Subgroups — #89686

Order 448.

90 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	D₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#6402	C₂	1	1	yes	maximal, characteristic, complemented
5	#6400	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₁₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#6372	C₂	1	1	yes	maximal, characteristic, complemented
8	#1161	—	2	1	no	—
9	D₄×D₇	—	2	1	no	—
10	C₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
11	C₇×D₈	C₂²	1	1	yes	characteristic
12	#1143	—	2	1	no	—
13	#1141	—	2	1	no	—
14	D₅₆	C₂²	1	1	yes	characteristic
15	D₅₆	C₂²	1	1	yes	characteristic, complemented
16	C₈×D₇	C₂²	1	1	yes	characteristic
17	C₇×C₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
19	#509	—	7	1	no	maximal, Sylow
20	C₂²×D₇	—	4	1	no	—
21	C₇×Q₈	—	2	1	no	—
22	C₇×D₄	—	2	1	no	—
23	C₂²⋊D₇	—	4	1	no	—
24	D₂₈	—	2	1	no	—
25	D₂₈	—	2	1	no	—
26	D₂₈	—	4	1	no	—
27	C₄×D₇	—	4	1	no	—
28	C₄×D₇	D₄	1	1	yes	characteristic
29	C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇⋊C₈	D₄	1	1	yes	characteristic
31	#135	—	7	1	no	—
32	C₂×D₈	—	7	1	no	—
33	SD₃₂	—	7	1	no	—
34	SD₃₂	D₇	1	1	yes	characteristic, complemented
35	D₁₆	—	7	1	no	—
36	D₁₆	—	7	1	no	—
37	M₃₂	—	7	1	no	—
38	C₂²×C₇	—	4	1	no	—
39	C₂×D₇	—	4	1	no	—
40	C₂×D₇	—	4	1	no	—
41	C₂×D₇	—	8	2	no	—
42	C₂×D₇	D₈	1	1	yes	characteristic
43	C₄×C₇	—	4	1	no	—
44	C₄×C₇	C₂×D₄	1	1	yes	characteristic
45	Dic₇	D₈	1	1	yes	characteristic
46	#41	—	14	1	no	—
47	C₂×D₄	—	14	1	no	—
48	Q₁₆	C₂×D₇	1	1	yes	characteristic, complemented
49	SD₁₆	—	14	1	no	—
50	D₈	—	7	1	no	—
51	D₈	—	7	1	no	—
52	D₈	—	14	1	no	—
53	D₈	C₂×D₇	1	1	yes	characteristic
54	C₂×C₈	—	7	1	no	—
55	C₁₆	—	7	1	no	—
56	C₁₆	C₂×D₇	1	1	yes	characteristic, complemented
57	C₂×C₇	—	8	1	no	—
58	C₂×C₇	C₂×D₈	1	1	yes	characteristic
59	D₇	—	2	1	no	—
60	D₇	—	8	1	no	—
61	D₇	—	8	1	no	—
62	C₂³	—	28	1	no	—
63	Q₈	—	2	1	no	—
64	D₄	—	2	1	no	—
65	D₄	—	14	1	no	—
66	D₄	—	14	1	no	—
67	D₄	—	28	1	no	—
68	D₄	—	28	1	no	—
69	C₂×C₄	—	7	1	no	—
70	C₂×C₄	—	28	1	no	—
71	C₈	—	7	1	no	—
72	C₈	C₂²×D₇	1	1	yes	characteristic
73	C₇	#509	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	4	1	no	—
75	C₂²	—	7	1	no	—
76	C₂²	—	28	1	no	—
77	C₂²	—	28	1	no	—
78	C₂²	—	56	2	no	—
79	C₄	—	4	1	no	—
80	C₄	—	7	1	no	—
81	C₄	D₄×D₇	1	1	yes	characteristic
82	C₂	—	8	1	no	—
83	C₂	—	14	1	no	—
84	C₂	—	56	1	no	—
85	C₂	—	56	1	no	—
86	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	#509
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. D₇×D₈	C₂	semidirect
3. C₇×SD₃₂	C₂	semidirect
4. #6402	C₂	semidirect
5. #6400	C₂	semidirect
6. D₁₁₂	C₂	semidirect
7. #6372	C₂	semidirect
10. C₇×Q₁₆	C₂²	semidirect
15. D₅₆	C₂²	semidirect
17. C₇×C₁₆	C₂²	semidirect
18. C₇⋊C₁₆	C₂²	semidirect
34. SD₃₂	D₇	semidirect
48. Q₁₆	C₂×D₇	semidirect
56. C₁₆	C₂×D₇	semidirect
73. C₇	#509	semidirect

Maximal subgroups

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

subgroup	index
#6482	2
D₇×D₈	2
C₇×SD₃₂	2
#6402	2
#6400	2
D₁₁₂	2
#6372	2
#509	7