Subgroups — #13070518

Order 1408.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₂₂	C₂	1	1	yes	maximal, characteristic
2	C₁₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	Dic₁₁⋊C₁₆	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₁	C₂	1	2	yes	maximal
5	C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₂	C₄	1	1	yes	characteristic
7	C₂×C₁₁×C₁₆	C₂²	1	2	yes	—
8	C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	Dic₁₁⋊C₈	C₄	1	2	yes	—
10	C₈×Dic₁₁	C₂²	1	2	yes	—
11	C₂×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄⋊(C₁₁⋊C₈)	C₄	1	1	yes	characteristic
14	C₂×Dic₂₂	C₈	1	1	yes	characteristic
15	C₂×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
17	C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₁₁	C₈	1	1	yes	characteristic
19	Dic₁₁⋊C₄	C₈	1	2	yes	—
20	C₄×Dic₁₁	C₂×C₄	1	2	yes	—
21	C₂×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₁₁×C₁₆	—	2	2	no	—
23	C₁₁⋊C₁₆	—	2	1	no	—
24	C₁₁⋊C₁₆	Q₈	1	2	yes	complemented
25	Q₈×C₁₆	—	11	1	no	maximal, Sylow
26	C₂×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×Dic₁₁	C₂×C₈	1	2	yes	—
30	C₂×Dic₁₁	C₂×C₈	1	2	yes	—
31	Dic₂₂	C₁₆	1	4	yes	complemented
32	C₈×C₁₁	—	2	1	no	—
33	C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₁₁	#41	1	1	yes	characteristic
35	C₁₁⋊C₈	—	2	2	no	—
36	C₈×Q₈	—	11	1	no	—
37	C₄⋊C₁₆	—	11	2	no	—
38	C₄⋊C₁₆	D₁₁	1	1	yes	characteristic
39	C₄×C₁₆	—	11	2	no	—
40	C₄×C₁₆	—	11	1	no	—
41	C₂²×C₁₁	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₁	—	2	1	no	—
43	C₄×C₁₁	C₂×C₁₆	1	2	yes	—
44	C₄×C₁₁	C₄×Q₈	1	1	yes	characteristic
45	C₄×C₁₁	#131	1	1	yes	characteristic
46	Dic₁₁	—	2	2	no	—
47	Dic₁₁	C₂×C₁₆	1	4	yes	—
48	C₄×Q₈	—	11	1	no	—
49	C₂×C₁₆	—	11	1	no	—
50	C₂×C₁₆	—	11	1	no	—
51	C₂×C₁₆	C₂×D₁₁	1	2	yes	—
52	C₄⋊C₈	—	11	2	no	—
53	C₄⋊C₈	—	11	1	no	—
54	C₄×C₈	—	11	2	no	—
55	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
56	C₂×C₁₁	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
58	C₂×C₁₁	#504	1	1	yes	characteristic
59	C₂×Q₈	—	11	1	no	—
60	C₂×C₈	—	11	2	no	—
61	C₂×C₈	C₄×D₁₁	1	1	yes	characteristic
62	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
63	C₄⋊C₄	—	11	2	no	—
64	C₄⋊C₄	—	11	1	no	—
65	C₄²	—	11	2	no	—
66	C₄²	C₄×D₁₁	1	1	yes	characteristic
67	C₁₆	—	2	2	no	—
68	C₁₆	—	11	2	no	—
69	C₁₆	—	22	1	no	—
70	C₁₁	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
71	Q₈	—	11	4	no	—
72	C₂×C₄	—	11	2	no	—
73	C₂×C₄	—	11	2	no	—
74	C₂×C₄	C₈×D₁₁	1	1	yes	characteristic
75	C₂×C₄	C₈×D₁₁	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	22	2	no	—
79	C₈	#4341	1	1	yes	characteristic
80	C₈	Q₈×D₁₁	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₁₁	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	11	4	no	—
84	C₄	—	22	2	no	—
85	C₄	C₁₆×D₁₁	1	2	yes	—
86	C₄	#66940	1	1	yes	characteristic
87	C₄	#66960	1	1	yes	characteristic
88	C₂	#10622443	1	1	yes	characteristic
89	C₂	C₂×C₁₆×D₁₁	1	1	yes	characteristic
90	C₂	#10622520	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₁
commutator	C₂×C₁₁	C₂²×C₁₆
Fitting	C₁₁×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂³
hypercenter	C₄⋊C₁₆	D₁₁
hypocenter	C₁₁	Q₈×C₁₆
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
24. C₁₁⋊C₁₆	Q₈	semidirect
31. Dic₂₂	C₁₆	semidirect
70. C₁₁	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₂₂	2
C₁₁×C₄⋊C₁₆	2
Dic₁₁⋊C₁₆	2
C₁₆×Dic₁₁	2
C₄×C₁₁⋊C₁₆	2
Q₈×C₁₆	11