Subgroups — #13089864

Order 1456.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#10624411	C₂	1	2	yes	maximal, complemented
5	#10624409	C₂	1	2	yes	maximal, complemented
6	C₂×D₉₁	C₂²	1	2	yes	—
7	C₂×C₇×D₁₃	C₂²	1	1	yes	characteristic
8	C₄×C₇×C₁₃	C₂²	1	1	yes	characteristic
9	C₇×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×Dic₇	C₂²	1	2	yes	—
11	#6131	—	7	1	no	maximal, Hall
12	C₂×C₇×C₁₃	C₂³	1	1	yes	characteristic
13	D₉₁	—	2	2	no	—
14	C₇×D₁₃	—	2	1	no	—
15	#1157	—	13	1	no	maximal, Hall
16	Q₈×C₁₃	—	7	1	no	—
17	D₅₂	—	7	2	no	—
18	D₅₂	—	7	1	no	—
19	C₄×D₁₃	—	7	2	no	—
20	C₄×D₁₃	D₇	1	1	yes	characteristic, complemented
21	C₇×C₁₃	#41	1	1	yes	characteristic, complemented, Hall
22	C₂×C₄×C₇	—	13	1	no	—
23	C₂²⋊D₇	—	13	2	no	—
24	D₂₈	—	13	1	no	—
25	C₄×D₇	—	13	2	no	—
26	Dic₁₄	D₁₃	1	1	yes	characteristic, complemented
27	C₂×D₁₃	—	7	2	no	—
28	C₂×D₁₃	C₂×D₇	1	1	yes	characteristic
29	C₄×C₁₃	—	7	2	no	—
30	C₄×C₁₃	C₂×D₇	1	1	yes	characteristic
31	Dic₁₃	C₂×D₇	1	1	yes	characteristic
32	C₂²×C₇	—	13	1	no	—
33	C₂×D₇	—	13	2	no	—
34	C₄×C₇	—	13	1	no	—
35	C₄×C₇	C₂×D₁₃	1	1	yes	characteristic
36	Dic₇	C₂×D₁₃	1	2	yes	—
37	C₂×C₁₃	C₂²×D₇	1	1	yes	characteristic
38	D₁₃	—	2	1	no	—
39	D₁₃	—	14	2	no	—
40	#41	—	91	1	no	Sylow
41	C₂×C₇	—	26	1	no	—
42	C₂×C₇	C₂²×D₁₃	1	1	yes	characteristic
43	D₇	—	26	2	no	—
44	C₁₃	#1157	1	1	yes	characteristic, complemented, Sylow
45	Q₈	—	7	1	no	—
46	D₄	—	91	2	no	—
47	D₄	—	91	1	no	—
48	C₂×C₄	—	13	1	no	—
49	C₂×C₄	—	91	2	no	—
50	C₇	#6131	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	13	1	no	—
52	C₂²	—	91	2	no	—
53	C₄	—	7	2	no	—
54	C₄	—	13	1	no	—
55	C₄	D₇×D₁₃	1	1	yes	characteristic
56	C₂	—	26	1	no	—
57	C₂	—	182	2	no	—
58	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	C₄×C₇×C₁₃	C₂²
hypercenter	C₄	D₇×D₁₃
hypocenter	C₇×C₁₃	#41
LCS	G ▷ C₂×C₇×C₁₃ ▷ C₇×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. D₃₆₄	C₂	semidirect
2. C₁₃×Dic₁₄	C₂	semidirect
3. C₄×C₇×D₁₃	C₂	semidirect
4. #10624411	C₂	semidirect
5. #10624409	C₂	semidirect
20. C₄×D₁₃	D₇	semidirect
21. C₇×C₁₃	#41	semidirect
26. Dic₁₄	D₁₃	semidirect
44. C₁₃	#1157	semidirect
50. C₇	#6131	semidirect

Maximal subgroups

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

subgroup	index
D₃₆₄	2
C₁₃×Dic₁₄	2
C₄×C₇×D₁₃	2
#10624411	2
#10624409	2
#6131	7
#1157	13