Subgroups — #13077697

Order 1408.

396 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₁⋊C₈	C₂	1	8	yes	maximal, complemented
3	#10622538	C₂	1	6	yes	maximal
4	C₂²×Q₈×C₁₁	C₄	1	1	yes	characteristic
5	C₄×Q₈×C₁₁	C₄	1	4	yes	—
6	C₄×Q₈×C₁₁	C₂²	1	4	yes	—
7	C₂×C₁₁×C₄⋊C₄	C₄	1	3	yes	—
8	C₂×C₄²×C₁₁	C₂²	1	3	yes	—
9	C₂²⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
10	C₄⋊(C₁₁⋊C₈)	C₂²	1	12	yes	—
11	C₄×C₁₁⋊C₈	C₂²	1	12	yes	—
12	C₂×Q₈×C₁₁	C₈	1	8	yes	complemented
13	C₂×Q₈×C₁₁	C₂×C₄	1	4	yes	—
14	C₂²×C₄×C₁₁	C₂×C₄	1	3	yes	—
15	C₂²×C₄×C₁₁	C₂×C₄	1	3	yes	—
16	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
17	C₁₁×C₄⋊C₄	C₂×C₄	1	12	yes	—
18	C₄²×C₁₁	C₂×C₄	1	6	yes	—
19	C₄²×C₁₁	C₂³	1	6	yes	—
20	C₂×C₁₁⋊C₈	C₂³	1	8	yes	—
21	#2982	—	11	1	no	maximal, Sylow
22	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
23	Q₈×C₁₁	C₂×C₈	1	16	yes	—
24	C₂×C₄×C₁₁	—	2	2	no	—
25	C₂×C₄×C₁₁	—	4	3	no	—
26	C₂×C₄×C₁₁	C₂×C₈	1	12	yes	—
27	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₁	C₂²×C₄	1	6	yes	—
29	C₂×C₄×C₁₁	C₂²×C₄	1	6	yes	—
30	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
31	C₁₁⋊C₈	—	2	8	no	—
32	C₂×C₄×Q₈	D₁₁	1	1	yes	characteristic
33	C₈×Q₈	—	11	8	no	—
34	#431	—	11	6	no	—
35	C₂²×C₁₁	—	2	1	no	—
36	C₂²×C₁₁	—	2	1	no	—
37	C₂²×C₁₁	C₂²×C₈	1	2	yes	—
38	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
39	C₄×C₁₁	—	2	1	no	—
40	C₄×C₁₁	—	2	6	no	—
41	C₄×C₁₁	C₂²×C₈	1	12	yes	—
42	C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
43	C₂²×Q₈	Dic₁₁	1	1	yes	characteristic
44	C₄×Q₈	Dic₁₁	1	4	yes	—
45	C₄×Q₈	C₂×D₁₁	1	4	yes	—
46	C₂×C₄⋊C₄	Dic₁₁	1	3	yes	—
47	C₂×C₄²	C₂×D₁₁	1	3	yes	—
48	C₄⋊C₈	—	11	12	no	—
49	C₂²⋊C₈	—	11	4	no	—
50	C₄×C₈	—	11	12	no	—
51	C₂×C₁₁	—	2	2	no	—
52	C₂×C₁₁	#519	1	1	yes	characteristic
53	C₂×C₁₁	C₂³×C₈	1	1	yes	characteristic
54	C₂×C₁₁	#568	1	1	yes	characteristic
55	C₂×Q₈	C₁₁⋊C₈	1	8	yes	complemented
56	C₂×Q₈	C₂×Dic₁₁	1	4	yes	—
57	C₂²×C₄	C₂×Dic₁₁	1	3	yes	—
58	C₂²×C₄	C₂×Dic₁₁	1	3	yes	—
59	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
60	C₂×C₈	—	11	8	no	—
61	C₄⋊C₄	C₂×Dic₁₁	1	12	yes	—
62	C₄²	C₂×Dic₁₁	1	6	yes	—
63	C₄²	C₂²×D₁₁	1	6	yes	—
64	C₁₁	#2982	1	1	yes	characteristic, complemented, Sylow
65	C₂³	C₂²×Dic₁₁	1	1	yes	characteristic
66	Q₈	C₂×C₁₁⋊C₈	1	16	yes	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	4	3	no	—
69	C₂×C₄	C₂×C₁₁⋊C₈	1	12	yes	—
70	C₂×C₄	C₂²×Dic₁₁	1	6	yes	—
71	C₂×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
72	C₂×C₄	C₂²×Dic₁₁	1	6	yes	—
73	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
74	C₈	—	22	8	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	C₂²×C₁₁⋊C₈	1	2	yes	—
78	C₂²	C₂³×Dic₁₁	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	2	6	no	—
81	C₄	C₂²×C₁₁⋊C₈	1	12	yes	—
82	C₄	#67040	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
85	C₂	#10623343	1	1	yes	characteristic
86	C₂	#10623350	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator	C₂×C₁₁	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₁₁	C₂
socle	C₂²×C₁₁	C₂³×C₄
prime radical	C₂³×C₁₁	C₂²×C₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₁
hypocenter	C₁₁	#2982
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×C₁₁⋊C₈	C₂	semidirect
12. C₂×Q₈×C₁₁	C₈	semidirect
55. C₂×Q₈	C₁₁⋊C₈	semidirect
64. C₁₁	#2982	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₁₁	2
Q₈×C₁₁⋊C₈	2
#10622538	2
#2982	11