Group #26030641 — subgroups

Subgroups — #26030641

Order 11392.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15668000	C₂	1	1	yes	maximal, characteristic
2	#19151960	C₂	1	4	yes	maximal, complemented
3	#19151955	C₂	1	4	yes	maximal, complemented
4	#19151606	C₂	1	2	yes	maximal, complemented
5	#19151602	C₂	1	2	yes	maximal
6	C₄×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
7	#533×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
10	C₂×#13083558	C₂²	1	2	yes	—
11	#15668000	—	2	2	no	—
12	#15668000	C₂²	1	2	yes	—
13	#15668000	C₂²	1	2	yes	—
14	C₂×#13083551	—	2	2	no	—
15	#15667935	C₂²	1	8	yes	complemented
16	#15667932	C₂²	1	8	yes	—
17	#15667923	C₂²	1	2	yes	complemented
18	C₄×#10623494	C₂²	1	2	yes	complemented
19	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	complemented
20	#125×C₈₉	C₂²	1	2	yes	complemented
21	C₂²⋊Q₈×C₈₉	—	2	2	no	—
22	C₄×Q₈×C₈₉	—	2	2	no	—
23	#117×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
25	C₂³×D₈₉	—	2	1	no	—
26	C₂×Dic₁₇₈	—	2	4	no	—
27	C₂×Dic₁₇₈	—	2	1	no	—
28	C₂×Dic₁₇₈	C₂³	1	2	yes	—
29	C₂×D₃₅₆	—	2	4	no	—
30	C₂×D₃₅₆	—	2	1	no	—
31	C₂×D₃₅₆	C₂³	1	2	yes	—
32	C₂²×Dic₈₉	—	2	1	no	—
33	#13083558	—	2	4	no	—
34	C₂×#10623494	C₂³	1	4	yes	—
35	#13083551	—	2	4	no	—
36	#13083551	—	2	4	no	—
37	C₂×Q₈×C₈₉	—	2	2	no	—
38	C₂²×C₄×C₈₉	D₄	1	2	yes	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₈₉	—	2	4	no	—
41	C₄⋊C₄×C₈₉	—	2	2	no	—
42	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₈₉	—	2	2	no	—
44	C₄²×C₈₉	—	2	2	no	—
45	C₄²×C₈₉	D₄	1	2	yes	—
46	C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₈₉	—	2	2	no	—
50	C₂²×D₈₉	—	4	4	no	—
51	C₂²×D₈₉	—	4	2	no	—
52	Q₈×C₈₉	—	4	4	no	—
53	C₂×C₄×C₈₉	—	2	4	no	—
54	C₂×C₄×C₈₉	—	4	2	no	—
55	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#41	1	4	yes	—
59	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₈₉	—	2	2	no	—
61	C₂×Dic₈₉	—	4	2	no	—
62	D₃₅₆	—	2	4	no	—
63	D₃₅₆	—	4	2	no	—
64	Dic₁₇₈	—	2	4	no	—
65	Dic₁₇₈	—	4	2	no	—
66	#10623494	—	2	4	no	—
67	C₂²×C₈₉	—	2	2	no	—
68	C₂²×C₈₉	#124	1	2	yes	—
69	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
70	C₂×D₈₉	—	4	2	no	—
71	C₂×D₈₉	—	4	4	no	—
72	C₂×D₈₉	—	8	4	no	—
73	C₄×C₈₉	—	2	2	no	—
74	C₄×C₈₉	—	2	2	no	—
75	C₄×C₈₉	—	4	4	no	—
76	C₄×C₈₉	#124	1	2	yes	—
77	C₄×C₈₉	C₂×#41	1	2	yes	—
78	Dic₈₉	—	4	2	no	—
79	C₂×C₈₉	—	2	2	no	—
80	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
81	C₂×C₈₉	#575	1	2	yes	—
82	D₈₉	—	8	2	no	—
83	#3140	—	89	1	no	maximal, Sylow
84	C₈₉	#3140	1	1	yes	characteristic, complemented, Sylow
85	#533	D₈₉	1	1	yes	characteristic, complemented
86	C₂×#124	—	89	1	no	—
87	#489	—	89	4	no	—
88	#488	—	89	4	no	—
89	#419	—	89	2	no	—
90	#418	—	89	2	no	—
91	C₄×M₁₆	—	89	1	no	—
92	C₂²×Q₈	—	89	1	no	—
93	C₂²×D₄	—	89	1	no	—
94	C₂×M₁₆	—	89	2	no	—
95	C₄⋊Q₈	C₂×D₈₉	1	2	yes	complemented
96	#125	C₂×D₈₉	1	2	yes	complemented
97	#124	—	89	2	no	—
98	#124	—	89	2	no	—
99	#124	—	178	2	no	—
100	C₂²⋊Q₈	—	2	2	no	—
101	C₄×Q₈	—	2	2	no	—
102	#117	C₂×D₈₉	1	2	yes	—
103	C₂×C₂²⋊C₄	—	178	2	no	—
104	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
105	#103	—	89	8	no	—
106	#102	—	89	8	no	—
107	#97	—	89	2	no	—
108	C₄×C₈	—	89	2	no	—
109	C₂⁴	—	178	1	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	89	2	no	—
112	C₂×Q₈	—	178	4	no	—
113	C₂×Q₈	—	178	1	no	—
114	C₂×D₄	—	89	2	no	—
115	C₂×D₄	—	178	1	no	—
116	C₂×D₄	—	178	4	no	—
117	C₂²×C₄	—	178	1	no	—
118	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
119	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
120	M₁₆	—	178	4	no	—
121	C₂×C₈	—	89	4	no	—
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	178	4	no	—
127	C₂²⋊C₄	—	178	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂²⋊D₈₉	1	2	yes	—
130	C₄²	C₂²×D₈₉	1	1	yes	characteristic
131	C₄²	C₂²×D₈₉	1	1	yes	characteristic
132	C₂³	—	356	4	no	—
133	C₂³	—	356	2	no	—
134	C₂³	—	178	2	no	—
135	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
136	Q₈	—	4	4	no	—
137	Q₈	—	356	2	no	—
138	Q₈	—	178	4	no	—
139	D₄	—	356	2	no	—
140	D₄	—	178	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	356	2	no	—
144	C₂×C₄	—	178	2	no	—
145	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
148	C₂×C₄	#13083575	1	4	yes	—
149	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
150	C₈	—	178	4	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	356	4	no	—
153	C₂²	—	356	2	no	—
154	C₂²	—	712	4	no	—
155	C₂²	#15668001	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	356	2	no	—
161	C₄	#15668001	1	2	yes	—
162	C₄	C₂×#13083575	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	712	2	no	—
165	C₂	C₂×#15668001	1	1	yes	characteristic
166	C₂	#19152396	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668000

C₂

yes

maximal, characteristic

#19151960

C₂

yes

maximal, complemented

#19151955

C₂

yes

maximal, complemented

#19151606

C₂

yes

maximal, complemented

#19151602

C₂

yes

maximal

C₄×#13083558

C₂

yes

maximal, characteristic, complemented

#533×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂²×D₃₅₆

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

—

#15668000

—

#15668000

C₂²

yes

—

#15668000

C₂²

yes

—

C₂×#13083551

—

#15667935

C₂²

yes

complemented

#15667932

C₂²

yes

—

#15667923

C₂²

yes

complemented

C₄×#10623494

C₂²

yes

complemented

C₄⋊Q₈×C₈₉

C₂²

yes

complemented

#125×C₈₉

C₂²

yes

complemented

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂³

yes

—

C₂²×Dic₈₉

—

#13083558

—

C₂×#10623494

C₂³

yes

—

#13083551

—

#13083551

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#124

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#124

yes

—

C₄×C₈₉

C₂×#41

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#124

yes

characteristic

C₂×C₈₉

#575

yes

—

D₈₉

—

#3140

—

maximal, Sylow

C₈₉

#3140

yes

characteristic, complemented, Sylow

#533

D₈₉

yes

characteristic, complemented

—

—

—

—

—

C₄×M₁₆

—

C₂²×Q₈

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₈₉

yes

complemented

#125

C₂×D₈₉

yes

—

—

—

178

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

C₂×D₈₉

yes

—

103

C₂×C₂²⋊C₄

—

178

—

104

C₂×C₄²

C₂×D₈₉

yes

characteristic

105

#103

—

106

#102

—

107

#97

—

108

C₄×C₈

—

109

C₂⁴

—

178

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

178

—

113

C₂×Q₈

—

178

—

114

C₂×D₄

—

115

C₂×D₄

—

178

—

116

C₂×D₄

—

178

—

117

C₂²×C₄

—

178

—

118

C₂²×C₄

C₂²⋊D₈₉

yes

—

119

C₂²×C₄

C₂²×D₈₉

yes

characteristic

120

M₁₆

—

178

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂²×D₈₉

yes

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

178

—

127

C₂²⋊C₄

—

178

—

128

C₄²

—

129

C₄²

C₂²⋊D₈₉

yes

—

130

C₄²

C₂²×D₈₉

yes

characteristic

131

C₄²

C₂²×D₈₉

yes

characteristic

132

C₂³

—

356

—

133

C₂³

—

356

—

134

C₂³

—

178

—

135

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

136

Q₈

—

137

Q₈

—

356

—

138

Q₈

—

178

—

139

D₄

—

356

—

140

D₄

—

178

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

356

—

144

C₂×C₄

—

178

—

145

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

146

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

147

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

148

C₂×C₄

#13083575

yes

—

149

C₂×C₄

C₂³×D₈₉

yes

characteristic

150

C₈

—

178

—

151

C₂²

—

152

C₂²

—

356

—

153

C₂²

—

356

—

154

C₂²

—

712

—

155

C₂²

#15668001

yes

—

156

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

356

—

161

C₄

#15668001

yes

—

162

C₄

C₂×#13083575

yes

—

163

C₂

—

164

C₂

—

712

—

165

C₂

C₂×#15668001

yes

characteristic

166

C₂

#19152396

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#533×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	#533	D₈₉
hypocenter	C₈₉	#3140
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#533×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂²×D₃₅₆

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #19151960	C₂	semidirect
3. #19151955	C₂	semidirect
4. #19151606	C₂	semidirect
6. C₄×#13083558	C₂	semidirect
7. #533×C₈₉	C₂	semidirect
15. #15667935	C₂²	semidirect
17. #15667923	C₂²	semidirect
18. C₄×#10623494	C₂²	semidirect
19. C₄⋊Q₈×C₈₉	C₂²	semidirect
20. #125×C₈₉	C₂²	semidirect
84. C₈₉	#3140	semidirect
85. #533	D₈₉	semidirect
95. C₄⋊Q₈	C₂×D₈₉	semidirect
96. #125	C₂×D₈₉	semidirect

H ≅ G/N

type

2. #19151960

C₂

semidirect

3. #19151955

C₂

semidirect

4. #19151606

C₂

semidirect

6. C₄×#13083558

C₂

semidirect

7. #533×C₈₉

C₂

semidirect

15. #15667935

C₂²

semidirect

17. #15667923

C₂²

semidirect

18. C₄×#10623494

C₂²

semidirect

19. C₄⋊Q₈×C₈₉

C₂²

semidirect

20. #125×C₈₉

C₂²

semidirect

84. C₈₉

#3140

semidirect

85. #533

D₈₉

semidirect

95. C₄⋊Q₈

C₂×D₈₉

semidirect

96. #125

C₂×D₈₉

semidirect

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

subgroup	index
C₂×#15668000	2
#19151960	2
#19151955	2
#19151606	2
#19151602	2
C₄×#13083558	2
#533×C₈₉	2
#3140	89

subgroup

index

#533×C₈₉

Subgroups — #26030641

Conjugacy classes of subgroups

Complements

Maximal subgroups