Subgroups — #89564

Order 448.

98 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	#6451	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	C₈×Dic₇	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	—	2	1	no	—
10	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	#487	—	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	Dic₁₄	—	4	2	no	—
32	C₇×C₈	—	2	1	no	—
33	C₇⋊C₈	—	2	1	no	—
34	C₄⋊Q₈	—	7	1	no	—
35	C₄⋊Q₈	—	7	1	no	—
36	#103	—	7	1	no	—
37	#103	—	7	1	no	—
38	#103	—	7	1	no	—
39	#103	D₇	1	1	yes	characteristic
40	C₄×C₈	—	7	1	no	—
41	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₇	—	4	1	no	—
43	C₄×C₇	—	4	1	no	—
44	C₄×C₇	#41	1	1	yes	characteristic
45	C₄×C₇	#41	1	1	yes	characteristic
46	Dic₇	—	4	1	no	—
47	Dic₇	—	4	1	no	—
48	Dic₇	SD₁₆	1	2	yes	—
49	Dic₇	Q₁₆	1	2	yes	—
50	C₂×Q₈	—	7	1	no	—
51	C₂×Q₈	—	14	1	no	—
52	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
53	C₂×C₈	—	7	1	no	—
54	C₂×C₈	C₂×D₇	1	1	yes	characteristic
55	C₄⋊C₄	—	7	1	no	—
56	C₄⋊C₄	—	14	1	no	—
57	C₄⋊C₄	—	14	1	no	—
58	C₄⋊C₄	—	14	1	no	—
59	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
60	C₄²	—	7	1	no	—
61	C₂×C₇	#124	1	1	yes	characteristic
62	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
64	Q₈	—	2	1	no	—
65	Q₈	—	2	1	no	—
66	Q₈	—	14	1	no	—
67	Q₈	—	14	1	no	—
68	Q₈	—	28	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	7	1	no	—
72	C₂×C₄	—	7	1	no	—
73	C₂×C₄	—	14	1	no	—
74	C₂×C₄	—	14	1	no	—
75	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	14	1	no	—
78	C₇	#487	1	1	yes	characteristic, complemented, Sylow
79	C₂²	D₄×D₇	1	1	yes	characteristic
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	—	7	2	no	—
83	C₄	—	7	2	no	—
84	C₄	—	28	1	no	—
85	C₄	—	28	1	no	—
86	C₄	#1157	1	1	yes	characteristic
87	C₄	#1159	1	1	yes	characteristic
88	C₂	#6448	1	1	yes	characteristic
89	C₂	D₇×SD₁₆	1	1	yes	characteristic
90	C₂	D₇×Q₁₆	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	#487
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
78. C₇	#487	semidirect

Maximal subgroups

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

subgroup	index
#6507	2
#6451	2
C₇×#103	2
#6409	2
#6390	2
C₈×Dic₇	2
#6384	2
#487	7