Group #90326 — subgroups

Subgroups — #90326

Order 448.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#6509	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#6460	C₂	1	2	yes	maximal, complemented
7	D₇×C₄⋊C₄	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	(C₂²⋊D₇)⋊C₄	C₂	1	2	yes	maximal
11	#6441	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₇	C₂²	1	2	yes	—
16	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₇	C₂²	1	2	yes	—
18	C₂×Dic₁₄	C₂²	1	2	yes	—
19	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
23	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₇	C₂²	1	2	yes	—
26	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₇	C₂²	1	2	yes	—
28	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
29	Dic₇⋊C₄	C₂²	1	2	yes	—
30	Dic₇⋊C₄	C₂²	1	2	yes	—
31	Dic₇⋊C₄	C₂²	1	2	yes	—
32	Dic₇⋊C₄	C₂²	1	1	yes	characteristic
33	C₄×Dic₇	C₂²	1	2	yes	—
34	C₄×Dic₇	C₂²	1	1	yes	characteristic
35	#554	—	7	1	no	maximal, Sylow
36	C₂³×C₇	C₂³	1	1	yes	characteristic
37	C₂²×D₇	C₂³	1	1	yes	characteristic
38	C₇×Q₈	—	2	2	no	—
39	C₂×C₄×C₇	—	2	2	no	—
40	C₂×C₄×C₇	C₂³	1	2	yes	—
41	C₂×C₄×C₇	C₂³	1	2	yes	—
42	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²⋊D₇	Q₈	1	4	yes	complemented
45	C₂×Dic₇	—	4	2	no	—
46	C₂×Dic₇	C₂³	1	2	yes	—
47	C₂×Dic₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₇	C₂³	1	1	yes	characteristic
49	C₂×Dic₇	C₂³	1	2	yes	—
50	C₂×Dic₇	C₂³	1	1	yes	characteristic
51	C₄×D₇	—	2	2	no	—
52	C₄×D₇	—	4	2	no	—
53	Dic₁₄	—	4	2	no	—
54	C₄⋊Q₈	—	7	1	no	—
55	#125	—	7	2	no	—
56	C₂²⋊Q₈	—	7	1	no	—
57	C₂²⋊Q₈	—	7	2	no	—
58	C₂²⋊Q₈	—	7	2	no	—
59	C₂²⋊Q₈	D₇	1	1	yes	characteristic, complemented
60	C₄×Q₈	—	7	1	no	—
61	C₄×D₄	—	7	1	no	—
62	C₄×D₄	—	7	2	no	—
63	C₂×C₄⋊C₄	—	7	1	no	—
64	C₂×C₄⋊C₄	—	7	1	no	—
65	C₂²×C₇	—	2	1	no	—
66	C₂²×C₇	—	2	1	no	—
67	C₂²×C₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₇	C₂⁴	1	1	yes	characteristic
69	C₂×D₇	—	2	1	no	—
70	C₂×D₇	—	2	1	no	—
71	C₂×D₇	C₂×Q₈	1	2	yes	—
72	C₄×C₇	—	2	1	no	—
73	C₄×C₇	—	2	2	no	—
74	C₄×C₇	—	2	2	no	—
75	C₄×C₇	#41	1	2	yes	—
76	Dic₇	—	2	1	no	—
77	Dic₇	—	2	1	no	—
78	Dic₇	—	2	2	no	—
79	Dic₇	—	2	2	no	—
80	Dic₇	C₂×Q₈	1	2	yes	—
81	C₂×Q₈	—	7	2	no	—
82	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
83	C₂×D₄	—	7	1	no	—
84	C₂²×C₄	—	7	2	no	—
85	C₂²×C₄	—	7	1	no	—
86	C₂²×C₄	—	7	2	no	—
87	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
88	C₄⋊C₄	—	7	2	no	—
89	C₄⋊C₄	—	7	1	no	—
90	C₄⋊C₄	—	7	2	no	—
91	C₄⋊C₄	—	7	2	no	—
92	C₄⋊C₄	—	7	1	no	—
93	C₄⋊C₄	—	7	2	no	—
94	C₄⋊C₄	—	7	1	no	—
95	C₄⋊C₄	—	7	2	no	—
96	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₇	1	2	yes	—
98	C₂²⋊C₄	—	7	1	no	—
99	C₂²⋊C₄	—	7	1	no	—
100	C₂²⋊C₄	—	7	2	no	—
101	C₂²⋊C₄	C₂×D₇	1	2	yes	—
102	C₄²	—	7	2	no	—
103	C₄²	—	7	1	no	—
104	C₂×C₇	—	2	2	no	—
105	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
108	D₇	—	2	2	no	—
109	C₂³	—	7	1	no	—
110	C₂³	C₂²×D₇	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	Q₈	—	28	2	no	—
113	D₄	—	7	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	7	1	no	—
116	C₂×C₄	—	7	1	no	—
117	C₂×C₄	—	7	2	no	—
118	C₂×C₄	—	7	1	no	—
119	C₂×C₄	—	7	2	no	—
120	C₂×C₄	—	14	2	no	—
121	C₂×C₄	—	28	2	no	—
122	C₂×C₄	—	28	2	no	—
123	C₂×C₄	C₂²×D₇	1	2	yes	—
124	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₇	1	2	yes	—
126	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
127	C₇	#554	1	1	yes	characteristic, complemented, Sylow
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	7	2	no	—
131	C₂²	—	14	1	no	—
132	C₂²	—	14	1	no	—
133	C₂²	Q₈×D₇	1	2	yes	—
134	C₂²	C₂³×D₇	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	7	2	no	—
139	C₄	—	14	1	no	—
140	C₄	—	14	2	no	—
141	C₄	—	14	2	no	—
142	C₄	—	14	1	no	—
143	C₄	#1159	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	14	2	no	—
146	C₂	C₂×#1159	1	1	yes	characteristic
147	C₂	C₂×Q₈×D₇	1	1	yes	characteristic
148	C₂	#6553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×C₂²⋊Q₈

C₂

yes

maximal, characteristic, complemented

#6509

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₇

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇

C₂

yes

maximal, characteristic, complemented

#6460

C₂

yes

maximal, complemented

D₇×C₄⋊C₄

C₂

yes

maximal, characteristic, complemented

#6452

C₂

yes

maximal, complemented

#6451

C₂

yes

maximal, characteristic, complemented

(C₂²⋊D₇)⋊C₄

C₂

yes

maximal

#6441

C₂

yes

maximal, complemented

C₂×C₇×Q₈

C₂²

yes

characteristic

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

characteristic

C₂²×Dic₇

C₂²

yes

—

C₂×C₄×D₇

C₂²

yes

characteristic

C₂×C₄×D₇

C₂²

yes

—

C₂×Dic₁₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₄⋊C₄

C₂²

yes

—

C₇×C₂²⋊C₄

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

characteristic

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

characteristic

C₄×Dic₇

C₂²

yes

—

C₄×Dic₇

C₂²

yes

characteristic

#554

—

maximal, Sylow

C₂³×C₇

C₂³

yes

characteristic

C₂²×D₇

C₂³

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

Q₈

yes

complemented

C₂×Dic₇

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

characteristic

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₇

yes

characteristic, complemented

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×Q₈

yes

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂×Q₈

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#41

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

C₂×Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×D₇

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₇

yes

characteristic

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

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

C₂×D₇

yes

—

102

C₄²

—

103

C₄²

—

104

C₂×C₇

—

105

C₂×C₇

C₂²×Q₈

yes

characteristic

106

C₂×C₇

C₂×#41

yes

characteristic

107

C₂×C₇

H(5,𝔽₂)

yes

characteristic

108

D₇

—

109

C₂³

—

110

C₂³

C₂²×D₇

yes

characteristic

111

Q₈

—

112

Q₈

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂²×D₇

yes

—

124

C₂×C₄

C₂²×D₇

yes

characteristic

125

C₂×C₄

C₂²×D₇

yes

—

126

C₂×C₄

C₂²×D₇

yes

characteristic

127

C₇

#554

yes

characteristic, complemented, Sylow

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

Q₈×D₇

yes

—

134

C₂²

C₂³×D₇

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#1159

yes

—

144

C₂

—

145

C₂

—

146

C₂

C₂×#1159

yes

characteristic

147

C₂

C₂×Q₈×D₇

yes

characteristic

148

C₂

#6553

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×C₂²⋊Q₈	C₂
prime radical	C₂×C₂²⋊D₇	C₂²
hypercenter	C₂²⋊Q₈	D₇
hypocenter	C₇	#554
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

Fitting

C₇×C₂²⋊Q₈

C₂

prime radical

C₂×C₂²⋊D₇

C₂²

hypercenter

C₂²⋊Q₈

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. C₇×C₂²⋊Q₈	C₂	semidirect
2. #6509	C₂	semidirect
3. Q₈×Dic₇	C₂	semidirect
5. C₂×C₄⋊Dic₇	C₂	semidirect
6. #6460	C₂	semidirect
7. D₇×C₄⋊C₄	C₂	semidirect
8. #6452	C₂	semidirect
9. #6451	C₂	semidirect
11. #6441	C₂	semidirect
44. C₂²⋊D₇	Q₈	semidirect
59. C₂²⋊Q₈	D₇	semidirect
127. C₇	#554	semidirect

H ≅ G/N

type

1. C₇×C₂²⋊Q₈

C₂

semidirect

2. #6509

C₂

semidirect

3. Q₈×Dic₇

C₂

semidirect

5. C₂×C₄⋊Dic₇

C₂

semidirect

6. #6460

C₂

semidirect

7. D₇×C₄⋊C₄

C₂

semidirect

8. #6452

C₂

semidirect

9. #6451

C₂

semidirect

11. #6441

C₂

semidirect

44. C₂²⋊D₇

Q₈

semidirect

59. C₂²⋊Q₈

D₇

semidirect

127. C₇

#554

semidirect

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

subgroup	index
C₇×C₂²⋊Q₈	2
#6509	2
Q₈×Dic₇	2
C₄×C₂²⋊D₇	2
C₂×C₄⋊Dic₇	2
#6460	2
D₇×C₄⋊C₄	2
#6452	2
#6451	2
(C₂²⋊D₇)⋊C₄	2
#6441	2
#554	7

subgroup

index

C₇×C₂²⋊Q₈

#6509

Q₈×Dic₇

C₄×C₂²⋊D₇

C₂×C₄⋊Dic₇

D₇×C₄⋊C₄

(C₂²⋊D₇)⋊C₄

Subgroups — #90326

Conjugacy classes of subgroups

Complements

Maximal subgroups