Subgroups — #89629

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6453	C₂	1	1	yes	maximal, characteristic
2	#6450	C₂	1	1	yes	maximal, characteristic
3	C₇×#106	C₂	1	1	yes	maximal, characteristic
4	#6390	C₂	1	1	yes	maximal, characteristic
5	Dic₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#6384	C₂	1	1	yes	maximal, characteristic
7	#6381	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₇	—	2	1	no	—
13	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
14	Dic₇⋊C₄	—	2	1	no	—
15	Dic₇⋊C₄	—	2	1	no	—
16	C₄×Dic₇	—	2	1	no	—
17	C₄×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#479	—	7	1	no	maximal, Sylow
20	C₂×C₄×C₇	—	2	1	no	—
21	C₂×C₄×C₇	—	2	1	no	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₇	—	2	1	no	—
24	C₂×Dic₇	—	2	1	no	—
25	C₂×Dic₇	D₄	1	1	yes	characteristic
26	C₂×Dic₇	D₄	1	1	yes	characteristic
27	Dic₁₄	—	2	1	no	—
28	Dic₁₄	Q₈	1	2	yes	—
29	C₇×C₈	—	2	1	no	—
30	C₇⋊C₈	—	2	1	no	—
31	#125	—	7	1	no	—
32	C₄×Q₈	—	7	1	no	—
33	#107	—	7	1	no	—
34	#106	D₇	1	1	yes	characteristic
35	C₄⋊C₈	—	7	1	no	—
36	#103	—	7	1	no	—
37	#103	—	7	1	no	—
38	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
39	C₄×C₇	—	4	1	no	—
40	C₄×C₇	—	4	1	no	—
41	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
42	C₄×C₇	#41	1	1	yes	characteristic
43	Dic₇	—	2	2	no	—
44	Dic₇	—	2	1	no	—
45	Dic₇	—	2	1	no	—
46	Dic₇	—	4	1	no	—
47	C₂×Q₈	—	7	1	no	—
48	C₂×C₈	—	7	1	no	—
49	C₂×C₈	C₂×D₇	1	1	yes	characteristic
50	C₄⋊C₄	—	7	1	no	—
51	C₄⋊C₄	—	14	1	no	—
52	C₄⋊C₄	—	14	1	no	—
53	C₄⋊C₄	—	14	1	no	—
54	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
55	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
56	C₄²	—	7	1	no	—
57	C₄²	—	14	1	no	—
58	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₇	#135	1	1	yes	characteristic
60	C₂×C₇	#137	1	1	yes	characteristic
61	Q₈	—	7	2	no	—
62	Q₈	—	14	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	7	1	no	—
66	C₂×C₄	—	7	1	no	—
67	C₂×C₄	—	14	1	no	—
68	C₂×C₄	—	14	1	no	—
69	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	14	1	no	—
72	C₇	#479	1	1	yes	characteristic, complemented, Sylow
73	C₂²	D₄×D₇	1	1	yes	characteristic
74	C₄	—	4	1	no	—
75	C₄	—	4	1	no	—
76	C₄	—	14	2	no	—
77	C₄	—	14	1	no	—
78	C₄	—	14	1	no	—
79	C₄	—	28	1	no	—
80	C₄	#1157	1	1	yes	characteristic
81	C₄	Q₈×D₇	1	1	yes	characteristic
82	C₂	#6459	1	1	yes	characteristic
83	C₂	#6478	1	1	yes	characteristic
84	C₂	#6479	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#106	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	#106	D₇
hypocenter	C₇	#479
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
72. C₇	#479	semidirect

Maximal subgroups

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

subgroup	index
#6453	2
#6450	2
C₇×#106	2
#6390	2
Dic₇⋊C₈	2
#6384	2
#6381	2
#479	7