Subgroups — #89569

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6507	C₂	1	1	yes	maximal, characteristic
2	#6453	C₂	1	1	yes	maximal, characteristic
3	C₇×#103	C₂	1	1	yes	maximal, characteristic
4	#6409	C₂	1	1	yes	maximal, characteristic
5	#6390	C₂	1	1	yes	maximal, characteristic
6	#6389	C₂	1	1	yes	maximal, characteristic
7	#6384	C₂	1	1	yes	maximal, characteristic
8	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	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	Dic₇⋊C₄	—	2	1	no	—
17	C₄×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#491	—	7	1	no	maximal, Sylow
20	C₇×Q₈	—	2	1	no	—
21	C₇×Q₈	—	2	1	no	—
22	C₂×C₄×C₇	—	2	1	no	—
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₇	—	2	1	no	—
26	C₂×Dic₇	—	2	1	no	—
27	C₂×Dic₇	D₄	1	1	yes	characteristic
28	C₂×Dic₇	D₄	1	1	yes	characteristic
29	Dic₁₄	—	2	1	no	—
30	Dic₁₄	—	2	1	no	—
31	C₇×C₈	—	2	1	no	—
32	C₇⋊C₈	—	2	1	no	—
33	C₄⋊Q₈	—	7	1	no	—
34	#125	—	7	1	no	—
35	#103	—	7	1	no	—
36	#103	—	7	1	no	—
37	#103	—	7	1	no	—
38	#103	D₇	1	1	yes	characteristic
39	#97	—	7	1	no	—
40	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
41	C₄×C₇	—	4	1	no	—
42	C₄×C₇	—	4	1	no	—
43	C₄×C₇	#41	1	1	yes	characteristic
44	C₄×C₇	#41	1	1	yes	characteristic
45	Dic₇	—	2	1	no	—
46	Dic₇	—	2	1	no	—
47	Dic₇	—	4	1	no	—
48	Dic₇	—	4	1	no	—
49	C₂×Q₈	—	7	1	no	—
50	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
51	C₂×C₈	—	7	1	no	—
52	C₂×C₈	C₂×D₇	1	1	yes	characteristic
53	C₄⋊C₄	—	7	1	no	—
54	C₄⋊C₄	—	14	1	no	—
55	C₄⋊C₄	—	14	1	no	—
56	C₄⋊C₄	—	14	1	no	—
57	C₄⋊C₄	—	14	1	no	—
58	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
59	C₄²	—	7	1	no	—
60	C₂×C₇	#124	1	1	yes	characteristic
61	C₂×C₇	#137	1	1	yes	characteristic
62	C₂×C₇	#137	1	1	yes	characteristic
63	Q₈	—	2	1	no	—
64	Q₈	—	2	1	no	—
65	Q₈	—	14	1	no	—
66	Q₈	—	14	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	7	1	no	—
70	C₂×C₄	—	7	1	no	—
71	C₂×C₄	—	14	1	no	—
72	C₂×C₄	—	14	1	no	—
73	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	14	1	no	—
76	C₇	#491	1	1	yes	characteristic, complemented, Sylow
77	C₂²	D₄×D₇	1	1	yes	characteristic
78	C₄	—	4	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	14	1	no	—
81	C₄	—	14	1	no	—
82	C₄	—	28	1	no	—
83	C₄	—	28	1	no	—
84	C₄	#1157	1	1	yes	characteristic
85	C₄	#1159	1	1	yes	characteristic
86	C₂	#6448	1	1	yes	characteristic
87	C₂	#6478	1	1	yes	characteristic
88	C₂	#6481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#103	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	#103	D₇
hypocenter	C₇	#491
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
76. C₇	#491	semidirect

Maximal subgroups

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

subgroup	index
#6507	2
#6453	2
C₇×#103	2
#6409	2
#6390	2
#6389	2
#6384	2
#491	7