Subgroups — C₇×#7538 #13184754

Order 1792.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₆₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#1423	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×Q₃₂	—	2	1	no	—
5	C₇×Q₆₄	—	2	1	no	—
6	C₇×Q₆₄	C₄	1	2	yes	complemented
7	C₂×C₇×C₃₂	C₂²	1	1	yes	characteristic
8	C₇×#366	—	2	1	no	—
9	C₇×C₆₄	—	2	1	no	—
10	#7538	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₇×Q₁₆	—	4	1	no	—
12	C₇×Q₃₂	—	2	2	no	—
13	C₇×Q₃₂	—	4	1	no	—
14	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
15	C₇×#107	—	4	1	no	—
16	C₇×C₃₂	C₂×C₄	1	1	yes	characteristic
17	C₇×C₃₂	D₄	1	1	yes	characteristic
18	C₂×Q₆₄	C₂×C₇	1	1	yes	characteristic
19	C₂×C₆₄	C₂×C₇	1	1	yes	characteristic
20	#1423	C₂×C₇	1	1	yes	characteristic
21	C₂×C₇×Q₈	—	8	1	no	—
22	C₇×Q₁₆	—	4	2	no	—
23	C₇×Q₁₆	—	8	1	no	—
24	C₂×C₇×C₈	D₈	1	1	yes	characteristic
25	C₇×C₄⋊C₄	—	8	1	no	—
26	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
27	C₇×C₁₆	SD₁₆	1	1	yes	characteristic
28	C₂×Q₃₂	—	2	1	no	—
29	Q₆₄	—	2	1	no	—
30	Q₆₄	C₄×C₇	1	2	yes	complemented
31	C₂×C₃₂	C₂²×C₇	1	1	yes	characteristic
32	#366	—	2	1	no	—
33	C₆₄	—	2	1	no	—
34	C₇×Q₈	—	8	2	no	—
35	C₇×Q₈	—	16	1	no	—
36	C₂×C₄×C₇	—	16	1	no	—
37	C₂×C₄×C₇	—	16	1	no	—
38	C₂×C₄×C₇	D₁₆	1	1	yes	characteristic
39	C₇×C₈	#102	1	1	yes	characteristic
40	C₇×C₈	SD₃₂	1	1	yes	characteristic
41	C₂×Q₁₆	—	4	1	no	—
42	Q₃₂	—	2	2	no	—
43	Q₃₂	—	4	1	no	—
44	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
45	#107	—	4	1	no	—
46	C₃₂	C₂×C₄×C₇	1	1	yes	characteristic
47	C₃₂	C₇×D₄	1	1	yes	characteristic
48	C₂²×C₇	D₃₂	1	1	yes	characteristic
49	C₄×C₇	—	16	2	no	—
50	C₄×C₇	—	32	1	no	—
51	C₄×C₇	#357	1	1	yes	characteristic
52	C₄×C₇	SD₆₄	1	1	yes	characteristic
53	C₂×Q₈	—	8	1	no	—
54	Q₁₆	—	4	2	no	—
55	Q₁₆	—	8	1	no	—
56	C₂×C₈	C₇×D₈	1	1	yes	characteristic
57	C₄⋊C₄	—	8	1	no	—
58	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
59	C₁₆	C₇×SD₁₆	1	1	yes	characteristic
60	C₂×C₇	#1415	1	1	yes	characteristic
61	C₂×C₇	SD₁₂₈	1	1	yes	characteristic
62	C₂×C₇	Q₁₂₈	1	1	yes	characteristic
63	Q₈	—	8	2	no	—
64	Q₈	—	16	1	no	—
65	C₂×C₄	—	16	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	C₇×D₁₆	1	1	yes	characteristic
68	C₈	C₇×#102	1	1	yes	characteristic
69	C₈	C₇×SD₃₂	1	1	yes	characteristic
70	C₇	#7538	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²	C₇×D₃₂	1	1	yes	characteristic
72	C₄	—	16	2	no	—
73	C₄	—	32	1	no	—
74	C₄	C₇×#357	1	1	yes	characteristic
75	C₄	C₇×SD₆₄	1	1	yes	characteristic
76	C₂	C₇×#1415	1	1	yes	characteristic
77	C₂	C₇×SD₁₂₈	1	1	yes	characteristic
78	C₂	C₇×Q₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	D₃₂
commutator	C₃₂	C₂×C₄×C₇
Frattini, cosocle	C₂×C₃₂	C₂²×C₇
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₂×C₇×C₁₆ ◁ C₂×C₇×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₇×Q₆₄	C₄	semidirect
10. #7538	C₇	direct (N × H)
30. Q₆₄	C₄×C₇	semidirect
70. C₇	#7538	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×Q₆₄	2
C₂×C₇×C₆₄	2
C₇×#1423	2
#7538	7

Subgroups — C7×#7538 #13184754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7538 #13184754