Subgroups — #89847

Order 448.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×#1142	C₂	1	2	yes	maximal, complemented
3	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#6406	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	—	2	1	no	—
7	C₂×C₇×D₄	—	2	1	no	—
8	C₂×C₇×D₄	C₂²	1	2	yes	—
9	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	#1142	—	2	4	no	—
12	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
13	Dic₇⋊C₄	—	2	1	no	—
14	C₄×Dic₇	—	2	1	no	—
15	C₂×C₇⋊C₈	C₂²	1	2	yes	—
16	#460	—	7	1	no	maximal, Sylow
17	C₂³×C₇	—	2	2	no	—
18	C₇×D₄	—	2	2	no	—
19	C₇×D₄	—	2	2	no	—
20	C₇×D₄	—	4	2	no	—
21	C₇×D₄	—	4	2	no	—
22	C₂×C₄×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₇	—	2	1	no	—
26	C₂×Dic₇	—	2	1	no	—
27	Dic₁₄	—	2	1	no	—
28	Dic₁₄	D₄	1	2	yes	complemented
29	C₇⋊C₈	—	2	2	no	—
30	C₂×SD₁₆	—	7	2	no	—
31	#127	D₇	1	1	yes	characteristic
32	C₄×Q₈	—	7	1	no	—
33	C₄⋊C₈	—	7	1	no	—
34	#102	—	7	2	no	—
35	C₂²×C₇	—	4	2	no	—
36	C₂²×C₇	—	4	2	no	—
37	C₂²×C₇	—	4	2	no	—
38	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
39	C₄×C₇	—	2	1	no	—
40	C₄×C₇	SD₁₆	1	2	yes	complemented
41	C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₇	#41	1	1	yes	characteristic
43	Dic₇	—	2	2	no	—
44	Dic₇	—	4	1	no	—
45	C₂×Q₈	—	7	1	no	—
46	C₂×D₄	—	2	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	C₂×D₇	1	2	yes	—
49	SD₁₆	—	14	4	no	—
50	C₂×C₈	—	7	2	no	—
51	C₄⋊C₄	—	7	1	no	—
52	C₄⋊C₄	—	14	1	no	—
53	C₄²	—	14	1	no	—
54	C₄²	C₂×D₇	1	1	yes	characteristic
55	C₂×C₇	—	8	2	no	—
56	C₂×C₇	#121	1	1	yes	characteristic
57	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂³	—	2	2	no	—
60	Q₈	—	7	2	no	—
61	Q₈	—	14	1	no	—
62	D₄	—	2	2	no	—
63	D₄	—	2	2	no	—
64	D₄	—	4	2	no	—
65	D₄	—	4	2	no	—
66	C₂×C₄	—	14	1	no	—
67	C₂×C₄	—	14	1	no	—
68	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
71	C₈	—	14	2	no	—
72	C₇	#460	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	4	2	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	4	2	no	—
76	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	14	2	no	—
79	C₄	—	28	1	no	—
80	C₄	#1142	1	2	yes	complemented
81	C₄	D₄×D₇	1	1	yes	characteristic
82	C₄	#1159	1	1	yes	characteristic
83	C₂	—	8	2	no	—
84	C₂	#6495	1	1	yes	characteristic
85	C₂	C₂×#1142	1	1	yes	characteristic
86	C₂	#6501	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×#127	C₂
socle	C₂²×C₇	C₂×D₄
hypercenter	#127	D₇
hypocenter	C₇	#460
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. C₂×#1142	C₂	semidirect
3. C₄×Dic₁₄	C₂	semidirect
4. #6406	C₂	semidirect
5. C₄⋊(C₇⋊C₈)	C₂	semidirect
28. Dic₁₄	D₄	semidirect
40. C₄×C₇	SD₁₆	semidirect
72. C₇	#460	semidirect
80. C₄	#1142	semidirect

Maximal subgroups

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

subgroup	index
C₇×#127	2
C₂×#1142	2
C₄×Dic₁₄	2
#6406	2
C₄⋊(C₇⋊C₈)	2
#460	7