Group #90420 — subgroups

Subgroups — #90420

Order 448.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#6509	C₂	1	2	yes	maximal
3	#6507	C₂	1	2	yes	maximal, complemented
4	#6459	C₂	1	2	yes	maximal
5	#6455	C₂	1	1	yes	maximal, characteristic
6	D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	#6453	C₂	1	1	yes	maximal, characteristic, complemented
8	#6452	C₂	1	2	yes	maximal, complemented
9	#6451	C₂	1	1	yes	maximal, characteristic, complemented
10	#6435	C₂	1	1	yes	maximal, characteristic
11	#6433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×Q₈	C₂²	1	2	yes	—
13	C₂×C₄×D₇	C₂²	1	2	yes	—
14	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄	C₂²	1	2	yes	—
16	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄²×C₇	C₂²	1	1	yes	characteristic
20	(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
21	(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
22	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
23	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₇	C₂²	1	2	yes	—
26	Dic₇⋊C₄	C₂²	1	2	yes	—
27	Dic₇⋊C₄	C₂²	1	2	yes	—
28	Dic₇⋊C₄	C₂²	1	1	yes	characteristic
29	Dic₇⋊C₄	C₂²	1	1	yes	characteristic
30	Dic₇⋊C₄	C₂²	1	2	yes	—
31	Dic₇⋊C₄	C₂²	1	2	yes	—
32	Dic₇⋊C₄	C₂²	1	2	yes	—
33	C₄×Dic₇	C₂²	1	1	yes	characteristic
34	C₄×Dic₇	C₂²	1	2	yes	—
35	#544	—	7	1	no	maximal, Sylow
36	C₂²×D₇	C₂³	1	1	yes	characteristic
37	C₇×Q₈	—	4	2	no	—
38	C₂×C₄×C₇	C₂³	1	2	yes	—
39	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₇	C₂³	1	2	yes	—
42	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₇	C₂³	1	2	yes	—
45	C₂×Dic₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₇	C₂³	1	2	yes	—
47	C₂×Dic₇	C₂³	1	1	yes	characteristic
48	C₄×D₇	—	4	2	no	—
49	C₄×D₇	Q₈	1	4	yes	complemented
50	Dic₁₄	—	4	2	no	—
51	C₄⋊Q₈	—	7	2	no	—
52	C₄⋊Q₈	—	7	1	no	—
53	C₄⋊Q₈	D₇	1	1	yes	characteristic, complemented
54	#125	—	7	2	no	—
55	#125	—	7	1	no	—
56	#125	—	7	1	no	—
57	C₂²⋊Q₈	—	7	2	no	—
58	C₂²⋊Q₈	—	7	2	no	—
59	#117	—	7	1	no	—
60	#117	—	7	1	no	—
61	C₂×C₄⋊C₄	—	7	1	no	—
62	C₂²×C₇	C₂⁴	1	1	yes	characteristic
63	C₂×D₇	—	2	1	no	—
64	C₂×D₇	—	2	1	no	—
65	C₂×D₇	C₂×Q₈	1	2	yes	—
66	C₄×C₇	—	2	2	no	—
67	C₄×C₇	—	2	2	no	—
68	C₄×C₇	—	2	1	no	—
69	C₄×C₇	—	2	1	no	—
70	C₄×C₇	C₂×Q₈	1	2	yes	—
71	Dic₇	—	2	1	no	—
72	Dic₇	—	2	2	no	—
73	Dic₇	—	2	1	no	—
74	Dic₇	—	2	2	no	—
75	Dic₇	C₂×Q₈	1	2	yes	—
76	C₂×Q₈	—	7	2	no	—
77	C₂×Q₈	C₂×D₇	1	2	yes	—
78	C₂²×C₄	—	7	1	no	—
79	C₂²×C₄	—	7	2	no	—
80	C₄⋊C₄	—	7	2	no	—
81	C₄⋊C₄	—	7	1	no	—
82	C₄⋊C₄	—	7	1	no	—
83	C₄⋊C₄	—	7	2	no	—
84	C₄⋊C₄	—	7	2	no	—
85	C₄⋊C₄	—	7	2	no	—
86	C₄⋊C₄	—	7	2	no	—
87	C₄⋊C₄	—	7	1	no	—
88	C₄⋊C₄	—	7	1	no	—
89	C₄⋊C₄	—	7	2	no	—
90	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
91	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
92	C₄⋊C₄	C₂×D₇	1	2	yes	—
93	C₂²⋊C₄	—	7	2	no	—
94	C₂²⋊C₄	—	7	1	no	—
95	C₂²⋊C₄	—	7	1	no	—
96	C₄²	—	7	2	no	—
97	C₄²	—	7	1	no	—
98	C₄²	C₂×D₇	1	1	yes	characteristic
99	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
102	D₇	—	2	2	no	—
103	C₂³	—	7	1	no	—
104	Q₈	—	4	2	no	—
105	Q₈	—	28	2	no	—
106	C₂×C₄	—	7	1	no	—
107	C₂×C₄	—	7	4	no	—
108	C₂×C₄	—	7	1	no	—
109	C₂×C₄	—	7	2	no	—
110	C₂×C₄	—	7	1	no	—
111	C₂×C₄	—	7	2	no	—
112	C₂×C₄	—	28	2	no	—
113	C₂×C₄	C₂²×D₇	1	2	yes	—
114	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₇	1	2	yes	—
117	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
118	C₇	#544	1	1	yes	characteristic, complemented, Sylow
119	C₂²	—	7	2	no	—
120	C₂²	—	14	1	no	—
121	C₂²	—	14	1	no	—
122	C₂²	C₂³×D₇	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	7	2	no	—
128	C₄	—	14	2	no	—
129	C₄	—	14	1	no	—
130	C₄	—	14	1	no	—
131	C₄	—	14	2	no	—
132	C₄	Q₈×D₇	1	2	yes	—
133	C₂	—	14	2	no	—
134	C₂	#6548	1	1	yes	characteristic
135	C₂	C₂×Q₈×D₇	1	1	yes	characteristic
136	C₂	#6551	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

#6509

C₂

yes

maximal

#6507

C₂

yes

maximal, complemented

#6459

C₂

yes

maximal

#6455

C₂

yes

maximal, characteristic

D₇×C₄⋊C₄

C₂

yes

maximal, characteristic

#6453

C₂

yes

maximal, characteristic, complemented

#6452

C₂

yes

maximal, complemented

#6451

C₂

yes

maximal, characteristic, complemented

#6435

C₂

yes

maximal, characteristic

#6433

C₂

yes

maximal, characteristic, complemented

C₂×C₇×Q₈

C₂²

yes

—

C₂×C₄×D₇

C₂²

yes

—

C₂×C₄×D₇

C₂²

yes

characteristic

C₂×Dic₁₄

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

characteristic

(C₂×D₇)⋊C₄

C₂²

yes

characteristic

(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

characteristic

Dic₇⋊C₄

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

characteristic

C₄×Dic₇

C₂²

yes

—

#544

—

maximal, Sylow

C₂²×D₇

C₂³

yes

characteristic

C₇×Q₈

—

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

characteristic

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

characteristic

C₄×D₇

—

C₄×D₇

Q₈

yes

complemented

Dic₁₄

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₇

yes

characteristic, complemented

#125

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂×Q₈

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×Q₈

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

C₂×Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×D₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

C₂×D₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₇

yes

characteristic

C₂×C₇

C₂²×Q₈

yes

characteristic

100

C₂×C₇

H(5,𝔽₂)

yes

characteristic

101

C₂×C₇

2₋¹⁺⁴

yes

characteristic

102

D₇

—

103

C₂³

—

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×D₇

yes

—

114

C₂×C₄

C₂²×D₇

yes

characteristic

115

C₂×C₄

C₂²×D₇

yes

characteristic

116

C₂×C₄

C₂²×D₇

yes

—

117

C₂×C₄

C₂²×D₇

yes

characteristic

118

C₇

#544

yes

characteristic, complemented, Sylow

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂³×D₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

Q₈×D₇

yes

—

133

C₂

—

134

C₂

#6548

yes

characteristic

135

C₂

C₂×Q₈×D₇

yes

characteristic

136

C₂

#6551

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×C₄⋊Q₈	C₂
prime radical	C₂²×D₇	C₂³
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	#544
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

Fitting

C₇×C₄⋊Q₈

C₂

prime radical

C₂²×D₇

C₂³

hypercenter

C₄⋊Q₈

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₇×C₄⋊Q₈	C₂	semidirect
3. #6507	C₂	semidirect
7. #6453	C₂	semidirect
8. #6452	C₂	semidirect
9. #6451	C₂	semidirect
11. #6433	C₂	semidirect
49. C₄×D₇	Q₈	semidirect
53. C₄⋊Q₈	D₇	semidirect
118. C₇	#544	semidirect

H ≅ G/N

type

1. C₇×C₄⋊Q₈

C₂

semidirect

3. #6507

C₂

semidirect

7. #6453

C₂

semidirect

8. #6452

C₂

semidirect

9. #6451

C₂

semidirect

11. #6433

C₂

semidirect

49. C₄×D₇

Q₈

semidirect

53. C₄⋊Q₈

D₇

semidirect

118. C₇

#544

semidirect

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

subgroup	index
C₇×C₄⋊Q₈	2
#6509	2
#6507	2
#6459	2
#6455	2
D₇×C₄⋊C₄	2
#6453	2
#6452	2
#6451	2
#6435	2
#6433	2
#544	7

subgroup

index

C₇×C₄⋊Q₈

D₇×C₄⋊C₄

Subgroups — #90420

Conjugacy classes of subgroups

Complements

Maximal subgroups