Subgroups — #89873

Order 448.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	#6489	C₂	1	1	yes	maximal, characteristic
3	C₂×#1136	C₂	1	1	yes	maximal, characteristic
4	#6389	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₈	C₄	1	4	yes	complemented
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂²⋊Dic₇	C₄	1	2	yes	—
11	C₄⋊Dic₇	C₄	1	2	yes	—
12	C₄×Dic₇	C₂²	1	2	yes	—
13	#1136	C₄	1	4	yes	complemented
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	#405	—	7	1	no	maximal, Sylow
16	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₇	C₂×C₄	1	4	yes	—
22	C₇×C₈	C₂×C₄	1	2	yes	—
23	C₇×C₈	C₂×C₄	1	2	yes	—
24	C₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₂×M₁₆	—	7	1	no	—
26	C₂²×C₈	D₇	1	1	yes	characteristic
27	#117	—	7	1	no	—
28	#97	—	7	2	no	—
29	C₄×C₈	—	7	2	no	—
30	C₂²×C₇	—	2	2	no	—
31	C₂²×C₇	C₄²	1	2	yes	—
32	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₇	C₄²	1	2	yes	—
34	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	Dic₇	—	2	4	no	—
37	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
38	M₁₆	—	7	4	no	—
39	C₂×C₈	—	7	2	no	—
40	C₂×C₈	Dic₇	1	4	yes	complemented
41	C₂×C₈	C₂×D₇	1	1	yes	characteristic
42	C₂×C₈	C₂×D₇	1	1	yes	characteristic
43	C₄⋊C₄	—	7	2	no	—
44	C₂²⋊C₄	—	7	2	no	—
45	C₄²	—	7	2	no	—
46	C₂×C₇	—	2	2	no	—
47	C₂×C₇	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₇	#131	1	2	yes	—
49	C₂³	C₄×D₇	1	1	yes	characteristic
50	C₂×C₄	—	7	4	no	—
51	C₂×C₄	C₄×D₇	1	1	yes	characteristic
52	C₂×C₄	C₄×D₇	1	2	yes	—
53	C₂×C₄	C₂×Dic₇	1	2	yes	—
54	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
55	C₈	—	7	4	no	—
56	C₈	C₂×Dic₇	1	2	yes	—
57	C₈	C₂×Dic₇	1	2	yes	—
58	C₇	#405	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	2	2	no	—
60	C₂²	C₄×Dic₇	1	2	yes	—
61	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
62	C₄	—	14	4	no	—
63	C₄	C₄×Dic₇	1	2	yes	—
64	C₄	C₂×C₄×D₇	1	1	yes	characteristic
65	C₄	C₂²×Dic₇	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	#6464	1	2	yes	—
68	C₂	C₂×C₄×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₇
commutator	C₂×C₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂²×C₇×C₈	C₂
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂²×C₈	D₇
hypocenter	C₇	#405
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. #6389	C₂	semidirect
5. C₈×Dic₇	C₂	semidirect
7. C₂×C₇×C₈	C₄	semidirect
13. #1136	C₄	semidirect
40. C₂×C₈	Dic₇	semidirect
58. C₇	#405	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₈	2
#6489	2
C₂×#1136	2
#6389	2
C₈×Dic₇	2
#405	7