Group #26028969 — subgroups

Subgroups — #26028969

Order 11392.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×#15668020	C₂	1	2	yes	maximal, complemented
2	#19152148	C₂	1	1	yes	maximal, characteristic
3	#19151668	C₂	1	8	yes	maximal, complemented
4	#19151566	C₂	1	2	yes	maximal, complemented
5	C₂×#15667923	C₂	1	1	yes	maximal, characteristic, complemented
6	#423×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13083574	C₂²	1	2	yes	—
8	#15668020	—	2	8	no	—
9	C₂×#13083557	C₂²	1	4	yes	—
10	C₂²×#10623494	C₂²	1	1	yes	characteristic
11	C₂²×#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₈₉	C₂²	1	4	yes	—
13	#15668012	C₄	1	1	yes	characteristic
14	#15668009	C₄	1	1	yes	characteristic
15	#15668000	C₄	1	2	yes	—
16	#15667986	C₄	1	4	yes	—
17	C₄×D₃₅₆	C₂²	1	4	yes	—
18	#15667924	C₂²	1	8	yes	—
19	#15667923	C₂²	1	4	yes	—
20	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
23	#13083574	—	2	4	no	—
24	#13083574	—	2	4	no	—
25	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
26	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
27	C₂×D₃₅₆	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊D₈₉	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₈₉	C₂³	1	4	yes	—
30	#13083557	—	2	8	no	—
31	C₂×#10623494	—	2	2	no	—
32	C₂×#10623494	D₄	1	4	yes	complemented
33	C₂×#10623494	C₂³	1	2	yes	—
34	C₂×#10623494	C₂³	1	2	yes	—
35	C₈×D₈₉	—	2	8	no	—
36	C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
37	#13083551	C₂×C₄	1	8	yes	—
38	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	M₁₆×C₈₉	—	2	4	no	—
41	C₂×C₈×C₈₉	C₂³	1	4	yes	—
42	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₈₉	C₂³	1	2	yes	—
45	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₈₉	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₈₉	—	4	2	no	—
48	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₈₉	#41	1	2	yes	—
53	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₈₉	—	2	8	no	—
55	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
56	D₃₅₆	—	2	4	no	—
57	D₃₅₆	—	4	4	no	—
58	C₄×D₈₉	—	2	4	no	—
59	C₄×D₈₉	—	2	4	no	—
60	Dic₁₇₈	—	2	4	no	—
61	C₈×C₈₉	—	2	4	no	—
62	#10623494	—	2	2	no	—
63	#10623494	C₂×D₄	1	4	yes	—
64	C₂²×C₈₉	—	2	2	no	—
65	C₂²×C₈₉	C₄×D₄	1	2	yes	—
66	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₈₉	—	2	4	no	—
68	C₂×D₈₉	—	2	8	no	—
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	C₄×D₄	1	2	yes	—
72	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
74	Dic₈₉	—	2	4	no	—
75	C₂×C₈₉	—	2	2	no	—
76	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
77	C₂×C₈₉	#568	1	2	yes	—
78	D₈₉	—	4	4	no	—
79	#2932	—	89	1	no	maximal, Sylow
80	C₈₉	#2932	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	89	2	no	—
82	#532	—	89	1	no	—
83	#435	—	89	8	no	—
84	#423	D₈₉	1	1	yes	characteristic, complemented
85	#408	—	89	2	no	—
86	C₂×#97	—	89	1	no	—
87	C₂×#41	—	89	2	no	—
88	#131	—	178	8	no	—
89	C₂×M₁₆	—	89	4	no	—
90	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
91	C₂²×C₈	—	89	1	no	—
92	C₂²×C₈	—	89	1	no	—
93	C₂²×C₈	—	89	4	no	—
94	C₄⋊Q₈	—	89	1	no	—
95	#127	—	89	1	no	—
96	#124	—	89	2	no	—
97	#121	—	89	4	no	—
98	C₄×D₄	—	89	4	no	—
99	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
100	C₄⋊C₈	C₂×D₈₉	1	4	yes	—
101	C₂²⋊C₈	—	89	8	no	—
102	#97	—	89	4	no	—
103	#41	—	178	4	no	—
104	#41	—	178	4	no	—
105	C₂×Q₈	—	89	2	no	—
106	C₂×D₄	—	89	4	no	—
107	C₂×D₄	—	89	4	no	—
108	C₂×D₄	—	89	2	no	—
109	C₂²×C₄	—	89	4	no	—
110	C₂²×C₄	C₄×D₈₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	178	8	no	—
114	C₂×C₈	—	89	4	no	—
115	C₂×C₈	—	89	2	no	—
116	C₂×C₈	—	89	2	no	—
117	C₂×C₈	—	178	8	no	—
118	C₂×C₈	—	178	2	no	—
119	C₂×C₈	C₂²×D₈₉	1	4	yes	—
120	C₄⋊C₄	—	89	4	no	—
121	C₂²⋊C₄	—	89	8	no	—
122	C₄²	C₄×D₈₉	1	1	yes	characteristic
123	C₄²	C₄×D₈₉	1	1	yes	characteristic
124	C₄²	C₂²×D₈₉	1	2	yes	—
125	C₂³	—	89	4	no	—
126	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
127	Q₈	—	178	4	no	—
128	D₄	—	356	4	no	—
129	D₄	—	178	8	no	—
130	D₄	—	178	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	89	4	no	—
133	C₂×C₄	—	178	4	no	—
134	C₂×C₄	—	178	4	no	—
135	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
138	C₂×C₄	D₄×D₈₉	1	2	yes	—
139	C₂×C₄	#13083575	1	2	yes	—
140	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	89	4	no	—
143	C₈	—	178	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	178	4	no	—
146	C₂²	—	178	8	no	—
147	C₂²	#15667974	1	2	yes	—
148	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	178	4	no	—
152	C₄	#15667974	1	2	yes	—
153	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
154	C₄	C₂×#13083575	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	356	4	no	—
157	C₂	C₂×#15667974	1	1	yes	characteristic
158	C₂	#19152364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668020

C₂

yes

maximal, complemented

#19152148

C₂

yes

maximal, characteristic

#19151668

C₂

yes

maximal, complemented

#19151566

C₂

yes

maximal, complemented

C₂×#15667923

C₂

yes

maximal, characteristic, complemented

#423×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

—

#15668020

—

C₂×#13083557

C₂²

yes

—

C₂²×#10623494

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₂×C₈×D₈₉

C₂²

yes

—

#15668012

C₄

yes

characteristic

#15668009

C₄

yes

characteristic

#15668000

C₄

yes

—

#15667986

C₄

yes

—

C₄×D₃₅₆

C₂²

yes

—

#15667924

C₂²

yes

—

#15667923

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂×D₃₅₆

C₂×C₄

yes

—

C₂×D₃₅₆

C₂×C₄

yes

—

C₂×C₂²⋊D₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

C₂³

yes

—

#13083557

—

C₂×#10623494

—

C₂×#10623494

D₄

yes

complemented

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

—

C₈×D₈₉

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083551

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂²×D₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

C₂²×C₄

yes

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

C₂×D₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×D₄

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄×D₄

yes

characteristic

C₂×C₈₉

#568

yes

—

D₈₉

—

#2932

—

maximal, Sylow

C₈₉

#2932

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic, complemented

#408

—

C₂×#97

—

C₂×#41

—

#131

—

178

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

100

C₄⋊C₈

C₂×D₈₉

yes

—

101

C₂²⋊C₈

—

102

#97

—

103

#41

—

178

—

104

#41

—

178

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₈₉

yes

—

111

C₂²×C₄

C₂²×D₈₉

yes

characteristic

112

M₁₆

—

113

M₁₆

—

178

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

178

—

118

C₂×C₈

—

178

—

119

C₂×C₈

C₂²×D₈₉

yes

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₄×D₈₉

yes

characteristic

123

C₄²

C₄×D₈₉

yes

characteristic

124

C₄²

C₂²×D₈₉

yes

—

125

C₂³

—

126

C₂³

C₂×C₄×D₈₉

yes

characteristic

127

Q₈

—

178

—

128

D₄

—

356

—

129

D₄

—

178

—

130

D₄

—

178

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

178

—

134

C₂×C₄

—

178

—

135

C₂×C₄

C₂×C₄×D₈₉

yes

—

136

C₂×C₄

C₂×C₄×D₈₉

yes

—

137

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

138

C₂×C₄

D₄×D₈₉

yes

—

139

C₂×C₄

#13083575

yes

—

140

C₂×C₄

C₂³×D₈₉

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

178

—

144

C₂²

—

145

C₂²

—

178

—

146

C₂²

—

178

—

147

C₂²

#15667974

yes

—

148

C₂²

C₂²×C₄×D₈₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

178

—

152

C₄

#15667974

yes

—

153

C₄

C₂×D₄×D₈₉

yes

characteristic

154

C₄

C₂×#13083575

yes

characteristic

155

C₂

—

156

C₂

—

356

—

157

C₂

C₂×#15667974

yes

characteristic

158

C₂

#19152364

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#423×C₈₉	C₂
prime radical	#19152148	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#423	D₈₉
hypocenter	C₈₉	#2932
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

#423×C₈₉

C₂

prime radical

#19152148

C₂

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. C₂×#15668020	C₂	semidirect
3. #19151668	C₂	semidirect
4. #19151566	C₂	semidirect
5. C₂×#15667923	C₂	semidirect
6. #423×C₈₉	C₂	semidirect
32. C₂×#10623494	D₄	semidirect
80. C₈₉	#2932	semidirect
84. #423	D₈₉	semidirect

H ≅ G/N

type

1. C₂×#15668020

C₂

semidirect

3. #19151668

C₂

semidirect

4. #19151566

C₂

semidirect

5. C₂×#15667923

C₂

semidirect

6. #423×C₈₉

C₂

semidirect

32. C₂×#10623494

D₄

semidirect

80. C₈₉

#2932

semidirect

84. #423

D₈₉

semidirect

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

subgroup	index
C₂×#15668020	2
#19152148	2
#19151668	2
#19151566	2
C₂×#15667923	2
#423×C₈₉	2
#2932	89

subgroup

index

#423×C₈₉

Subgroups — #26028969

Conjugacy classes of subgroups

Complements

Maximal subgroups