Subgroups — C₇×#540 #90561

Order 448.

240 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, complemented
2	C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#123	C₂	1	4	yes	maximal, complemented
4	C₇×#121	C₂	1	4	yes	maximal, complemented
5	C₇×#121	C₂	1	2	yes	maximal, complemented
6	C₄×C₇×D₄	C₂	1	2	yes	maximal, complemented
7	C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×D₄	C₂²	1	2	yes	—
9	C₂×C₇×D₄	C₂²	1	4	yes	—
10	C₂×C₇×D₄	C₂²	1	4	yes	—
11	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₇	C₂²	1	4	yes	—
13	C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₇×C₄⋊C₄	C₂²	1	4	yes	—
16	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
18	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₄²×C₇	C₂²	1	1	yes	characteristic
21	#540	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₇	C₂³	1	1	yes	characteristic
23	C₂³×C₇	C₂³	1	4	yes	—
24	C₇×D₄	—	2	4	no	—
25	C₇×D₄	—	4	4	no	—
26	C₇×D₄	—	4	4	no	—
27	C₂×C₄×C₇	—	2	2	no	—
28	C₂×C₄×C₇	—	4	4	no	—
29	C₂×C₄×C₇	C₂³	1	2	yes	—
30	C₂×C₄×C₇	C₂³	1	2	yes	—
31	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₇	C₂³	1	4	yes	—
33	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
34	#127	C₂×C₇	1	1	yes	characteristic, complemented
35	#125	C₂×C₇	1	1	yes	characteristic, complemented
36	#123	C₂×C₇	1	4	yes	complemented
37	#121	C₂×C₇	1	4	yes	complemented
38	#121	C₂×C₇	1	2	yes	complemented
39	C₄×D₄	C₂×C₇	1	2	yes	complemented
40	#117	C₂×C₇	1	1	yes	characteristic, complemented
41	C₂²×C₇	—	2	1	no	—
42	C₂²×C₇	—	2	4	no	—
43	C₂²×C₇	—	2	4	no	—
44	C₂²×C₇	—	2	2	no	—
45	C₂²×C₇	—	2	4	no	—
46	C₂²×C₇	C₂⁴	1	1	yes	characteristic
47	C₄×C₇	—	2	1	no	—
48	C₄×C₇	—	2	2	no	—
49	C₄×C₇	—	2	4	no	—
50	C₄×C₇	—	2	2	no	—
51	C₄×C₇	#41	1	2	yes	—
52	C₂×D₄	C₂²×C₇	1	2	yes	—
53	C₂×D₄	C₂²×C₇	1	4	yes	—
54	C₂×D₄	C₂²×C₇	1	4	yes	—
55	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
56	C₂²×C₄	C₂²×C₇	1	4	yes	—
57	C₄⋊C₄	C₂²×C₇	1	2	yes	—
58	C₄⋊C₄	C₂²×C₇	1	2	yes	—
59	C₄⋊C₄	C₂²×C₇	1	4	yes	—
60	C₂²⋊C₄	C₂²×C₇	1	2	yes	—
61	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
62	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
63	C₄²	C₂²×C₇	1	1	yes	characteristic
64	C₄²	C₂²×C₇	1	1	yes	characteristic
65	C₂×C₇	—	4	1	no	—
66	C₂×C₇	—	4	4	no	—
67	C₂×C₇	C₂×#41	1	1	yes	characteristic
68	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
69	C₂³	C₂³×C₇	1	1	yes	characteristic
70	C₂³	C₂³×C₇	1	4	yes	—
71	D₄	—	2	4	no	—
72	D₄	—	4	4	no	—
73	D₄	—	4	4	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	4	4	no	—
76	C₂×C₄	C₂³×C₇	1	2	yes	—
77	C₂×C₄	C₂³×C₇	1	2	yes	—
78	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
79	C₂×C₄	C₂³×C₇	1	4	yes	—
80	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
81	C₇	#540	1	1	yes	characteristic, direct product factor, Sylow
82	C₂²	—	2	1	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	4	no	—
87	C₂²	C₂⁴×C₇	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	2	no	—
90	C₄	—	2	4	no	—
91	C₄	—	2	2	no	—
92	C₄	C₇×#41	1	2	yes	—
93	C₂	—	4	1	no	—
94	C₂	—	4	4	no	—
95	C₂	C₂×C₇×#41	1	1	yes	characteristic
96	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. C₇×#127	C₂	semidirect
2. C₇×#125	C₂	semidirect
3. C₇×#123	C₂	semidirect
4. C₇×#121	C₂	semidirect
5. C₇×#121	C₂	semidirect
6. C₄×C₇×D₄	C₂	semidirect
7. C₇×#117	C₂	semidirect
21. #540	C₇	direct (N × H)
34. #127	C₂×C₇	semidirect
35. #125	C₂×C₇	semidirect
36. #123	C₂×C₇	semidirect
37. #121	C₂×C₇	semidirect
38. #121	C₂×C₇	semidirect
39. C₄×D₄	C₂×C₇	semidirect
40. #117	C₂×C₇	semidirect
81. C₇	#540	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×#125	2
C₇×#123	2
C₇×#121	2
C₄×C₇×D₄	2
C₇×#117	2
#540	7

Subgroups — C7×#540 #90561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#540 #90561