Subgroups — #13089870

Order 1456.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊D₉₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#10624413	C₂	1	1	yes	maximal, characteristic, complemented
5	#10624411	C₂	1	1	yes	maximal, characteristic, complemented
6	#10624409	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×D₉₁	C₂²	1	1	yes	characteristic
10	C₂×C₇×D₁₃	C₂²	1	1	yes	characteristic
11	Dic₉₁	C₂²	1	1	yes	characteristic
12	C₇×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×Dic₇	C₂²	1	1	yes	characteristic
14	C₁₃×Dic₇	C₂²	1	1	yes	characteristic
15	#6127	—	7	1	no	maximal, Hall
16	C₂×C₇×C₁₃	—	2	1	no	—
17	C₂×C₇×C₁₃	C₂³	1	1	yes	characteristic
18	D₉₁	—	2	1	no	—
19	C₇×D₁₃	—	2	1	no	—
20	#1159	—	13	1	no	maximal, Hall
21	C₂×C₄×C₁₃	—	7	1	no	—
22	C₂²⋊D₁₃	—	7	1	no	—
23	C₂²⋊D₁₃	D₇	1	1	yes	characteristic, complemented
24	D₅₂	—	7	1	no	—
25	C₄×D₁₃	—	7	1	no	—
26	C₄×D₁₃	—	7	1	no	—
27	Dic₂₆	—	7	1	no	—
28	C₇×C₁₃	#41	1	1	yes	characteristic, complemented, Hall
29	C₇×D₄	—	13	1	no	—
30	C₂²⋊D₇	—	13	1	no	—
31	C₂²⋊D₇	—	13	1	no	—
32	C₂×Dic₇	—	13	1	no	—
33	C₂×Dic₇	D₁₃	1	1	yes	characteristic, complemented
34	C₄×D₇	—	13	1	no	—
35	Dic₁₄	—	13	1	no	—
36	C₂²×C₁₃	C₂×D₇	1	1	yes	characteristic
37	C₂×D₁₃	—	7	1	no	—
38	C₂×D₁₃	C₂×D₇	1	1	yes	characteristic
39	C₄×C₁₃	—	7	1	no	—
40	C₄×C₁₃	—	7	1	no	—
41	Dic₁₃	—	7	1	no	—
42	Dic₁₃	C₂×D₇	1	1	yes	characteristic
43	C₂²×C₇	—	13	1	no	—
44	C₂²×C₇	C₂×D₁₃	1	1	yes	characteristic
45	C₂×D₇	—	13	1	no	—
46	C₄×C₇	—	13	1	no	—
47	Dic₇	—	13	1	no	—
48	Dic₇	C₂×D₁₃	1	1	yes	characteristic
49	Dic₇	C₂×D₁₃	1	1	yes	characteristic
50	C₂×C₁₃	—	2	1	no	—
51	C₂×C₁₃	C₂²×D₇	1	1	yes	characteristic
52	D₁₃	—	2	1	no	—
53	D₁₃	—	14	1	no	—
54	#41	—	91	1	no	Sylow
55	C₂×C₇	—	2	1	no	—
56	C₂×C₇	—	26	1	no	—
57	C₂×C₇	C₂²×D₁₃	1	1	yes	characteristic
58	D₇	—	26	1	no	—
59	C₁₃	#1159	1	1	yes	characteristic, complemented, Sylow
60	Q₈	—	91	1	no	—
61	D₄	—	13	1	no	—
62	D₄	—	91	1	no	—
63	D₄	—	91	1	no	—
64	C₂×C₄	—	7	1	no	—
65	C₂×C₄	—	91	1	no	—
66	C₂×C₄	—	91	1	no	—
67	C₇	#6127	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	13	1	no	—
69	C₂²	—	91	1	no	—
70	C₂²	D₇×D₁₃	1	1	yes	characteristic
71	C₄	—	7	1	no	—
72	C₄	—	7	1	no	—
73	C₄	—	13	1	no	—
74	C₄	—	91	1	no	—
75	C₂	—	2	1	no	—
76	C₂	—	26	1	no	—
77	C₂	—	182	1	no	—
78	C₂	C₂×D₇×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₇×D₁₃
commutator, socle, cosocle	C₂×C₇×C₁₃	C₂³
Fitting	C₂²×C₇×C₁₃	C₂²
hypercenter	C₂²	D₇×D₁₃
hypocenter	C₇×C₁₃	#41
LCS	G ▷ C₂×C₇×C₁₃ ▷ C₇×C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²⋊D₉₁	C₂	semidirect
2. C₂×C₁₃×Dic₇	C₂	semidirect
3. C₇×C₂²⋊D₁₃	C₂	semidirect
4. #10624413	C₂	semidirect
5. #10624411	C₂	semidirect
6. #10624409	C₂	semidirect
7. D₁₃×Dic₇	C₂	semidirect
23. C₂²⋊D₁₃	D₇	semidirect
28. C₇×C₁₃	#41	semidirect
33. C₂×Dic₇	D₁₃	semidirect
59. C₁₃	#1159	semidirect
67. C₇	#6127	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊D₉₁	2
C₂×C₁₃×Dic₇	2
C₇×C₂²⋊D₁₃	2
#10624413	2
#10624411	2
#10624409	2
D₁₃×Dic₇	2
#6127	7
#1159	13