Subgroups — #89479

Order 448.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1131	C₂	1	2	yes	maximal
2	C₄²×D₇	C₂	1	1	yes	maximal, characteristic
3	C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×Dic₇	C₂	1	2	yes	maximal, complemented
5	#6377	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₇	C₄	1	2	yes	—
7	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₈	C₂²	1	2	yes	—
9	C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₄×Dic₇	C₄	1	2	yes	—
11	C₄×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊C₈	C₂²	1	2	yes	—
13	#1131	C₄	1	8	yes	complemented
14	C₄×M₁₆	—	7	1	no	maximal, Sylow
15	C₂²×D₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₇	C₂×C₄	1	2	yes	—
20	C₄×D₇	—	2	4	no	—
21	C₄×D₇	C₂×C₄	1	4	yes	—
22	C₇×C₈	C₂×C₄	1	4	yes	complemented
23	C₇⋊C₈	C₂×C₄	1	4	yes	complemented
24	C₂×M₁₆	—	7	2	no	—
25	C₂×C₄²	—	7	1	no	—
26	#97	—	7	1	no	—
27	#97	D₇	1	1	yes	characteristic, complemented
28	C₄×C₈	—	7	2	no	—
29	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×D₇	—	2	2	no	—
31	C₂×D₇	C₄²	1	2	yes	—
32	C₄×C₇	—	2	2	no	—
33	C₄×C₇	C₂²×C₄	1	2	yes	—
34	Dic₇	C₄²	1	2	yes	—
35	Dic₇	M₁₆	1	4	yes	complemented
36	C₂²×C₄	—	7	1	no	—
37	C₂²×C₄	—	7	2	no	—
38	M₁₆	—	7	8	no	—
39	C₂×C₈	—	7	2	no	—
40	C₂×C₈	C₂×D₇	1	2	yes	—
41	C₄²	—	7	2	no	—
42	C₄²	—	7	1	no	—
43	C₄²	C₂×D₇	1	1	yes	characteristic
44	C₂×C₇	C₂×C₄²	1	1	yes	characteristic
45	C₂×C₇	C₂×M₁₆	1	2	yes	—
46	D₇	—	2	2	no	—
47	C₂³	—	7	1	no	—
48	C₂×C₄	—	7	1	no	—
49	C₂×C₄	—	7	2	no	—
50	C₂×C₄	—	7	4	no	—
51	C₂×C₄	—	14	4	no	—
52	C₂×C₄	C₄×D₇	1	2	yes	—
53	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
54	C₈	—	7	4	no	—
55	C₈	C₄×D₇	1	4	yes	complemented
56	C₇	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	7	2	no	—
58	C₂²	—	14	2	no	—
59	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	7	2	no	—
62	C₄	—	7	4	no	—
63	C₄	C₂×C₄×D₇	1	2	yes	—
64	C₂	—	14	2	no	—
65	C₂	C₄²×D₇	1	1	yes	characteristic
66	C₂	D₇×M₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator	C₂×C₇	C₂×C₄²
Fitting	C₇×#97	C₂
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂²×D₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#97	D₇
hypocenter	C₇	C₄×M₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
3. C₇×#97	C₂	semidirect
4. C₈×Dic₇	C₂	semidirect
5. #6377	C₂	semidirect
13. #1131	C₄	semidirect
22. C₇×C₈	C₂×C₄	semidirect
23. C₇⋊C₈	C₂×C₄	semidirect
27. #97	D₇	semidirect
35. Dic₇	M₁₆	semidirect
55. C₈	C₄×D₇	semidirect
56. C₇	C₄×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1131	2
C₄²×D₇	2
C₇×#97	2
C₈×Dic₇	2
#6377	2
C₄×M₁₆	7