Subgroups — #89964

Order 448.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#6505	C₂	1	2	yes	maximal
3	#6464	C₂	1	1	yes	maximal, characteristic
4	#6410	C₂	1	2	yes	maximal, complemented
5	#6393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	#1157	C₂²	1	1	yes	characteristic
8	C₇×Q₁₆	—	2	1	no	—
9	C₇×Q₁₆	—	2	1	no	—
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	#1144	—	2	2	no	—
12	#1143	—	2	2	no	—
13	#1136	C₂²	1	1	yes	characteristic
14	#1136	C₂²	1	2	yes	—
15	#1131	—	2	1	no	—
16	C₈×D₇	—	2	1	no	—
17	#473	—	7	1	no	maximal, Sylow
18	C₇×Q₈	—	2	2	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₂×C₄×C₇	—	2	2	no	—
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂²⋊D₇	—	2	1	no	—
23	D₂₈	D₄	1	1	yes	characteristic
24	C₄×D₇	—	2	1	no	—
25	Dic₁₄	D₄	1	1	yes	characteristic
26	C₇×C₈	D₄	1	1	yes	characteristic
27	C₇×C₈	D₄	1	1	yes	characteristic
28	C₇⋊C₈	—	2	1	no	—
29	C₇⋊C₈	—	2	2	no	—
30	#137	—	7	2	no	—
31	C₂×Q₁₆	D₇	1	1	yes	characteristic, complemented
32	#131	—	7	1	no	—
33	#108	—	7	1	no	—
34	#101	—	7	2	no	—
35	C₂²×C₇	#41	1	1	yes	characteristic
36	C₂×D₇	—	2	1	no	—
37	C₄×C₇	—	4	2	no	—
38	C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	Dic₇	—	2	1	no	—
41	#41	—	7	1	no	—
42	C₂×Q₈	C₂×D₇	1	2	yes	—
43	Q₁₆	—	2	1	no	—
44	Q₁₆	—	2	1	no	—
45	Q₁₆	—	14	2	no	—
46	SD₁₆	—	14	2	no	—
47	M₁₆	—	7	1	no	—
48	M₁₆	—	7	2	no	—
49	M₁₆	—	14	1	no	—
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	—	2	2	no	—
56	Q₈	—	2	2	no	—
57	Q₈	—	7	1	no	—
58	D₄	—	7	1	no	—
59	D₄	—	14	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	14	1	no	—
62	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
63	C₈	—	14	2	no	—
64	C₈	—	14	1	no	—
65	C₈	C₂²⋊D₇	1	1	yes	characteristic
66	C₈	C₂²⋊D₇	1	1	yes	characteristic
67	C₇	#473	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	14	1	no	—
69	C₂²	#1159	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	14	1	no	—
72	C₄	D₄×D₇	1	1	yes	characteristic
73	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	28	1	no	—
76	C₂	#6501	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6501
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×Q₁₆	C₂
socle	C₂×C₇	#121
prime radical	#1157	C₂²
hypercenter	C₂×Q₁₆	D₇
hypocenter	C₇	#473
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×Q₁₆	C₂	semidirect
4. #6410	C₂	semidirect
5. #6393	C₂	semidirect
31. C₂×Q₁₆	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
C₂×C₇×Q₁₆	2
#6505	2
#6464	2
#6410	2
#6393	2
#473	7