Subgroups — #6548

Order 224.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₇	C₂	1	4	yes	maximal, complemented
3	#1159	C₂	1	4	yes	maximal, complemented
4	D₄×D₇	C₂	1	4	yes	maximal, complemented
5	#1157	C₂	1	2	yes	maximal, complemented
6	C₂³×C₇	C₂²	1	2	yes	complemented
7	C₂²×D₇	C₂²	1	4	yes	complemented
8	C₇×D₄	C₂²	1	4	yes	complemented
9	C₂×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
10	C₂²⋊D₇	C₂²	1	4	yes	complemented
11	C₂²⋊D₇	C₂²	1	8	yes	complemented
12	C₂×Dic₇	C₂²	1	4	yes	complemented
13	D₂₈	C₂²	1	2	yes	complemented
14	C₄×D₇	C₂²	1	4	yes	complemented
15	Dic₁₄	C₂²	1	2	yes	complemented
16	H(5,𝔽₂)	—	7	1	no	maximal, Sylow
17	C₂²×C₇	—	4	2	no	—
18	C₂²×C₇	C₂³	1	4	yes	—
19	C₂²×C₇	C₂³	1	1	yes	characteristic
20	C₂×D₇	—	4	4	no	—
21	C₂×D₇	C₂³	1	4	yes	—
22	C₄×C₇	C₂³	1	2	yes	—
23	Dic₇	C₂³	1	4	yes	—
24	#41	—	7	4	no	—
25	#41	—	7	2	no	—
26	C₂×D₄	—	7	4	no	—
27	C₂×D₄	—	7	4	no	—
28	C₂×D₄	D₇	1	1	yes	characteristic, complemented
29	C₂×C₇	—	2	4	no	—
30	C₂×C₇	—	2	1	no	—
31	C₂×C₇	C₂⁴	1	1	yes	characteristic
32	D₇	—	2	4	no	—
33	C₂³	—	7	4	no	—
34	C₂³	C₂×D₇	1	2	yes	complemented
35	Q₈	—	7	2	no	—
36	D₄	—	7	8	no	—
37	D₄	—	7	4	no	—
38	D₄	—	7	2	no	—
39	D₄	C₂×D₇	1	4	yes	complemented
40	C₂×C₄	—	7	4	no	—
41	C₂×C₄	—	7	4	no	—
42	C₂×C₄	C₂×D₇	1	1	yes	characteristic, complemented
43	C₇	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
44	C₂²	—	4	2	no	—
45	C₂²	—	7	4	no	—
46	C₂²	—	28	4	no	—
47	C₂²	C₂²×D₇	1	4	yes	—
48	C₂²	C₂²×D₇	1	1	yes	characteristic
49	C₄	—	7	4	no	—
50	C₄	C₂²×D₇	1	2	yes	—
51	C₂	—	2	1	no	—
52	C₂	—	2	4	no	—
53	C₂	—	14	4	no	—
54	C₂	C₂³×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₇
commutator, socle, cosocle	C₂×C₇	C₂⁴
Fitting	C₂×C₇×D₄	C₂
hypercenter	C₂×D₄	D₇
hypocenter	C₇	H(5,𝔽₂)
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×D₄	C₂	semidirect
2. C₂×C₂²⋊D₇	C₂	semidirect
3. #1159	C₂	semidirect
4. D₄×D₇	C₂	semidirect
5. #1157	C₂	semidirect
6. C₂³×C₇	C₂²	semidirect
7. C₂²×D₇	C₂²	semidirect
8. C₇×D₄	C₂²	semidirect
9. C₂×C₄×C₇	C₂²	semidirect
10. C₂²⋊D₇	C₂²	semidirect
11. C₂²⋊D₇	C₂²	semidirect
12. C₂×Dic₇	C₂²	semidirect
13. D₂₈	C₂²	semidirect
14. C₄×D₇	C₂²	semidirect
15. Dic₁₄	C₂²	semidirect
28. C₂×D₄	D₇	semidirect
34. C₂³	C₂×D₇	semidirect
39. D₄	C₂×D₇	semidirect
42. C₂×C₄	C₂×D₇	semidirect
43. C₇	H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×D₄	2
C₂×C₂²⋊D₇	2
#1159	2
D₄×D₇	2
#1157	2
H(5,𝔽₂)	7