Subgroups — #89848

Order 448.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×#1142	C₂	1	4	yes	maximal, complemented
3	#6432	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×D₄	—	2	2	no	—
6	C₂×C₇×D₄	C₂²	1	2	yes	—
7	C₂×Dic₁₄	C₂²	1	2	yes	—
8	C₄²×C₇	C₂²	1	1	yes	characteristic
9	#1142	—	2	8	no	—
10	C₄⋊Dic₇	—	2	2	no	—
11	C₂×C₇⋊C₈	C₂²	1	2	yes	—
12	#492	—	7	1	no	maximal, Sylow
13	C₂³×C₇	—	2	2	no	—
14	C₇×D₄	—	2	4	no	—
15	C₇×D₄	—	4	4	no	—
16	C₂×C₄×C₇	D₄	1	2	yes	—
17	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₇	—	2	2	no	—
19	Dic₁₄	—	2	4	no	—
20	C₇⋊C₈	D₄	1	4	yes	complemented
21	C₂×SD₁₆	—	7	4	no	—
22	C₄⋊Q₈	—	7	1	no	—
23	#127	D₇	1	1	yes	characteristic
24	C₄×C₈	—	7	1	no	—
25	C₂²×C₇	—	4	2	no	—
26	C₂²×C₇	—	4	4	no	—
27	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
28	C₄×C₇	SD₁₆	1	4	yes	complemented
29	C₄×C₇	C₂×D₄	1	2	yes	—
30	Dic₇	—	4	2	no	—
31	C₂×Q₈	—	7	2	no	—
32	C₂×D₄	—	2	2	no	—
33	C₂×D₄	C₂×D₇	1	2	yes	—
34	SD₁₆	—	14	8	no	—
35	C₂×C₈	—	7	2	no	—
36	C₄⋊C₄	—	14	2	no	—
37	C₄²	C₂×D₇	1	1	yes	characteristic
38	C₂×C₇	—	8	2	no	—
39	C₂×C₇	#127	1	1	yes	characteristic
40	C₂×C₇	C₂×SD₁₆	1	2	yes	—
41	C₂³	—	2	2	no	—
42	Q₈	—	14	4	no	—
43	D₄	—	2	4	no	—
44	D₄	—	4	4	no	—
45	C₂×C₄	—	14	2	no	—
46	C₂×C₄	C₂²⋊D₇	1	2	yes	—
47	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
48	C₈	—	7	4	no	—
49	C₇	#492	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	4	4	no	—
51	C₂²	—	4	2	no	—
52	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
53	C₄	—	28	2	no	—
54	C₄	#1142	1	4	yes	complemented
55	C₄	D₄×D₇	1	2	yes	—
56	C₂	—	8	2	no	—
57	C₂	C₂×#1142	1	2	yes	—
58	C₂	#6503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting, prime radical	C₇×#127	C₂
socle	C₂²×C₇	C₂×D₄
hypercenter	#127	D₇
hypocenter	C₇	#492
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. C₂×#1142	C₂	semidirect
3. #6432	C₂	semidirect
4. C₄×C₇⋊C₈	C₂	semidirect
20. C₇⋊C₈	D₄	semidirect
28. C₄×C₇	SD₁₆	semidirect
49. C₇	#492	semidirect
54. C₄	#1142	semidirect

Maximal subgroups

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

subgroup	index
C₇×#127	2
C₂×#1142	2
#6432	2
C₄×C₇⋊C₈	2
#492	7