Subgroups — C₇×#559 #90580

Order 448.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#127	C₂	1	1	yes	maximal, characteristic
2	C₇×#124	C₂	1	4	yes	maximal, complemented
3	C₇×#123	C₂	1	4	yes	maximal, complemented
4	C₄×C₇×Q₈	C₂	1	2	yes	maximal, complemented
5	C₄×C₇×D₄	C₂	1	4	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×C₇×D₄	C₂²	1	4	yes	—
8	C₂×C₇×D₄	C₂²	1	2	yes	—
9	C₂²×C₄×C₇	C₂²	1	4	yes	—
10	C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	C₇×C₄⋊C₄	C₂²	1	4	yes	—
12	C₇×C₂²⋊C₄	C₂²	1	8	yes	—
13	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₄²×C₇	C₂²	1	4	yes	—
15	C₄²×C₇	C₂²	1	1	yes	characteristic
16	#559	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₇	C₂³	1	4	yes	—
18	C₇×Q₈	—	2	4	no	—
19	C₇×D₄	—	2	8	no	—
20	C₇×D₄	—	4	2	no	—
21	C₂×C₄×C₇	—	4	4	no	—
22	C₂×C₄×C₇	C₂³	1	4	yes	—
23	C₂×C₄×C₇	C₂³	1	4	yes	—
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂³	1	2	yes	—
26	#127	C₂×C₇	1	1	yes	characteristic
27	#124	C₂×C₇	1	4	yes	complemented
28	#123	C₂×C₇	1	4	yes	complemented
29	C₄×Q₈	C₂×C₇	1	2	yes	complemented
30	C₄×D₄	C₂×C₇	1	4	yes	complemented
31	C₂²×C₇	—	2	8	no	—
32	C₂²×C₇	—	2	4	no	—
33	C₂²×C₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₇	—	2	4	no	—
35	C₄×C₇	—	2	4	no	—
36	C₄×C₇	—	2	1	no	—
37	C₄×C₇	#41	1	4	yes	—
38	C₂×Q₈	C₂²×C₇	1	2	yes	—
39	C₂×D₄	C₂²×C₇	1	4	yes	—
40	C₂×D₄	C₂²×C₇	1	2	yes	—
41	C₂²×C₄	C₂²×C₇	1	4	yes	—
42	C₄⋊C₄	C₂²×C₇	1	2	yes	—
43	C₄⋊C₄	C₂²×C₇	1	4	yes	—
44	C₂²⋊C₄	C₂²×C₇	1	8	yes	—
45	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
46	C₄²	C₂²×C₇	1	4	yes	—
47	C₄²	C₂²×C₇	1	1	yes	characteristic
48	C₂×C₇	—	4	4	no	—
49	C₂×C₇	C₂×#41	1	2	yes	—
50	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂³	C₂³×C₇	1	4	yes	—
52	Q₈	—	2	4	no	—
53	D₄	—	2	8	no	—
54	D₄	—	4	2	no	—
55	C₂×C₄	—	4	4	no	—
56	C₂×C₄	C₂³×C₇	1	4	yes	—
57	C₂×C₄	C₂³×C₇	1	4	yes	—
58	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
59	C₂×C₄	C₂³×C₇	1	2	yes	—
60	C₇	#559	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂⁴×C₇	1	1	yes	characteristic
64	C₄	—	2	4	no	—
65	C₄	—	2	4	no	—
66	C₄	—	2	1	no	—
67	C₄	C₇×#41	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×C₇×#41	1	2	yes	—
70	C₂	C₇×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₇×#127	C₂
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
2. C₇×#124	C₂	semidirect
3. C₇×#123	C₂	semidirect
4. C₄×C₇×Q₈	C₂	semidirect
5. C₄×C₇×D₄	C₂	semidirect
16. #559	C₇	direct (N × H)
27. #124	C₂×C₇	semidirect
28. #123	C₂×C₇	semidirect
29. C₄×Q₈	C₂×C₇	semidirect
30. C₄×D₄	C₂×C₇	semidirect
60. C₇	#559	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×#127	2
C₇×#124	2
C₇×#123	2
C₄×C₇×Q₈	2
C₄×C₇×D₄	2
#559	7

Subgroups — C7×#559 #90580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#559 #90580