Subgroups — C₇×#455 #90101

Order 448.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₇×#137	C₂	1	2	yes	maximal, complemented
3	C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#104	C₂	1	2	yes	maximal, complemented
5	C₇×#100	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	—	2	2	no	—
7	C₇×#41	C₂²	1	2	yes	—
8	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₇×D₄	—	2	2	no	—
10	C₂×C₇×D₄	—	2	2	no	—
11	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₇×Q₁₆	—	2	2	no	—
13	C₇×SD₁₆	—	2	2	no	—
14	C₇×M₁₆	C₂²	1	2	yes	complemented
15	C₇×C₂²⋊C₄	—	2	2	no	—
16	C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
17	#455	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₇	—	2	2	no	—
19	C₂³×C₇	D₄	1	2	yes	—
20	C₇×Q₈	—	2	2	no	—
21	C₇×Q₈	D₄	1	2	yes	—
22	C₇×D₄	—	2	4	no	—
23	C₇×D₄	—	2	2	no	—
24	C₇×D₄	—	2	2	no	—
25	C₇×D₄	D₄	1	2	yes	—
26	C₂×C₄×C₇	—	2	2	no	—
27	C₂×C₄×C₇	—	2	2	no	—
28	C₂×C₄×C₇	—	2	1	no	—
29	C₂×C₄×C₇	—	2	1	no	—
30	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₇×C₈	—	2	2	no	—
32	H(5,𝔽₂)	C₂×C₇	1	1	yes	characteristic
33	#137	C₂×C₇	1	2	yes	complemented
34	#124	C₂×C₇	1	1	yes	characteristic, complemented
35	#104	C₂×C₇	1	2	yes	complemented
36	#100	C₂×C₇	1	1	yes	characteristic, complemented
37	C₂²×C₇	—	2	2	no	—
38	C₂²×C₇	—	2	2	no	—
39	C₂²×C₇	—	4	2	no	—
40	C₂²×C₇	—	8	2	no	—
41	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₇	—	2	2	no	—
43	C₄×C₇	—	4	1	no	—
44	C₄×C₇	—	4	1	no	—
45	C₄×C₇	C₂×D₄	1	2	yes	—
46	#41	—	2	2	no	—
47	#41	C₂²×C₇	1	2	yes	—
48	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
49	C₂×D₄	—	2	2	no	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
52	Q₁₆	—	2	2	no	—
53	SD₁₆	—	2	2	no	—
54	M₁₆	C₂²×C₇	1	2	yes	complemented
55	C₂²⋊C₄	—	2	2	no	—
56	C₄²	C₂²×C₇	1	1	yes	characteristic, complemented
57	C₂×C₇	—	2	1	no	—
58	C₂×C₇	—	4	2	no	—
59	C₂×C₇	—	4	2	no	—
60	C₂×C₇	#120	1	1	yes	characteristic
61	C₂³	—	2	2	no	—
62	C₂³	C₇×D₄	1	2	yes	—
63	Q₈	—	2	2	no	—
64	Q₈	C₇×D₄	1	2	yes	—
65	D₄	—	2	4	no	—
66	D₄	—	2	2	no	—
67	D₄	—	2	2	no	—
68	D₄	C₇×D₄	1	2	yes	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₇	#455	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²	—	2	2	no	—
77	C₂²	—	2	2	no	—
78	C₂²	—	4	2	no	—
79	C₂²	—	8	2	no	—
80	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	4	1	no	—
83	C₄	—	4	1	no	—
84	C₄	C₂×C₇×D₄	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	4	2	no	—
87	C₂	—	4	2	no	—
88	C₂	C₇×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₇×H(5,𝔽₂)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#137	C₂	semidirect
3. C₇×#124	C₂	semidirect
4. C₇×#104	C₂	semidirect
5. C₇×#100	C₂	semidirect
8. C₂×C₇×Q₈	C₂²	semidirect
14. C₇×M₁₆	C₂²	semidirect
16. C₄²×C₇	C₂²	semidirect
17. #455	C₇	direct (N × H)
33. #137	C₂×C₇	semidirect
34. #124	C₂×C₇	semidirect
35. #104	C₂×C₇	semidirect
36. #100	C₂×C₇	semidirect
48. C₂×Q₈	C₂²×C₇	semidirect
54. M₁₆	C₂²×C₇	semidirect
56. C₄²	C₂²×C₇	semidirect
75. C₇	#455	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×H(5,𝔽₂)	2
C₇×#137	2
C₇×#124	2
C₇×#104	2
C₇×#100	2
#455	7

Subgroups — C7×#455 #90101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#455 #90101