Subgroups — #90510

Order 448.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	#6511	C₂	1	8	yes	maximal, complemented
3	C₂×#1136	C₂	1	6	yes	maximal, complemented
4	C₇×#41	C₄	1	4	yes	—
5	C₇×#41	C₂²	1	4	yes	—
6	C₂×C₇×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₇×D₄	C₄	1	3	yes	—
8	C₂²×C₄×C₇	C₂²	1	3	yes	—
9	#1136	C₂²	1	4	yes	—
10	#1136	C₂²	1	12	yes	complemented
11	C₂×C₇⋊C₈	C₂²	1	12	yes	complemented
12	#568	—	7	1	no	maximal, Sylow
13	C₂³×C₇	C₂×C₄	1	3	yes	—
14	C₇×Q₈	C₂×C₄	1	4	yes	—
15	C₇×D₄	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₇	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₇	C₂×C₄	1	6	yes	—
18	C₂×C₄×C₇	C₂³	1	6	yes	—
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₇⋊C₈	C₂³	1	8	yes	—
21	C₂×#41	D₇	1	1	yes	characteristic
22	#131	—	7	8	no	—
23	C₂×M₁₆	—	7	6	no	—
24	C₂²×C₇	—	4	3	no	—
25	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₇	C₂²×C₄	1	6	yes	—
27	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₇	C₂²×C₄	1	6	yes	—
29	C₄×C₇	C₂⁴	1	1	yes	characteristic
30	#41	Dic₇	1	4	yes	—
31	#41	C₂×D₇	1	4	yes	—
32	C₂×Q₈	Dic₇	1	1	yes	characteristic
33	C₂×D₄	Dic₇	1	3	yes	—
34	C₂²×C₄	C₂×D₇	1	3	yes	—
35	M₁₆	—	7	4	no	—
36	M₁₆	—	7	12	no	—
37	C₂×C₈	—	7	12	no	—
38	C₂×C₇	—	2	1	no	—
39	C₂×C₇	—	2	6	no	—
40	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
41	C₂³	C₂×Dic₇	1	3	yes	—
42	Q₈	C₂×Dic₇	1	4	yes	—
43	D₄	C₂×Dic₇	1	12	yes	—
44	C₂×C₄	C₂×Dic₇	1	3	yes	—
45	C₂×C₄	C₂×Dic₇	1	6	yes	—
46	C₂×C₄	C₂²×D₇	1	6	yes	—
47	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
48	C₈	—	7	8	no	—
49	C₇	#568	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	4	3	no	—
51	C₂²	C₂²×Dic₇	1	1	yes	characteristic
52	C₂²	C₂²×Dic₇	1	6	yes	—
53	C₄	C₂²×Dic₇	1	1	yes	characteristic
54	C₄	C₂²×Dic₇	1	6	yes	—
55	C₄	C₂³×D₇	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	2	6	no	—
58	C₂	C₂³×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₇
commutator, socle	C₂×C₇	C₂³×C₄
Fitting, prime radical	C₂×C₇×#41	C₂
cosocle	C₄×C₇	C₂⁴
hypercenter	C₂×#41	D₇
hypocenter	C₇	#568
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. #6511	C₂	semidirect
3. C₂×#1136	C₂	semidirect
10. #1136	C₂²	semidirect
11. C₂×C₇⋊C₈	C₂²	semidirect
49. C₇	#568	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#41	2
#6511	2
C₂×#1136	2
#568	7