Subgroups — #14719494

Order 2816.

814 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077583	C₂	1	2	yes	maximal
2	#13066919	C₂	1	2	yes	maximal, complemented
3	#13066913	C₂	1	2	yes	maximal, complemented
4	#13066903	C₂	1	2	yes	maximal, complemented
5	#13066899	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#4318	C₂	1	4	yes	maximal, complemented
7	C₂×C₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₁×#131	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₁×#131	C₄	1	2	yes	—
10	C₄×C₁₁×#41	C₂²	1	1	yes	characteristic
11	C₁₁×#405	C₄	1	4	yes	—
12	C₁₁×#405	C₄	1	2	yes	—
13	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₁₁×M₁₆	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
17	#10622637	C₄	1	4	yes	—
18	#10622625	C₄	1	4	yes	—
19	#10622624	C₄	1	8	yes	—
20	#10622538	C₂²	1	4	yes	—
21	#10622538	C₂²	1	2	yes	—
22	C₄×#4318	C₂²	1	4	yes	—
23	C₂×C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
24	#10622093	C₂²	1	2	yes	—
25	#10622092	C₂²	1	4	yes	—
26	#10622092	C₂²	1	4	yes	—
27	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
28	C₈×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
29	C₈×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
30	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
31	C₂×C₁₁×#41	C₂×C₄	1	1	yes	characteristic
32	C₁₁×#131	—	2	8	no	—
33	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₁×M₁₆	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
37	C₄×Q₈×C₁₁	C₂×C₄	1	2	yes	—
38	C₄×D₄×C₁₁	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₁₁	C₂×C₄	1	2	yes	—
40	C₁₁×#117	C₂×C₄	1	2	yes	—
41	C₁₁×#117	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
44	C₂×#4318	C₂×C₄	1	8	yes	—
45	C₂²×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
46	C₁₁×#97	C₂×C₄	1	2	yes	—
47	C₁₁×#97	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₁	C₂³	1	2	yes	—
52	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
53	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
54	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	8	yes	—
55	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	8	yes	—
56	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
57	#66867	C₂×C₄	1	8	yes	—
58	C₄×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
59	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
60	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
61	C₄×C₁₁⋊C₈	C₂³	1	4	yes	—
62	#17335	—	11	1	no	maximal, Sylow
63	C₁₁×#41	—	2	4	no	—
64	C₂×Q₈×C₁₁	C₄²	1	1	yes	characteristic
65	C₂×D₄×C₁₁	C₄²	1	2	yes	—
66	C₂×D₄×C₁₁	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₁₁×M₁₆	—	2	8	no	—
72	C₁₁×M₁₆	—	2	4	no	—
73	C₂×C₈×C₁₁	—	2	8	no	—
74	C₂×C₈×C₁₁	M₁₆	1	8	yes	complemented
75	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
78	C₁₁×C₄⋊C₄	C₄²	1	4	yes	—
79	C₁₁×C₄⋊C₄	C₄²	1	2	yes	—
80	C₁₁×C₂²⋊C₄	C₄²	1	4	yes	—
81	C₁₁×C₂²⋊C₄	C₄²	1	2	yes	—
82	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₁	C₂²×C₄	1	4	yes	—
84	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
87	#4318	—	2	16	no	—
88	C₂×C₁₁⋊C₈	—	2	8	no	—
89	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
90	C₂×C₁₁⋊C₈	C₂²×C₄	1	8	yes	—
91	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
92	#2965	—	11	2	no	—
93	C₄×#131	D₁₁	1	1	yes	characteristic
94	#1455	—	11	2	no	—
95	#1453	—	11	2	no	—
96	#1451	—	11	2	no	—
97	#1450	—	11	1	no	—
98	C₈×M₁₆	—	11	4	no	—
99	C₂×C₈²	—	11	1	no	—
100	C₂³×C₁₁	C₂×C₄²	1	2	yes	—
101	C₂³×C₁₁	C₂×C₄²	1	1	yes	characteristic
102	Q₈×C₁₁	—	2	2	no	—
103	D₄×C₁₁	—	2	4	no	—
104	D₄×C₁₁	—	2	2	no	—
105	C₂×C₄×C₁₁	—	2	4	no	—
106	C₂×C₄×C₁₁	—	2	4	no	—
107	C₂×C₄×C₁₁	—	2	8	no	—
108	C₂×C₄×C₁₁	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₁₁	C₂×C₄²	1	4	yes	—
110	C₂×C₄×C₁₁	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
113	C₂×C₄×C₁₁	C₂³×C₄	1	2	yes	—
114	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
115	C₈×C₁₁	—	2	4	no	—
116	C₈×C₁₁	C₂×M₁₆	1	4	yes	—
117	C₈×C₁₁	C₂×M₁₆	1	4	yes	—
118	C₁₁⋊C₈	—	2	4	no	—
119	C₁₁⋊C₈	—	2	8	no	—
120	C₁₁⋊C₈	#131	1	8	yes	complemented
121	C₂×#131	Dic₁₁	1	2	yes	—
122	C₄×#41	C₂×D₁₁	1	1	yes	characteristic
123	#446	—	11	4	no	—
124	#436	—	11	4	no	—
125	#435	—	11	8	no	—
126	#431	—	11	4	no	—
127	#431	—	11	2	no	—
128	#405	Dic₁₁	1	4	yes	—
129	#405	Dic₁₁	1	2	yes	—
130	C₄×M₁₆	—	11	4	no	—
131	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
132	C₄×M₁₆	C₂×D₁₁	1	2	yes	—
133	C₂×C₄×C₈	—	11	2	no	—
134	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
136	#322	—	11	4	no	—
137	#322	—	11	4	no	—
138	#322	—	11	2	no	—
139	C₈²	—	11	1	no	—
140	C₈²	—	11	1	no	—
141	C₈²	—	11	2	no	—
142	C₈²	—	11	2	no	—
143	C₂²×C₁₁	—	2	2	no	—
144	C₂²×C₁₁	—	2	2	no	—
145	C₂²×C₁₁	—	2	4	no	—
146	C₂²×C₁₁	C₄×M₁₆	1	2	yes	—
147	C₂²×C₁₁	C₂²×C₄²	1	1	yes	characteristic
148	C₄×C₁₁	—	2	2	no	—
149	C₄×C₁₁	—	2	2	no	—
150	C₄×C₁₁	—	2	4	no	—
151	C₄×C₁₁	C₄×M₁₆	1	2	yes	—
152	C₄×C₁₁	C₂²×M₁₆	1	2	yes	—
153	C₄×C₁₁	C₂×#131	1	4	yes	—
154	C₂×#41	C₂×Dic₁₁	1	1	yes	characteristic
155	#131	—	2	8	no	—
156	C₂×M₁₆	—	11	8	no	—
157	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
158	C₂×M₁₆	C₂×Dic₁₁	1	4	yes	—
159	C₂²×C₈	—	11	4	no	—
160	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
161	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
162	C₄×Q₈	C₄×D₁₁	1	2	yes	—
163	C₄×D₄	C₄×D₁₁	1	4	yes	—
164	C₄×D₄	C₄×D₁₁	1	2	yes	—
165	#117	C₂×Dic₁₁	1	2	yes	—
166	#117	C₂×Dic₁₁	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
169	C₄⋊C₈	—	11	8	no	—
170	C₄⋊C₈	—	11	4	no	—
171	C₂²⋊C₈	—	11	4	no	—
172	C₂²⋊C₈	—	11	8	no	—
173	#97	—	11	8	no	—
174	#97	C₂×Dic₁₁	1	2	yes	—
175	#97	C₂×Dic₁₁	1	4	yes	—
176	C₄×C₈	—	11	4	no	—
177	C₄×C₈	—	11	2	no	—
178	C₄×C₈	—	11	2	no	—
179	C₄×C₈	—	11	4	no	—
180	C₄×C₈	C₂×Dic₁₁	1	4	yes	—
181	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
182	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
183	C₄×C₈	C₂²×D₁₁	1	2	yes	—
184	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
185	C₂×C₁₁	—	2	2	no	—
186	C₂×C₁₁	—	4	2	no	—
187	C₂×C₁₁	C₂×C₄×M₁₆	1	1	yes	characteristic
188	C₂×C₁₁	C₄×#131	1	2	yes	—
189	#41	—	2	4	no	—
190	C₂×Q₈	C₄×Dic₁₁	1	1	yes	characteristic
191	C₂×D₄	C₄×Dic₁₁	1	2	yes	—
192	C₂×D₄	C₄×Dic₁₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₁₁	1	4	yes	—
195	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
196	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	8	no	—
199	M₁₆	—	22	16	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	11	4	no	—
202	C₂×C₈	—	11	4	no	—
203	C₂×C₈	—	11	8	no	—
204	C₂×C₈	—	22	8	no	—
205	C₂×C₈	#4318	1	8	yes	complemented
206	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
207	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
208	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
209	C₄⋊C₄	C₄×Dic₁₁	1	4	yes	—
210	C₄⋊C₄	C₄×Dic₁₁	1	2	yes	—
211	C₂²⋊C₄	C₄×Dic₁₁	1	4	yes	—
212	C₂²⋊C₄	C₄×Dic₁₁	1	2	yes	—
213	C₄²	C₂×C₄×D₁₁	1	2	yes	—
214	C₄²	C₂×C₄×D₁₁	1	4	yes	—
215	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
216	C₄²	C₂²×Dic₁₁	1	2	yes	—
217	C₄²	C₂³×D₁₁	1	1	yes	characteristic
218	C₁₁	#17335	1	1	yes	characteristic, complemented, Sylow
219	C₂³	C₂×C₄×Dic₁₁	1	2	yes	—
220	C₂³	C₂×C₄×Dic₁₁	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	C₂×#4318	1	4	yes	—
228	C₂×C₄	C₂×C₄×Dic₁₁	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₁₁	1	2	yes	—
230	C₂×C₄	C₂×C₄×Dic₁₁	1	4	yes	—
231	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₄×D₁₁	1	2	yes	—
233	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
234	C₈	—	2	4	no	—
235	C₈	—	11	8	no	—
236	C₈	—	22	4	no	—
237	C₈	—	22	8	no	—
238	C₈	C₂×#4318	1	4	yes	—
239	C₈	C₂×#4318	1	4	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	C₄×#4318	1	2	yes	—
244	C₂²	C₂²×C₄×Dic₁₁	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	C₄×#4318	1	2	yes	—
249	C₄	D₁₁×#131	1	4	yes	—
250	C₄	C₂²×#4318	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	4	2	no	—
253	C₂	C₂×C₄×#4318	1	1	yes	characteristic
254	C₂	#131×Dic₁₁	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₁
commutator, socle	C₂²×C₁₁	C₂²×C₄²
Frattini	C₄²	C₂³×D₁₁
Fitting	C₄×C₁₁×#131	C₂
prime radical	C₂×C₁₁×#41	C₂×C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₄×#131	D₁₁
hypocenter	C₁₁	#17335
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #13066919	C₂	semidirect
3. #13066913	C₂	semidirect
4. #13066903	C₂	semidirect
5. #13066899	C₂	semidirect
6. C₈×#4318	C₂	semidirect
7. C₂×C₈×C₁₁⋊C₈	C₂	semidirect
74. C₂×C₈×C₁₁	M₁₆	semidirect
120. C₁₁⋊C₈	#131	semidirect
205. C₂×C₈	#4318	semidirect
218. C₁₁	#17335	semidirect

Maximal subgroups

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

subgroup	index
#13077583	2
#13066919	2
#13066913	2
#13066903	2
#13066899	2
C₈×#4318	2
C₂×C₈×C₁₁⋊C₈	2
C₄×C₁₁×#131	2
#17335	11