Group #22554000 — subgroups

Subgroups — #22554000

Order 8384.

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

normal subgroups only

This group has 214 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	#16862241	C₂	1	2	yes	maximal, complemented
2	#16862237	C₂	1	2	yes	maximal, complemented
3	#16862231	C₂	1	2	yes	maximal
4	#16862226	C₂	1	1	yes	maximal, characteristic, complemented
5	#16862226	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₅₂₄	C₂	1	1	yes	maximal, characteristic
8	#16862204	C₂	1	1	yes	maximal, characteristic, complemented
9	#16862204	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₄×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#125×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
13	C₂×D₅₂₄	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₃₁	C₂²	1	2	yes	—
17	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
20	C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
21	Dic₁₃₁⋊C₄	C₂²	1	2	yes	—
22	Dic₁₃₁⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₁₃₁)⋊C₄	C₂²	1	1	yes	characteristic
24	(C₂×D₁₃₁)⋊C₄	C₂²	1	1	yes	characteristic
25	(C₂×D₁₃₁)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₁₃₁)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₁₃₁)⋊C₄	C₂²	1	2	yes	—
28	C₄×Dic₁₃₁	C₂²	1	2	yes	—
29	C₄×Dic₁₃₁	C₂²	1	2	yes	—
30	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
33	C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
34	C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₂²×D₁₃₁	C₂³	1	1	yes	characteristic
36	C₂²×D₁₃₁	C₂³	1	1	yes	characteristic
37	Dic₂₆₂	—	2	2	no	—
38	D₅₂₄	—	2	2	no	—
39	C₂×Dic₁₃₁	C₂³	1	2	yes	—
40	C₂×Dic₁₃₁	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₃₁	C₂³	1	2	yes	—
42	C₂×Dic₁₃₁	C₂³	1	1	yes	characteristic
43	C₄×D₁₃₁	—	2	4	no	—
44	C₄×D₁₃₁	—	2	2	no	—
45	C₄×D₁₃₁	—	2	2	no	—
46	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₁₃₁	C₂³	1	2	yes	—
49	C₂×C₄×C₁₃₁	C₂³	1	2	yes	—
50	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₁₃₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₁₃₁	—	2	1	no	—
53	C₂×D₁₃₁	—	2	1	no	—
54	C₂×D₁₃₁	—	2	1	no	—
55	C₂×D₁₃₁	—	2	1	no	—
56	C₂×D₁₃₁	—	2	1	no	—
57	C₂×D₁₃₁	#41	1	2	yes	—
58	C₄×C₁₃₁	—	2	2	no	—
59	C₄×C₁₃₁	—	2	1	no	—
60	C₄×C₁₃₁	—	2	1	no	—
61	C₄×C₁₃₁	—	2	2	no	—
62	C₄×C₁₃₁	#41	1	2	yes	—
63	Dic₁₃₁	—	2	2	no	—
64	Dic₁₃₁	—	2	1	no	—
65	Dic₁₃₁	—	2	1	no	—
66	Dic₁₃₁	—	2	2	no	—
67	C₂×C₁₃₁	C₂×#41	1	1	yes	characteristic
68	C₂×C₁₃₁	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
70	D₁₃₁	—	2	2	no	—
71	D₁₃₁	—	4	1	no	—
72	C₁₃₁	#552	1	1	yes	characteristic, complemented, Sylow
73	#552	—	131	1	no	maximal, Sylow
74	#126	—	131	2	no	—
75	#125	—	131	2	no	—
76	#125	D₁₃₁	1	1	yes	characteristic, complemented
77	#123	—	131	2	no	—
78	C₂²⋊Q₈	—	131	1	no	—
79	C₂²⋊Q₈	—	131	1	no	—
80	C₄×Q₈	—	131	1	no	—
81	C₄×D₄	—	131	1	no	—
82	#117	—	131	1	no	—
83	#117	—	131	2	no	—
84	C₂×C₄⋊C₄	—	131	1	no	—
85	C₂×Q₈	—	131	1	no	—
86	C₂×D₄	—	131	1	no	—
87	C₂²×C₄	—	131	2	no	—
88	C₂²×C₄	—	131	1	no	—
89	C₂²×C₄	—	131	1	no	—
90	C₄⋊C₄	—	131	2	no	—
91	C₄⋊C₄	—	131	2	no	—
92	C₄⋊C₄	—	131	1	no	—
93	C₄⋊C₄	—	131	2	no	—
94	C₄⋊C₄	—	131	1	no	—
95	C₄⋊C₄	—	131	2	no	—
96	C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₁₃₁	1	2	yes	—
99	C₄⋊C₄	C₂×D₁₃₁	1	2	yes	—
100	C₂²⋊C₄	—	131	1	no	—
101	C₂²⋊C₄	—	131	2	no	—
102	C₂²⋊C₄	—	131	2	no	—
103	C₂²⋊C₄	—	131	1	no	—
104	C₂²⋊C₄	—	131	2	no	—
105	C₄²	—	131	2	no	—
106	C₄²	—	131	2	no	—
107	C₄²	C₂×D₁₃₁	1	1	yes	characteristic
108	C₂³	—	131	1	no	—
109	C₂³	—	131	1	no	—
110	Q₈	—	262	2	no	—
111	D₄	—	262	2	no	—
112	C₂×C₄	—	262	4	no	—
113	C₂×C₄	—	262	2	no	—
114	C₂×C₄	—	262	2	no	—
115	C₂×C₄	—	131	2	no	—
116	C₂×C₄	—	131	1	no	—
117	C₂×C₄	—	131	2	no	—
118	C₂×C₄	—	131	1	no	—
119	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₁₃₁	1	2	yes	—
121	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₁₃₁	1	2	yes	—
123	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
124	C₂²	—	262	1	no	—
125	C₂²	—	262	1	no	—
126	C₂²	—	262	1	no	—
127	C₂²	—	262	1	no	—
128	C₂²	—	262	1	no	—
129	C₂²	—	131	2	no	—
130	C₂²	C₂³×D₁₃₁	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	2	no	—
135	C₄	—	262	2	no	—
136	C₄	—	262	1	no	—
137	C₄	—	262	1	no	—
138	C₄	—	262	2	no	—
139	C₄	#14523514	1	2	yes	—
140	C₂	—	262	2	no	—
141	C₂	—	524	1	no	—
142	C₂	#41×D₁₃₁	1	1	yes	characteristic
143	C₂	C₂×#14523514	1	1	yes	characteristic
144	C₂	#16862293	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16862241

C₂

yes

maximal, complemented

#16862237

C₂

yes

maximal, complemented

#16862231

C₂

yes

maximal

#16862226

C₂

yes

maximal, characteristic, complemented

#16862226

C₂

yes

maximal, characteristic, complemented

C₄×Dic₂₆₂

C₂

yes

maximal, characteristic, complemented

C₄×D₅₂₄

C₂

yes

maximal, characteristic

#16862204

C₂

yes

maximal, characteristic, complemented

#16862204

C₂

yes

maximal, complemented

C₄⋊C₄×D₁₃₁

C₂

yes

maximal, characteristic, complemented

#125×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₂

C₂²

yes

characteristic

C₂×D₅₂₄

C₂²

yes

characteristic

C₂×C₄×D₁₃₁

C₂²

yes

characteristic

C₂×C₄×D₁₃₁

C₂²

yes

characteristic

C₂×C₄×D₁₃₁

C₂²

yes

—

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

C₂²

yes

—

C₄⋊Dic₁₃₁

C₂²

yes

—

Dic₁₃₁⋊C₄

C₂²

yes

—

Dic₁₃₁⋊C₄

C₂²

yes

—

(C₂×D₁₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₁)⋊C₄

C₂²

yes

—

(C₂×D₁₃₁)⋊C₄

C₂²

yes

—

(C₂×D₁₃₁)⋊C₄

C₂²

yes

—

C₄×Dic₁₃₁

C₂²

yes

—

C₄×Dic₁₃₁

C₂²

yes

—

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₄²×C₁₃₁

C₂²

yes

characteristic

C₂²×D₁₃₁

C₂³

yes

characteristic

C₂²×D₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

D₅₂₄

—

C₂×Dic₁₃₁

C₂³

yes

—

C₂×Dic₁₃₁

C₂³

yes

characteristic

C₂×Dic₁₃₁

C₂³

yes

—

C₂×Dic₁₃₁

C₂³

yes

characteristic

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₄×C₁₃₁

C₂³

yes

—

C₂×C₄×C₁₃₁

C₂³

yes

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂²×C₁₃₁

C₂⁴

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

#41

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#41

yes

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

C₂×#41

yes

characteristic

C₂×C₁₃₁

C₂×#41

yes

characteristic

C₂×C₁₃₁

2₋¹⁺⁴

yes

characteristic

D₁₃₁

—

D₁₃₁

—

C₁₃₁

#552

yes

characteristic, complemented, Sylow

#552

—

131

maximal, Sylow

#126

—

131

—

#125

—

131

—

#125

D₁₃₁

yes

characteristic, complemented

#123

—

131

—

C₂²⋊Q₈

—

131

—

C₂²⋊Q₈

—

131

—

C₄×Q₈

—

131

—

C₄×D₄

—

131

—

#117

—

131

—

#117

—

131

—

C₂×C₄⋊C₄

—

131

—

C₂×Q₈

—

131

—

C₂×D₄

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

C₂×D₁₃₁

yes

—

C₄⋊C₄

C₂×D₁₃₁

yes

—

100

C₂²⋊C₄

—

131

—

101

C₂²⋊C₄

—

131

—

102

C₂²⋊C₄

—

131

—

103

C₂²⋊C₄

—

131

—

104

C₂²⋊C₄

—

131

—

105

C₄²

—

131

—

106

C₄²

—

131

—

107

C₄²

C₂×D₁₃₁

yes

characteristic

108

C₂³

—

131

—

109

C₂³

—

131

—

110

Q₈

—

262

—

111

D₄

—

262

—

112

C₂×C₄

—

262

—

113

C₂×C₄

—

262

—

114

C₂×C₄

—

262

—

115

C₂×C₄

—

131

—

116

C₂×C₄

—

131

—

117

C₂×C₄

—

131

—

118

C₂×C₄

—

131

—

119

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

120

C₂×C₄

C₂²×D₁₃₁

yes

—

121

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

122

C₂×C₄

C₂²×D₁₃₁

yes

—

123

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

124

C₂²

—

262

—

125

C₂²

—

262

—

126

C₂²

—

262

—

127

C₂²

—

262

—

128

C₂²

—

262

—

129

C₂²

—

131

—

130

C₂²

C₂³×D₁₃₁

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

262

—

136

C₄

—

262

—

137

C₄

—

262

—

138

C₄

—

262

—

139

C₄

#14523514

yes

—

140

C₂

—

262

—

141

C₂

—

524

—

142

C₂

#41×D₁₃₁

yes

characteristic

143

C₂

C₂×#14523514

yes

characteristic

144

C₂

#16862293

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂⁴
Fitting	#125×C₁₃₁	C₂
hypercenter	#125	D₁₃₁
hypocenter	C₁₃₁	#552
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₁

commutator, socle

C₂²×C₁₃₁

C₂⁴

Fitting

#125×C₁₃₁

C₂

D₁₃₁

C₁₃₁

G ▷ C₂²×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #16862241	C₂	semidirect
2. #16862237	C₂	semidirect
4. #16862226	C₂	semidirect
5. #16862226	C₂	semidirect
6. C₄×Dic₂₆₂	C₂	semidirect
8. #16862204	C₂	semidirect
9. #16862204	C₂	semidirect
10. C₄⋊C₄×D₁₃₁	C₂	semidirect
11. #125×C₁₃₁	C₂	semidirect
72. C₁₃₁	#552	semidirect
76. #125	D₁₃₁	semidirect

H ≅ G/N

type

1. #16862241

C₂

semidirect

2. #16862237

C₂

semidirect

4. #16862226

C₂

semidirect

5. #16862226

C₂

semidirect

6. C₄×Dic₂₆₂

C₂

semidirect

8. #16862204

C₂

semidirect

9. #16862204

C₂

semidirect

10. C₄⋊C₄×D₁₃₁

C₂

semidirect

11. #125×C₁₃₁

C₂

semidirect

72. C₁₃₁

#552

semidirect

76. #125

D₁₃₁

semidirect

Subgroups — #22554000

Conjugacy classes of subgroups

Complements