Subgroups — #89768

Order 448.

96 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	#6487	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#6384	C₂	1	1	yes	maximal, characteristic, complemented
5	#6384	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	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₇×C₂²⋊C₄	—	2	1	no	—
13	C₄²×C₇	—	2	1	no	—
14	C₂²⋊Dic₇	—	2	1	no	—
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	Dic₇⋊C₄	—	2	1	no	—
17	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#485	—	7	1	no	maximal, Sylow
20	C₂³×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₇	—	2	1	no	—
22	C₂×C₄×C₇	—	2	1	no	—
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	—	2	1	no	—
25	C₂×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₇	—	2	1	no	—
28	C₂×Dic₇	—	2	1	no	—
29	Dic₁₄	—	2	1	no	—
30	Dic₁₄	—	2	1	no	—
31	C₇⋊C₈	—	2	1	no	—
32	C₇⋊C₈	—	2	1	no	—
33	C₂²⋊Q₈	—	7	1	no	—
34	#117	D₇	1	1	yes	characteristic
35	#107	—	7	1	no	—
36	#106	—	7	1	no	—
37	#103	—	7	1	no	—
38	#103	—	7	1	no	—
39	C₂²⋊C₈	—	7	1	no	—
40	C₂²×C₇	—	2	1	no	—
41	C₂²×C₇	—	2	1	no	—
42	C₂²×C₇	—	2	1	no	—
43	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
44	C₄×C₇	—	2	1	no	—
45	C₄×C₇	—	4	1	no	—
46	C₄×C₇	—	4	1	no	—
47	C₄×C₇	#41	1	1	yes	characteristic
48	C₄×C₇	#41	1	1	yes	characteristic
49	Dic₇	—	4	1	no	—
50	Dic₇	—	4	1	no	—
51	C₂×Q₈	—	7	1	no	—
52	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
53	C₂×C₈	—	7	1	no	—
54	C₂×C₈	—	7	1	no	—
55	C₄⋊C₄	—	7	1	no	—
56	C₄⋊C₄	—	14	1	no	—
57	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
58	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
59	C₂²⋊C₄	—	2	1	no	—
60	C₂²⋊C₄	—	14	1	no	—
61	C₄²	—	2	1	no	—
62	C₂×C₇	—	4	1	no	—
63	C₂×C₇	#123	1	1	yes	characteristic
64	C₂×C₇	#135	1	1	yes	characteristic
65	C₂×C₇	#137	1	1	yes	characteristic
66	C₂³	C₂²⋊D₇	1	1	yes	characteristic
67	Q₈	—	14	1	no	—
68	Q₈	—	14	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	14	1	no	—
74	C₂×C₄	—	14	1	no	—
75	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
77	C₈	—	14	1	no	—
78	C₈	—	14	1	no	—
79	C₇	#485	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	28	1	no	—
88	C₄	—	28	1	no	—
89	C₄	#1157	1	1	yes	characteristic
90	C₄	#1157	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#6492	1	1	yes	characteristic
93	C₂	#6513	1	1	yes	characteristic
94	C₂	#6514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂³×C₇	D₄
hypercenter	#117	D₇
hypocenter	C₇	#485
LCS	G ▷ C₂×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. #6384	C₂	semidirect
5. #6384	C₂	semidirect
6. #6382	C₂	semidirect
7. #6381	C₂	semidirect
79. C₇	#485	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
#6487	2
C₂²⋊(C₇⋊C₈)	2
#6384	2
#6382	2
#6381	2
#485	7