Group #26028970 — subgroups

Subgroups — #26028970

Order 11392.

484 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 221 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×#15668019	C₂	1	2	yes	maximal, complemented
2	#19151670	C₂	1	8	yes	maximal, complemented
3	#19151622	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151565	C₂	1	2	yes	maximal, complemented
5	C₂×#15667923	C₂	1	1	yes	maximal, characteristic, complemented
6	#532×C₈₉	C₂	1	1	yes	maximal, characteristic
7	#15668019	—	2	8	no	—
8	C₂×#13083558	C₂²	1	2	yes	—
9	C₂×#13083558	C₂²	1	4	yes	—
10	C₂²×#10623494	C₂²	1	4	yes	—
11	C₂²×#10623494	C₂²	1	1	yes	characteristic
12	C₂²×#10623494	C₂²	1	1	yes	characteristic
13	C₄⋊#10623494	C₂²	1	4	yes	—
14	C₂²⋊#10623494	C₂²	1	8	yes	—
15	#15667923	C₂²	1	4	yes	—
16	C₂×#41×C₈₉	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
18	#127×C₈₉	C₄	1	1	yes	characteristic
19	#124×C₈₉	C₄	1	2	yes	—
20	#121×C₈₉	C₄	1	4	yes	—
21	C₄×D₄×C₈₉	C₂²	1	4	yes	—
22	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	#13083558	—	2	4	no	—
24	#13083558	—	2	8	no	—
25	C₂×#10623494	—	2	2	no	—
26	C₂×#10623494	—	2	8	no	—
27	C₂×#10623494	D₄	1	4	yes	complemented
28	C₂×#10623494	C₂³	1	4	yes	—
29	C₂×#10623494	C₂³	1	2	yes	—
30	C₂×#10623494	C₂³	1	2	yes	—
31	#41×C₈₉	—	2	4	no	—
32	#41×C₈₉	—	2	4	no	—
33	C₂×Q₈×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₈₉	C₂×C₄	1	4	yes	—
35	C₂×D₄×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₈₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₈₉	C₂×C₄	1	8	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	4	yes	—
46	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₈₉	—	2	4	no	—
48	D₄×C₈₉	—	2	8	no	—
49	D₄×C₈₉	—	2	4	no	—
50	D₄×C₈₉	—	4	4	no	—
51	C₂×C₄×C₈₉	—	2	4	no	—
52	C₂×C₄×C₈₉	—	2	4	no	—
53	C₂×C₄×C₈₉	—	4	2	no	—
54	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₈₉	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₈₉	#41	1	2	yes	—
60	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
61	#10623494	—	2	4	no	—
62	#10623494	—	2	2	no	—
63	#10623494	C₂×D₄	1	4	yes	—
64	C₂²×C₈₉	—	2	2	no	—
65	C₂²×C₈₉	—	2	4	no	—
66	C₂²×C₈₉	—	2	8	no	—
67	C₂²×C₈₉	C₄×D₄	1	2	yes	—
68	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₈₉	—	2	4	no	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	—	2	2	no	—
72	C₄×C₈₉	C₄×D₄	1	2	yes	—
73	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
75	C₂×C₈₉	—	2	2	no	—
76	C₂×C₈₉	—	4	4	no	—
77	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂×C₈₉	#568	1	2	yes	—
79	#2932	—	89	1	no	maximal, Sylow
80	C₈₉	#2932	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	89	2	no	—
82	#532	D₈₉	1	1	yes	characteristic
83	#435	—	89	8	no	—
84	#423	—	89	1	no	—
85	#408	—	89	2	no	—
86	C₂×#97	—	89	1	no	—
87	C₂×#41	C₂×D₈₉	1	2	yes	—
88	#131	—	178	8	no	—
89	C₂×M₁₆	—	89	2	no	—
90	C₂×M₁₆	—	89	4	no	—
91	C₂²×C₈	—	89	1	no	—
92	C₂²×C₈	—	89	4	no	—
93	C₂²×C₈	—	89	1	no	—
94	C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
95	#127	Dic₈₉	1	1	yes	characteristic
96	#124	Dic₈₉	1	2	yes	—
97	#121	Dic₈₉	1	4	yes	—
98	C₄×D₄	C₂×D₈₉	1	4	yes	—
99	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
100	C₄⋊C₈	—	89	4	no	—
101	C₂²⋊C₈	—	89	8	no	—
102	#97	—	89	4	no	—
103	#41	—	2	4	no	—
104	#41	—	2	4	no	—
105	C₂×Q₈	C₂×Dic₈₉	1	2	yes	—
106	C₂×D₄	C₂×Dic₈₉	1	4	yes	—
107	C₂×D₄	C₂×Dic₈₉	1	2	yes	—
108	C₂×D₄	C₂×Dic₈₉	1	4	yes	—
109	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
110	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
111	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
112	M₁₆	—	178	8	no	—
113	M₁₆	—	178	4	no	—
114	C₂×C₈	—	89	4	no	—
115	C₂×C₈	—	89	2	no	—
116	C₂×C₈	—	89	2	no	—
117	C₂×C₈	—	89	4	no	—
118	C₂×C₈	—	178	8	no	—
119	C₂×C₈	—	178	2	no	—
120	C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
121	C₂²⋊C₄	C₂×Dic₈₉	1	8	yes	—
122	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
123	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
124	C₄²	C₂²×D₈₉	1	2	yes	—
125	C₂³	C₂²×Dic₈₉	1	4	yes	—
126	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	D₄	—	2	8	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
135	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
136	C₂×C₄	C₂²×Dic₈₉	1	4	yes	—
137	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
138	C₂×C₄	D₄×D₈₉	1	2	yes	—
139	C₂×C₄	#13083573	1	2	yes	—
140	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
141	C₈	—	89	4	no	—
142	C₈	—	178	4	no	—
143	C₈	—	178	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	D₄×Dic₈₉	1	2	yes	—
148	C₂²	C₂³×Dic₈₉	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	D₄×Dic₈₉	1	2	yes	—
153	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
154	C₄	C₂×#13083573	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	4	4	no	—
157	C₂	C₂×D₄×Dic₈₉	1	1	yes	characteristic
158	C₂	#19152365	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668019

C₂

yes

maximal, complemented

#19151670

C₂

yes

maximal, complemented

#19151622

C₂

yes

maximal, characteristic, complemented

#19151565

C₂

yes

maximal, complemented

C₂×#15667923

C₂

yes

maximal, characteristic, complemented

#532×C₈₉

C₂

yes

maximal, characteristic

#15668019

—

C₂×#13083558

C₂²

yes

—

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

C₂²×#10623494

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

C₂²⋊#10623494

C₂²

yes

—

#15667923

C₂²

yes

—

C₂×#41×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₄

yes

characteristic

#127×C₈₉

C₄

yes

characteristic

#124×C₈₉

C₄

yes

—

#121×C₈₉

C₄

yes

—

C₄×D₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

—

—

—

D₄

yes

complemented

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂²⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂²×C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

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

—

—

C₂×D₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×D₄

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄×D₄

yes

characteristic

C₂×C₈₉

#568

yes

—

#2932

—

maximal, Sylow

C₈₉

#2932

yes

characteristic, complemented, Sylow

C₂×#131

—

#532

D₈₉

yes

—

—

—

C₂×#97

—

C₂×#41

C₂×D₈₉

yes

—

#131

—

178

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₈₉

yes

characteristic

#127

Dic₈₉

yes

characteristic

#124

Dic₈₉

yes

—

#121

Dic₈₉

yes

—

C₄×D₄

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

#41

—

104

#41

—

105

C₂×Q₈

C₂×Dic₈₉

yes

—

106

C₂×D₄

C₂×Dic₈₉

yes

—

107

C₂×D₄

C₂×Dic₈₉

yes

—

108

C₂×D₄

C₂×Dic₈₉

yes

—

109

C₂²×C₄

C₂×Dic₈₉

yes

—

110

C₂²×C₄

C₂²×D₈₉

yes

—

111

C₂²×C₄

C₂²×D₈₉

yes

characteristic

112

M₁₆

—

178

—

113

M₁₆

—

178

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

178

—

119

C₂×C₈

—

178

—

120

C₄⋊C₄

C₂×Dic₈₉

yes

—

121

C₂²⋊C₄

C₂×Dic₈₉

yes

—

122

C₄²

C₂×Dic₈₉

yes

characteristic

123

C₄²

C₂×Dic₈₉

yes

characteristic

124

C₄²

C₂²×D₈₉

yes

—

125

C₂³

C₂²×Dic₈₉

yes

—

126

C₂³

C₂²×Dic₈₉

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₂²×Dic₈₉

yes

—

135

C₂×C₄

C₂²×Dic₈₉

yes

—

136

C₂×C₄

C₂²×Dic₈₉

yes

—

137

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

138

C₂×C₄

D₄×D₈₉

yes

—

139

C₂×C₄

#13083573

yes

—

140

C₂×C₄

C₂³×D₈₉

yes

characteristic

141

C₈

—

142

C₈

—

178

—

143

C₈

—

178

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

D₄×Dic₈₉

yes

—

148

C₂²

C₂³×Dic₈₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

D₄×Dic₈₉

yes

—

153

C₄

C₂×D₄×D₈₉

yes

characteristic

154

C₄

C₂×#13083573

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

C₂×D₄×Dic₈₉

yes

characteristic

158

C₂

#19152365

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting, prime radical	#532×C₈₉	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#532	D₈₉
hypocenter	C₈₉	#2932
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting, prime radical

#532×C₈₉

C₂

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. C₂×#15668019	C₂	semidirect
2. #19151670	C₂	semidirect
3. #19151622	C₂	semidirect
4. #19151565	C₂	semidirect
5. C₂×#15667923	C₂	semidirect
27. C₂×#10623494	D₄	semidirect
80. C₈₉	#2932	semidirect

H ≅ G/N

type

1. C₂×#15668019

C₂

semidirect

2. #19151670

C₂

semidirect

3. #19151622

C₂

semidirect

4. #19151565

C₂

semidirect

5. C₂×#15667923

C₂

semidirect

27. C₂×#10623494

D₄

semidirect

80. C₈₉

#2932

semidirect

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

subgroup	index
C₂×#15668019	2
#19151670	2
#19151622	2
#19151565	2
C₂×#15667923	2
#532×C₈₉	2
#2932	89

subgroup

index

#532×C₈₉

Subgroups — #26028970

Conjugacy classes of subgroups

Complements

Maximal subgroups