Subgroups — #89332

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#121	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×D₄	—	2	1	no	—
5	C₂×C₇×D₄	C₄	1	1	yes	characteristic, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₇	—	2	1	no	—
8	C₇×C₄⋊C₄	C₄	1	1	yes	characteristic, complemented
9	C₇×C₂²⋊C₄	—	2	1	no	—
10	C₂×C₇⋊C₈	—	2	1	no	—
11	#327	—	7	1	no	maximal, Sylow
12	C₂³×C₇	—	2	1	no	—
13	C₂³×C₇	D₄	1	1	yes	characteristic
14	C₇×D₄	—	4	1	no	—
15	C₇×D₄	—	4	1	no	—
16	C₇×D₄	—	4	1	no	—
17	C₂×C₄×C₇	—	2	1	no	—
18	C₂×C₄×C₇	—	4	1	no	—
19	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×Dic₇	—	2	2	no	—
22	C₂×Dic₇	—	4	1	no	—
23	C₂×Dic₇	—	4	1	no	—
24	C₇⋊C₈	—	4	1	no	—
25	#121	D₇	1	1	yes	characteristic
26	C₂²⋊C₈	—	7	1	no	—
27	#95	—	7	1	no	—
28	C₂²×C₇	—	2	1	no	—
29	C₂²×C₇	—	2	1	no	—
30	C₂²×C₇	—	4	1	no	—
31	C₂²×C₇	—	4	1	no	—
32	C₂²×C₇	—	4	1	no	—
33	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₇	D₈	1	1	yes	characteristic
35	C₂²×C₇	SD₁₆	1	1	yes	characteristic
36	C₄×C₇	—	2	1	no	—
37	C₄×C₇	—	2	1	no	—
38	C₄×C₇	—	4	1	no	—
39	Dic₇	—	4	2	no	—
40	C₂×D₄	—	2	1	no	—
41	C₂×D₄	Dic₇	1	1	yes	characteristic, complemented
42	C₂²×C₄	—	14	1	no	—
43	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
44	C₂×C₈	—	14	1	no	—
45	C₄⋊C₄	Dic₇	1	1	yes	characteristic, complemented
46	C₂²⋊C₄	—	2	1	no	—
47	C₂×C₇	—	2	1	no	—
48	C₂×C₇	—	2	1	no	—
49	C₂×C₇	—	8	1	no	—
50	C₂×C₇	#99	1	1	yes	characteristic
51	C₂×C₇	#102	1	1	yes	characteristic
52	C₂×C₇	#104	1	1	yes	characteristic
53	C₂³	—	2	1	no	—
54	C₂³	C₂²⋊D₇	1	1	yes	characteristic
55	D₄	—	4	1	no	—
56	D₄	—	4	1	no	—
57	D₄	—	4	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	14	2	no	—
61	C₂×C₄	—	28	1	no	—
62	C₂×C₄	—	28	1	no	—
63	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
65	C₈	—	28	1	no	—
66	C₇	#327	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	#1141	1	1	yes	characteristic
73	C₂²	#1142	1	1	yes	characteristic
74	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	28	2	no	—
79	C₂	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#6406	1	1	yes	characteristic
83	C₂	#6408	1	1	yes	characteristic
84	C₂	#6411	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂×C₄
Frattini	C₂²×C₄	C₂×D₇
Fitting, prime radical	C₇×#121	C₂
socle	C₂²×C₇	C₂²⋊C₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	#121	D₇
hypocenter	C₇	#327
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

Complements

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

N	H ≅ G/N	type
2. C₇⋊#95	C₂	semidirect
3. C₂²⋊(C₇⋊C₈)	C₂	semidirect
5. C₂×C₇×D₄	C₄	semidirect
8. C₇×C₄⋊C₄	C₄	semidirect
41. C₂×D₄	Dic₇	semidirect
45. C₄⋊C₄	Dic₇	semidirect
66. C₇	#327	semidirect

Maximal subgroups

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

subgroup	index
C₇×#121	2
C₇⋊#95	2
C₂²⋊(C₇⋊C₈)	2
#327	7