Subgroups — #89767

Order 448.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#1136	C₂	1	1	yes	maximal, characteristic
4	#6382	C₂	1	2	yes	maximal, complemented
5	#6381	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₇	—	2	1	no	—
8	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₇×C₂²⋊C₄	—	2	1	no	—
11	C₄²×C₇	—	2	1	no	—
12	C₄⋊Dic₇	—	2	1	no	—
13	C₄⋊Dic₇	C₂²	1	2	yes	—
14	#1136	C₄	1	4	yes	complemented
15	C₂×C₇⋊C₈	C₂²	1	2	yes	—
16	#428	—	7	1	no	maximal, Sylow
17	C₂³×C₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₇	—	2	1	no	—
19	C₂×C₄×C₇	—	2	1	no	—
20	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₇	Q₈	1	2	yes	—
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₇	—	2	2	no	—
25	C₂×Dic₇	—	4	2	no	—
26	C₇⋊C₈	C₂×C₄	1	4	yes	complemented
27	C₂×M₁₆	—	7	1	no	—
28	#117	D₇	1	1	yes	characteristic
29	C₂×C₄⋊C₄	—	7	1	no	—
30	#107	—	7	2	no	—
31	#106	—	7	2	no	—
32	C₂²×C₇	—	2	1	no	—
33	C₂²×C₇	—	2	1	no	—
34	C₂²×C₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₇	—	4	1	no	—
37	C₄×C₇	—	4	1	no	—
38	C₄×C₇	C₄⋊C₄	1	2	yes	—
39	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
41	Dic₇	—	4	2	no	—
42	C₂²×C₄	—	14	1	no	—
43	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
44	M₁₆	—	7	4	no	—
45	C₂×C₈	—	7	2	no	—
46	C₄⋊C₄	—	7	2	no	—
47	C₄⋊C₄	—	14	1	no	—
48	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
49	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
50	C₂²⋊C₄	—	2	1	no	—
51	C₄²	—	2	1	no	—
52	C₂×C₇	—	2	2	no	—
53	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×C₇	#137	1	1	yes	characteristic
56	C₂³	C₂²⋊D₇	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	14	2	no	—
60	C₂×C₄	—	28	2	no	—
61	C₂×C₄	Dic₁₄	1	2	yes	—
62	C₂×C₄	C₄×D₇	1	2	yes	—
63	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
65	C₈	—	7	4	no	—
66	C₇	#428	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	Dic₇⋊C₄	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	28	2	no	—
74	C₄	Dic₇⋊C₄	1	2	yes	—
75	C₄	C₂×Dic₁₄	1	1	yes	characteristic
76	C₄	C₂×C₄×D₇	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×Dic₇⋊C₄	1	1	yes	characteristic
79	C₂	#6512	1	1	yes	characteristic
80	C₂	#6514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂³×C₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#428
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #6382	C₂	semidirect
5. #6381	C₂	semidirect
14. #1136	C₄	semidirect
26. C₇⋊C₈	C₂×C₄	semidirect
66. C₇	#428	semidirect

Maximal subgroups

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

subgroup	index
C₇×#117	2
C₂×C₄⋊Dic₇	2
C₂×#1136	2
#6382	2
#6381	2
#428	7