Subgroups — #89688

Order 448.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6482	C₂	1	1	yes	maximal, characteristic, complemented
2	#6475	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#6403	C₂	1	1	yes	maximal, characteristic, complemented
5	#6400	C₂	1	1	yes	maximal, characteristic, complemented
6	#6374	C₂	1	1	yes	maximal, characteristic, complemented
7	D₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#1161	—	2	1	no	—
9	#1159	—	2	1	no	—
10	C₇×Q₁₆	C₂²	1	1	yes	characteristic
11	C₇×D₈	C₂²	1	1	yes	characteristic
12	#1143	—	2	1	no	—
13	#1142	—	2	1	no	—
14	Dic₂₈	C₂²	1	1	yes	characteristic
15	D₅₆	C₂²	1	1	yes	characteristic
16	C₈×D₇	C₂²	1	1	yes	characteristic
17	C₇×C₁₆	C₂²	1	1	yes	characteristic
18	C₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	#508	—	7	1	no	maximal, Sylow
20	C₇×Q₈	—	2	1	no	—
21	C₇×D₄	—	2	1	no	—
22	C₂²⋊D₇	—	4	1	no	—
23	C₂×Dic₇	—	4	1	no	—
24	D₂₈	—	2	1	no	—
25	D₂₈	—	4	1	no	—
26	C₄×D₇	—	4	1	no	—
27	C₄×D₇	D₄	1	1	yes	characteristic
28	Dic₁₄	—	2	1	no	—
29	C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇⋊C₈	D₄	1	1	yes	characteristic
31	#135	—	7	1	no	—
32	#135	—	7	1	no	—
33	Q₃₂	—	7	1	no	—
34	SD₃₂	—	7	1	no	—
35	SD₃₂	D₇	1	1	yes	characteristic, complemented
36	D₁₆	—	7	1	no	—
37	C₂×C₁₆	—	7	1	no	—
38	C₂²×C₇	—	4	1	no	—
39	C₂×D₇	—	4	1	no	—
40	C₂×D₇	D₈	1	1	yes	characteristic
41	C₄×C₇	—	4	1	no	—
42	C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	Dic₇	—	4	1	no	—
44	Dic₇	D₈	1	1	yes	characteristic
45	#41	—	14	1	no	—
46	#41	—	14	1	no	—
47	Q₁₆	—	7	1	no	—
48	Q₁₆	C₂×D₇	1	1	yes	characteristic
49	SD₁₆	—	14	1	no	—
50	SD₁₆	—	14	1	no	—
51	D₈	—	7	1	no	—
52	D₈	C₂×D₇	1	1	yes	characteristic
53	C₂×C₈	—	7	1	no	—
54	C₁₆	—	7	1	no	—
55	C₁₆	C₂×D₇	1	1	yes	characteristic
56	C₂×C₇	—	8	1	no	—
57	C₂×C₇	C₂×D₈	1	1	yes	characteristic
58	D₇	—	2	1	no	—
59	D₇	—	8	1	no	—
60	Q₈	—	2	1	no	—
61	Q₈	—	14	1	no	—
62	D₄	—	2	1	no	—
63	D₄	—	14	1	no	—
64	D₄	—	28	1	no	—
65	D₄	—	28	1	no	—
66	C₂×C₄	—	7	1	no	—
67	C₂×C₄	—	28	1	no	—
68	C₂×C₄	—	28	1	no	—
69	C₈	—	7	1	no	—
70	C₈	C₂²×D₇	1	1	yes	characteristic
71	C₇	#508	1	1	yes	characteristic, complemented, Sylow
72	C₂²	—	4	1	no	—
73	C₂²	—	7	1	no	—
74	C₂²	—	28	1	no	—
75	C₄	—	4	1	no	—
76	C₄	—	7	1	no	—
77	C₄	—	28	1	no	—
78	C₄	D₄×D₇	1	1	yes	characteristic
79	C₂	—	8	1	no	—
80	C₂	—	14	1	no	—
81	C₂	—	56	1	no	—
82	C₂	D₇×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₇×D₈
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₇×SD₃₂	C₂
socle	C₂×C₇	C₂×D₈
hypercenter	SD₃₂	D₇
hypocenter	C₇	#508
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #6482	C₂	semidirect
2. #6475	C₂	semidirect
3. C₇×SD₃₂	C₂	semidirect
4. #6403	C₂	semidirect
5. #6400	C₂	semidirect
6. #6374	C₂	semidirect
7. D₇×C₁₆	C₂	semidirect
35. SD₃₂	D₇	semidirect
71. C₇	#508	semidirect

Maximal subgroups

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

subgroup	index
#6482	2
#6475	2
C₇×SD₃₂	2
#6403	2
#6400	2
#6374	2
D₇×C₁₆	2
#508	7