Group #90294 — subgroups

Subgroups — #90294

Order 448.

216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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×#121	C₂	1	1	yes	maximal, characteristic
2	#6499	C₂	1	2	yes	maximal, complemented
3	#6499	C₂	1	1	yes	maximal, characteristic, complemented
4	#6498	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#6489	C₂	1	1	yes	maximal, characteristic, complemented
7	#6487	C₂	1	1	yes	maximal, characteristic, complemented
8	#6451	C₂	1	1	yes	maximal, characteristic, complemented
9	#6450	C₂	1	1	yes	maximal, characteristic, complemented
10	#6442	C₂	1	2	yes	maximal, complemented
11	#6441	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₇×D₄	C₂²	1	2	yes	—
14	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₇	C₂²	1	2	yes	—
16	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄	C₂²	1	2	yes	—
18	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂²⋊Dic₇	C₂²	1	2	yes	—
21	C₂²⋊Dic₇	C₂²	1	2	yes	—
22	C₂²⋊Dic₇	C₂²	1	2	yes	—
23	C₂²⋊Dic₇	C₂²	1	2	yes	—
24	C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
25	C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
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	C₄×Dic₇	C₂²	1	2	yes	—
33	C₄×Dic₇	C₂²	1	1	yes	characteristic
34	C₄×Dic₇	C₂²	1	1	yes	characteristic
35	#542	—	7	1	no	maximal, Sylow
36	C₂³×C₇	C₂³	1	1	yes	characteristic
37	C₂³×C₇	C₂³	1	2	yes	—
38	C₇×D₄	—	2	2	no	—
39	C₇×D₄	—	4	2	no	—
40	C₂×C₄×C₇	—	2	2	no	—
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	C₂×Dic₇	—	4	2	no	—
45	C₂×Dic₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₇	C₂³	1	1	yes	characteristic
47	C₂×Dic₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₇	C₂³	1	2	yes	—
49	C₂×Dic₇	C₂³	1	1	yes	characteristic
50	C₂×Dic₇	C₂³	1	2	yes	—
51	Dic₁₄	—	2	2	no	—
52	Dic₁₄	—	4	2	no	—
53	C₄⋊Q₈	—	7	1	no	—
54	#126	—	7	2	no	—
55	#124	—	7	1	no	—
56	#124	—	7	2	no	—
57	#123	—	7	2	no	—
58	C₂²⋊Q₈	—	7	1	no	—
59	C₂²⋊Q₈	—	7	2	no	—
60	#121	D₇	1	1	yes	characteristic
61	C₄×Q₈	—	7	1	no	—
62	C₄×D₄	—	7	1	no	—
63	#117	—	7	1	no	—
64	C₂²×C₇	—	2	1	no	—
65	C₂²×C₇	—	2	2	no	—
66	C₂²×C₇	—	2	2	no	—
67	C₂²×C₇	—	2	1	no	—
68	C₂²×C₇	—	2	1	no	—
69	C₂²×C₇	—	2	2	no	—
70	C₂²×C₇	C₂⁴	1	1	yes	characteristic
71	C₄×C₇	—	2	1	no	—
72	C₄×C₇	—	2	2	no	—
73	C₄×C₇	#41	1	2	yes	—
74	Dic₇	—	2	2	no	—
75	Dic₇	—	2	1	no	—
76	Dic₇	—	2	2	no	—
77	Dic₇	—	2	1	no	—
78	Dic₇	—	2	1	no	—
79	Dic₇	—	2	1	no	—
80	C₂×Q₈	—	7	1	no	—
81	C₂×Q₈	—	7	2	no	—
82	C₂×D₄	C₂×D₇	1	1	yes	characteristic
83	C₂×D₄	C₂×D₇	1	2	yes	—
84	C₂²×C₄	—	7	2	no	—
85	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
86	C₄⋊C₄	—	7	2	no	—
87	C₄⋊C₄	—	7	2	no	—
88	C₄⋊C₄	—	7	2	no	—
89	C₄⋊C₄	—	7	1	no	—
90	C₄⋊C₄	—	7	1	no	—
91	C₄⋊C₄	—	7	1	no	—
92	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
93	C₂²⋊C₄	—	7	1	no	—
94	C₂²⋊C₄	—	7	1	no	—
95	C₂²⋊C₄	—	7	2	no	—
96	C₂²⋊C₄	—	7	2	no	—
97	C₂²⋊C₄	—	7	2	no	—
98	C₂²⋊C₄	—	7	2	no	—
99	C₂²⋊C₄	C₂×D₇	1	2	yes	—
100	C₄²	—	7	1	no	—
101	C₄²	—	7	1	no	—
102	C₄²	—	7	2	no	—
103	C₂×C₇	—	4	1	no	—
104	C₂×C₇	—	4	2	no	—
105	C₂×C₇	C₂×#41	1	1	yes	characteristic
106	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂³	C₂²×D₇	1	1	yes	characteristic
109	C₂³	C₂²×D₇	1	2	yes	—
110	Q₈	—	14	2	no	—
111	Q₈	—	28	2	no	—
112	D₄	—	2	2	no	—
113	D₄	—	4	2	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	1	no	—
120	C₂×C₄	—	7	2	no	—
121	C₂×C₄	—	28	2	no	—
122	C₂×C₄	C₂²×D₇	1	2	yes	—
123	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
125	C₇	#542	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	2	1	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	2	no	—
132	C₂²	C₂³×D₇	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	2	no	—
135	C₄	—	14	2	no	—
136	C₄	—	14	1	no	—
137	C₄	—	14	2	no	—
138	C₄	—	14	1	no	—
139	C₄	—	14	1	no	—
140	C₄	—	14	1	no	—
141	C₄	#1159	1	2	yes	—
142	C₂	—	4	1	no	—
143	C₂	—	4	2	no	—
144	C₂	C₂×#1159	1	1	yes	characteristic
145	C₂	#6548	1	1	yes	characteristic
146	C₂	#6554	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#121

C₂

yes

maximal, characteristic

#6499

C₂

yes

maximal, complemented

#6499

C₂

yes

maximal, characteristic, complemented

#6498

C₂

yes

maximal, complemented

D₄×Dic₇

C₂

yes

maximal, characteristic, complemented

#6489

C₂

yes

maximal, characteristic, complemented

#6487

C₂

yes

maximal, characteristic, complemented

#6451

C₂

yes

maximal, characteristic, complemented

#6450

C₂

yes

maximal, characteristic, complemented

#6442

C₂

yes

maximal, complemented

#6441

C₂

yes

maximal, complemented

C₂×C₇×D₄

C₂²

yes

characteristic

C₂×C₇×D₄

C₂²

yes

—

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂²×Dic₇

C₂²

yes

—

C₂×Dic₁₄

C₂²

yes

characteristic

C₂×Dic₁₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₂²⋊C₄

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

—

C₂²⋊Dic₇

C₂²

yes

characteristic

C₂²⋊Dic₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

—

C₄×Dic₇

C₂²

yes

characteristic

C₄×Dic₇

C₂²

yes

characteristic

#542

—

maximal, Sylow

C₂³×C₇

C₂³

yes

characteristic

C₂³×C₇

C₂³

yes

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

—

Dic₁₄

—

Dic₁₄

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

D₇

yes

characteristic

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂⁴

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#41

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₇

yes

characteristic

C₂×D₄

C₂×D₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₇

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₂×C₇

—

104

C₂×C₇

—

105

C₂×C₇

C₂×#41

yes

characteristic

106

C₂×C₇

H(5,𝔽₂)

yes

characteristic

107

C₂×C₇

2₋¹⁺⁴

yes

characteristic

108

C₂³

C₂²×D₇

yes

characteristic

109

C₂³

C₂²×D₇

yes

—

110

Q₈

—

111

Q₈

—

112

D₄

—

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

C₂²×D₇

yes

—

123

C₂×C₄

C₂²×D₇

yes

characteristic

124

C₂×C₄

C₂²×D₇

yes

characteristic

125

C₇

#542

yes

characteristic, complemented, Sylow

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×D₇

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#1159

yes

—

142

C₂

—

143

C₂

—

144

C₂

C₂×#1159

yes

characteristic

145

C₂

#6548

yes

characteristic

146

C₂

#6554

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting, prime radical	C₇×#121	C₂
hypercenter	#121	D₇
hypocenter	C₇	#542
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

Fitting, prime radical

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
2. #6499	C₂	semidirect
3. #6499	C₂	semidirect
4. #6498	C₂	semidirect
5. D₄×Dic₇	C₂	semidirect
6. #6489	C₂	semidirect
7. #6487	C₂	semidirect
8. #6451	C₂	semidirect
9. #6450	C₂	semidirect
10. #6442	C₂	semidirect
11. #6441	C₂	semidirect
125. C₇	#542	semidirect

H ≅ G/N

type

2. #6499

C₂

semidirect

3. #6499

C₂

semidirect

4. #6498

C₂

semidirect

5. D₄×Dic₇

C₂

semidirect

6. #6489

C₂

semidirect

7. #6487

C₂

semidirect

8. #6451

C₂

semidirect

9. #6450

C₂

semidirect

10. #6442

C₂

semidirect

11. #6441

C₂

semidirect

125. C₇

#542

semidirect

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

subgroup	index
C₇×#121	2
#6499	2
#6498	2
D₄×Dic₇	2
#6489	2
#6487	2
#6451	2
#6450	2
#6442	2
#6441	2
#542	7

subgroup

index

D₄×Dic₇

Subgroups — #90294

Conjugacy classes of subgroups

Complements

Maximal subgroups