Subgroups — #89690

Order 448.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6482	C₂	1	1	yes	maximal, characteristic
2	D₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#6403	C₂	1	1	yes	maximal, characteristic, complemented
5	#6402	C₂	1	1	yes	maximal, characteristic, complemented
6	#6374	C₂	1	1	yes	maximal, characteristic, complemented
7	#6372	C₂	1	1	yes	maximal, characteristic, complemented
8	#1161	—	2	1	no	—
9	Q₈×D₇	—	2	1	no	—
10	C₇×Q₁₆	C₂²	1	1	yes	characteristic
11	C₇×Q₁₆	C₂²	1	1	yes	characteristic
12	#1144	—	2	1	no	—
13	#1143	—	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	#510	—	7	1	no	maximal, Sylow
20	C₇×Q₈	—	2	1	no	—
21	C₇×Q₈	—	2	1	no	—
22	D₂₈	—	2	1	no	—
23	D₂₈	—	4	1	no	—
24	C₄×D₇	—	4	1	no	—
25	C₄×D₇	—	4	1	no	—
26	C₄×D₇	D₄	1	1	yes	characteristic
27	Dic₁₄	—	2	1	no	—
28	Dic₁₄	—	4	1	no	—
29	C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇⋊C₈	D₄	1	1	yes	characteristic
31	#135	—	7	1	no	—
32	C₂×Q₁₆	—	7	1	no	—
33	Q₃₂	—	7	1	no	—
34	Q₃₂	D₇	1	1	yes	characteristic, complemented
35	SD₃₂	—	7	1	no	—
36	SD₃₂	—	7	1	no	—
37	M₃₂	—	7	1	no	—
38	C₂×D₇	—	4	1	no	—
39	C₂×D₇	D₈	1	1	yes	characteristic
40	C₄×C₇	—	4	1	no	—
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	C₂×Q₈	—	14	1	no	—
47	Q₁₆	—	7	1	no	—
48	Q₁₆	—	14	1	no	—
49	Q₁₆	C₂×D₇	1	1	yes	characteristic
50	Q₁₆	C₂×D₇	1	1	yes	characteristic
51	SD₁₆	—	14	1	no	—
52	D₈	—	7	1	no	—
53	C₂×C₈	—	7	1	no	—
54	C₁₆	—	7	1	no	—
55	C₁₆	C₂×D₇	1	1	yes	characteristic
56	C₂×C₇	C₂×D₈	1	1	yes	characteristic
57	D₇	—	2	1	no	—
58	D₇	—	8	1	no	—
59	Q₈	—	2	1	no	—
60	Q₈	—	2	1	no	—
61	Q₈	—	14	1	no	—
62	Q₈	—	28	1	no	—
63	D₄	—	14	1	no	—
64	D₄	—	28	1	no	—
65	C₂×C₄	—	7	1	no	—
66	C₂×C₄	—	28	1	no	—
67	C₂×C₄	—	28	1	no	—
68	C₈	—	7	1	no	—
69	C₈	C₂²×D₇	1	1	yes	characteristic
70	C₇	#510	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	7	1	no	—
72	C₂²	—	28	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	7	1	no	—
76	C₄	—	28	1	no	—
77	C₄	D₄×D₇	1	1	yes	characteristic
78	C₂	—	14	1	no	—
79	C₂	—	56	1	no	—
80	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×Q₃₂	C₂
socle	C₂×C₇	C₂×D₈
prime radical	#6482	C₂
hypercenter	Q₃₂	D₇
hypocenter	C₇	#510
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. D₇×Q₁₆	C₂	semidirect
3. C₇×Q₃₂	C₂	semidirect
4. #6403	C₂	semidirect
5. #6402	C₂	semidirect
6. #6374	C₂	semidirect
7. #6372	C₂	semidirect
34. Q₃₂	D₇	semidirect
70. C₇	#510	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×Q₁₆	2
C₇×Q₃₂	2
#6403	2
#6402	2
#6374	2
#6372	2
#510	7