Subgroups — #90518

Order 448.

294 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	Q₈×Dic₇	C₂	1	2	yes	maximal, complemented
3	D₄×Dic₇	C₂	1	6	yes	maximal, complemented
4	#6489	C₂	1	3	yes	maximal, complemented
5	C₂×C₄⋊Dic₇	C₂	1	3	yes	maximal, complemented
6	C₇×#41	C₄	1	8	yes	complemented
7	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×D₄	C₂²	1	3	yes	—
9	C₂²×C₄×C₇	C₂²	1	3	yes	—
10	C₂²×Dic₇	C₂²	1	6	yes	—
11	C₂²⋊Dic₇	C₂²	1	6	yes	—
12	C₄⋊Dic₇	C₂²	1	3	yes	—
13	C₄⋊Dic₇	C₂²	1	6	yes	—
14	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	C₄×Dic₇	C₂²	1	6	yes	—
16	#520	—	7	1	no	maximal, Sylow
17	C₂³×C₇	C₂³	1	3	yes	—
18	C₇×Q₈	C₂×C₄	1	4	yes	complemented
19	C₇×D₄	C₂×C₄	1	12	yes	complemented
20	C₂×C₄×C₇	C₂×C₄	1	12	yes	complemented
21	C₂×C₄×C₇	C₂³	1	3	yes	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₇	—	4	6	no	—
24	C₂×Dic₇	C₂³	1	2	yes	—
25	C₂×Dic₇	C₂³	1	6	yes	—
26	C₂×#41	D₇	1	1	yes	characteristic
27	C₄×Q₈	—	7	2	no	—
28	C₄×D₄	—	7	6	no	—
29	#117	—	7	3	no	—
30	C₂×C₄⋊C₄	—	7	3	no	—
31	C₂²×C₇	—	2	3	no	—
32	C₂²×C₇	—	2	3	no	—
33	C₂²×C₇	C₂²×C₄	1	6	yes	—
34	C₂²×C₇	C₂⁴	1	1	yes	characteristic
35	C₄×C₇	C₂²×C₄	1	2	yes	—
36	C₄×C₇	C₂²×C₄	1	6	yes	—
37	Dic₇	—	2	6	no	—
38	Dic₇	—	2	2	no	—
39	#41	Dic₇	1	8	yes	complemented
40	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
41	C₂×D₄	C₂×D₇	1	3	yes	—
42	C₂²×C₄	—	7	6	no	—
43	C₂²×C₄	C₂×D₇	1	3	yes	—
44	C₄⋊C₄	—	7	3	no	—
45	C₄⋊C₄	—	7	6	no	—
46	C₄⋊C₄	—	7	1	no	—
47	C₂²⋊C₄	—	7	6	no	—
48	C₄²	—	7	6	no	—
49	C₂×C₇	—	2	6	no	—
50	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
51	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
53	C₂³	C₂²×D₇	1	3	yes	—
54	Q₈	C₂×Dic₇	1	4	yes	complemented
55	D₄	C₂×Dic₇	1	12	yes	complemented
56	C₂×C₄	—	7	2	no	—
57	C₂×C₄	—	7	6	no	—
58	C₂×C₄	—	28	6	no	—
59	C₂×C₄	C₂×Dic₇	1	12	yes	complemented
60	C₂×C₄	C₂²×D₇	1	3	yes	—
61	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
62	C₇	#520	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	2	3	no	—
64	C₂²	—	2	3	no	—
65	C₂²	C₂²×Dic₇	1	6	yes	—
66	C₂²	C₂³×D₇	1	1	yes	characteristic
67	C₄	—	14	6	no	—
68	C₄	—	14	2	no	—
69	C₄	C₂²×Dic₇	1	2	yes	—
70	C₄	C₂²×Dic₇	1	6	yes	—
71	C₂	—	2	6	no	—
72	C₂	#6553	1	1	yes	characteristic
73	C₂	#6554	1	1	yes	characteristic
74	C₂	C₂³×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator	C₂×C₇	C₂³×C₄
Fitting, prime radical	C₂×C₇×#41	C₂
socle, cosocle	C₂²×C₇	C₂⁴
hypercenter	C₂×#41	D₇
hypocenter	C₇	#520
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₇	C₂	semidirect
3. D₄×Dic₇	C₂	semidirect
4. #6489	C₂	semidirect
5. C₂×C₄⋊Dic₇	C₂	semidirect
6. C₇×#41	C₄	semidirect
18. C₇×Q₈	C₂×C₄	semidirect
19. C₇×D₄	C₂×C₄	semidirect
20. C₂×C₄×C₇	C₂×C₄	semidirect
39. #41	Dic₇	semidirect
54. Q₈	C₂×Dic₇	semidirect
55. D₄	C₂×Dic₇	semidirect
59. C₂×C₄	C₂×Dic₇	semidirect
62. C₇	#520	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
Q₈×Dic₇	2
D₄×Dic₇	2
#6489	2
C₂×C₄⋊Dic₇	2
#520	7