Group #26033266 — subgroups

Subgroups — #26033266

Order 11392.

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

normal subgroups only

This group has 384 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	#19152334	C₂	1	2	yes	maximal, complemented
2	#19151981	C₂	1	2	yes	maximal, complemented
3	#19151811	C₂	1	4	yes	maximal, complemented
4	#19151795	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₈₉	C₂	1	2	yes	maximal, complemented
8	#493×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
10	C₂×#13083562	C₂²	1	4	yes	—
11	#15667999	—	2	2	no	—
12	#15667999	C₂²	1	4	yes	—
13	#15667993	—	2	4	no	—
14	C₄×Dic₁₇₈	C₂²	1	2	yes	—
15	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
16	D₄×Dic₈₉	—	2	2	no	—
17	D₄×Dic₈₉	C₂²	1	4	yes	—
18	#15667946	C₂²	1	1	yes	characteristic
19	C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	#15667941	C₂²	1	2	yes	—
21	#15667937	C₂²	1	2	yes	—
22	#15667933	C₂²	1	4	yes	—
23	C₈×Dic₈₉	C₂²	1	2	yes	—
24	C₂×D₈×C₈₉	C₂²	1	2	yes	—
25	C₂×D₈×C₈₉	C₂²	1	2	yes	—
26	#127×C₈₉	C₂²	1	2	yes	—
27	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
29	C₂×Dic₁₇₈	C₂³	1	2	yes	—
30	C₂²×Dic₈₉	—	2	4	no	—
31	Dic₃₅₆	—	2	2	no	—
32	#13083562	—	4	4	no	—
33	C₂×#10623494	C₂³	1	2	yes	—
34	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
35	C₄⋊Dic₈₉	C₂³	1	2	yes	—
36	#13083553	—	2	2	no	—
37	#13083553	D₄	1	2	yes	—
38	C₂²⋊Dic₈₉	—	2	4	no	—
39	C₂²⋊Dic₈₉	—	2	4	no	—
40	C₂²⋊Dic₈₉	—	2	4	no	—
41	C₄×Dic₈₉	—	2	2	no	—
42	C₄×Dic₈₉	C₂³	1	2	yes	—
43	C₂×D₄×C₈₉	—	2	2	no	—
44	C₂×D₄×C₈₉	—	2	2	no	—
45	C₂×D₄×C₈₉	C₂³	1	4	yes	—
46	D₈×C₈₉	—	2	4	no	—
47	D₈×C₈₉	—	4	2	no	—
48	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₂³×C₈₉	—	2	4	no	—
52	D₄×C₈₉	—	2	4	no	—
53	D₄×C₈₉	—	2	4	no	—
54	D₄×C₈₉	—	4	4	no	—
55	D₄×C₈₉	—	8	2	no	—
56	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₈₉	—	2	2	no	—
60	C₂×Dic₈₉	—	2	2	no	—
61	C₂×Dic₈₉	—	8	4	no	—
62	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
63	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
64	Dic₁₇₈	—	2	2	no	—
65	Dic₁₇₈	—	2	2	no	—
66	Dic₁₇₈	—	2	2	no	—
67	C₈×C₈₉	—	2	1	no	—
68	C₈×C₈₉	#41	1	2	yes	—
69	#10623494	—	2	2	no	—
70	C₂²×C₈₉	—	4	4	no	—
71	C₂²×C₈₉	—	4	4	no	—
72	C₂²×C₈₉	—	4	4	no	—
73	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₈₉	—	2	1	no	—
75	C₄×C₈₉	#135	1	2	yes	—
76	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
77	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
78	Dic₈₉	—	2	2	no	—
79	Dic₈₉	—	2	2	no	—
80	Dic₈₉	—	4	2	no	—
81	Dic₈₉	—	4	2	no	—
82	C₂×C₈₉	—	8	4	no	—
83	C₂×C₈₉	#550	1	1	yes	characteristic
84	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
85	C₂×C₈₉	#576	1	1	yes	characteristic
86	#3396	—	89	1	no	maximal, Sylow
87	C₈₉	#3396	1	1	yes	characteristic, complemented, Sylow
88	#559	—	89	2	no	—
89	#495	—	89	2	no	—
90	#493	D₈₉	1	1	yes	characteristic
91	#463	—	89	4	no	—
92	#460	—	89	2	no	—
93	C₈×Q₈	—	89	1	no	—
94	C₄×Q₁₆	—	89	1	no	—
95	C₄×D₈	—	89	2	no	—
96	C₂×Q₁₆	—	89	1	no	—
97	C₂×SD₁₆	—	89	4	no	—
98	C₂×D₈	C₂×D₈₉	1	2	yes	—
99	C₂×D₈	C₂×D₈₉	1	2	yes	—
100	#127	C₂×D₈₉	1	2	yes	—
101	#124	—	89	4	no	—
102	#124	—	178	2	no	—
103	#123	—	178	4	no	—
104	C₄×Q₈	—	89	1	no	—
105	C₄×Q₈	—	89	2	no	—
106	C₄×D₄	—	89	4	no	—
107	C₄×D₄	—	178	2	no	—
108	#107	—	89	1	no	—
109	C₄⋊C₈	—	89	2	no	—
110	C₄⋊C₈	—	89	1	no	—
111	#103	—	89	2	no	—
112	#102	—	89	4	no	—
113	C₄×C₈	—	89	2	no	—
114	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
115	C₂×Q₈	—	89	1	no	—
116	C₂×Q₈	—	89	2	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	C₂²×D₈₉	1	4	yes	—
120	C₂²×C₄	—	178	4	no	—
121	Q₁₆	—	178	2	no	—
122	SD₁₆	—	356	4	no	—
123	D₈	—	2	4	no	—
124	D₈	—	4	2	no	—
125	C₂×C₈	—	89	2	no	—
126	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
127	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
128	C₄⋊C₄	—	89	1	no	—
129	C₄⋊C₄	—	89	2	no	—
130	C₄⋊C₄	—	89	2	no	—
131	C₄⋊C₄	—	178	2	no	—
132	C₂²⋊C₄	—	178	4	no	—
133	C₂²⋊C₄	—	178	4	no	—
134	C₂²⋊C₄	—	178	4	no	—
135	C₄²	—	89	2	no	—
136	C₄²	—	178	2	no	—
137	C₄²	C₂²×D₈₉	1	1	yes	characteristic
138	C₂³	—	2	4	no	—
139	Q₈	—	178	2	no	—
140	Q₈	—	178	2	no	—
141	Q₈	—	178	2	no	—
142	D₄	—	2	4	no	—
143	D₄	—	2	4	no	—
144	D₄	—	4	4	no	—
145	D₄	—	8	2	no	—
146	C₂×C₄	—	89	2	no	—
147	C₂×C₄	—	89	2	no	—
148	C₂×C₄	—	178	2	no	—
149	C₂×C₄	—	178	2	no	—
150	C₂×C₄	—	712	4	no	—
151	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
152	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	178	2	no	—
156	C₈	#13083573	1	2	yes	—
157	C₂²	—	4	4	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	356	2	no	—
163	C₄	—	356	2	no	—
164	C₄	—	178	2	no	—
165	C₄	—	178	2	no	—
166	C₄	#15668031	1	2	yes	—
167	C₄	C₂×#13083573	1	1	yes	characteristic
168	C₄	#15668057	1	1	yes	characteristic
169	C₂	—	8	4	no	—
170	C₂	#19152268	1	1	yes	characteristic
171	C₂	C₂×#15668031	1	1	yes	characteristic
172	C₂	#19152399	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152334

C₂

yes

maximal, complemented

#19151981

C₂

yes

maximal, complemented

#19151811

C₂

yes

maximal, complemented

#19151795

C₂

yes

maximal, complemented

C₈×Dic₁₇₈

C₂

yes

maximal, characteristic, complemented

C₄×Dic₃₅₆

C₂

yes

maximal, characteristic, complemented

D₈×Dic₈₉

C₂

yes

maximal, complemented

#493×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

characteristic

C₂×#13083562

C₂²

yes

—

#15667999

—

#15667999

C₂²

yes

—

#15667993

—

C₄×Dic₁₇₈

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

characteristic

D₄×Dic₈₉

—

D₄×Dic₈₉

C₂²

yes

—

#15667946

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

characteristic

#15667941

C₂²

yes

—

#15667937

C₂²

yes

—

#15667933

C₂²

yes

—

C₈×Dic₈₉

C₂²

yes

—

C₂×D₈×C₈₉

C₂²

yes

—

C₂×D₈×C₈₉

C₂²

yes

—

#127×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×Dic₁₇₈

C₂³

yes

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

#13083562

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

D₄

yes

characteristic

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

D₄

yes

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂³

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂³

yes

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

#41

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

C₂×#41

yes

characteristic

C₄×C₈₉

H(5,𝔽₂)

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#559

—

#495

—

#493

D₈₉

yes

characteristic

#463

—

#460

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×D₈

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

C₂×D₈₉

yes

—

C₂×D₈

C₂×D₈₉

yes

—

100

#127

C₂×D₈₉

yes

—

101

#124

—

102

#124

—

178

—

103

#123

—

178

—

104

C₄×Q₈

—

105

C₄×Q₈

—

106

C₄×D₄

—

107

C₄×D₄

—

178

—

108

#107

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

—

112

#102

—

113

C₄×C₈

—

114

C₄×C₈

C₂×D₈₉

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

C₂²×D₈₉

yes

—

120

C₂²×C₄

—

178

—

121

Q₁₆

—

178

—

122

SD₁₆

—

356

—

123

D₈

—

124

D₈

—

125

C₂×C₈

—

126

C₂×C₈

C₂²×D₈₉

yes

characteristic

127

C₂×C₈

C₂²×D₈₉

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

178

—

132

C₂²⋊C₄

—

178

—

133

C₂²⋊C₄

—

178

—

134

C₂²⋊C₄

—

178

—

135

C₄²

—

136

C₄²

—

178

—

137

C₄²

C₂²×D₈₉

yes

characteristic

138

C₂³

—

139

Q₈

—

178

—

140

Q₈

—

178

—

141

Q₈

—

178

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

178

—

149

C₂×C₄

—

178

—

150

C₂×C₄

—

712

—

151

C₂×C₄

D₄×D₈₉

yes

characteristic

152

C₂×C₄

D₄×D₈₉

yes

characteristic

153

C₂×C₄

C₂³×D₈₉

yes

characteristic

154

C₈

—

155

C₈

—

178

—

156

C₈

#13083573

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

C₂×D₄×D₈₉

yes

characteristic

161

C₄

—

162

C₄

—

356

—

163

C₄

—

356

—

164

C₄

—

178

—

165

C₄

—

178

—

166

C₄

#15668031

yes

—

167

C₄

C₂×#13083573

yes

characteristic

168

C₄

#15668057

yes

characteristic

169

C₂

—

170

C₂

#19152268

yes

characteristic

171

C₂

C₂×#15668031

yes

characteristic

172

C₂

#19152399

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting, prime radical	#493×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#493	D₈₉
hypocenter	C₈₉	#3396
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting, prime radical

#493×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #19152334	C₂	semidirect
2. #19151981	C₂	semidirect
3. #19151811	C₂	semidirect
4. #19151795	C₂	semidirect
5. C₈×Dic₁₇₈	C₂	semidirect
6. C₄×Dic₃₅₆	C₂	semidirect
7. D₈×Dic₈₉	C₂	semidirect
87. C₈₉	#3396	semidirect

H ≅ G/N

type

1. #19152334

C₂

semidirect

2. #19151981

C₂

semidirect

3. #19151811

C₂

semidirect

4. #19151795

C₂

semidirect

5. C₈×Dic₁₇₈

C₂

semidirect

6. C₄×Dic₃₅₆

C₂

semidirect

7. D₈×Dic₈₉

C₂

semidirect

87. C₈₉

#3396

semidirect

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

subgroup	index
#19152334	2
#19151981	2
#19151811	2
#19151795	2
C₈×Dic₁₇₈	2
C₄×Dic₃₅₆	2
D₈×Dic₈₉	2
#493×C₈₉	2
#3396	89

subgroup

index

C₈×Dic₁₇₈

C₄×Dic₃₅₆

D₈×Dic₈₉

#493×C₈₉

#3396

Subgroups — #26033266

Conjugacy classes of subgroups

Complements

Maximal subgroups