Subgroups — #89864

Order 448.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#1144	C₂	1	4	yes	maximal
3	#6432	C₂	1	1	yes	maximal, characteristic
4	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×Q₈	C₂²	1	2	yes	—
6	C₂×Dic₁₄	C₂²	1	2	yes	—
7	C₇×C₄⋊C₄	—	2	2	no	—
8	C₄²×C₇	C₂²	1	1	yes	characteristic
9	#1144	—	2	8	no	—
10	C₄⋊Dic₇	—	2	2	no	—
11	C₂×C₇⋊C₈	C₂²	1	2	yes	—
12	#494	—	7	1	no	maximal, Sylow
13	C₇×Q₈	—	2	4	no	—
14	C₂×C₄×C₇	—	2	2	no	—
15	C₂×C₄×C₇	D₄	1	2	yes	—
16	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂×Dic₇	—	2	2	no	—
18	Dic₁₄	—	2	4	no	—
19	C₇⋊C₈	D₄	1	4	yes	—
20	C₂×Q₁₆	—	7	4	no	—
21	C₄⋊Q₈	—	7	1	no	—
22	C₄⋊Q₈	D₇	1	1	yes	characteristic
23	C₄×C₈	—	7	1	no	—
24	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
25	C₄×C₇	—	4	2	no	—
26	C₄×C₇	Q₁₆	1	4	yes	complemented
27	C₄×C₇	C₂×D₄	1	2	yes	—
28	Dic₇	—	4	2	no	—
29	C₂×Q₈	—	7	2	no	—
30	C₂×Q₈	C₂×D₇	1	2	yes	—
31	Q₁₆	—	14	8	no	—
32	C₂×C₈	—	7	2	no	—
33	C₄⋊C₄	—	2	2	no	—
34	C₄⋊C₄	—	14	2	no	—
35	C₄²	C₂×D₇	1	1	yes	characteristic
36	C₂×C₇	#127	1	1	yes	characteristic
37	C₂×C₇	C₂×Q₁₆	1	2	yes	—
38	Q₈	—	2	4	no	—
39	Q₈	—	14	4	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	14	2	no	—
42	C₂×C₄	C₂²⋊D₇	1	2	yes	—
43	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
44	C₈	—	7	4	no	—
45	C₇	#494	1	1	yes	characteristic, complemented, Sylow
46	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	28	2	no	—
49	C₄	#1144	1	4	yes	complemented
50	C₄	D₄×D₇	1	2	yes	—
51	C₂	#6503	1	1	yes	characteristic
52	C₂	C₂×#1144	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	#494
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
26. C₄×C₇	Q₁₆	semidirect
45. C₇	#494	semidirect
49. C₄	#1144	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₄⋊Q₈	2
C₂×#1144	2
#6432	2
C₄×C₇⋊C₈	2
#494	7