Subgroups — #89849

Order 448.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6548	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	#6495	C₂	1	2	yes	maximal, complemented
4	#6407	C₂	1	1	yes	maximal, characteristic, complemented
5	#6379	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×D₄	—	2	2	no	—
7	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₂²⋊D₇	—	2	2	no	—
10	#1159	—	2	2	no	—
11	D₄×D₇	—	2	2	no	—
12	#1157	C₂²	1	2	yes	—
13	C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
14	#1142	—	2	2	no	—
15	#1141	—	2	2	no	—
16	#1136	C₂²	1	2	yes	complemented
17	#453	—	7	1	no	maximal, Sylow
18	C₂³×C₇	—	2	1	no	—
19	C₂³×C₇	D₄	1	2	yes	—
20	C₂²×D₇	—	2	2	no	—
21	C₇×D₄	—	2	2	no	—
22	C₇×D₄	—	2	2	no	—
23	C₇×D₄	—	4	4	no	—
24	C₂×C₄×C₇	—	2	1	no	—
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂²⋊D₇	—	2	4	no	—
27	C₂²⋊D₇	—	2	2	no	—
28	C₂×Dic₇	—	2	2	no	—
29	D₂₈	D₄	1	2	yes	complemented
30	C₄×D₇	—	2	2	no	—
31	Dic₁₄	D₄	1	2	yes	complemented
32	C₇⋊C₈	—	2	2	no	—
33	H(5,𝔽₂)	—	7	1	no	—
34	AGL(1,ℤ/8)	—	7	2	no	—
35	#127	D₇	1	1	yes	characteristic, complemented
36	#104	—	7	2	no	—
37	#100	—	7	1	no	—
38	C₂²×C₇	—	2	2	no	—
39	C₂²×C₇	—	4	2	no	—
40	C₂²×C₇	—	4	2	no	—
41	C₂²×C₇	—	4	1	no	—
42	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂×D₇	—	2	2	no	—
44	C₂×D₇	—	8	2	no	—
45	C₄×C₇	—	2	2	no	—
46	C₄×C₇	C₂×D₄	1	2	yes	—
47	Dic₇	—	2	2	no	—
48	#41	—	7	2	no	—
49	#41	—	14	2	no	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	—	14	2	no	—
52	C₂×D₄	—	14	2	no	—
53	C₂×D₄	C₂×D₇	1	1	yes	characteristic
54	C₂×D₄	C₂×D₇	1	1	yes	characteristic, complemented
55	SD₁₆	—	14	2	no	—
56	D₈	—	14	2	no	—
57	M₁₆	—	7	2	no	—
58	C₄²	C₂×D₇	1	1	yes	characteristic, complemented
59	C₂×C₇	—	2	1	no	—
60	C₂×C₇	—	4	2	no	—
61	C₂×C₇	—	8	1	no	—
62	C₂×C₇	#120	1	1	yes	characteristic
63	D₇	—	4	2	no	—
64	C₂³	—	2	1	no	—
65	C₂³	—	14	2	no	—
66	C₂³	C₂²⋊D₇	1	2	yes	—
67	Q₈	—	7	2	no	—
68	D₄	—	2	2	no	—
69	D₄	—	2	2	no	—
70	D₄	—	4	4	no	—
71	D₄	—	7	2	no	—
72	D₄	—	14	4	no	—
73	D₄	—	14	2	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	14	2	no	—
76	C₂×C₄	—	14	2	no	—
77	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
78	C₈	—	14	2	no	—
79	C₇	#453	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	2	2	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	14	2	no	—
85	C₂²	—	56	2	no	—
86	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	14	2	no	—
89	C₄	D₄×D₇	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	8	1	no	—
93	C₂	—	28	2	no	—
94	C₂	#6500	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6500
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#127	C₂
socle	C₂×C₇	#120
hypercenter	#127	D₇
hypocenter	C₇	#453
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #127

Complements

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

N	H ≅ G/N	type
1. #6548	C₂	semidirect
2. C₇×#127	C₂	semidirect
3. #6495	C₂	semidirect
4. #6407	C₂	semidirect
5. #6379	C₂	semidirect
8. C₂×C₇×D₄	C₂²	semidirect
13. C₄²×C₇	C₂²	semidirect
16. #1136	C₂²	semidirect
29. D₂₈	D₄	semidirect
31. Dic₁₄	D₄	semidirect
35. #127	D₇	semidirect
54. C₂×D₄	C₂×D₇	semidirect
58. C₄²	C₂×D₇	semidirect
79. C₇	#453	semidirect

Maximal subgroups

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

subgroup	index
#6548	2
C₇×#127	2
#6495	2
#6407	2
#6379	2
#453	7