Group #90390 — subgroups

Subgroups — #90390

Order 448.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#6461	C₂	1	2	yes	maximal, complemented
3	#6461	C₂	1	2	yes	maximal, complemented
4	#6460	C₂	1	2	yes	maximal, complemented
5	#6459	C₂	1	2	yes	maximal, complemented
6	#6457	C₂	1	2	yes	maximal
7	#6451	C₂	1	2	yes	maximal, complemented
8	#6438	C₂	1	1	yes	maximal, characteristic
9	#6433	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₂₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₇	C₂²	1	2	yes	—
12	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₄	C₂²	1	2	yes	—
14	C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₄²×C₇	C₂²	1	1	yes	characteristic
19	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
24	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₇	C₂²	1	2	yes	—
27	Dic₇⋊C₄	C₂²	1	2	yes	—
28	Dic₇⋊C₄	C₂²	1	2	yes	—
29	Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₄×Dic₇	C₂²	1	2	yes	—
31	#563	—	7	1	no	maximal, Sylow
32	C₂²×D₇	C₂³	1	2	yes	—
33	C₂×C₄×C₇	C₂³	1	2	yes	—
34	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₇	C₂³	1	2	yes	—
37	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂×Dic₇	C₂³	1	2	yes	—
39	C₂×Dic₇	C₂³	1	2	yes	—
40	C₂×Dic₇	C₂³	1	2	yes	—
41	D₂₈	—	4	1	no	—
42	C₄×D₇	—	4	2	no	—
43	Dic₁₄	—	4	1	no	—
44	Dic₁₄	—	4	2	no	—
45	C₄⋊Q₈	—	7	2	no	—
46	#126	—	7	2	no	—
47	#126	—	7	2	no	—
48	#125	—	7	1	no	—
49	#125	D₇	1	1	yes	characteristic, complemented
50	#124	—	7	1	no	—
51	#123	—	7	2	no	—
52	C₂²⋊Q₈	—	7	2	no	—
53	C₂²⋊Q₈	—	7	2	no	—
54	C₂²×C₇	C₂⁴	1	1	yes	characteristic
55	C₂×D₇	—	2	2	no	—
56	C₂×D₇	—	2	2	no	—
57	C₂×D₇	—	2	2	no	—
58	C₄×C₇	—	2	2	no	—
59	C₄×C₇	—	2	1	no	—
60	C₄×C₇	—	2	1	no	—
61	C₄×C₇	—	2	2	no	—
62	C₄×C₇	—	2	1	no	—
63	Dic₇	—	2	2	no	—
64	Dic₇	—	2	2	no	—
65	Dic₇	—	2	2	no	—
66	C₂×Q₈	—	7	2	no	—
67	C₂×Q₈	—	7	1	no	—
68	C₂×D₄	—	7	1	no	—
69	C₂²×C₄	—	7	2	no	—
70	C₄⋊C₄	—	7	2	no	—
71	C₄⋊C₄	—	7	2	no	—
72	C₄⋊C₄	—	7	2	no	—
73	C₄⋊C₄	—	7	1	no	—
74	C₄⋊C₄	—	7	2	no	—
75	C₄⋊C₄	—	7	1	no	—
76	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₇	1	2	yes	—
78	C₄⋊C₄	C₂×D₇	1	2	yes	—
79	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
80	C₂²⋊C₄	—	7	2	no	—
81	C₂²⋊C₄	—	7	2	no	—
82	C₂²⋊C₄	—	7	2	no	—
83	C₂²⋊C₄	—	7	2	no	—
84	C₂²⋊C₄	—	7	2	no	—
85	C₄²	—	7	2	no	—
86	C₄²	C₂×D₇	1	1	yes	characteristic
87	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
89	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
90	D₇	—	4	2	no	—
91	C₂³	—	7	2	no	—
92	Q₈	—	28	2	no	—
93	Q₈	—	28	1	no	—
94	D₄	—	28	1	no	—
95	C₂×C₄	—	7	2	no	—
96	C₂×C₄	—	7	2	no	—
97	C₂×C₄	—	7	2	no	—
98	C₂×C₄	—	28	2	no	—
99	C₂×C₄	C₂²×D₇	1	2	yes	—
100	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₇	1	2	yes	—
104	C₇	#563	1	1	yes	characteristic, complemented, Sylow
105	C₂²	—	14	2	no	—
106	C₂²	—	14	2	no	—
107	C₂²	—	14	2	no	—
108	C₂²	C₂³×D₇	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	2	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	14	2	no	—
115	C₄	—	14	2	no	—
116	C₄	—	14	2	no	—
117	C₂	—	28	2	no	—
118	C₂	#6551	1	1	yes	characteristic
119	C₂	#6553	1	1	yes	characteristic
120	C₂	#6554	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#125

C₂

yes

maximal, characteristic, complemented

#6461

C₂

yes

maximal, complemented

#6461

C₂

yes

maximal, complemented

#6460

C₂

yes

maximal, complemented

#6459

C₂

yes

maximal, complemented

#6457

C₂

yes

maximal

#6451

C₂

yes

maximal, complemented

#6438

C₂

yes

maximal, characteristic

#6433

C₂

yes

maximal, characteristic, complemented

C₂×D₂₈

C₂²

yes

characteristic

C₂×C₄×D₇

C₂²

yes

—

C₂×Dic₁₄

C₂²

yes

characteristic

C₂×Dic₁₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₄⋊C₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₄²×C₇

C₂²

yes

characteristic

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

—

#563

—

maximal, Sylow

C₂²×D₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

D₂₈

—

C₄×D₇

—

Dic₁₄

—

Dic₁₄

—

C₄⋊Q₈

—

—

—

—

D₇

yes

characteristic, complemented

#124

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

C₂×D₇

yes

—

C₄⋊C₄

C₂×D₇

yes

—

C₄⋊C₄

C₂×D₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₇

yes

characteristic

C₂×C₇

H(5,𝔽₂)

yes

characteristic

C₂×C₇

2₋¹⁺⁴

yes

characteristic

C₂×C₇

2₋¹⁺⁴

yes

characteristic

D₇

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₇

yes

—

100

C₂×C₄

C₂²×D₇

yes

characteristic

101

C₂×C₄

C₂²×D₇

yes

characteristic

102

C₂×C₄

C₂²×D₇

yes

characteristic

103

C₂×C₄

C₂²×D₇

yes

—

104

C₇

#563

yes

characteristic, complemented, Sylow

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂³×D₇

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₂

—

118

C₂

#6551

yes

characteristic

119

C₂

#6553

yes

characteristic

120

C₂

#6554

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#125	C₂
prime radical	C₂×D₂₈	C₂²
hypercenter	#125	D₇
hypocenter	C₇	#563
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

Fitting

C₇×#125

C₂

prime radical

C₂×D₂₈

C₂²

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. C₇×#125	C₂	semidirect
2. #6461	C₂	semidirect
3. #6461	C₂	semidirect
4. #6460	C₂	semidirect
5. #6459	C₂	semidirect
7. #6451	C₂	semidirect
9. #6433	C₂	semidirect
49. #125	D₇	semidirect
104. C₇	#563	semidirect

H ≅ G/N

type

1. C₇×#125

C₂

semidirect

2. #6461

C₂

semidirect

3. #6461

C₂

semidirect

4. #6460

C₂

semidirect

5. #6459

C₂

semidirect

7. #6451

C₂

semidirect

9. #6433

C₂

semidirect

49. #125

D₇

semidirect

104. C₇

#563

semidirect

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

subgroup	index
C₇×#125	2
#6461	2
#6460	2
#6459	2
#6457	2
#6451	2
#6438	2
#6433	2
#563	7

subgroup

index

Subgroups — #90390

Conjugacy classes of subgroups

Complements

Maximal subgroups