Subgroups — #14907423

Order 2816.

494 conjugacy classes of subgroups.

The subgroup lattice has 290 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	#13070522	C₂	1	4	yes	maximal, complemented
3	#13070521	C₂	1	2	yes	maximal, complemented
4	#13070518	C₂	1	2	yes	maximal, complemented
5	#13070416	C₂	1	2	yes	maximal
6	#13070415	C₂	1	1	yes	maximal, characteristic
7	#13070412	C₂	1	2	yes	maximal
8	C₁₁×#2163	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₂	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₁⋊C₈	C₄	1	2	yes	—
12	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	1	yes	characteristic
14	C₈×Dic₂₂	C₄	1	1	yes	characteristic
15	C₈×Dic₂₂	C₄	1	1	yes	characteristic
16	C₈×Dic₂₂	C₄	1	2	yes	—
17	C₈×Dic₂₂	C₂²	1	1	yes	characteristic
18	C₈×Dic₂₂	C₂²	1	1	yes	characteristic
19	C₈×Dic₂₂	C₂²	1	2	yes	—
20	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₁₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₁₁×#346	C₂²	1	2	yes	—
23	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
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	C₄⋊(C₁₁⋊C₁₆)	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	C₂×C₁₁×C₁₆	C₂³	1	4	yes	—
46	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
50	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
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	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
56	C₈×Dic₁₁	C₂³	1	2	yes	—
57	C₈×Dic₁₁	C₂³	1	2	yes	—
58	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
60	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
61	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
62	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
63	#33498	—	11	1	no	maximal, Sylow
64	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
68	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
69	C₂×Dic₂₂	—	2	4	no	—
70	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
71	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
72	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
73	C₂×C₈×C₁₁	—	2	2	no	—
74	C₂×C₈×C₁₁	—	4	1	no	—
75	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
79	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
84	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
85	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
86	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
87	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
88	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
89	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
90	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
91	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
92	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
93	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
94	C₂×C₁₁⋊C₈	—	4	2	no	—
95	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
97	C₁₁×C₁₆	—	2	4	no	—
98	C₁₁⋊C₁₆	—	2	2	no	—
99	C₁₁⋊C₁₆	—	2	2	no	—
100	C₂×C₈×Q₈	—	11	1	no	—
101	#2183	—	11	2	no	—
102	#2183	—	11	4	no	—
103	Q₈×C₁₆	—	11	2	no	—
104	#2163	—	11	2	no	—
105	#2163	D₁₁	1	1	yes	characteristic
106	#2162	—	11	1	no	—
107	#2162	—	11	2	no	—
108	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₁	—	2	2	no	—
110	C₂×C₄×C₁₁	—	2	1	no	—
111	C₂×C₄×C₁₁	—	2	1	no	—
112	C₂×C₄×C₁₁	—	2	2	no	—
113	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₁₁	—	2	4	no	—
120	C₂×Dic₁₁	—	2	4	no	—
121	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
122	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
123	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
124	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
125	Dic₂₂	—	2	2	no	—
126	Dic₂₂	—	2	4	no	—
127	Dic₂₂	M₃₂	1	4	yes	—
128	C₈×C₁₁	—	2	1	no	—
129	C₈×C₁₁	—	2	1	no	—
130	C₈×C₁₁	—	2	1	no	—
131	C₈×C₁₁	2₋¹⁺⁴	1	2	yes	—
132	C₁₁⋊C₈	—	2	2	no	—
133	C₁₁⋊C₈	—	2	2	no	—
134	C₂×C₄×Q₈	—	11	1	no	—
135	C₈×Q₈	—	11	1	no	—
136	C₈×Q₈	—	11	2	no	—
137	C₈×Q₈	—	11	1	no	—
138	C₈×Q₈	—	11	2	no	—
139	C₈×Q₈	—	11	1	no	—
140	C₈×Q₈	—	11	1	no	—
141	C₂×C₄⋊C₈	—	11	2	no	—
142	C₂×C₄⋊C₈	—	11	1	no	—
143	C₂×C₄×C₈	—	11	2	no	—
144	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
145	C₄⋊C₁₆	—	11	4	no	—
146	C₄⋊C₁₆	—	11	4	no	—
147	C₄⋊C₁₆	—	11	2	no	—
148	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
149	C₂²⋊C₁₆	—	11	1	no	—
150	C₂²⋊C₁₆	—	11	1	no	—
151	C₂²⋊C₁₆	C₂×D₁₁	1	2	yes	—
152	#346	—	11	4	no	—
153	#346	C₂×D₁₁	1	2	yes	—
154	C₄×C₁₆	—	11	4	no	—
155	C₄×C₁₆	—	11	2	no	—
156	C₂²×C₁₁	—	2	1	no	—
157	C₂²×C₁₁	—	2	1	no	—
158	C₂²×C₁₁	#504	1	2	yes	—
159	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
160	C₄×C₁₁	—	2	1	no	—
161	C₄×C₁₁	—	2	1	no	—
162	C₄×C₁₁	—	2	1	no	—
163	C₄×C₁₁	—	2	1	no	—
164	C₄×C₁₁	C₂×M₃₂	1	2	yes	—
165	C₄×C₁₁	#519	1	1	yes	characteristic
166	C₄×C₁₁	#568	1	1	yes	characteristic
167	Dic₁₁	—	2	2	no	—
168	Dic₁₁	—	2	2	no	—
169	Dic₁₁	—	2	2	no	—
170	Dic₁₁	C₂×M₃₂	1	4	yes	—
171	C₂²×Q₈	—	11	1	no	—
172	C₂²×C₈	—	11	2	no	—
173	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
175	C₄×Q₈	—	11	1	no	—
176	C₄×Q₈	—	11	2	no	—
177	C₄×Q₈	—	11	1	no	—
178	C₄×Q₈	—	11	1	no	—
179	C₄×Q₈	—	11	2	no	—
180	C₄×Q₈	—	11	1	no	—
181	C₂×C₄⋊C₄	—	11	2	no	—
182	C₂×C₄⋊C₄	—	11	1	no	—
183	C₂×C₄²	—	11	2	no	—
184	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
185	C₂×C₁₆	—	11	2	no	—
186	C₂×C₁₆	—	11	2	no	—
187	C₂×C₁₆	C₂²×D₁₁	1	4	yes	—
188	C₄⋊C₈	—	11	2	no	—
189	C₄⋊C₈	—	11	2	no	—
190	C₄⋊C₈	—	11	2	no	—
191	C₄⋊C₈	—	11	2	no	—
192	C₄⋊C₈	—	11	1	no	—
193	C₄⋊C₈	—	11	2	no	—
194	C₄⋊C₈	—	11	1	no	—
195	C₄×C₈	—	11	2	no	—
196	C₄×C₈	—	11	2	no	—
197	C₄×C₈	—	11	2	no	—
198	C₄×C₈	—	11	2	no	—
199	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
200	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
203	C₂×C₁₁	—	2	2	no	—
204	C₂×C₁₁	#2982	1	1	yes	characteristic
205	C₂×C₁₁	C₂²×M₃₂	1	1	yes	characteristic
206	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
207	C₂×Q₈	—	11	1	no	—
208	C₂×Q₈	—	11	2	no	—
209	C₂×Q₈	—	11	1	no	—
210	C₂×Q₈	—	22	4	no	—
211	C₂²×C₄	—	11	2	no	—
212	C₂²×C₄	—	11	2	no	—
213	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
214	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	11	2	no	—
219	C₂×C₈	—	11	2	no	—
220	C₂×C₈	—	44	2	no	—
221	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
224	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
225	C₄⋊C₄	—	11	2	no	—
226	C₄⋊C₄	—	11	2	no	—
227	C₄⋊C₄	—	11	2	no	—
228	C₄⋊C₄	—	11	2	no	—
229	C₄⋊C₄	—	11	2	no	—
230	C₄⋊C₄	—	11	1	no	—
231	C₄⋊C₄	—	11	1	no	—
232	C₄²	—	11	2	no	—
233	C₄²	—	11	2	no	—
234	C₄²	—	11	2	no	—
235	C₄²	—	11	2	no	—
236	C₄²	C₈×D₁₁	1	1	yes	characteristic
237	C₄²	C₈×D₁₁	1	1	yes	characteristic
238	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
239	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
240	C₁₆	—	2	4	no	—
241	C₁₆	—	22	2	no	—
242	C₁₆	—	22	2	no	—
243	C₁₁	#33498	1	1	yes	characteristic, complemented, Sylow
244	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
245	Q₈	—	11	4	no	—
246	Q₈	—	22	4	no	—
247	Q₈	—	22	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	11	2	no	—
253	C₂×C₄	—	11	2	no	—
254	C₂×C₄	—	11	2	no	—
255	C₂×C₄	—	11	2	no	—
256	C₂×C₄	—	22	4	no	—
257	C₂×C₄	—	22	4	no	—
258	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
260	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
262	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	22	2	no	—
268	C₈	—	22	2	no	—
269	C₈	#67043	1	2	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	#10622520	1	2	yes	—
273	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	11	4	no	—
279	C₄	—	22	2	no	—
280	C₄	—	22	2	no	—
281	C₄	—	22	2	no	—
282	C₄	D₁₁×M₃₂	1	2	yes	—
283	C₄	#10623047	1	1	yes	characteristic
284	C₄	#10623276	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	#13077696	1	1	yes	characteristic
287	C₂	C₂×D₁₁×M₃₂	1	1	yes	characteristic
288	C₂	C₂×#10622520	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2163	C₂
prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2163	D₁₁
hypocenter	C₁₁	#33498
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #13070522	C₂	semidirect
3. #13070521	C₂	semidirect
4. #13070518	C₂	semidirect
243. C₁₁	#33498	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
#13070522	2
#13070521	2
#13070518	2
#13070416	2
#13070415	2
#13070412	2
C₁₁×#2163	2
#33498	11