Subgroups — #10622113

Order 704.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66990	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#100	C₂	1	1	yes	maximal, characteristic, complemented
3	#66898	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₁₁	C₄	1	1	yes	characteristic, complemented
5	C₂×D₄×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊D₁₁	—	2	1	no	—
7	C₁₁×M₁₆	—	2	1	no	—
8	C₂²⋊Dic₁₁	—	2	1	no	—
9	C₂²⋊Dic₁₁	C₄	1	1	yes	characteristic, complemented
10	(C₂×D₁₁)⋊C₄	—	2	1	no	—
11	C₂³×C₁₁	C₂×C₄	1	1	yes	characteristic
12	C₂³×C₁₁	D₄	1	1	yes	characteristic
13	C₂²×D₁₁	—	2	1	no	—
14	C₂²×D₁₁	—	4	2	no	—
15	C₂²×D₁₁	—	4	1	no	—
16	D₄×C₁₁	—	4	1	no	—
17	C₂×C₄×C₁₁	D₄	1	1	yes	characteristic
18	C₂²⋊D₁₁	—	4	2	no	—
19	C₂×Dic₁₁	—	2	1	no	—
20	C₂×Dic₁₁	—	4	1	no	—
21	C₈×C₁₁	—	4	1	no	—
22	AΣL(1,𝔽₁₆)	—	11	1	no	maximal, Sylow
23	C₂²×C₁₁	—	2	1	no	—
24	C₂²×C₁₁	—	2	1	no	—
25	C₂²×C₁₁	—	4	1	no	—
26	C₂²×C₁₁	—	4	1	no	—
27	C₂²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×D₁₁	—	2	2	no	—
29	C₂×D₁₁	—	4	1	no	—
30	C₂×D₁₁	—	4	2	no	—
31	C₂×D₁₁	—	4	1	no	—
32	C₂×D₁₁	—	4	2	no	—
33	C₄×C₁₁	—	2	1	no	—
34	Dic₁₁	—	4	1	no	—
35	Dic₁₁	—	8	1	no	—
36	#120	—	11	1	no	—
37	#100	D₁₁	1	1	yes	characteristic, complemented
38	#99	—	11	1	no	—
39	C₂×C₁₁	—	2	1	no	—
40	C₂×C₁₁	—	4	1	no	—
41	C₂×C₁₁	—	4	1	no	—
42	C₂×C₁₁	#99	1	1	yes	characteristic
43	D₁₁	—	4	2	no	—
44	C₂⁴	—	11	1	no	—
45	C₂×D₄	—	22	1	no	—
46	C₂×D₄	C₂×D₁₁	1	1	yes	characteristic
47	M₁₆	—	2	1	no	—
48	C₂²⋊C₄	—	11	1	no	—
49	C₂²⋊C₄	—	22	1	no	—
50	C₂²⋊C₄	—	22	1	no	—
51	C₁₁	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
52	C₂³	—	22	1	no	—
53	C₂³	—	44	1	no	—
54	C₂³	—	44	2	no	—
55	C₂³	C₄×D₁₁	1	1	yes	characteristic
56	C₂³	D₄₄	1	1	yes	characteristic
57	D₄	—	4	1	no	—
58	D₄	—	44	2	no	—
59	C₂×C₄	—	22	1	no	—
60	C₂×C₄	—	44	1	no	—
61	C₂×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
62	C₈	—	4	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	22	2	no	—
68	C₂²	—	44	1	no	—
69	C₂²	—	44	2	no	—
70	C₂²	—	44	2	no	—
71	C₂²	—	44	1	no	—
72	C₂²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	44	1	no	—
75	C₄	—	88	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	4	1	no	—
79	C₂	—	44	2	no	—
80	C₂	#66870	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#66870
commutator	C₂³×C₁₁	C₂×C₄
Frattini	C₂×D₄	C₂×D₁₁
Fitting	C₁₁×#100	C₂
socle	C₂×C₁₁	#99
prime radical	#66990	C₂
cosocle	C₂×D₄×C₁₁	C₂²
hypercenter	#100	D₁₁
hypocenter	C₁₁	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₁₁ ▷ C₂²×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #100

Complements

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

N	H ≅ G/N	type
2. C₁₁×#100	C₂	semidirect
3. #66898	C₂	semidirect
4. C₂³×D₁₁	C₄	semidirect
9. C₂²⋊Dic₁₁	C₄	semidirect
37. #100	D₁₁	semidirect
51. C₁₁	AΣL(1,𝔽₁₆)	semidirect

Maximal subgroups

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

subgroup	index
#66990	2
C₁₁×#100	2
#66898	2
AΣL(1,𝔽₁₆)	11