Subgroups — C₇×#550 #90571

Order 448.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#41	C₂	1	2	yes	maximal, complemented
2	C₇×#127	C₂	1	3	yes	maximal, complemented
3	C₇×#121	C₂	1	6	yes	maximal, complemented
4	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₇×D₄	C₂	1	3	yes	maximal, complemented
6	C₇×#41	—	2	8	no	—
7	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×D₄	C₂²	1	3	yes	—
9	C₂×C₇×D₄	C₂²	1	6	yes	—
10	C₂×C₇×D₄	C₂²	1	6	yes	—
11	C₂²×C₄×C₇	C₂²	1	6	yes	—
12	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₇×C₄⋊C₄	C₂²	1	3	yes	—
14	C₇×C₂²⋊C₄	C₂²	1	6	yes	—
15	C₄²×C₇	C₂²	1	3	yes	—
16	#550	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₇	C₂³	1	6	yes	—
18	C₇×Q₈	D₄	1	4	yes	complemented
19	C₇×D₄	—	2	12	no	—
20	C₇×D₄	—	2	6	no	—
21	C₇×D₄	—	4	6	no	—
22	C₂×C₄×C₇	—	2	12	no	—
23	C₂×C₄×C₇	C₂³	1	3	yes	—
24	C₂×C₄×C₇	C₂³	1	2	yes	—
25	C₂×C₄×C₇	C₂³	1	3	yes	—
26	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×#41	C₂×C₇	1	2	yes	complemented
28	#127	C₂×C₇	1	3	yes	complemented
29	#121	C₂×C₇	1	6	yes	complemented
30	C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
31	C₄×D₄	C₂×C₇	1	3	yes	complemented
32	C₂²×C₇	—	2	6	no	—
33	C₂²×C₇	—	2	6	no	—
34	C₂²×C₇	—	2	6	no	—
35	C₂²×C₇	C₂⁴	1	1	yes	characteristic
36	C₄×C₇	—	2	2	no	—
37	C₄×C₇	—	2	3	no	—
38	C₄×C₇	C₂×D₄	1	6	yes	—
39	C₄×C₇	#41	1	2	yes	complemented
40	#41	—	2	8	no	—
41	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
42	C₂×D₄	C₂²×C₇	1	3	yes	—
43	C₂×D₄	C₂²×C₇	1	6	yes	—
44	C₂×D₄	C₂²×C₇	1	6	yes	—
45	C₂²×C₄	C₂²×C₇	1	6	yes	—
46	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
47	C₄⋊C₄	C₂²×C₇	1	3	yes	—
48	C₂²⋊C₄	C₂²×C₇	1	6	yes	—
49	C₄²	C₂²×C₇	1	3	yes	—
50	C₂×C₇	—	4	6	no	—
51	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×C₇	C₂×#41	1	1	yes	characteristic
53	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
54	C₂³	C₂³×C₇	1	6	yes	—
55	Q₈	C₇×D₄	1	4	yes	complemented
56	D₄	—	2	12	no	—
57	D₄	—	2	6	no	—
58	D₄	—	4	6	no	—
59	C₂×C₄	—	2	12	no	—
60	C₂×C₄	C₂³×C₇	1	3	yes	—
61	C₂×C₄	C₂³×C₇	1	2	yes	—
62	C₂×C₄	C₂³×C₇	1	3	yes	—
63	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
64	C₇	#550	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	—	2	6	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	6	no	—
68	C₂²	C₂⁴×C₇	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	2	3	no	—
71	C₄	C₂×C₇×D₄	1	6	yes	—
72	C₄	C₇×#41	1	2	yes	complemented
73	C₂	—	4	6	no	—
74	C₂	C₂²×C₇×D₄	1	1	yes	characteristic
75	C₂	C₂×C₇×#41	1	1	yes	characteristic
76	C₂	C₇×H(5,𝔽₂)	1	1	yes	characteristic

	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. C₂×C₇×#41	C₂	semidirect
2. C₇×#127	C₂	semidirect
3. C₇×#121	C₂	semidirect
4. C₄×C₇×Q₈	C₂	semidirect
5. C₄×C₇×D₄	C₂	semidirect
16. #550	C₇	direct (N × H)
18. C₇×Q₈	D₄	semidirect
27. C₂×#41	C₂×C₇	semidirect
28. #127	C₂×C₇	semidirect
29. #121	C₂×C₇	semidirect
30. C₄×Q₈	C₂×C₇	semidirect
31. C₄×D₄	C₂×C₇	semidirect
39. C₄×C₇	#41	semidirect
55. Q₈	C₇×D₄	semidirect
64. C₇	#550	direct (N × H)
72. C₄	C₇×#41	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#41	2
C₇×#127	2
C₇×#121	2
C₄×C₇×Q₈	2
C₄×C₇×D₄	2
#550	7

Subgroups — C7×#550 #90571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#550 #90571