Group #36467834 — subgroups

Subgroups — #36467834

Order 22784.

356 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 59 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	#26032056	C₂	1	1	yes	maximal, characteristic
2	#26022330	C₂	1	2	yes	maximal, complemented
3	#26018291	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018226	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152411	—	2	2	no	—
7	#19152039	—	2	2	no	—
8	#19151870	—	2	2	no	—
9	#19151780	—	2	2	no	—
10	#19151716	C₂²	1	2	yes	—
11	#19151419	C₂²	1	2	yes	—
12	#19151414	—	2	2	no	—
13	#500×C₈₉	—	2	2	no	—
14	#494×C₈₉	C₂²	1	1	yes	characteristic
15	#494×C₈₉	C₂²	1	1	yes	characteristic
16	C₈²×C₈₉	C₂²	1	1	yes	characteristic
17	#15668060	—	2	2	no	—
18	#15668044	—	4	4	no	—
19	#15668043	—	2	4	no	—
20	#15668035	—	4	2	no	—
21	#15668034	D₄	1	2	yes	—
22	Q₁₆×D₈₉	—	4	2	no	—
23	#15668018	—	2	2	no	—
24	#15667958	—	4	2	no	—
25	#15667955	—	4	2	no	—
26	#15667952	—	2	2	no	—
27	#15667948	D₄	1	2	yes	—
28	#15667940	—	2	2	no	—
29	#15667931	—	2	2	no	—
30	C₂×Q₁₆×C₈₉	—	2	2	no	—
31	C₂×Q₁₆×C₈₉	—	2	2	no	—
32	C₄⋊Q₈×C₈₉	—	2	1	no	—
33	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
34	#107×C₈₉	—	4	4	no	—
35	C₄×C₈×C₈₉	—	2	1	no	—
36	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	#13083575	—	4	4	no	—
38	#13083574	—	2	2	no	—
39	#13083574	—	8	2	no	—
40	Q₈×D₈₉	—	4	4	no	—
41	Dic₃₅₆	D₈	1	2	yes	—
42	#13083564	—	4	4	no	—
43	#13083563	—	2	2	no	—
44	#13083561	—	4	4	no	—
45	D₇₁₂	D₈	1	2	yes	—
46	#13083558	—	2	2	no	—
47	#13083557	—	4	2	no	—
48	C₈×D₈₉	—	4	2	no	—
49	#13083549	—	4	2	no	—
50	C₂×Q₈×C₈₉	—	2	2	no	—
51	C₂×Q₈×C₈₉	—	4	1	no	—
52	Q₁₆×C₈₉	—	4	2	no	—
53	Q₁₆×C₈₉	—	4	2	no	—
54	Q₁₆×C₈₉	—	4	2	no	—
55	Q₁₆×C₈₉	—	4	2	no	—
56	C₂×C₈×C₈₉	—	2	2	no	—
57	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
58	C₄⋊C₄×C₈₉	—	4	2	no	—
59	C₄⋊C₄×C₈₉	—	4	2	no	—
60	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
61	Q₈×C₈₉	—	4	4	no	—
62	Q₈×C₈₉	—	8	2	no	—
63	C₂×C₄×C₈₉	—	2	1	no	—
64	C₂×C₄×C₈₉	—	4	2	no	—
65	C₂×C₄×C₈₉	—	8	1	no	—
66	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
67	C₂²⋊D₈₉	—	4	2	no	—
68	D₃₅₆	—	2	2	no	—
69	D₃₅₆	—	8	2	no	—
70	C₄×D₈₉	—	4	2	no	—
71	C₄×D₈₉	—	8	4	no	—
72	Dic₁₇₈	—	2	2	no	—
73	Dic₁₇₈	—	8	2	no	—
74	C₈×C₈₉	—	2	4	no	—
75	C₈×C₈₉	C₂×D₈	1	2	yes	—
76	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
77	#10623494	—	4	2	no	—
78	C₂²×C₈₉	#453	1	1	yes	characteristic
79	C₂×D₈₉	—	4	2	no	—
80	C₄×C₈₉	—	2	2	no	—
81	C₄×C₈₉	—	8	2	no	—
82	C₄×C₈₉	—	16	1	no	—
83	C₄×C₈₉	#447	1	2	yes	—
84	Dic₈₉	—	4	2	no	—
85	#12313	—	89	1	no	maximal, Sylow
86	C₂×C₈₉	—	2	1	no	—
87	C₂×C₈₉	#1619	1	1	yes	characteristic
88	D₈₉	—	8	2	no	—
89	#3293	—	89	1	no	—
90	#2214	—	89	2	no	—
91	#1713	D₈₉	1	1	yes	characteristic, complemented
92	#1358	—	89	1	no	—
93	#1335	—	89	2	no	—
94	C₈₉	#12313	1	1	yes	characteristic, complemented, Sylow
95	#578	—	178	2	no	—
96	#510	—	178	2	no	—
97	#500	—	2	2	no	—
98	#494	C₂×D₈₉	1	1	yes	characteristic
99	#494	C₂×D₈₉	1	1	yes	characteristic
100	#473	—	178	2	no	—
101	#456	—	178	2	no	—
102	#443	—	89	2	no	—
103	#364	—	89	2	no	—
104	#362	—	178	2	no	—
105	C₈²	C₂×D₈₉	1	1	yes	characteristic
106	2₋¹⁺⁴	—	178	2	no	—
107	#137	—	356	4	no	—
108	#137	—	178	4	no	—
109	#135	—	89	2	no	—
110	#135	—	356	2	no	—
111	C₂×Q₁₆	—	2	2	no	—
112	C₂×Q₁₆	—	2	2	no	—
113	C₂×Q₁₆	—	356	2	no	—
114	#131	—	178	2	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	C₂²⋊D₈₉	1	2	yes	—
117	Q₃₂	—	356	2	no	—
118	SD₃₂	—	356	2	no	—
119	M₃₂	—	178	2	no	—
120	#108	—	89	2	no	—
121	#107	—	4	4	no	—
122	#104	—	178	2	no	—
123	#101	—	178	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
126	#41	—	356	4	no	—
127	#41	—	178	2	no	—
128	#41	—	712	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	4	1	no	—
131	C₂×Q₈	—	356	4	no	—
132	Q₁₆	—	4	2	no	—
133	Q₁₆	—	4	2	no	—
134	Q₁₆	—	4	2	no	—
135	Q₁₆	—	4	2	no	—
136	Q₁₆	—	89	2	no	—
137	Q₁₆	—	356	4	no	—
138	SD₁₆	—	356	4	no	—
139	SD₁₆	—	178	2	no	—
140	D₈	—	89	2	no	—
141	M₁₆	—	356	2	no	—
142	M₁₆	—	178	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	356	2	no	—
145	C₂×C₈	D₄×D₈₉	1	2	yes	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	C₁₆	—	356	2	no	—
150	Q₈	—	4	4	no	—
151	Q₈	—	8	2	no	—
152	Q₈	—	178	2	no	—
153	Q₈	—	712	2	no	—
154	D₄	—	356	2	no	—
155	D₄	—	178	2	no	—
156	D₄	—	712	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	356	2	no	—
161	C₂×C₄	—	712	4	no	—
162	C₂×C₄	#15667980	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₈	—	356	2	no	—
165	C₈	D₈×D₈₉	1	2	yes	—
166	C₈	#15668036	1	2	yes	—
167	C₂²	—	356	2	no	—
168	C₂²	#19151765	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	8	2	no	—
171	C₄	—	16	1	no	—
172	C₄	—	356	2	no	—
173	C₄	#19151730	1	2	yes	—
174	C₂	—	2	1	no	—
175	C₂	—	712	2	no	—
176	C₂	#26019304	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26032056

C₂

yes

maximal, characteristic

#26022330

C₂

yes

maximal, complemented

#26018291

C₂

yes

maximal, characteristic, complemented

#26018226

C₂

yes

maximal, complemented

C₈₉×#1713

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

—

#19151419

C₂²

yes

—

#19151414

—

#500×C₈₉

—

#494×C₈₉

C₂²

yes

characteristic

#494×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₂²

yes

—

—

—

—

D₄

yes

—

Q₁₆×D₈₉

—

—

—

—

—

D₄

yes

—

#15667940

—

#15667931

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

D₄

yes

—

#107×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

—

—

—

Q₈×D₈₉

—

Dic₃₅₆

D₈

yes

—

#13083564

—

#13083563

—

#13083561

—

D₇₁₂

D₈

yes

—

#13083558

—

#13083557

—

C₈×D₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂²⋊D₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

#10623494

—

C₂²×C₈₉

#453

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#447

yes

—

Dic₈₉

—

#12313

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1619

yes

characteristic

D₈₉

—

#3293

—

#2214

—

#1713

D₈₉

yes

characteristic, complemented

#1358

—

#1335

—

C₈₉

#12313

yes

characteristic, complemented, Sylow

#578

—

178

—

#510

—

178

—

#500

—

#494

C₂×D₈₉

yes

characteristic

#494

C₂×D₈₉

yes

characteristic

100

#473

—

178

—

101

#456

—

178

—

102

#443

—

103

#364

—

104

#362

—

178

—

105

C₈²

C₂×D₈₉

yes

characteristic

106

2₋¹⁺⁴

—

178

—

107

#137

—

356

—

108

#137

—

178

—

109

#135

—

110

#135

—

356

—

111

C₂×Q₁₆

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

356

—

114

#131

—

178

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

C₂²⋊D₈₉

yes

—

117

Q₃₂

—

356

—

118

SD₃₂

—

356

—

119

M₃₂

—

178

—

120

#108

—

121

#107

—

122

#104

—

178

—

123

#101

—

178

—

124

C₄×C₈

—

125

C₄×C₈

C₂²×D₈₉

yes

characteristic

126

#41

—

356

—

127

#41

—

178

—

128

#41

—

712

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

356

—

132

Q₁₆

—

133

Q₁₆

—

134

Q₁₆

—

135

Q₁₆

—

136

Q₁₆

—

137

Q₁₆

—

356

—

138

SD₁₆

—

356

—

139

SD₁₆

—

178

—

140

D₈

—

141

M₁₆

—

356

—

142

M₁₆

—

178

—

143

C₂×C₈

—

144

C₂×C₈

—

356

—

145

C₂×C₈

D₄×D₈₉

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

149

C₁₆

—

356

—

150

Q₈

—

151

Q₈

—

152

Q₈

—

178

—

153

Q₈

—

712

—

154

D₄

—

356

—

155

D₄

—

178

—

156

D₄

—

712

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

356

—

161

C₂×C₄

—

712

—

162

C₂×C₄

#15667980

yes

characteristic

163

C₈

—

164

C₈

—

356

—

165

C₈

D₈×D₈₉

yes

—

166

C₈

#15668036

yes

—

167

C₂²

—

356

—

168

C₂²

#19151765

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

356

—

173

C₄

#19151730

yes

—

174

C₂

—

175

C₂

—

712

—

176

C₂

#26019304

yes

characteristic

	subgroup	G/H
center	C₂	#26019304
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1713	C₂
socle	C₂×C₈₉	#1619
prime radical	#26032056	C₂
hypercenter	#1713	D₈₉
hypocenter	C₈₉	#12313
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1713

subgroup

G/H

center

C₂

#26019304

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1713

C₂

C₂×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1713

N	H ≅ G/N	type
2. #26022330	C₂	semidirect
3. #26018291	C₂	semidirect
4. #26018226	C₂	semidirect
5. C₈₉×#1713	C₂	semidirect
91. #1713	D₈₉	semidirect
94. C₈₉	#12313	semidirect

H ≅ G/N

type

2. #26022330

C₂

semidirect

3. #26018291

C₂

semidirect

4. #26018226

C₂

semidirect

5. C₈₉×#1713

C₂

semidirect

91. #1713

D₈₉

semidirect

94. C₈₉

#12313

semidirect

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

subgroup	index
#26032056	2
#26022330	2
#26018291	2
#26018226	2
C₈₉×#1713	2
#12313	89

subgroup

index

C₈₉×#1713

Subgroups — #36467834

Conjugacy classes of subgroups

Complements

Maximal subgroups