—
162	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
163	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
164	#435	—	11	4	no	—
165	C₈×D₄	—	11	4	no	—
166	#424	—	11	1	no	—
167	#408	—	11	1	no	—
168	#407	—	11	1	no	—
169	#405	—	11	1	no	—
170	C₄⋊C₁₆	—	11	4	no	—
171	C₂²⋊C₁₆	—	11	4	no	—
172	C₂²⋊C₁₆	—	11	4	no	—
173	#346	—	11	2	no	—
174	C₄×C₁₆	—	11	2	no	—
175	C₂²×C₁₁	—	2	2	no	—
176	C₂²×C₁₁	—	2	2	no	—
177	C₂²×C₁₁	—	2	2	no	—
178	C₂²×C₁₁	—	2	2	no	—
179	C₂²×C₁₁	—	4	4	no	—
180	C₂²×C₁₁	C₈×D₄	1	2	yes	—
181	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
182	C₂×D₁₁	—	2	4	no	—
183	C₂×D₁₁	—	2	2	no	—
184	C₄×C₁₁	—	2	2	no	—
185	C₄×C₁₁	C₈×D₄	1	2	yes	—
186	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
188	Dic₁₁	—	2	4	no	—
189	Dic₁₁	—	2	2	no	—
190	C₂×#41	—	11	1	no	—
191	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
192	#131	—	22	2	no	—
193	#131	—	22	2	no	—
194	C₂×M₁₆	—	11	1	no	—
195	C₂×M₁₆	—	11	1	no	—
196	C₂×M₁₆	—	11	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	11	2	no	—
200	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
201	C₂²×C₈	C₄×D₁₁	1	2	yes	—
202	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
203	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
204	#123	—	11	2	no	—
205	C₂²⋊Q₈	—	11	2	no	—
206	#121	—	11	2	no	—
207	#120	—	11	2	no	—
208	C₄×D₄	—	11	4	no	—
209	#117	—	11	1	no	—
210	M₃₂	—	2	4	no	—
211	M₃₂	—	2	4	no	—
212	M₃₂	—	22	2	no	—
213	M₃₂	—	22	2	no	—
214	M₃₂	—	22	4	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	—	11	2	no	—
217	C₂×C₁₆	—	11	2	no	—
218	C₂×C₁₆	—	22	4	no	—
219	C₂×C₁₆	C₂²⋊D₁₁	1	4	yes	complemented
220	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
221	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
222	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
223	C₄⋊C₈	—	11	4	no	—
224	C₂²⋊C₈	—	11	4	no	—
225	C₂²⋊C₈	—	11	4	no	—
226	#97	—	11	2	no	—
227	C₄×C₈	—	11	2	no	—
228	C₂×C₁₁	—	2	2	no	—
229	C₂×C₁₁	—	4	2	no	—
230	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
231	C₂×C₁₁	#3407	1	2	yes	—
232	D₁₁	—	4	2	no	—
233	C₂⁴	C₈×D₁₁	1	1	yes	characteristic
234	#41	—	22	2	no	—
235	#41	—	22	2	no	—
236	C₂×Q₈	—	11	1	no	—
237	C₂×D₄	—	11	2	no	—
238	C₂×D₄	—	11	4	no	—
239	C₂×D₄	—	11	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	11	2	no	—
243	C₂²×C₄	C₈×D₁₁	1	2	yes	—
244	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
246	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
247	M₁₆	—	22	2	no	—
248	M₁₆	—	22	2	no	—
249	M₁₆	—	22	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	11	2	no	—
255	C₂×C₈	—	11	2	no	—
256	C₂×C₈	—	22	2	no	—
257	C₂×C₈	—	22	2	no	—
258	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
259	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
260	C₂×C₈	#4339	1	2	yes	—
261	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
262	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
263	C₄⋊C₄	—	11	2	no	—
264	C₄⋊C₄	—	11	4	no	—
265	C₂²⋊C₄	—	11	2	no	—
266	C₂²⋊C₄	—	11	4	no	—
267	C₂²⋊C₄	—	11	4	no	—
268	C₄²	—	11	2	no	—
269	C₁₆	—	2	2	no	—
270	C₁₆	—	22	2	no	—
271	C₁₆	—	22	2	no	—
272	C₁₆	C₂×C₂²⋊D₁₁	1	2	yes	—
273	C₁₆	C₂×C₂²⋊D₁₁	1	2	yes	—
274	C₁₁	#33433	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	2	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	11	2	no	—
278	C₂³	C₂×C₈×D₁₁	1	2	yes	—
279	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
280	Q₈	—	22	2	no	—
281	D₄	—	22	4	no	—
282	D₄	—	22	2	no	—
283	D₄	—	44	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	11	4	no	—
289	C₂×C₄	—	11	2	no	—
290	C₂×C₄	—	22	2	no	—
291	C₂×C₄	—	22	2	no	—
292	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
293	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
294	C₂×C₄	#66954	1	2	yes	—
295	C₂×C₄	C₄×C₂²⋊D₁₁	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	22	2	no	—
299	C₈	—	22	2	no	—
300	C₈	C₄×C₂²⋊D₁₁	1	2	yes	—
301	C₈	C₂×#4339	1	1	yes	characteristic
302	C₈	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	22	2	no	—
309	C₂²	—	22	4	no	—
310	C₂²	C₈×C₂²⋊D₁₁	1	2	yes	—
311	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	22	2	no	—
314	C₄	—	22	4	no	—
315	C₄	C₈×C₂²⋊D₁₁	1	2	yes	—
316	C₄	C₂×#66954	1	1	yes	characteristic
317	C₄	C₂×C₄×C₂²⋊D₁₁	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	44	2	no	—
321	C₂	C₂×C₈×C₂²⋊D₁₁	1	1	yes	characteristic
322	C₂	#13081718	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₂²×C₁₁×M₃₂	C₂
prime radical	#10623322	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂²×M₃₂	D₁₁
hypocenter	C₁₁	#33433
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#10622514	C₂	semidirect
3. #13070446	C₂	semidirect
4. #13070439	C₂	semidirect
5. #13070374	C₂	semidirect
6. #13070229	C₂	semidirect
7. #13070227	C₂	semidirect
8. #13070212	C₂	semidirect
9. C₂²×C₁₁×M₃₂	C₂	semidirect
27. C₂×#66862	C₂²	semidirect
28. C₂×C₁₆×D₁₁	C₂²	semidirect
30. (C₂×D₁₁)⋊C₁₆	C₂²	semidirect
31. #10622140	C₂²	semidirect
32. Dic₁₁⋊C₁₆	C₂²	semidirect
33. C₁₆×Dic₁₁	C₂²	semidirect
57. C₂×C₁₁×C₁₆	D₄	semidirect
115. C₂²×M₃₂	D₁₁	semidirect
219. C₂×C₁₆	C₂²⋊D₁₁	semidirect
274. C₁₁	#33433	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622514	2
#13077310	2
#13070446	2
#13070439	2
#13070374	2
#13070229	2
#13070227	2
#13070212	2
C₂²×C₁₁×M₃₂	2
#33433	11