Subgroups — #89668

Order 448.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6472	C₂	1	2	yes	maximal
2	C₂×Dic₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#6464	C₂	1	1	yes	maximal, characteristic
4	C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	#6398	C₂	1	2	yes	maximal, complemented
6	#1157	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₄	C₂²	1	2	yes	—
8	C₇×M₁₆	C₂²	1	2	yes	—
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	#1136	C₂²	1	1	yes	characteristic
11	Dic₂₈	—	2	2	no	—
12	Dic₂₈	—	2	1	no	—
13	Dic₂₈	—	2	1	no	—
14	#1132	—	2	2	no	—
15	#1131	—	2	1	no	—
16	C₈×D₇	—	2	1	no	—
17	#473	—	7	1	no	maximal, Sylow
18	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂²⋊D₇	—	2	1	no	—
20	C₂×Dic₇	—	2	2	no	—
21	D₂₈	D₄	1	1	yes	characteristic
22	C₄×D₇	—	2	1	no	—
23	Dic₁₄	—	2	2	no	—
24	Dic₁₄	—	2	2	no	—
25	Dic₁₄	D₄	1	1	yes	characteristic
26	C₇×C₈	—	2	2	no	—
27	C₇×C₈	D₄	1	1	yes	characteristic
28	C₇×C₈	D₄	1	1	yes	characteristic
29	C₇⋊C₈	—	2	1	no	—
30	#137	—	7	2	no	—
31	C₂×Q₁₆	—	7	1	no	—
32	#131	—	7	1	no	—
33	#108	D₇	1	1	yes	characteristic, complemented
34	#101	—	7	2	no	—
35	C₂²×C₇	#41	1	1	yes	characteristic
36	C₂×D₇	—	2	1	no	—
37	C₄×C₇	C₂×D₄	1	1	yes	characteristic
38	C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	Dic₇	—	2	1	no	—
40	Dic₇	—	4	2	no	—
41	#41	—	7	1	no	—
42	C₂×Q₈	—	7	2	no	—
43	Q₁₆	—	14	2	no	—
44	Q₁₆	—	14	1	no	—
45	Q₁₆	—	14	1	no	—
46	SD₁₆	—	14	2	no	—
47	M₁₆	—	7	1	no	—
48	M₁₆	—	14	1	no	—
49	M₁₆	C₂×D₇	1	2	yes	—
50	C₂×C₈	—	14	1	no	—
51	C₂×C₈	C₂×D₇	1	1	yes	characteristic
52	C₂×C₇	—	2	1	no	—
53	C₂×C₇	#121	1	1	yes	characteristic
54	D₇	—	4	1	no	—
55	Q₈	—	7	1	no	—
56	Q₈	—	14	2	no	—
57	Q₈	—	14	2	no	—
58	D₄	—	7	1	no	—
59	D₄	—	14	1	no	—
60	C₂×C₄	—	14	1	no	—
61	C₂×C₄	—	14	2	no	—
62	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	14	1	no	—
65	C₈	D₂₈	1	1	yes	characteristic
66	C₈	D₂₈	1	1	yes	characteristic
67	C₇	#473	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	14	1	no	—
69	C₂²	#1161	1	1	yes	characteristic
70	C₄	—	14	1	no	—
71	C₄	—	28	2	no	—
72	C₄	C₂×D₂₈	1	1	yes	characteristic
73	C₄	D₄×D₇	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	28	1	no	—
76	C₂	#6458	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6458
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#108	C₂
socle	C₂×C₇	#121
prime radical	#1157	C₂²
hypercenter	#108	D₇
hypocenter	C₇	#473
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₂₈	C₂	semidirect
4. C₇×#108	C₂	semidirect
5. #6398	C₂	semidirect
33. #108	D₇	semidirect
67. C₇	#473	semidirect

Maximal subgroups

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

subgroup	index
#6472	2
C₂×Dic₂₈	2
#6464	2
C₇×#108	2
#6398	2
#473	7