Subgroups — C₇×#15692569 #37524328

Order 22932.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#11717941	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₁₃×Dic₇	C₃	1	1	yes	maximal, characteristic
3	C₇×#11717941	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic
5	C₄×C₇×C₁₃⋊C₉	—	7	1	no	maximal
6	#15692569	C₇	1	1	yes	maximal, characteristic, direct product factor
7	C₇×C₁₃×Dic₇	C₉	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic
9	C₇×Dic₇⋊C₉	—	13	1	no	maximal, Hall
10	C₂×#11717941	C₂×C₇	1	1	yes	characteristic
11	C₂×C₇×C₁₃⋊C₉	—	7	1	no	—
12	C₂×C₇²×C₁₃	C₂×C₉	1	1	yes	characteristic
13	C₃×C₄×C₇×C₁₃	—	21	1	no	—
14	C₃×C₁₃×Dic₇	C₃×C₇	1	1	yes	characteristic
15	C₂×C₇×C₇⋊C₉	—	13	1	no	—
16	#11717941	C₄×C₇	1	1	yes	characteristic, complemented
17	C₇×C₁₃⋊C₉	—	7	1	no	—
18	C₇²×C₁₃	C₄×C₉	1	1	yes	characteristic, complemented, Hall
19	C₃×C₇×Dic₇	C₁₃⋊C₃	1	1	yes	characteristic
20	C₂×C₃×C₇×C₁₃	—	6	1	no	—
21	C₂×C₃×C₇×C₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic
22	C₂×C₃×C₇×C₁₃	C₂×C₃×C₇	1	1	yes	characteristic
23	C₄×C₁₃⋊C₉	—	7	1	no	Hall
24	C₇×C₇⋊C₉	—	13	1	no	Hall
25	C₄×C₇×C₁₃	—	21	1	no	—
26	C₁₃×Dic₇	C₇×C₉	1	1	yes	characteristic, complemented
27	C₂×C₃×C₇²	C₂×C₁₃⋊C₃	1	1	yes	characteristic
28	C₃×C₇×C₁₃	—	6	1	no	—
29	C₃×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
30	C₃×C₇×C₁₃	C₃×C₄×C₇	1	1	yes	characteristic
31	C₄×C₇×C₉	—	91	1	no	—
32	Dic₇⋊C₉	—	13	1	no	—
33	C₂×C₁₃⋊C₉	—	7	1	no	—
34	C₇×Dic₇	C₁₃⋊C₉	1	1	yes	characteristic, complemented, Hall
35	C₂×C₇×C₁₃	—	6	1	no	—
36	C₂×C₇×C₁₃	D₇⋊C₉	1	1	yes	characteristic
37	C₂×C₇×C₁₃	C₂×C₇×C₉	1	1	yes	characteristic
38	C₃×C₄×C₁₃	—	21	1	no	—
39	C₃×C₇²	C₄×C₁₃⋊C₃	1	1	yes	characteristic
40	C₂×C₇×C₉	—	91	1	no	—
41	C₂×C₇⋊C₉	—	13	1	no	—
42	C₁₃⋊C₉	—	7	1	no	Hall
43	C₂×C₇²	C₂×C₁₃⋊C₉	1	1	yes	characteristic
44	C₇×C₁₃	—	6	1	no	—
45	C₇×C₁₃	Dic₇⋊C₉	1	1	yes	characteristic, complemented
46	C₇×C₁₃	C₄×C₇×C₉	1	1	yes	characteristic, complemented
47	C₃×C₄×C₇	—	21	1	no	—
48	C₃×Dic₇	C₇×C₁₃⋊C₃	1	1	yes	characteristic
49	C₂×C₃×C₁₃	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic
50	C₇×C₉	—	91	1	no	—
51	C₇⋊C₉	—	13	1	no	—
52	C₄×C₁₃	—	21	1	no	Hall
53	C₇²	C₄×C₁₃⋊C₉	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₃×C₇	—	6	1	no	—
55	C₂×C₃×C₇	#10587720	1	1	yes	characteristic
56	C₂×C₃×C₇	C₂×C₇×C₁₃⋊C₃	1	1	yes	characteristic
57	C₃×C₁₃	C₇×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
58	C₄×C₉	—	91	1	no	Hall
59	C₄×C₇	—	21	1	no	—
60	Dic₇	C₇×C₁₃⋊C₉	1	1	yes	characteristic, complemented
61	C₂×C₁₃	C₇×D₇⋊C₉	1	1	yes	characteristic
62	C₃×C₇	—	6	1	no	—
63	C₃×C₇	#11763961	1	1	yes	characteristic
64	C₃×C₇	C₄×C₇×C₁₃⋊C₃	1	1	yes	characteristic
65	C₂×C₉	—	91	1	no	—
66	C₂×C₇	—	6	1	no	—
67	C₂×C₇	#13109900	1	1	yes	characteristic
68	C₂×C₇	C₂×C₇×C₁₃⋊C₉	1	1	yes	characteristic
69	C₁₃	C₇×Dic₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
70	C₃×C₄	—	7	1	no	—
71	C₉	—	91	1	no	Sylow
72	C₇	—	6	1	no	—
73	C₇	#15692569	1	1	yes	characteristic, direct product factor
74	C₇	C₄×C₇×C₁₃⋊C₉	1	1	yes	characteristic, complemented
75	C₂×C₃	C₇×#10587720	1	1	yes	characteristic
76	C₄	—	7	1	no	Sylow
77	C₃	C₇×#11763961	1	1	yes	characteristic
78	C₂	C₇×#13109900	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇	#10587720
commutator, hypocenter	C₇×C₁₃	C₄×C₇×C₉
Frattini	C₂×C₃	C₇×#10587720
Fitting, socle, prime radical	C₂×C₃×C₇²×C₁₃	C₂×C₃
cosocle	C₂×C₃×C₇×C₁₃	C₂×C₃×C₇
LCS	G ▷ C₇×C₁₃
UCS	C₁ ◁ C₂×C₃×C₇

Complements

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

N	H ≅ G/N	type
3. C₇×#11717941	C₄	semidirect
6. #15692569	C₇	direct (N × H)
7. C₇×C₁₃×Dic₇	C₉	semidirect
16. #11717941	C₄×C₇	semidirect
18. C₇²×C₁₃	C₄×C₉	semidirect
26. C₁₃×Dic₇	C₇×C₉	semidirect
34. C₇×Dic₇	C₁₃⋊C₉	semidirect
45. C₇×C₁₃	Dic₇⋊C₉	semidirect
46. C₇×C₁₃	C₄×C₇×C₉	semidirect
53. C₇²	C₄×C₁₃⋊C₉	semidirect
60. Dic₇	C₇×C₁₃⋊C₉	semidirect
69. C₁₃	C₇×Dic₇⋊C₉	semidirect
73. C₇	#15692569	direct (N × H)
74. C₇	C₄×C₇×C₁₃⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#11717941	2
C₃×C₇×C₁₃×Dic₇	3
C₄×C₇×C₁₃⋊C₉	7
#15692569	7
C₇×Dic₇⋊C₉	13

Subgroups — C7×#15692569 #37524328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#15692569 #37524328