Subgroups — #14644355

Order 2816.

494 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622301	C₂	1	1	yes	maximal, characteristic
2	C₄×#66887	C₂	1	1	yes	maximal, characteristic
3	#13066936	C₂	1	1	yes	maximal, characteristic
4	C₈×#4318	C₂	1	1	yes	maximal, characteristic, complemented
5	#13066467	C₂	1	1	yes	maximal, characteristic, complemented
6	#13066465	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#4313	C₄	1	2	yes	—
9	C₂×C₄×D₄₄	C₂²	1	1	yes	characteristic
10	C₁₁×#431	C₂²	1	1	yes	characteristic
11	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×#66887	C₄	1	2	yes	—
14	C₂×(C₂×D₁₁)⋊C₈	C₄	1	2	yes	—
15	#10622538	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄×#4318	C₂²	1	1	yes	characteristic
18	#10622301	—	2	1	no	—
19	#10622301	—	2	1	no	—
20	#10622301	—	2	1	no	—
21	#10622301	—	2	1	no	—
22	C₁₁×#322	—	2	1	no	—
23	C₁₁×#322	—	2	1	no	—
24	#10622192	C₄	1	2	yes	—
25	#10622106	C₄	1	4	yes	—
26	#10622092	—	2	1	no	—
27	C₈×C₁₁⋊C₈	—	2	1	no	—
28	C₂²×D₄₄	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₁	C₂×C₄	1	2	yes	—
30	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
34	C₂×(C₂×D₁₁)⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂×#4318	C₂×C₄	1	2	yes	—
37	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×#4313	—	2	2	no	—
39	C₂×#4313	—	2	2	no	—
40	C₂×#4313	—	4	4	no	—
41	C₄×D₄₄	—	2	1	no	—
42	C₄×D₄₄	—	2	1	no	—
43	C₄×D₄₄	—	2	1	no	—
44	C₄×D₄₄	—	2	1	no	—
45	C₁₁×C₄⋊C₈	—	2	1	no	—
46	C₁₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₁×#97	—	2	1	no	—
48	C₄×C₈×C₁₁	—	2	1	no	—
49	C₄×C₈×C₁₁	—	2	1	no	—
50	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
54	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
55	#66887	—	2	4	no	—
56	(C₂×D₁₁)⋊C₈	—	2	2	no	—
57	(C₂×D₁₁)⋊C₈	—	2	2	no	—
58	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
61	#66867	—	2	1	no	—
62	C₄×C₁₁⋊C₈	—	2	1	no	—
63	C₄×C₁₁⋊C₈	—	2	1	no	—
64	#8166	—	11	1	no	maximal, Sylow
65	C₂³×D₁₁	C₄²	1	2	yes	—
66	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₁₁	C₄²	1	2	yes	—
70	C₂×D₄₄	—	2	1	no	—
71	C₂×D₄₄	—	2	1	no	—
72	C₂×D₄₄	—	2	2	no	—
73	C₂×D₄₄	—	2	2	no	—
74	C₂×C₄×D₁₁	—	2	2	no	—
75	C₂×C₄×D₁₁	—	2	2	no	—
76	C₂×C₄×D₁₁	—	4	2	no	—
77	C₂×C₄×D₁₁	—	4	2	no	—
78	C₁₁×M₁₆	—	2	2	no	—
79	C₁₁×M₁₆	—	2	2	no	—
80	C₂×C₈×C₁₁	—	2	2	no	—
81	C₂×C₈×C₁₁	—	2	2	no	—
82	C₂×C₈×C₁₁	—	2	2	no	—
83	C₂×C₈×C₁₁	—	4	1	no	—
84	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₁	#41	1	1	yes	characteristic
91	C₄²×C₁₁	#41	1	1	yes	characteristic
92	(C₂×D₁₁)⋊C₄	—	2	2	no	—
93	(C₂×D₁₁)⋊C₄	—	2	2	no	—
94	C₄⋊Dic₁₁	—	2	1	no	—
95	C₄⋊Dic₁₁	—	2	1	no	—
96	#4318	—	2	2	no	—
97	#4318	M₁₆	1	4	yes	complemented
98	C₂×C₁₁⋊C₈	—	2	2	no	—
99	C₂×C₁₁⋊C₈	—	2	2	no	—
100	C₂×C₁₁⋊C₈	—	2	2	no	—
101	C₂×C₁₁⋊C₈	—	4	2	no	—
102	#4313	—	4	4	no	—
103	#4313	—	4	4	no	—
104	C₂×#436	—	11	1	no	—
105	C₄×#100	—	11	1	no	—
106	#1460	—	11	1	no	—
107	#1451	D₁₁	1	1	yes	characteristic, complemented
108	C₈×M₁₆	—	11	1	no	—
109	#1283	—	11	1	no	—
110	#1283	—	11	1	no	—
111	C₂³×C₁₁	C₂×C₄²	1	1	yes	characteristic
112	C₂²×D₁₁	—	2	2	no	—
113	C₂²×D₁₁	—	2	2	no	—
114	C₂²×D₁₁	—	2	2	no	—
115	C₂²×D₁₁	—	4	2	no	—
116	C₂²×D₁₁	—	4	2	no	—
117	C₂×C₄×C₁₁	—	2	1	no	—
118	C₂×C₄×C₁₁	—	2	1	no	—
119	C₂×C₄×C₁₁	—	4	1	no	—
120	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₁₁	#131	1	2	yes	—
128	C₂×Dic₁₁	—	2	2	no	—
129	C₂×Dic₁₁	—	4	2	no	—
130	D₄₄	—	4	2	no	—
131	D₄₄	—	4	2	no	—
132	C₄×D₁₁	—	4	4	no	—
133	C₄×D₁₁	—	4	2	no	—
134	C₄×D₁₁	—	4	2	no	—
135	C₈×C₁₁	—	2	1	no	—
136	C₈×C₁₁	—	2	1	no	—
137	C₈×C₁₁	—	2	1	no	—
138	C₈×C₁₁	—	4	2	no	—
139	C₈×C₁₁	—	4	2	no	—
140	C₈×C₁₁	#100	1	2	yes	complemented
141	C₁₁⋊C₈	—	2	4	no	—
142	C₁₁⋊C₈	—	4	2	no	—
143	C₁₁⋊C₈	—	4	2	no	—
144	C₂²×M₁₆	—	11	2	no	—
145	C₂×C₄×D₄	—	11	1	no	—
146	#436	—	22	1	no	—
147	#436	—	22	1	no	—
148	#436	—	22	1	no	—
149	#436	—	22	1	no	—
150	#431	—	11	1	no	—
151	#431	C₂×D₁₁	1	1	yes	characteristic
152	C₂×C₄⋊C₈	—	11	1	no	—
153	C₂×#100	—	11	2	no	—
154	C₂×C₂²⋊C₈	—	11	2	no	—
155	C₄×M₁₆	—	11	1	no	—
156	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
158	#343	—	11	2	no	—
159	#323	—	11	4	no	—
160	#322	—	2	1	no	—
161	#322	—	2	1	no	—
162	#322	—	22	1	no	—
163	C₈²	—	22	1	no	—
164	C₂²×C₁₁	—	2	1	no	—
165	C₂²×C₁₁	—	2	1	no	—
166	C₂²×C₁₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₁₁	C₄×M₁₆	1	1	yes	characteristic
168	C₂²×C₁₁	#405	1	1	yes	characteristic
169	C₂×D₁₁	—	4	2	no	—
170	C₂×D₁₁	—	4	2	no	—
171	C₂×D₁₁	—	4	2	no	—
172	C₂×D₁₁	—	8	2	no	—
173	C₂×D₁₁	—	8	2	no	—
174	C₄×C₁₁	—	2	1	no	—
175	C₄×C₁₁	—	2	1	no	—
176	C₄×C₁₁	—	2	1	no	—
177	C₄×C₁₁	C₂×#100	1	1	yes	characteristic
178	C₄×C₁₁	#413	1	1	yes	characteristic
179	C₄×C₁₁	#435	1	2	yes	—
180	C₄×C₁₁	#436	1	2	yes	—
181	Dic₁₁	—	4	2	no	—
182	C₂²×D₄	—	11	1	no	—
183	C₂³×C₄	—	11	2	no	—
184	C₂×M₁₆	—	11	2	no	—
185	C₂×M₁₆	—	22	2	no	—
186	C₂×M₁₆	—	22	2	no	—
187	C₂×M₁₆	—	44	4	no	—
188	C₂×M₁₆	C₄×D₁₁	1	2	yes	—
189	C₂²×C₈	—	11	2	no	—
190	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
191	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
192	C₄×D₄	—	22	1	no	—
193	C₄×D₄	—	22	1	no	—
194	C₄×D₄	—	22	1	no	—
195	C₄×D₄	—	22	1	no	—
196	C₂×C₄⋊C₄	—	11	1	no	—
197	C₂×C₂²⋊C₄	—	11	2	no	—
198	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	22	1	no	—
201	C₄⋊C₈	—	22	1	no	—
202	C₄⋊C₈	—	22	1	no	—
203	#100	—	22	4	no	—
204	C₂²⋊C₈	—	11	2	no	—
205	C₂²⋊C₈	—	22	2	no	—
206	C₂²⋊C₈	—	22	2	no	—
207	C₂²⋊C₈	C₄×D₁₁	1	2	yes	—
208	#97	—	2	1	no	—
209	#97	—	22	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	22	1	no	—
213	C₄×C₈	—	22	1	no	—
214	C₄×C₈	D₄₄	1	1	yes	characteristic
215	C₄×C₈	D₄₄	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
218	C₂×C₁₁	—	2	1	no	—
219	C₂×C₁₁	—	2	1	no	—
220	C₂×C₁₁	#1753	1	1	yes	characteristic
221	C₂×C₁₁	C₄×#100	1	1	yes	characteristic
222	C₂×C₁₁	#1757	1	1	yes	characteristic
223	D₁₁	—	8	2	no	—
224	C₂⁴	—	11	2	no	—
225	C₂×D₄	—	22	1	no	—
226	C₂×D₄	—	22	2	no	—
227	C₂×D₄	—	22	2	no	—
228	C₂×D₄	—	22	1	no	—
229	C₂²×C₄	—	11	2	no	—
230	C₂²×C₄	—	22	2	no	—
231	C₂²×C₄	—	22	2	no	—
232	C₂²×C₄	—	44	2	no	—
233	C₂²×C₄	—	44	2	no	—
234	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
237	M₁₆	—	2	2	no	—
238	M₁₆	—	2	2	no	—
239	M₁₆	—	11	4	no	—
240	M₁₆	—	22	2	no	—
241	M₁₆	—	44	4	no	—
242	M₁₆	—	44	4	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	22	2	no	—
248	C₂×C₈	—	22	2	no	—
249	C₂×C₈	—	22	2	no	—
250	C₂×C₈	—	44	2	no	—
251	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
252	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
253	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
254	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
255	C₄⋊C₄	—	22	1	no	—
256	C₄⋊C₄	—	22	1	no	—
257	C₂²⋊C₄	—	22	2	no	—
258	C₂²⋊C₄	—	22	2	no	—
259	C₄²	C₂×D₄₄	1	1	yes	characteristic
260	C₄²	#4339	1	1	yes	characteristic
261	C₄²	#4339	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
263	C₁₁	#8166	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	22	2	no	—
265	C₂³	—	22	2	no	—
266	C₂³	—	22	2	no	—
267	C₂³	—	44	2	no	—
268	C₂³	—	44	2	no	—
269	C₂³	C₄²×D₁₁	1	1	yes	characteristic
270	D₄	—	44	2	no	—
271	D₄	—	44	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	22	2	no	—
276	C₂×C₄	—	44	4	no	—
277	C₂×C₄	—	44	2	no	—
278	C₂×C₄	—	44	2	no	—
279	C₂×C₄	—	44	2	no	—
280	C₂×C₄	#66925	1	1	yes	characteristic
281	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
283	C₂×C₄	M₁₆×D₁₁	1	2	yes	—
284	C₂×C₄	#66960	1	2	yes	—
285	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₄×C₂²⋊D₁₁	1	1	yes	characteristic
287	C₂×C₄	C₄×C₂²⋊D₁₁	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	22	4	no	—
294	C₈	—	44	2	no	—
295	C₈	—	44	2	no	—
296	C₈	#66887	1	2	yes	complemented
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	44	2	no	—
300	C₂²	—	44	2	no	—
301	C₂²	—	44	2	no	—
302	C₂²	—	88	2	no	—
303	C₂²	—	88	2	no	—
304	C₂²	#10622320	1	1	yes	characteristic
305	C₂²	#10622321	1	1	yes	characteristic
306	C₂²	C₄×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	44	2	no	—
311	C₄	#10622319	1	2	yes	—
312	C₄	#10622741	1	2	yes	—
313	C₄	C₂×#66887	1	1	yes	characteristic
314	C₄	#10622745	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	88	2	no	—
318	C₂	#13068401	1	1	yes	characteristic
319	C₂	C₄×#66887	1	1	yes	characteristic
320	C₂	#13068417	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄
commutator	C₂³×C₁₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₁₁×#1451	C₂
socle	C₂²×C₁₁	C₄×C₂²⋊C₄
prime radical	C₂²×D₄₄	C₂×C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	#1451	D₁₁
hypocenter	C₁₁	#8166
LCS	G ▷ C₂³×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
4. C₈×#4318	C₂	semidirect
5. #13066467	C₂	semidirect
6. #13066465	C₂	semidirect
7. C₁₁×#1451	C₂	semidirect
97. #4318	M₁₆	semidirect
107. #1451	D₁₁	semidirect
140. C₈×C₁₁	#100	semidirect
263. C₁₁	#8166	semidirect
296. C₈	#66887	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622301	2
C₄×#66887	2
#13066936	2
C₈×#4318	2
#13066467	2
#13066465	2
C₁₁×#1451	2
#8166	11