Subgroups — #14907284

Order 2816.

530 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₂₂	C₂	1	1	yes	maximal, characteristic
2	#13070523	C₂	1	4	yes	maximal, complemented
3	#13070521	C₂	1	2	yes	maximal, complemented
4	#13070518	C₂	1	2	yes	maximal, complemented
5	#13070371	C₂	1	2	yes	maximal
6	C₄×#66876	C₂	1	1	yes	maximal, characteristic
7	M₃₂×Dic₁₁	C₂	1	2	yes	maximal
8	C₁₁×#2150	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₂	C₄	1	1	yes	characteristic
10	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₁⋊C₈	C₄	1	2	yes	—
13	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
14	C₂×#66876	C₂²	1	1	yes	characteristic
15	C₂×#66876	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	1	yes	characteristic
17	C₈×Dic₂₂	C₄	1	1	yes	characteristic
18	C₈×Dic₂₂	C₄	1	1	yes	characteristic
19	C₈×Dic₂₂	C₄	1	2	yes	—
20	C₈×Dic₂₂	C₂²	1	1	yes	characteristic
21	C₈×Dic₂₂	C₂²	1	1	yes	characteristic
22	C₈×Dic₂₂	C₂²	1	2	yes	—
23	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
25	#10622140	C₂²	1	4	yes	—
26	Dic₁₁⋊C₁₆	C₂²	1	4	yes	—
27	Dic₁₁⋊C₁₆	C₂²	1	4	yes	—
28	C₁₆×Dic₁₁	C₂²	1	4	yes	—
29	#10622099	C₂²	1	2	yes	—
30	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
31	C₂²×Dic₂₂	C₈	1	1	yes	characteristic
32	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₁	C₈	1	1	yes	characteristic
36	C₂×Dic₁₁⋊C₄	C₈	1	2	yes	—
37	C₂×C₄×Dic₁₁	C₂×C₄	1	2	yes	—
38	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×Dic₂₂	C₈	1	1	yes	characteristic
40	C₄×Dic₂₂	C₈	1	2	yes	—
41	C₄×Dic₂₂	C₈	1	1	yes	characteristic
42	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
44	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
45	C₁₁×M₃₂	—	2	4	no	—
46	C₂×C₁₁×C₁₆	C₂³	1	4	yes	—
47	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
51	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
52	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
53	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
54	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
56	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₁	C₂³	1	2	yes	—
58	C₈×Dic₁₁	C₂³	1	2	yes	—
59	#66876	—	2	2	no	—
60	#66876	Q₈	1	4	yes	complemented
61	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
62	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
63	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
64	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
65	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
66	Q₈×M₃₂	—	11	1	no	maximal, Sylow
67	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
71	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
72	C₂×Dic₂₂	—	2	4	no	—
73	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
74	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
75	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₁₁	#41	1	2	yes	—
82	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
83	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
90	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
91	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
92	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
93	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
94	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
95	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
96	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
97	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
98	C₂×C₁₁⋊C₈	—	2	4	no	—
99	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
101	C₁₁×C₁₆	—	2	4	no	—
102	C₁₁⋊C₁₆	—	2	2	no	—
103	C₁₁⋊C₁₆	C₂×Q₈	1	4	yes	complemented
104	C₂×C₈×Q₈	—	11	1	no	—
105	#2183	—	11	2	no	—
106	#2183	—	11	4	no	—
107	Q₈×C₁₆	—	11	2	no	—
108	#2150	—	11	2	no	—
109	#2150	D₁₁	1	1	yes	characteristic
110	C₄×M₃₂	—	11	2	no	—
111	C₄×M₃₂	—	11	1	no	—
112	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₁	—	2	2	no	—
114	C₂×C₄×C₁₁	—	4	1	no	—
115	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₁	#131	1	2	yes	—
122	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₁₁	—	2	4	no	—
124	C₂×Dic₁₁	—	4	2	no	—
125	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
126	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
127	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
128	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
129	Dic₂₂	—	2	2	no	—
130	Dic₂₂	—	2	4	no	—
131	Dic₂₂	M₃₂	1	4	yes	complemented
132	C₈×C₁₁	—	2	1	no	—
133	C₈×C₁₁	—	2	1	no	—
134	C₈×C₁₁	C₄×Q₈	1	2	yes	—
135	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
136	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
137	C₁₁⋊C₈	—	2	2	no	—
138	C₁₁⋊C₈	—	2	2	no	—
139	C₂×C₄×Q₈	—	11	1	no	—
140	C₂×M₃₂	—	11	1	no	—
141	C₂×M₃₂	—	11	1	no	—
142	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
143	C₈×Q₈	—	11	1	no	—
144	C₈×Q₈	—	11	2	no	—
145	C₈×Q₈	—	11	1	no	—
146	C₈×Q₈	—	11	2	no	—
147	C₈×Q₈	—	11	1	no	—
148	C₈×Q₈	—	11	1	no	—
149	C₂×C₄⋊C₈	—	11	2	no	—
150	C₂×C₄⋊C₈	—	11	1	no	—
151	C₂×C₄×C₈	—	11	2	no	—
152	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
153	C₄⋊C₁₆	—	11	4	no	—
154	C₄⋊C₁₆	—	11	4	no	—
155	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
156	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
157	#346	—	11	4	no	—
158	#346	—	11	2	no	—
159	C₄×C₁₆	—	11	4	no	—
160	C₄×C₁₆	—	11	2	no	—
161	C₂²×C₁₁	—	2	1	no	—
162	C₂²×C₁₁	—	2	1	no	—
163	C₂²×C₁₁	C₈×Q₈	1	2	yes	—
164	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
165	C₄×C₁₁	—	2	1	no	—
166	C₄×C₁₁	—	2	1	no	—
167	C₄×C₁₁	—	2	1	no	—
168	C₄×C₁₁	C₈×Q₈	1	2	yes	—
169	C₄×C₁₁	C₂×M₃₂	1	2	yes	—
170	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
172	Dic₁₁	—	2	2	no	—
173	Dic₁₁	—	2	2	no	—
174	Dic₁₁	—	2	2	no	—
175	Dic₁₁	C₂×M₃₂	1	4	yes	—
176	C₂²×Q₈	—	11	1	no	—
177	C₂²×C₈	—	11	2	no	—
178	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
180	C₄×Q₈	—	11	1	no	—
181	C₄×Q₈	—	11	2	no	—
182	C₄×Q₈	—	11	1	no	—
183	C₄×Q₈	—	11	1	no	—
184	C₄×Q₈	—	11	2	no	—
185	C₄×Q₈	—	11	1	no	—
186	C₂×C₄⋊C₄	—	11	2	no	—
187	C₂×C₄⋊C₄	—	11	1	no	—
188	C₂×C₄²	—	11	2	no	—
189	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
190	M₃₂	—	2	4	no	—
191	M₃₂	—	11	4	no	—
192	M₃₂	—	22	2	no	—
193	C₂×C₁₆	—	11	2	no	—
194	C₂×C₁₆	—	11	2	no	—
195	C₂×C₁₆	C₂²×D₁₁	1	4	yes	—
196	C₄⋊C₈	—	11	2	no	—
197	C₄⋊C₈	—	11	2	no	—
198	C₄⋊C₈	—	11	2	no	—
199	C₄⋊C₈	—	11	2	no	—
200	C₄⋊C₈	—	11	1	no	—
201	C₄⋊C₈	—	11	2	no	—
202	C₄⋊C₈	—	11	1	no	—
203	C₄×C₈	—	11	2	no	—
204	C₄×C₈	—	11	2	no	—
205	C₄×C₈	—	11	2	no	—
206	C₄×C₈	—	11	2	no	—
207	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
208	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
211	C₂×C₁₁	—	2	2	no	—
212	C₂×C₁₁	C₂×C₈×Q₈	1	1	yes	characteristic
213	C₂×C₁₁	C₂²×M₃₂	1	1	yes	characteristic
214	C₂×C₁₁	#3407	1	1	yes	characteristic
215	C₂×Q₈	—	11	1	no	—
216	C₂×Q₈	—	11	2	no	—
217	C₂×Q₈	—	11	1	no	—
218	C₂×Q₈	—	22	4	no	—
219	C₂²×C₄	—	11	2	no	—
220	C₂²×C₄	—	11	2	no	—
221	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
222	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	11	2	no	—
226	C₂×C₈	—	11	2	no	—
227	C₂×C₈	—	22	4	no	—
228	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
231	C₂×C₈	#4341	1	2	yes	—
232	C₂×C₈	Q₈×D₁₁	1	2	yes	—
233	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
234	C₄⋊C₄	—	11	2	no	—
235	C₄⋊C₄	—	11	2	no	—
236	C₄⋊C₄	—	11	2	no	—
237	C₄⋊C₄	—	11	2	no	—
238	C₄⋊C₄	—	11	2	no	—
239	C₄⋊C₄	—	11	1	no	—
240	C₄⋊C₄	—	11	1	no	—
241	C₄²	—	11	2	no	—
242	C₄²	—	11	2	no	—
243	C₄²	—	11	2	no	—
244	C₄²	—	11	2	no	—
245	C₄²	C₈×D₁₁	1	1	yes	characteristic
246	C₄²	C₈×D₁₁	1	1	yes	characteristic
247	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
249	C₁₆	—	2	4	no	—
250	C₁₆	—	11	4	no	—
251	C₁₆	—	22	2	no	—
252	C₁₁	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
253	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
254	Q₈	—	11	4	no	—
255	Q₈	—	22	4	no	—
256	Q₈	—	22	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	11	2	no	—
260	C₂×C₄	—	11	2	no	—
261	C₂×C₄	—	11	2	no	—
262	C₂×C₄	—	11	2	no	—
263	C₂×C₄	—	22	4	no	—
264	C₂×C₄	—	44	2	no	—
265	C₂×C₄	#66940	1	2	yes	—
266	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
268	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
269	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
270	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
271	C₂×C₄	#66960	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	22	2	no	—
276	C₈	—	22	2	no	—
277	C₈	#66940	1	2	yes	—
278	C₈	C₂×#4341	1	1	yes	characteristic
279	C₈	C₂×Q₈×D₁₁	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	#10622443	1	2	yes	—
283	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	11	4	no	—
288	C₄	—	22	2	no	—
289	C₄	—	22	2	no	—
290	C₄	—	22	2	no	—
291	C₄	#10622443	1	2	yes	—
292	C₄	D₁₁×M₃₂	1	2	yes	—
293	C₄	C₂×#66940	1	1	yes	characteristic
294	C₄	C₂×#66960	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	C₂×#10622443	1	1	yes	characteristic
297	C₂	C₂×D₁₁×M₃₂	1	1	yes	characteristic
298	C₂	#13081718	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2150	C₂
prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2150	D₁₁
hypocenter	C₁₁	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #13070523	C₂	semidirect
3. #13070521	C₂	semidirect
4. #13070518	C₂	semidirect
60. #66876	Q₈	semidirect
103. C₁₁⋊C₁₆	C₂×Q₈	semidirect
131. Dic₂₂	M₃₂	semidirect
252. C₁₁	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₂₂	2
#13070523	2
#13070521	2
#13070518	2
#13070371	2
C₄×#66876	2
M₃₂×Dic₁₁	2
C₁₁×#2150	2
Q₈×M₃₂	11