Subgroups — #89806

Order 448.

114 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	C₂×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#6406	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#6382	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×D₄	—	2	1	no	—
7	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₇	—	2	1	no	—
10	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₇×C₂²⋊C₄	—	2	1	no	—
12	C₄⋊Dic₇	—	2	1	no	—
13	C₄⋊Dic₇	C₂²	1	2	yes	—
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	#481	—	7	1	no	maximal, Sylow
16	C₂³×C₇	—	2	1	no	—
17	C₂³×C₇	D₄	1	1	yes	characteristic
18	C₇×D₄	—	2	2	no	—
19	C₇×D₄	—	4	2	no	—
20	C₂×C₄×C₇	—	2	2	no	—
21	C₂×C₄×C₇	—	2	1	no	—
22	C₂×C₄×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₇	—	2	2	no	—
25	C₂×Dic₇	—	4	2	no	—
26	C₇⋊C₈	—	2	2	no	—
27	#121	D₇	1	1	yes	characteristic
28	C₂×C₄⋊C₄	—	7	1	no	—
29	#106	—	7	2	no	—
30	#102	—	7	2	no	—
31	C₂²⋊C₈	—	7	1	no	—
32	C₂²×C₇	—	2	1	no	—
33	C₂²×C₇	—	2	1	no	—
34	C₂²×C₇	—	4	1	no	—
35	C₂²×C₇	—	4	1	no	—
36	C₂²×C₇	—	4	1	no	—
37	C₂²×C₇	SD₁₆	1	2	yes	—
38	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
39	C₄×C₇	—	2	1	no	—
40	C₄×C₇	—	4	1	no	—
41	C₄×C₇	#41	1	2	yes	—
42	Dic₇	—	4	2	no	—
43	C₂×D₄	—	2	1	no	—
44	C₂×D₄	C₂×D₇	1	1	yes	characteristic
45	C₂²×C₄	—	14	1	no	—
46	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
47	C₂×C₈	—	7	2	no	—
48	C₄⋊C₄	—	7	2	no	—
49	C₄⋊C₄	—	14	1	no	—
50	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
51	C₂²⋊C₄	—	2	1	no	—
52	C₂×C₇	—	2	2	no	—
53	C₂×C₇	—	8	1	no	—
54	C₂×C₇	#123	1	1	yes	characteristic
55	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₂³	—	2	1	no	—
58	C₂³	C₂²⋊D₇	1	1	yes	characteristic
59	D₄	—	2	2	no	—
60	D₄	—	4	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	14	2	no	—
64	C₂×C₄	—	28	2	no	—
65	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
67	C₈	—	14	2	no	—
68	C₇	#481	1	1	yes	characteristic, complemented, Sylow
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	#1142	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	28	2	no	—
79	C₄	#1159	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	8	1	no	—
82	C₂	C₂×#1142	1	1	yes	characteristic
83	C₂	#6498	1	1	yes	characteristic
84	C₂	#6512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting, prime radical	C₇×#121	C₂
socle	C₂²×C₇	C₂×D₄
hypercenter	#121	D₇
hypocenter	C₇	#481
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×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. C₂×C₄⋊Dic₇	C₂	semidirect
3. #6406	C₂	semidirect
4. C₂²⋊(C₇⋊C₈)	C₂	semidirect
5. #6382	C₂	semidirect
68. C₇	#481	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
C₂×C₄⋊Dic₇	2
#6406	2
C₂²⋊(C₇⋊C₈)	2
#6382	2
#481	7