Group #90225 — subgroups

Subgroups — #90225

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×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	#6493	C₂	1	1	yes	maximal, characteristic
3	#6487	C₂	1	1	yes	maximal, characteristic, complemented
4	#6461	C₂	1	2	yes	maximal, complemented
5	#6459	C₂	1	2	yes	maximal, complemented
6	#6448	C₂	1	2	yes	maximal, complemented
7	#6446	C₂	1	2	yes	maximal, complemented
8	#6438	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#6432	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₇	C₂²	1	2	yes	—
14	C₂×D₂₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₇	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×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₄²×C₇	C₂²	1	1	yes	characteristic
23	C₄²×C₇	C₂²	1	1	yes	characteristic
24	C₂²⋊Dic₇	C₂²	1	2	yes	—
25	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
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	1	yes	characteristic
30	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
31	C₄⋊Dic₇	C₂²	1	2	yes	—
32	Dic₇⋊C₄	C₂²	1	2	yes	—
33	Dic₇⋊C₄	C₂²	1	2	yes	—
34	C₄×Dic₇	C₂²	1	2	yes	—
35	#542	—	7	1	no	maximal, Sylow
36	C₂³×C₇	C₂³	1	1	yes	characteristic
37	C₂²×D₇	C₂³	1	2	yes	—
38	C₂×C₄×C₇	—	2	2	no	—
39	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂²⋊D₇	—	4	2	no	—
46	C₂×Dic₇	C₂³	1	2	yes	—
47	C₂×Dic₇	C₂³	1	2	yes	—
48	C₂×Dic₇	C₂³	1	2	yes	—
49	D₂₈	—	2	2	no	—
50	C₄×D₇	—	4	2	no	—
51	Dic₁₄	—	2	2	no	—
52	Dic₁₄	—	4	2	no	—
53	C₄⋊Q₈	—	7	1	no	—
54	#126	—	7	2	no	—
55	#124	—	7	2	no	—
56	#124	—	7	1	no	—
57	#123	—	7	2	no	—
58	C₂²⋊Q₈	—	7	1	no	—
59	C₂²⋊Q₈	—	7	2	no	—
60	#121	—	7	1	no	—
61	C₄×Q₈	—	7	1	no	—
62	C₄×D₄	—	7	1	no	—
63	#117	D₇	1	1	yes	characteristic, complemented
64	C₂²×C₇	—	2	1	no	—
65	C₂²×C₇	—	2	1	no	—
66	C₂²×C₇	—	2	1	no	—
67	C₂²×C₇	C₂⁴	1	1	yes	characteristic
68	C₂×D₇	—	2	2	no	—
69	C₂×D₇	—	2	2	no	—
70	C₂×D₇	—	2	2	no	—
71	C₄×C₇	—	2	1	no	—
72	C₄×C₇	—	2	1	no	—
73	C₄×C₇	—	2	1	no	—
74	C₄×C₇	—	2	1	no	—
75	C₄×C₇	—	2	1	no	—
76	C₄×C₇	#41	1	2	yes	—
77	Dic₇	—	2	2	no	—
78	Dic₇	—	2	2	no	—
79	Dic₇	—	2	2	no	—
80	C₂×Q₈	—	7	1	no	—
81	C₂×Q₈	—	7	2	no	—
82	C₂×D₄	—	7	2	no	—
83	C₂×D₄	—	7	1	no	—
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	1	no	—
89	C₄⋊C₄	—	7	1	no	—
90	C₄⋊C₄	—	7	2	no	—
91	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
92	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
93	C₂²⋊C₄	—	7	2	no	—
94	C₂²⋊C₄	—	7	2	no	—
95	C₂²⋊C₄	—	7	2	no	—
96	C₂²⋊C₄	—	7	2	no	—
97	C₂²⋊C₄	—	7	2	no	—
98	C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
99	C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
100	C₄²	—	7	2	no	—
101	C₄²	C₂×D₇	1	1	yes	characteristic
102	C₄²	C₂×D₇	1	1	yes	characteristic
103	C₂×C₇	—	4	1	no	—
104	C₂×C₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
107	D₇	—	4	2	no	—
108	C₂³	—	7	2	no	—
109	C₂³	C₂²×D₇	1	1	yes	characteristic
110	Q₈	—	14	2	no	—
111	Q₈	—	28	2	no	—
112	D₄	—	14	2	no	—
113	D₄	—	28	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	7	2	no	—
116	C₂×C₄	—	7	2	no	—
117	C₂×C₄	—	7	2	no	—
118	C₂×C₄	—	28	2	no	—
119	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
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	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	14	2	no	—
130	C₂²	—	14	2	no	—
131	C₂²	—	14	2	no	—
132	C₂²	C₂³×D₇	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	14	2	no	—
139	C₄	—	14	2	no	—
140	C₄	—	14	2	no	—
141	C₄	#1157	1	2	yes	—
142	C₂	—	4	1	no	—
143	C₂	—	28	2	no	—
144	C₂	C₂×#1157	1	1	yes	characteristic
145	C₂	#6553	1	1	yes	characteristic
146	C₂	#6554	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#117

C₂

yes

maximal, characteristic, complemented

#6493

C₂

yes

maximal, characteristic

#6487

C₂

yes

maximal, characteristic, complemented

#6461

C₂

yes

maximal, complemented

#6459

C₂

yes

maximal, complemented

#6448

C₂

yes

maximal, complemented

#6446

C₂

yes

maximal, complemented

#6438

C₂

yes

maximal, characteristic, complemented

C₄×D₂₈

C₂

yes

maximal, characteristic, complemented

#6432

C₂

yes

maximal, characteristic, complemented

C₄×Dic₁₄

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂×C₂²⋊D₇

C₂²

yes

—

C₂×D₂₈

C₂²

yes

characteristic

C₂×C₄×D₇

C₂²

yes

—

C₂×Dic₁₄

C₂²

yes

characteristic

C₂×Dic₁₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₄⋊C₄

C₂²

yes

characteristic

C₇×C₂²⋊C₄

C₂²

yes

characteristic

C₇×C₂²⋊C₄

C₂²

yes

characteristic

C₄²×C₇

C₂²

yes

characteristic

C₄²×C₇

C₂²

yes

characteristic

C₂²⋊Dic₇

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

—

C₄×Dic₇

C₂²

yes

—

#542

—

maximal, Sylow

C₂³×C₇

C₂³

yes

characteristic

C₂²×D₇

C₂³

yes

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

D₂₈

—

C₄×D₇

—

Dic₁₄

—

Dic₁₄

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

D₇

yes

characteristic, complemented

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#41

yes

—

Dic₇

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

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

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₇

yes

characteristic

C₂²⋊C₄

C₂×D₇

yes

characteristic

100

C₄²

—

101

C₄²

C₂×D₇

yes

characteristic

102

C₄²

C₂×D₇

yes

characteristic

103

C₂×C₇

—

104

C₂×C₇

C₂×#41

yes

characteristic

105

C₂×C₇

H(5,𝔽₂)

yes

characteristic

106

C₂×C₇

2₋¹⁺⁴

yes

characteristic

107

D₇

—

108

C₂³

—

109

C₂³

C₂²×D₇

yes

characteristic

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

C₂²×D₇

yes

characteristic

120

C₂×C₄

C₂²×D₇

yes

characteristic

121

C₂×C₄

C₂²×D₇

yes

characteristic

122

C₂×C₄

C₂²×D₇

yes

characteristic

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

#1157

yes

—

142

C₂

—

143

C₂

—

144

C₂

C₂×#1157

yes

characteristic

145

C₂

#6553

yes

characteristic

146

C₂

#6554

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#117	C₂
prime radical	#6493	C₂
hypercenter	#117	D₇
hypocenter	C₇	#542
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

C₂

C₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₇×#117	C₂	semidirect
3. #6487	C₂	semidirect
4. #6461	C₂	semidirect
5. #6459	C₂	semidirect
6. #6448	C₂	semidirect
7. #6446	C₂	semidirect
8. #6438	C₂	semidirect
9. C₄×D₂₈	C₂	semidirect
10. #6432	C₂	semidirect
11. C₄×Dic₁₄	C₂	semidirect
63. #117	D₇	semidirect
125. C₇	#542	semidirect

H ≅ G/N

type

1. C₇×#117

C₂

semidirect

3. #6487

C₂

semidirect

4. #6461

C₂

semidirect

5. #6459

C₂

semidirect

6. #6448

C₂

semidirect

7. #6446

C₂

semidirect

8. #6438

C₂

semidirect

9. C₄×D₂₈

C₂

semidirect

10. #6432

C₂

semidirect

11. C₄×Dic₁₄

C₂

semidirect

63. #117

D₇

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×#117	2
#6493	2
#6487	2
#6461	2
#6459	2
#6448	2
#6446	2
#6438	2
C₄×D₂₈	2
#6432	2
C₄×Dic₁₄	2
#542	7

subgroup

index

C₄×D₂₈

C₄×Dic₁₄

Subgroups — #90225

Conjugacy classes of subgroups

Complements

Maximal subgroups