Group #90221 — subgroups

Subgroups — #90221

Order 448.

238 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 97 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×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	#6493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#6458	C₂	1	1	yes	maximal, characteristic, complemented
5	#6456	C₂	1	1	yes	maximal, characteristic, complemented
6	D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#6453	C₂	1	1	yes	maximal, characteristic, complemented
8	#6447	C₂	1	2	yes	maximal, complemented
9	#6446	C₂	1	2	yes	maximal, complemented
10	#6439	C₂	1	2	yes	maximal, complemented
11	C₄×D₂₈	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
15	C₂×D₂₈	C₂²	1	2	yes	—
16	C₂×D₂₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₇	C₂²	1	2	yes	—
19	C₂×C₄×D₇	C₂²	1	2	yes	—
20	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₄²×C₇	C₂²	1	2	yes	—
24	C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
25	(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
28	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
29	C₄⋊Dic₇	C₂²	1	2	yes	—
30	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
31	Dic₇⋊C₄	C₂²	1	1	yes	characteristic
32	Dic₇⋊C₄	C₂²	1	2	yes	—
33	Dic₇⋊C₄	C₂²	1	2	yes	—
34	C₄×Dic₇	C₂²	1	1	yes	characteristic
35	#553	—	7	1	no	maximal, Sylow
36	C₂³×C₇	C₂³	1	1	yes	characteristic
37	C₂²×D₇	C₂³	1	2	yes	—
38	C₂²×D₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₇	—	2	2	no	—
40	C₂×C₄×C₇	C₂³	1	2	yes	—
41	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₇	C₂³	1	2	yes	—
43	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²⋊D₇	—	2	2	no	—
45	C₂²⋊D₇	—	4	2	no	—
46	C₂×Dic₇	C₂³	1	1	yes	characteristic
47	C₂×Dic₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₇	C₂³	1	1	yes	characteristic
49	C₂×Dic₇	C₂³	1	2	yes	—
50	D₂₈	—	2	4	no	—
51	D₂₈	—	2	2	no	—
52	C₄×D₇	—	2	2	no	—
53	C₄×D₇	—	4	2	no	—
54	C₄×D₇	—	4	2	no	—
55	#126	—	7	2	no	—
56	#125	—	7	1	no	—
57	#123	—	7	2	no	—
58	#121	—	7	1	no	—
59	#121	—	7	1	no	—
60	#121	—	7	2	no	—
61	C₄×D₄	—	7	1	no	—
62	C₄×D₄	—	7	1	no	—
63	C₄×D₄	—	7	2	no	—
64	#117	D₇	1	1	yes	characteristic, complemented
65	C₂×C₄⋊C₄	—	7	1	no	—
66	C₂²×C₇	—	2	1	no	—
67	C₂²×C₇	—	2	1	no	—
68	C₂²×C₇	—	2	1	no	—
69	C₂²×C₇	C₂⁴	1	1	yes	characteristic
70	C₂×D₇	—	2	1	no	—
71	C₂×D₇	—	2	1	no	—
72	C₂×D₇	—	2	2	no	—
73	C₂×D₇	—	2	2	no	—
74	C₂×D₇	—	2	2	no	—
75	C₂×D₇	#41	1	2	yes	—
76	C₄×C₇	—	2	2	no	—
77	C₄×C₇	—	2	1	no	—
78	C₄×C₇	—	2	2	no	—
79	C₄×C₇	#41	1	2	yes	—
80	Dic₇	—	2	1	no	—
81	Dic₇	—	2	1	no	—
82	Dic₇	—	2	1	no	—
83	Dic₇	—	2	2	no	—
84	C₂×D₄	—	7	1	no	—
85	C₂×D₄	—	7	2	no	—
86	C₂×D₄	—	7	2	no	—
87	C₂×D₄	—	7	1	no	—
88	C₂²×C₄	—	7	1	no	—
89	C₂²×C₄	—	7	2	no	—
90	C₂²×C₄	—	7	2	no	—
91	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
92	C₄⋊C₄	—	7	2	no	—
93	C₄⋊C₄	—	7	1	no	—
94	C₄⋊C₄	—	7	2	no	—
95	C₄⋊C₄	—	7	1	no	—
96	C₄⋊C₄	—	7	2	no	—
97	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
99	C₂²⋊C₄	—	7	1	no	—
100	C₂²⋊C₄	—	7	1	no	—
101	C₂²⋊C₄	—	7	2	no	—
102	C₂²⋊C₄	—	7	2	no	—
103	C₂²⋊C₄	—	7	2	no	—
104	C₂²⋊C₄	C₂×D₇	1	2	yes	—
105	C₄²	—	7	1	no	—
106	C₄²	C₂×D₇	1	2	yes	—
107	C₂×C₇	—	4	1	no	—
108	C₂×C₇	C₂×#41	1	1	yes	characteristic
109	C₂×C₇	C₂×#41	1	1	yes	characteristic
110	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
111	D₇	—	2	2	no	—
112	D₇	—	4	2	no	—
113	C₂³	—	7	1	no	—
114	C₂³	—	7	2	no	—
115	C₂³	C₂²×D₇	1	1	yes	characteristic
116	D₄	—	14	4	no	—
117	D₄	—	14	2	no	—
118	D₄	—	14	2	no	—
119	D₄	—	28	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	7	1	no	—
122	C₂×C₄	—	7	1	no	—
123	C₂×C₄	—	7	1	no	—
124	C₂×C₄	—	7	2	no	—
125	C₂×C₄	—	14	2	no	—
126	C₂×C₄	—	28	2	no	—
127	C₂×C₄	—	28	2	no	—
128	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
129	C₂×C₄	C₂²×D₇	1	2	yes	—
130	C₂×C₄	C₂²×D₇	1	2	yes	—
131	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
132	C₇	#553	1	1	yes	characteristic, complemented, Sylow
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	7	2	no	—
137	C₂²	—	14	1	no	—
138	C₂²	—	14	1	no	—
139	C₂²	—	14	2	no	—
140	C₂²	—	14	2	no	—
141	C₂²	—	14	2	no	—
142	C₂²	C₂³×D₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	14	1	no	—
147	C₄	—	14	1	no	—
148	C₄	—	14	1	no	—
149	C₄	—	14	2	no	—
150	C₄	#1157	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	—	14	2	no	—
153	C₂	—	28	2	no	—
154	C₂	C₂×#1157	1	1	yes	characteristic
155	C₂	D₇×#41	1	1	yes	characteristic
156	C₂	#6553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#117

C₂

yes

maximal, characteristic, complemented

#6493

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₇

C₂

yes

maximal, characteristic, complemented

#6458

C₂

yes

maximal, characteristic, complemented

#6456

C₂

yes

maximal, characteristic, complemented

D₇×C₄⋊C₄

C₂

yes

maximal, characteristic, complemented

#6453

C₂

yes

maximal, characteristic, complemented

#6447

C₂

yes

maximal, complemented

#6446

C₂

yes

maximal, complemented

#6439

C₂

yes

maximal, complemented

C₄×D₂₈

C₂

yes

maximal, complemented

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

—

C₂×D₂₈

C₂²

yes

—

C₂×D₂₈

C₂²

yes

characteristic

C₂×C₄×D₇

C₂²

yes

characteristic

C₂×C₄×D₇

C₂²

yes

—

C₂×C₄×D₇

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₂²⋊C₄

C₂²

yes

—

C₄²×C₇

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

characteristic

(C₂×D₇)⋊C₄

C₂²

yes

characteristic

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

characteristic

#553

—

maximal, Sylow

C₂³×C₇

C₂³

yes

characteristic

C₂²×D₇

C₂³

yes

—

C₂²×D₇

C₂³

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

—

C₂²⋊D₇

—

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

—

D₂₈

—

D₂₈

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

—

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

#117

D₇

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

#41

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#41

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

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

C₂×D₇

yes

characteristic

C₄⋊C₄

C₂×D₇

yes

characteristic

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

C₂×D₇

yes

—

105

C₄²

—

106

C₄²

C₂×D₇

yes

—

107

C₂×C₇

—

108

C₂×C₇

C₂×#41

yes

characteristic

109

C₂×C₇

C₂×#41

yes

characteristic

110

C₂×C₇

H(5,𝔽₂)

yes

characteristic

111

D₇

—

112

D₇

—

113

C₂³

—

114

C₂³

—

115

C₂³

C₂²×D₇

yes

characteristic

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂²×D₇

yes

characteristic

129

C₂×C₄

C₂²×D₇

yes

—

130

C₂×C₄

C₂²×D₇

yes

—

131

C₂×C₄

C₂²×D₇

yes

characteristic

132

C₇

#553

yes

characteristic, complemented, Sylow

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×D₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#1157

yes

—

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

C₂×#1157

yes

characteristic

155

C₂

D₇×#41

yes

characteristic

156

C₂

#6553

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#117	C₂
hypercenter	#117	D₇
hypocenter	C₇	#553
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₇×#117	C₂	semidirect
2. #6493	C₂	semidirect
3. C₄×C₂²⋊D₇	C₂	semidirect
4. #6458	C₂	semidirect
5. #6456	C₂	semidirect
6. D₇×C₄⋊C₄	C₂	semidirect
7. #6453	C₂	semidirect
8. #6447	C₂	semidirect
9. #6446	C₂	semidirect
10. #6439	C₂	semidirect
11. C₄×D₂₈	C₂	semidirect
64. #117	D₇	semidirect
132. C₇	#553	semidirect

H ≅ G/N

type

1. C₇×#117

C₂

semidirect

2. #6493

C₂

semidirect

3. C₄×C₂²⋊D₇

C₂

semidirect

4. #6458

C₂

semidirect

5. #6456

C₂

semidirect

6. D₇×C₄⋊C₄

C₂

semidirect

7. #6453

C₂

semidirect

8. #6447

C₂

semidirect

9. #6446

C₂

semidirect

10. #6439

C₂

semidirect

11. C₄×D₂₈

C₂

semidirect

64. #117

D₇

semidirect

132. C₇

#553

semidirect

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

subgroup	index
C₇×#117	2
#6493	2
C₄×C₂²⋊D₇	2
#6458	2
#6456	2
D₇×C₄⋊C₄	2
#6453	2
#6447	2
#6446	2
#6439	2
C₄×D₂₈	2
#553	7

subgroup

index

C₄×C₂²⋊D₇

D₇×C₄⋊C₄

C₄×D₂₈

Subgroups — #90221

Conjugacy classes of subgroups

Complements

Maximal subgroups