Subgroups — #89633

Order 448.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1131	C₂	1	1	yes	maximal, characteristic
2	#6455	C₂	1	1	yes	maximal, characteristic
3	D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	#6392	C₂	1	1	yes	maximal, characteristic, complemented
6	#6382	C₂	1	1	yes	maximal, characteristic, complemented
7	#6381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇	—	2	1	no	—
9	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	(C₂×D₇)⋊C₄	—	2	1	no	—
14	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	Dic₇⋊C₄	—	2	1	no	—
17	C₄×Dic₇	—	2	1	no	—
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#1131	C₄	1	4	yes	complemented
20	#428	—	7	1	no	maximal, Sylow
21	C₂²×D₇	D₄	1	1	yes	characteristic
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	C₄×D₇	—	4	2	no	—
29	C₄×D₇	C₂×C₄	1	2	yes	—
30	C₄×D₇	Q₈	1	2	yes	—
31	C₇×C₈	C₂×C₄	1	2	yes	complemented
32	C₇⋊C₈	C₂×C₄	1	2	yes	complemented
33	C₂×M₁₆	—	7	1	no	—
34	#117	—	7	1	no	—
35	C₂×C₄⋊C₄	—	7	1	no	—
36	#107	—	7	1	no	—
37	#107	—	7	1	no	—
38	#106	—	7	1	no	—
39	#106	D₇	1	1	yes	characteristic, complemented
40	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂×D₇	—	2	1	no	—
42	C₂×D₇	—	2	1	no	—
43	C₂×D₇	C₄⋊C₄	1	2	yes	—
44	C₄×C₇	—	4	1	no	—
45	C₄×C₇	—	4	1	no	—
46	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
48	Dic₇	—	4	1	no	—
49	Dic₇	—	4	1	no	—
50	Dic₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄	—	7	1	no	—
52	C₂²×C₄	—	14	1	no	—
53	M₁₆	—	7	4	no	—
54	C₂×C₈	—	7	1	no	—
55	C₂×C₈	C₂×D₇	1	1	yes	characteristic
56	C₄⋊C₄	—	7	1	no	—
57	C₄⋊C₄	—	7	1	no	—
58	C₄⋊C₄	—	14	1	no	—
59	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
60	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
61	C₂²⋊C₄	—	14	1	no	—
62	C₄²	—	14	1	no	—
63	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₇	#137	1	1	yes	characteristic
66	D₇	—	2	2	no	—
67	C₂³	—	7	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	7	1	no	—
71	C₂×C₄	—	7	2	no	—
72	C₂×C₄	—	7	2	no	—
73	C₂×C₄	—	14	1	no	—
74	C₂×C₄	—	14	1	no	—
75	C₂×C₄	—	28	2	no	—
76	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
77	C₈	—	7	2	no	—
78	C₈	C₄×D₇	1	2	yes	complemented
79	C₇	#428	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	7	2	no	—
81	C₂²	—	14	1	no	—
82	C₂²	—	14	1	no	—
83	C₂²	D₄×D₇	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	7	2	no	—
87	C₄	—	28	1	no	—
88	C₄	—	28	1	no	—
89	C₄	C₂×C₄×D₇	1	1	yes	characteristic
90	C₄	Q₈×D₇	1	1	yes	characteristic
91	C₂	—	14	2	no	—
92	C₂	D₇×C₄⋊C₄	1	1	yes	characteristic
93	C₂	#6477	1	1	yes	characteristic
94	C₂	#6478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#106	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂²×D₇	D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#106	D₇
hypocenter	C₇	#428
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
4. C₇×#106	C₂	semidirect
5. #6392	C₂	semidirect
6. #6382	C₂	semidirect
7. #6381	C₂	semidirect
19. #1131	C₄	semidirect
31. C₇×C₈	C₂×C₄	semidirect
32. C₇⋊C₈	C₂×C₄	semidirect
39. #106	D₇	semidirect
78. C₈	C₄×D₇	semidirect
79. C₇	#428	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1131	2
#6455	2
D₇×C₄⋊C₄	2
C₇×#106	2
#6392	2
#6382	2
#6381	2
#428	7