Subgroups — #6487

Order 224.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator, socle, cosocle	C₂²×C₇	C₂³
Fitting	C₂²×C₄×C₇	C₂
prime radical	C₂³×C₇	C₂²
hypercenter	C₂²×C₄	D₇
hypocenter	C₇	C₂²⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₄	C₂	semidirect
4. C₄⋊Dic₇	C₂	semidirect
5. Dic₇⋊C₄	C₂	semidirect
16. C₂²×C₇	Q₈	semidirect
38. C₇	C₂²⋊Q₈	semidirect
41. C₂²	Dic₁₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇	2
C₂×Dic₁₄	2
C₂²⋊Dic₇	2
C₄⋊Dic₇	2
Dic₇⋊C₄	2
C₂²⋊Q₈	7