Subgroups — #90489

Order 448.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₇	C₂	1	4	yes	maximal, complemented
3	D₄×Dic₇	C₂	1	8	yes	maximal, complemented
4	#6489	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₇	C₂²	1	2	yes	—
6	C₂×C₇×D₄	C₄	1	8	yes	complemented
7	C₂×C₇×D₄	C₂²	1	4	yes	—
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₇	C₂²	1	8	yes	complemented
10	C₂²⋊Dic₇	C₂²	1	4	yes	—
11	C₂²⋊Dic₇	C₂²	1	8	yes	complemented
12	C₄⋊Dic₇	C₂²	1	4	yes	complemented
13	C₄×Dic₇	C₂²	1	4	yes	complemented
14	#518	—	7	1	no	maximal, Sylow
15	C₂³×C₇	—	4	4	no	—
16	C₂³×C₇	C₂×C₄	1	8	yes	complemented
17	C₂³×C₇	C₂³	1	1	yes	characteristic
18	C₂³×C₇	C₂³	1	4	yes	—
19	C₇×D₄	C₂×C₄	1	16	yes	complemented
20	C₂×C₄×C₇	C₂×C₄	1	4	yes	complemented
21	C₂×C₄×C₇	C₂³	1	2	yes	—
22	C₂×Dic₇	—	4	8	no	—
23	C₂×Dic₇	C₂³	1	8	yes	—
24	C₂²×D₄	D₇	1	1	yes	characteristic
25	C₄×D₄	—	7	8	no	—
26	#117	—	7	2	no	—
27	C₂×C₂²⋊C₄	—	7	4	no	—
28	C₂²×C₇	—	2	8	no	—
29	C₂²×C₇	—	2	2	no	—
30	C₂²×C₇	—	4	8	no	—
31	C₂²×C₇	C₂²×C₄	1	2	yes	—
32	C₂²×C₇	C₂²×C₄	1	8	yes	—
33	C₂²×C₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₇	C₂²×C₄	1	4	yes	—
35	Dic₇	—	2	8	no	—
36	C₂⁴	C₂×D₇	1	2	yes	—
37	C₂×D₄	Dic₇	1	8	yes	complemented
38	C₂×D₄	C₂×D₇	1	4	yes	—
39	C₂²×C₄	—	7	8	no	—
40	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
41	C₄⋊C₄	—	7	4	no	—
42	C₂²⋊C₄	—	7	4	no	—
43	C₂²⋊C₄	—	7	8	no	—
44	C₄²	—	7	4	no	—
45	C₂×C₇	—	2	2	no	—
46	C₂×C₇	—	2	8	no	—
47	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
48	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
49	C₂³	—	4	4	no	—
50	C₂³	C₂×Dic₇	1	8	yes	complemented
51	C₂³	C₂²×D₇	1	1	yes	characteristic
52	C₂³	C₂²×D₇	1	4	yes	—
53	D₄	C₂×Dic₇	1	16	yes	complemented
54	C₂×C₄	—	7	8	no	—
55	C₂×C₄	—	28	8	no	—
56	C₂×C₄	C₂×Dic₇	1	4	yes	complemented
57	C₂×C₄	C₂²×D₇	1	2	yes	—
58	C₇	#518	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	2	8	no	—
60	C₂²	—	2	2	no	—
61	C₂²	—	4	8	no	—
62	C₂²	C₂²×Dic₇	1	2	yes	—
63	C₂²	C₂²×Dic₇	1	8	yes	—
64	C₂²	C₂³×D₇	1	1	yes	characteristic
65	C₄	—	14	8	no	—
66	C₄	C₂²×Dic₇	1	4	yes	—
67	C₂	—	2	2	no	—
68	C₂	—	2	8	no	—
69	C₂	#6548	1	2	yes	—
70	C₂	C₂³×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator	C₂×C₇	C₂³×C₄
Fitting, prime radical	C₂²×C₇×D₄	C₂
socle, cosocle	C₂²×C₇	C₂⁴
hypercenter	C₂²×D₄	D₇
hypocenter	C₇	#518
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₂²⋊Dic₇	C₂	semidirect
3. D₄×Dic₇	C₂	semidirect
4. #6489	C₂	semidirect
6. C₂×C₇×D₄	C₄	semidirect
9. C₂²×Dic₇	C₂²	semidirect
11. C₂²⋊Dic₇	C₂²	semidirect
12. C₄⋊Dic₇	C₂²	semidirect
13. C₄×Dic₇	C₂²	semidirect
16. C₂³×C₇	C₂×C₄	semidirect
19. C₇×D₄	C₂×C₄	semidirect
20. C₂×C₄×C₇	C₂×C₄	semidirect
37. C₂×D₄	Dic₇	semidirect
50. C₂³	C₂×Dic₇	semidirect
53. D₄	C₂×Dic₇	semidirect
56. C₂×C₄	C₂×Dic₇	semidirect
58. C₇	#518	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×D₄	2
C₂×C₂²⋊Dic₇	2
D₄×Dic₇	2
#6489	2
#518	7