Subgroups — C₇×#473 #90119

Order 448.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#137	C₂	1	2	yes	maximal
2	C₂×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#131	C₂	1	1	yes	maximal, characteristic
4	C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#101	C₂	1	2	yes	maximal, complemented
6	C₇×#41	C₂²	1	1	yes	characteristic
7	C₂×C₇×Q₈	C₂²	1	2	yes	—
8	C₇×Q₁₆	—	2	2	no	—
9	C₇×Q₁₆	—	2	1	no	—
10	C₇×Q₁₆	—	2	1	no	—
11	C₇×SD₁₆	—	2	2	no	—
12	C₇×M₁₆	—	2	1	no	—
13	C₇×M₁₆	C₂²	1	1	yes	characteristic
14	C₇×M₁₆	C₂²	1	2	yes	—
15	C₂×C₇×C₈	—	2	1	no	—
16	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
17	#473	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₇×Q₈	—	2	2	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₇×Q₈	D₄	1	1	yes	characteristic
21	C₇×D₄	—	2	1	no	—
22	C₇×D₄	D₄	1	1	yes	characteristic
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	—	2	2	no	—
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₇×C₈	—	2	1	no	—
27	C₇×C₈	—	2	2	no	—
28	C₇×C₈	D₄	1	1	yes	characteristic
29	C₇×C₈	D₄	1	1	yes	characteristic
30	#137	C₂×C₇	1	2	yes	—
31	C₂×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
32	#131	C₂×C₇	1	1	yes	characteristic
33	#108	C₂×C₇	1	1	yes	characteristic, complemented
34	#101	C₂×C₇	1	2	yes	complemented
35	C₂²×C₇	—	2	1	no	—
36	C₂²×C₇	#41	1	1	yes	characteristic
37	C₄×C₇	—	2	1	no	—
38	C₄×C₇	—	4	2	no	—
39	C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	#41	C₂²×C₇	1	1	yes	characteristic
42	C₂×Q₈	C₂²×C₇	1	2	yes	—
43	Q₁₆	—	2	2	no	—
44	Q₁₆	—	2	1	no	—
45	Q₁₆	—	2	1	no	—
46	SD₁₆	—	2	2	no	—
47	M₁₆	—	2	1	no	—
48	M₁₆	C₂²×C₇	1	1	yes	characteristic
49	M₁₆	C₂²×C₇	1	2	yes	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
52	C₂×C₇	—	2	1	no	—
53	C₂×C₇	—	4	1	no	—
54	C₂×C₇	#121	1	1	yes	characteristic
55	Q₈	—	2	2	no	—
56	Q₈	—	2	2	no	—
57	Q₈	C₇×D₄	1	1	yes	characteristic
58	D₄	—	2	1	no	—
59	D₄	C₇×D₄	1	1	yes	characteristic
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	2	no	—
65	C₈	C₇×D₄	1	1	yes	characteristic
66	C₈	C₇×D₄	1	1	yes	characteristic
67	C₇	#473	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²	—	2	1	no	—
69	C₂²	C₇×#41	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	2	no	—
72	C₄	C₂×C₇×D₄	1	1	yes	characteristic
73	C₄	C₂×C₇×D₄	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	C₇×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#121
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₇×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×Q₁₆	C₂	semidirect
4. C₇×#108	C₂	semidirect
5. C₇×#101	C₂	semidirect
17. #473	C₇	direct (N × H)
31. C₂×Q₁₆	C₂×C₇	semidirect
33. #108	C₂×C₇	semidirect
34. #101	C₂×C₇	semidirect
67. C₇	#473	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×#137	2
C₂×C₇×Q₁₆	2
C₇×#131	2
C₇×#108	2
C₇×#101	2
#473	7

Subgroups — C7×#473 #90119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#473 #90119