Group #26021616 — subgroups

Subgroups — #26021616

Order 11392.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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×#15667947	C₂	1	1	yes	maximal, characteristic
2	#19151625	C₂	1	1	yes	maximal, characteristic
3	#19151606	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667935	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667930	C₂	1	1	yes	maximal, characteristic
6	#19151367	C₂	1	1	yes	maximal, characteristic
7	#428×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	C₂×#13083558	C₂²	1	1	yes	characteristic
10	C₂×#13083558	C₂²	1	1	yes	characteristic
11	C₂²×#10623494	C₂²	1	1	yes	characteristic
12	#15667947	—	2	2	no	—
13	C₄⋊#10623494	—	2	2	no	—
14	#15667935	—	2	2	no	—
15	#15667935	—	2	1	no	—
16	#15667935	—	2	1	no	—
17	#15667930	—	2	2	no	—
18	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	#117×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
21	#107×C₈₉	—	2	1	no	—
22	#106×C₈₉	—	2	1	no	—
23	C₂×Dic₁₇₈	—	2	1	no	—
24	C₂×Dic₁₇₈	—	2	1	no	—
25	C₂×Dic₁₇₈	—	2	1	no	—
26	C₂×Dic₁₇₈	—	2	1	no	—
27	C₂×Dic₁₇₈	—	2	1	no	—
28	C₂×Dic₁₇₈	Q₈	1	2	yes	—
29	C₂²×Dic₈₉	—	2	1	no	—
30	#13083558	—	2	1	no	—
31	#13083558	—	2	1	no	—
32	#13083558	—	2	1	no	—
33	#13083558	—	2	1	no	—
34	C₂×#10623494	—	2	1	no	—
35	C₂×#10623494	—	2	1	no	—
36	C₂×#10623494	—	2	1	no	—
37	C₂×#10623494	—	2	1	no	—
38	C₂×#10623494	D₄	1	2	yes	—
39	C₂²×C₄×C₈₉	—	2	1	no	—
40	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	M₁₆×C₈₉	—	2	1	no	—
42	M₁₆×C₈₉	D₄	1	2	yes	—
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₄⋊C₄×C₈₉	—	2	1	no	—
45	C₄⋊C₄×C₈₉	—	2	1	no	—
46	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₈₉	—	2	1	no	—
49	C₄²×C₈₉	—	2	1	no	—
50	C₂³×C₈₉	#41	1	1	yes	characteristic
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	—	4	1	no	—
54	C₂×C₄×C₈₉	—	8	1	no	—
55	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
61	C₂×Dic₈₉	—	2	2	no	—
62	C₂×Dic₈₉	—	4	1	no	—
63	C₂×Dic₈₉	—	4	1	no	—
64	Dic₁₇₈	—	2	2	no	—
65	Dic₁₇₈	—	2	2	no	—
66	Dic₁₇₈	—	4	1	no	—
67	Dic₁₇₈	—	4	1	no	—
68	C₈×C₈₉	—	2	2	no	—
69	#10623494	—	2	2	no	—
70	#10623494	—	4	1	no	—
71	#10623494	—	4	1	no	—
72	C₂²×C₈₉	—	2	1	no	—
73	C₂²×C₈₉	—	2	1	no	—
74	C₂²×C₈₉	#121	1	1	yes	characteristic
75	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₈₉	#126	1	1	yes	characteristic
77	C₄×C₈₉	—	4	1	no	—
78	C₄×C₈₉	—	4	1	no	—
79	C₄×C₈₉	—	8	1	no	—
80	C₄×C₈₉	#121	1	1	yes	characteristic
81	C₄×C₈₉	#121	1	1	yes	characteristic
82	C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
83	C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
84	Dic₈₉	—	4	2	no	—
85	C₂×C₈₉	—	2	1	no	—
86	C₂×C₈₉	—	2	1	no	—
87	C₂×C₈₉	#396	1	1	yes	characteristic
88	C₂×C₈₉	#471	1	1	yes	characteristic
89	C₂×C₈₉	#473	1	1	yes	characteristic
90	#2064	—	89	1	no	maximal, Sylow
91	C₈₉	#2064	1	1	yes	characteristic, complemented, Sylow
92	C₂×#108	—	89	1	no	—
93	#428	D₈₉	1	1	yes	characteristic
94	#424	—	89	1	no	—
95	#419	—	89	1	no	—
96	C₂×#103	—	89	1	no	—
97	C₂×#101	—	89	1	no	—
98	#343	—	89	1	no	—
99	C₂²×Q₈	—	89	1	no	—
100	C₂×M₁₆	—	89	1	no	—
101	C₂×M₁₆	—	89	1	no	—
102	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
103	C₂²×C₈	—	89	1	no	—
104	#117	C₂×D₈₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
106	#108	—	178	2	no	—
107	#107	—	2	1	no	—
108	#106	—	2	1	no	—
109	C₄⋊C₈	—	178	2	no	—
110	#103	—	178	2	no	—
111	#103	—	178	1	no	—
112	#103	—	178	1	no	—
113	#101	—	178	2	no	—
114	C₂×Q₈	—	89	2	no	—
115	C₂×Q₈	—	178	1	no	—
116	C₂×Q₈	—	178	1	no	—
117	C₂×Q₈	—	178	1	no	—
118	C₂×Q₈	—	178	1	no	—
119	C₂×Q₈	—	178	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	178	1	no	—
122	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
123	M₁₆	—	2	1	no	—
124	M₁₆	—	178	1	no	—
125	M₁₆	—	178	1	no	—
126	M₁₆	—	178	1	no	—
127	M₁₆	—	178	1	no	—
128	M₁₆	D₃₅₆	1	2	yes	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	89	2	no	—
131	C₂×C₈	—	178	1	no	—
132	C₂×C₈	—	178	1	no	—
133	C₂×C₈	—	178	1	no	—
134	C₂×C₈	—	178	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
138	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
139	C₂²⋊C₄	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₂³	#13083573	1	1	yes	characteristic
142	Q₈	—	356	1	no	—
143	Q₈	—	356	1	no	—
144	Q₈	—	178	2	no	—
145	Q₈	—	178	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	356	1	no	—
151	C₂×C₄	—	356	1	no	—
152	C₂×C₄	—	178	2	no	—
153	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
155	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
156	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
157	C₂×C₄	#13083574	1	1	yes	characteristic
158	C₂×C₄	#13083575	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	356	1	no	—
161	C₈	—	356	1	no	—
162	C₈	—	178	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#15667984	1	1	yes	characteristic
166	C₂²	#15667992	1	1	yes	characteristic
167	C₂²	#15668007	1	1	yes	characteristic
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	8	1	no	—
171	C₄	—	356	2	no	—
172	C₄	#15667986	1	1	yes	characteristic
173	C₄	#15667987	1	1	yes	characteristic
174	C₄	#15667988	1	1	yes	characteristic
175	C₄	#15667990	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#19151520	1	1	yes	characteristic
179	C₂	#19151859	1	1	yes	characteristic
180	C₂	#19151871	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667947

C₂

yes

maximal, characteristic

#19151625

C₂

yes

maximal, characteristic

#19151606

C₂

yes

maximal, characteristic

C₂×#15667935

C₂

yes

maximal, characteristic

C₂×#15667930

C₂

yes

maximal, characteristic

#19151367

C₂

yes

maximal, characteristic

#428×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

—

—

—

—

—

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

#107×C₈₉

—

#106×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

Q₈

yes

—

C₂²×Dic₈₉

—

—

—

—

—

—

—

—

—

D₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₉

#126

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×#108

—

#428

D₈₉

yes

—

—

—

—

—

C₂²×Q₈

—

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂×M₁₆

C₂×D₈₉

yes

characteristic

103

C₂²×C₈

—

104

#117

C₂×D₈₉

yes

characteristic

105

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

106

#108

—

178

—

107

#107

—

108

#106

—

109

C₄⋊C₈

—

178

—

110

#103

—

178

—

111

#103

—

178

—

112

#103

—

178

—

113

#101

—

178

—

114

C₂×Q₈

—

115

C₂×Q₈

—

178

—

116

C₂×Q₈

—

178

—

117

C₂×Q₈

—

178

—

118

C₂×Q₈

—

178

—

119

C₂×Q₈

—

178

—

120

C₂²×C₄

—

121

C₂²×C₄

—

178

—

122

C₂²×C₄

C₂²×D₈₉

yes

characteristic

123

M₁₆

—

124

M₁₆

—

178

—

125

M₁₆

—

178

—

126

M₁₆

—

178

—

127

M₁₆

—

178

—

128

M₁₆

D₃₅₆

yes

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

178

—

132

C₂×C₈

—

178

—

133

C₂×C₈

—

178

—

134

C₂×C₈

—

178

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

C₂²⋊D₈₉

yes

characteristic

138

C₄⋊C₄

C₂²⋊D₈₉

yes

characteristic

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

#13083573

yes

characteristic

142

Q₈

—

356

—

143

Q₈

—

356

—

144

Q₈

—

178

—

145

Q₈

—

178

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

356

—

151

C₂×C₄

—

356

—

152

C₂×C₄

—

178

—

153

C₂×C₄

D₄×D₈₉

yes

characteristic

154

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

155

C₂×C₄

C₂×D₃₅₆

yes

characteristic

156

C₂×C₄

Q₈×D₈₉

yes

characteristic

157

C₂×C₄

#13083574

yes

characteristic

158

C₂×C₄

#13083575

yes

characteristic

159

C₈

—

160

C₈

—

356

—

161

C₈

—

356

—

162

C₈

—

178

—

163

C₂²

—

164

C₂²

—

165

C₂²

#15667984

yes

characteristic

166

C₂²

#15667992

yes

characteristic

167

C₂²

#15668007

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

356

—

172

C₄

#15667986

yes

characteristic

173

C₄

#15667987

yes

characteristic

174

C₄

#15667988

yes

characteristic

175

C₄

#15667990

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#19151520

yes

characteristic

179

C₂

#19151859

yes

characteristic

180

C₂

#19151871

yes

characteristic

	subgroup	G/H
center	C₂²	#15667984
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#428×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	C₂³×C₈₉	#41
hypercenter	#428	D₈₉
hypocenter	C₈₉	#2064
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

subgroup

G/H

center

C₂²

#15667984

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#428×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

N	H ≅ G/N	type
91. C₈₉	#2064	semidirect

H ≅ G/N

type

91. C₈₉

#2064

semidirect

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

subgroup	index
C₂×#15667947	2
#19151625	2
#19151606	2
C₂×#15667935	2
C₂×#15667930	2
#19151367	2
#428×C₈₉	2
#2064	89

subgroup

index

#428×C₈₉

Subgroups — #26021616

Conjugacy classes of subgroups

Complements

Maximal subgroups