Subgroups — C₇×#413 #90059

Order 448.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₇×#101	C₂	1	2	yes	maximal, complemented
4	C₇×#100	C₂	1	2	yes	maximal, complemented
5	C₇×#41	—	2	2	no	—
6	C₇×#41	—	2	2	no	—
7	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×D₄	C₄	1	2	yes	—
9	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₇	C₄	1	2	yes	—
11	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₇×M₁₆	—	2	2	no	—
13	C₇×M₁₆	C₂²	1	4	yes	—
14	C₂×C₇×C₈	—	2	2	no	—
15	#413	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	C₂×C₄	1	2	yes	—
17	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
18	C₇×Q₈	—	2	2	no	—
19	C₇×D₄	—	2	4	no	—
20	C₇×D₄	—	2	2	no	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	—	2	2	no	—
23	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₇	D₄	1	2	yes	—
26	C₂×C₄×C₇	D₄	1	2	yes	—
27	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₇×C₈	—	2	4	no	—
29	C₂×#41	C₂×C₇	1	1	yes	characteristic
30	C₂×M₁₆	C₂×C₇	1	2	yes	complemented
31	#101	C₂×C₇	1	2	yes	complemented
32	#100	C₂×C₇	1	2	yes	complemented
33	C₂²×C₇	—	2	2	no	—
34	C₂²×C₇	—	4	1	no	—
35	C₂²×C₇	—	4	2	no	—
36	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₇	—	2	2	no	—
39	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₇	C₂×D₄	1	2	yes	—
42	#41	—	2	2	no	—
43	#41	—	2	2	no	—
44	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
45	C₂×D₄	C₄×C₇	1	2	yes	—
46	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
47	C₂²×C₄	C₄×C₇	1	2	yes	—
48	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
49	M₁₆	—	2	2	no	—
50	M₁₆	C₂²×C₇	1	4	yes	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₇	—	2	1	no	—
53	C₂×C₇	—	2	2	no	—
54	C₂×C₇	—	4	2	no	—
55	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂³	C₂×C₄×C₇	1	2	yes	—
57	C₂³	C₂×C₄×C₇	1	1	yes	characteristic
58	Q₈	—	2	2	no	—
59	D₄	—	2	4	no	—
60	D₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
64	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
65	C₂×C₄	C₇×D₄	1	2	yes	—
66	C₂×C₄	C₇×D₄	1	2	yes	—
67	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
68	C₈	—	2	4	no	—
69	C₇	#413	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²	—	2	2	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	2	no	—
73	C₂²	C₇×C₂²⋊C₄	1	2	yes	—
74	C₂²	C₂²×C₄×C₇	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
77	C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
78	C₄	C₂×C₇×D₄	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	2	2	no	—
81	C₂	—	4	2	no	—
82	C₂	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₇
socle	C₂×C₇	C₂×C₂²⋊C₄
prime radical	C₂×C₇×#41	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×M₁₆	C₂	semidirect
3. C₇×#101	C₂	semidirect
4. C₇×#100	C₂	semidirect
15. #413	C₇	direct (N × H)
30. C₂×M₁₆	C₂×C₇	semidirect
31. #101	C₂×C₇	semidirect
32. #100	C₂×C₇	semidirect
69. C₇	#413	direct (N × H)

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
C₂×C₇×M₁₆	2
C₇×#101	2
C₇×#100	2
#413	7

Subgroups — C7×#413 #90059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#413 #90059