Subgroups — C₇×#534 #90555

Order 448.

346 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×D₄	C₂	1	2	yes	maximal, complemented
2	C₇×#123	C₂	1	4	yes	maximal, complemented
3	C₇×#121	C₂	1	4	yes	maximal, complemented
4	C₇×#120	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₇	C₂²	1	1	yes	characteristic
7	C₂⁴×C₇	C₂²	1	2	yes	complemented
8	C₂×C₇×D₄	—	2	8	no	—
9	C₂×C₇×D₄	C₂²	1	4	yes	—
10	C₂×C₇×D₄	C₂²	1	4	yes	complemented
11	C₂×C₇×D₄	C₂²	1	4	yes	complemented
12	C₂²×C₄×C₇	C₂²	1	2	yes	complemented
13	C₂²×C₄×C₇	C₂²	1	2	yes	complemented
14	C₇×C₄⋊C₄	C₂²	1	4	yes	complemented
15	C₇×C₂²⋊C₄	C₂²	1	4	yes	complemented
16	C₇×C₂²⋊C₄	C₂²	1	8	yes	complemented
17	#534	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₇	—	2	4	no	—
19	C₂³×C₇	—	4	2	no	—
20	C₂³×C₇	—	4	4	no	—
21	C₂³×C₇	D₄	1	4	yes	complemented
22	C₂³×C₇	C₂³	1	4	yes	—
23	C₂³×C₇	C₂³	1	2	yes	—
24	C₂³×C₇	C₂³	1	1	yes	characteristic
25	C₇×D₄	—	2	8	no	—
26	C₇×D₄	—	2	8	no	—
27	C₇×D₄	—	4	4	no	—
28	C₂×C₄×C₇	—	2	4	no	—
29	C₂×C₄×C₇	—	4	2	no	—
30	C₂×C₄×C₇	C₂³	1	4	yes	—
31	C₂×C₄×C₇	C₂³	1	4	yes	—
32	C₂²×D₄	C₂×C₇	1	2	yes	complemented
33	#123	C₂×C₇	1	4	yes	complemented
34	#121	C₂×C₇	1	4	yes	complemented
35	#120	C₂×C₇	1	4	yes	complemented
36	C₂×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
37	C₂²×C₇	—	2	2	no	—
38	C₂²×C₇	—	2	4	no	—
39	C₂²×C₇	—	2	4	no	—
40	C₂²×C₇	—	2	8	no	—
41	C₂²×C₇	—	4	4	no	—
42	C₂²×C₇	—	4	8	no	—
43	C₂²×C₇	C₂×D₄	1	4	yes	—
44	C₂²×C₇	C₂×D₄	1	2	yes	—
45	C₂²×C₇	C₂⁴	1	1	yes	characteristic
46	C₄×C₇	—	2	4	no	—
47	C₄×C₇	—	2	4	no	—
48	C₂⁴	C₂²×C₇	1	1	yes	characteristic
49	C₂⁴	C₂²×C₇	1	2	yes	complemented
50	C₂×D₄	—	2	8	no	—
51	C₂×D₄	C₂²×C₇	1	4	yes	—
52	C₂×D₄	C₂²×C₇	1	4	yes	complemented
53	C₂×D₄	C₂²×C₇	1	4	yes	complemented
54	C₂²×C₄	C₂²×C₇	1	2	yes	complemented
55	C₂²×C₄	C₂²×C₇	1	2	yes	complemented
56	C₄⋊C₄	C₂²×C₇	1	4	yes	complemented
57	C₂²⋊C₄	C₂²×C₇	1	4	yes	complemented
58	C₂²⋊C₄	C₂²×C₇	1	8	yes	complemented
59	C₂×C₇	—	2	4	no	—
60	C₂×C₇	—	2	2	no	—
61	C₂×C₇	—	4	4	no	—
62	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
64	C₂³	—	2	4	no	—
65	C₂³	—	4	2	no	—
66	C₂³	—	4	4	no	—
67	C₂³	C₇×D₄	1	4	yes	complemented
68	C₂³	C₂³×C₇	1	4	yes	—
69	C₂³	C₂³×C₇	1	2	yes	—
70	C₂³	C₂³×C₇	1	1	yes	characteristic
71	D₄	—	2	8	no	—
72	D₄	—	2	8	no	—
73	D₄	—	4	4	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	C₂³×C₇	1	4	yes	—
77	C₂×C₄	C₂³×C₇	1	4	yes	—
78	C₇	#534	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²	—	2	2	no	—
80	C₂²	—	2	4	no	—
81	C₂²	—	2	4	no	—
82	C₂²	—	2	8	no	—
83	C₂²	—	4	4	no	—
84	C₂²	—	4	8	no	—
85	C₂²	C₂×C₇×D₄	1	4	yes	—
86	C₂²	C₂×C₇×D₄	1	2	yes	—
87	C₂²	C₂⁴×C₇	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	2	4	no	—
90	C₂	—	2	4	no	—
91	C₂	—	2	2	no	—
92	C₂	—	4	4	no	—
93	C₂	C₂²×C₇×D₄	1	1	yes	characteristic
94	C₂	C₇×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×D₄	C₂	semidirect
2. C₇×#123	C₂	semidirect
3. C₇×#121	C₂	semidirect
4. C₇×#120	C₂	semidirect
5. C₂×C₇×C₂²⋊C₄	C₂	semidirect
7. C₂⁴×C₇	C₂²	semidirect
10. C₂×C₇×D₄	C₂²	semidirect
11. C₂×C₇×D₄	C₂²	semidirect
12. C₂²×C₄×C₇	C₂²	semidirect
13. C₂²×C₄×C₇	C₂²	semidirect
14. C₇×C₄⋊C₄	C₂²	semidirect
15. C₇×C₂²⋊C₄	C₂²	semidirect
16. C₇×C₂²⋊C₄	C₂²	semidirect
17. #534	C₇	direct (N × H)
21. C₂³×C₇	D₄	semidirect
32. C₂²×D₄	C₂×C₇	semidirect
33. #123	C₂×C₇	semidirect
34. #121	C₂×C₇	semidirect
35. #120	C₂×C₇	semidirect
36. C₂×C₂²⋊C₄	C₂×C₇	semidirect
49. C₂⁴	C₂²×C₇	semidirect
52. C₂×D₄	C₂²×C₇	semidirect
53. C₂×D₄	C₂²×C₇	semidirect
54. C₂²×C₄	C₂²×C₇	semidirect
55. C₂²×C₄	C₂²×C₇	semidirect
56. C₄⋊C₄	C₂²×C₇	semidirect
57. C₂²⋊C₄	C₂²×C₇	semidirect
58. C₂²⋊C₄	C₂²×C₇	semidirect
67. C₂³	C₇×D₄	semidirect
78. C₇	#534	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×D₄	2
C₇×#123	2
C₇×#121	2
C₇×#120	2
C₂×C₇×C₂²⋊C₄	2
#534	7

Subgroups — C7×#534 #90555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#534 #90555