Group #26031147 — subgroups

Subgroups — #26031147

Order 11392.

378 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19152169	C₂	1	2	yes	maximal
2	#19151940	C₂	1	4	yes	maximal, complemented
3	#19151856	C₂	1	2	yes	maximal, complemented
4	#19151839	C₂	1	2	yes	maximal, complemented
5	#19151756	C₂	1	4	yes	maximal, complemented
6	#408×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
8	C₂×Dic₃₅₆	C₂²	1	2	yes	—
9	C₂×Dic₃₅₆	C₂²	1	2	yes	—
10	#15668010	—	2	2	no	—
11	#15667999	—	2	2	no	—
12	#15667993	—	2	2	no	—
13	#15667993	—	2	2	no	—
14	#15667991	C₂²	1	4	yes	—
15	#15667991	C₂²	1	4	yes	—
16	#15667971	C₂²	1	2	yes	—
17	#15667946	C₂²	1	2	yes	—
18	#15667944	C₂²	1	2	yes	—
19	#15667937	C₂²	1	4	yes	—
20	#15667937	C₂²	1	4	yes	—
21	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²⋊C₈×C₈₉	C₂²	1	4	yes	—
25	C₂×Dic₁₇₈	—	2	2	no	—
26	C₂×Dic₁₇₈	—	4	2	no	—
27	C₂×Dic₁₇₈	—	4	2	no	—
28	C₂×Dic₁₇₈	C₂³	1	4	yes	—
29	C₂²×Dic₈₉	—	2	2	no	—
30	Dic₃₅₆	—	4	2	no	—
31	Dic₃₅₆	—	4	2	no	—
32	C₄⋊Dic₈₉	C₂³	1	4	yes	—
33	#13083553	—	2	4	no	—
34	#13083553	—	2	4	no	—
35	C₂²⋊Dic₈₉	—	2	2	no	—
36	C₂²⋊Dic₈₉	—	2	4	no	—
37	C₂²⋊Dic₈₉	—	2	4	no	—
38	C₄×Dic₈₉	—	2	2	no	—
39	#41×C₈₉	—	4	1	no	—
40	#41×C₈₉	—	4	1	no	—
41	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
43	C₂×D₄×C₈₉	D₄	1	2	yes	—
44	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
46	M₁₆×C₈₉	—	4	1	no	—
47	C₂×C₈×C₈₉	—	4	1	no	—
48	C₂×C₈×C₈₉	C₂³	1	2	yes	—
49	C₂×C₈×C₈₉	C₂³	1	2	yes	—
50	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂³×C₈₉	C₂×D₄	1	2	yes	—
52	Q₈×C₈₉	—	4	1	no	—
53	D₄×C₈₉	—	4	2	no	—
54	D₄×C₈₉	—	4	1	no	—
55	C₂×C₄×C₈₉	—	2	1	no	—
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₈₉	—	2	4	no	—
63	C₂×Dic₈₉	—	2	4	no	—
64	C₂×Dic₈₉	—	4	4	no	—
65	Dic₁₇₈	—	2	4	no	—
66	Dic₁₇₈	—	2	4	no	—
67	Dic₁₇₈	—	4	4	no	—
68	C₈×C₈₉	—	2	2	no	—
69	C₈×C₈₉	—	2	2	no	—
70	C₂²×C₈₉	—	2	2	no	—
71	C₂²×C₈₉	—	2	1	no	—
72	C₂²×C₈₉	—	2	2	no	—
73	C₂²×C₈₉	—	2	4	no	—
74	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	—	2	1	no	—
77	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
78	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
79	Dic₈₉	—	4	4	no	—
80	Dic₈₉	—	4	4	no	—
81	C₂×C₈₉	—	4	2	no	—
82	C₂×C₈₉	—	4	1	no	—
83	C₂×C₈₉	#534	1	1	yes	characteristic
84	C₂×C₈₉	#578	1	2	yes	—
85	#3202	—	89	1	no	maximal, Sylow
86	C₈₉	#3202	1	1	yes	characteristic, complemented, Sylow
87	#536	—	89	2	no	—
88	#485	—	89	4	no	—
89	#470	—	89	2	no	—
90	#467	—	89	2	no	—
91	#451	—	89	4	no	—
92	#408	D₈₉	1	1	yes	characteristic
93	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
94	C₂²×Q₈	—	89	2	no	—
95	C₂×Q₁₆	—	89	2	no	—
96	C₂×Q₁₆	—	89	2	no	—
97	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
99	C₄⋊Q₈	—	178	2	no	—
100	#124	—	178	2	no	—
101	#123	—	178	2	no	—
102	#123	—	178	2	no	—
103	C₂²⋊Q₈	—	89	4	no	—
104	C₂²⋊Q₈	—	89	4	no	—
105	#117	—	89	2	no	—
106	#107	—	89	2	no	—
107	#106	—	89	2	no	—
108	#103	—	89	4	no	—
109	#103	—	89	4	no	—
110	C₂²⋊C₈	C₂×D₈₉	1	4	yes	—
111	#41	—	4	1	no	—
112	#41	—	4	1	no	—
113	C₂×Q₈	—	89	4	no	—
114	C₂×Q₈	—	356	2	no	—
115	C₂×Q₈	—	356	2	no	—
116	C₂×Q₈	—	178	2	no	—
117	C₂×Q₈	D₃₅₆	1	1	yes	characteristic
118	C₂×D₄	D₃₅₆	1	1	yes	characteristic
119	C₂×D₄	D₃₅₆	1	2	yes	—
120	C₂²×C₄	—	178	2	no	—
121	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
122	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
123	Q₁₆	—	356	2	no	—
124	Q₁₆	—	356	2	no	—
125	M₁₆	—	4	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	C₂²×D₈₉	1	2	yes	—
128	C₂×C₈	C₂²×D₈₉	1	2	yes	—
129	C₄⋊C₄	—	89	4	no	—
130	C₄⋊C₄	—	178	4	no	—
131	C₄⋊C₄	—	178	4	no	—
132	C₂²⋊C₄	—	178	2	no	—
133	C₂²⋊C₄	—	178	4	no	—
134	C₂²⋊C₄	—	178	4	no	—
135	C₄²	—	178	2	no	—
136	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
137	C₂³	C₂×D₃₅₆	1	2	yes	—
138	Q₈	—	4	1	no	—
139	Q₈	—	356	4	no	—
140	Q₈	—	178	4	no	—
141	Q₈	—	178	4	no	—
142	D₄	—	4	2	no	—
143	D₄	—	4	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	356	4	no	—
149	C₂×C₄	—	178	4	no	—
150	C₂×C₄	—	178	4	no	—
151	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
152	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	4	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	1	no	—
163	C₄	—	356	4	no	—
164	C₄	—	356	4	no	—
165	C₄	#15668057	1	1	yes	characteristic
166	C₄	#15668057	1	1	yes	characteristic
167	C₂	—	4	2	no	—
168	C₂	—	4	1	no	—
169	C₂	#19152160	1	1	yes	characteristic
170	C₂	#19152415	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152169

C₂

yes

maximal

#19151940

C₂

yes

maximal, complemented

#19151856

C₂

yes

maximal, complemented

#19151839

C₂

yes

maximal, complemented

#19151756

C₂

yes

maximal, complemented

#408×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×Dic₃₅₆

C₂²

yes

—

C₂×Dic₃₅₆

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#15667991

C₂²

yes

—

#15667971

C₂²

yes

—

#15667946

C₂²

yes

—

#15667944

C₂²

yes

—

#15667937

C₂²

yes

—

#15667937

C₂²

yes

—

C₂×#41×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂²⋊C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×D₄×C₈₉

D₄

yes

characteristic

C₂×D₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#534

yes

characteristic

C₂×C₈₉

#578

yes

—

#3202

—

maximal, Sylow

C₈₉

#3202

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₈₉

yes

characteristic

C₂×#41

C₂×D₈₉

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

178

—

100

#124

—

178

—

101

#123

—

178

—

102

#123

—

178

—

103

C₂²⋊Q₈

—

104

C₂²⋊Q₈

—

105

#117

—

106

#107

—

107

#106

—

108

#103

—

109

#103

—

110

C₂²⋊C₈

C₂×D₈₉

yes

—

111

#41

—

112

#41

—

113

C₂×Q₈

—

114

C₂×Q₈

—

356

—

115

C₂×Q₈

—

356

—

116

C₂×Q₈

—

178

—

117

C₂×Q₈

D₃₅₆

yes

characteristic

118

C₂×D₄

D₃₅₆

yes

characteristic

119

C₂×D₄

D₃₅₆

yes

—

120

C₂²×C₄

—

178

—

121

C₂²×C₄

C₂²×D₈₉

yes

—

122

C₂²×C₄

C₂²×D₈₉

yes

characteristic

123

Q₁₆

—

356

—

124

Q₁₆

—

356

—

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

C₂²×D₈₉

yes

—

128

C₂×C₈

C₂²×D₈₉

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

178

—

131

C₄⋊C₄

—

178

—

132

C₂²⋊C₄

—

178

—

133

C₂²⋊C₄

—

178

—

134

C₂²⋊C₄

—

178

—

135

C₄²

—

178

—

136

C₂³

C₂×D₃₅₆

yes

characteristic

137

C₂³

C₂×D₃₅₆

yes

—

138

Q₈

—

139

Q₈

—

356

—

140

Q₈

—

178

—

141

Q₈

—

178

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

356

—

149

C₂×C₄

—

178

—

150

C₂×C₄

—

178

—

151

C₂×C₄

C₂×D₃₅₆

yes

—

152

C₂×C₄

C₂×D₃₅₆

yes

characteristic

153

C₂×C₄

C₂³×D₈₉

yes

characteristic

154

C₈

—

155

C₈

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

C₂²×D₃₅₆

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

356

—

164

C₄

—

356

—

165

C₄

#15668057

yes

characteristic

166

C₄

#15668057

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#19152160

yes

characteristic

170

C₂

#19152415

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#408×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#41×C₈₉	C₂²
hypercenter	#408	D₈₉
hypocenter	C₈₉	#3202
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #408

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#408×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂×#41×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #408

N	H ≅ G/N	type
2. #19151940	C₂	semidirect
3. #19151856	C₂	semidirect
4. #19151839	C₂	semidirect
5. #19151756	C₂	semidirect
86. C₈₉	#3202	semidirect

H ≅ G/N

type

2. #19151940

C₂

semidirect

3. #19151856

C₂

semidirect

4. #19151839

C₂

semidirect

5. #19151756

C₂

semidirect

86. C₈₉

#3202

semidirect

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

subgroup	index
#19152169	2
#19151940	2
#19151856	2
#19151839	2
#19151756	2
#408×C₈₉	2
#3202	89

subgroup

index

#408×C₈₉

Subgroups — #26031147

Conjugacy classes of subgroups

Complements

Maximal subgroups