Group #36467836 — subgroups

Subgroups — #36467836

Order 22784.

356 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26022333	C₂	1	2	yes	maximal, complemented
2	#26019868	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018291	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018225	C₂	1	2	yes	maximal, complemented
5	C₈₉×#3293	C₂	1	1	yes	maximal, characteristic
6	#19152038	—	2	2	no	—
7	#19152001	—	2	2	no	—
8	#19151978	C₂²	1	1	yes	characteristic
9	#19151419	C₂²	1	2	yes	—
10	#19151413	—	2	2	no	—
11	C₈×#10623494	C₂²	1	1	yes	characteristic
12	#578×C₈₉	—	2	2	no	—
13	#494×C₈₉	C₂²	1	1	yes	characteristic
14	#473×C₈₉	—	2	2	no	—
15	#456×C₈₉	—	2	2	no	—
16	#443×C₈₉	C₂²	1	2	yes	—
17	C₂×Dic₃₅₆	—	2	2	no	—
18	#15668012	—	2	1	no	—
19	#15667958	—	4	2	no	—
20	#15667956	—	4	2	no	—
21	#15667952	—	2	2	no	—
22	#15667945	—	4	4	no	—
23	C₄×#10623494	—	2	1	no	—
24	2₋¹⁺⁴×C₈₉	—	2	2	no	—
25	#137×C₈₉	—	2	4	no	—
26	#137×C₈₉	—	4	4	no	—
27	#135×C₈₉	—	4	2	no	—
28	#135×C₈₉	D₄	1	2	yes	—
29	C₂×Q₁₆×C₈₉	—	2	2	no	—
30	C₂×Q₁₆×C₈₉	—	4	2	no	—
31	#131×C₈₉	—	2	2	no	—
32	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
33	#108×C₈₉	D₄	1	2	yes	—
34	#104×C₈₉	—	2	2	no	—
35	#101×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×Dic₁₇₈	—	4	1	no	—
38	Dic₃₅₆	—	4	2	no	—
39	Dic₃₅₆	—	4	2	no	—
40	C₂×#10623494	—	2	2	no	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	#13083549	—	4	2	no	—
43	#41×C₈₉	—	2	2	no	—
44	#41×C₈₉	—	4	4	no	—
45	#41×C₈₉	—	8	2	no	—
46	C₂×Q₈×C₈₉	—	2	2	no	—
47	C₂×Q₈×C₈₉	—	4	4	no	—
48	Q₁₆×C₈₉	—	4	4	no	—
49	Q₁₆×C₈₉	—	4	2	no	—
50	Q₁₆×C₈₉	—	4	2	no	—
51	Q₁₆×C₈₉	D₈	1	2	yes	—
52	SD₁₆×C₈₉	—	2	2	no	—
53	SD₁₆×C₈₉	—	4	4	no	—
54	D₈×C₈₉	D₈	1	2	yes	—
55	M₁₆×C₈₉	—	2	2	no	—
56	M₁₆×C₈₉	—	4	2	no	—
57	C₂×C₈×C₈₉	—	4	2	no	—
58	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
59	C₄⋊C₄×C₈₉	—	4	2	no	—
60	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
61	Q₈×C₈₉	—	2	2	no	—
62	Q₈×C₈₉	—	4	4	no	—
63	Q₈×C₈₉	—	8	2	no	—
64	D₄×C₈₉	—	2	2	no	—
65	D₄×C₈₉	—	4	2	no	—
66	D₄×C₈₉	—	8	2	no	—
67	C₂×C₄×C₈₉	—	2	1	no	—
68	C₂×C₄×C₈₉	—	4	2	no	—
69	C₂×C₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	—	8	4	no	—
71	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
72	C₂×Dic₈₉	—	8	1	no	—
73	Dic₁₇₈	—	8	2	no	—
74	C₈×C₈₉	—	4	2	no	—
75	C₈×C₈₉	C₂×D₈	1	2	yes	—
76	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
77	#10623494	—	2	4	no	—
78	C₂²×C₈₉	—	4	2	no	—
79	C₂²×C₈₉	#453	1	1	yes	characteristic
80	C₄×C₈₉	—	2	2	no	—
81	C₄×C₈₉	—	4	2	no	—
82	C₄×C₈₉	—	8	2	no	—
83	C₄×C₈₉	#447	1	2	yes	—
84	Dic₈₉	—	16	1	no	—
85	#12313	—	89	1	no	maximal, Sylow
86	C₂×C₈₉	—	2	1	no	—
87	C₂×C₈₉	—	8	2	no	—
88	C₂×C₈₉	#1619	1	1	yes	characteristic
89	#3293	D₈₉	1	1	yes	characteristic
90	#2214	—	89	2	no	—
91	#1713	—	89	1	no	—
92	#1358	—	89	1	no	—
93	#1335	—	89	2	no	—
94	C₈₉	#12313	1	1	yes	characteristic, complemented, Sylow
95	#578	—	2	2	no	—
96	#510	—	178	2	no	—
97	#500	—	178	2	no	—
98	#494	—	89	1	no	—
99	#494	C₂×D₈₉	1	1	yes	characteristic
100	#473	—	2	2	no	—
101	#456	—	2	2	no	—
102	#443	C₂×D₈₉	1	2	yes	—
103	#364	—	89	2	no	—
104	#362	—	178	2	no	—
105	C₈²	—	89	1	no	—
106	2₋¹⁺⁴	—	2	2	no	—
107	#137	—	2	4	no	—
108	#137	—	4	4	no	—
109	#135	—	4	2	no	—
110	#135	C₂²⋊D₈₉	1	2	yes	—
111	C₂×Q₁₆	—	2	2	no	—
112	C₂×Q₁₆	—	4	2	no	—
113	C₂×Q₁₆	—	178	2	no	—
114	#131	—	2	2	no	—
115	C₄⋊Q₈	—	178	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	C₂²⋊D₈₉	1	2	yes	—
121	#107	—	356	4	no	—
122	#104	—	2	2	no	—
123	#101	—	2	2	no	—
124	C₄×C₈	—	178	1	no	—
125	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
126	#41	—	2	2	no	—
127	#41	—	4	4	no	—
128	#41	—	8	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	4	4	no	—
131	C₂×Q₈	—	356	1	no	—
132	Q₁₆	—	4	4	no	—
133	Q₁₆	—	4	2	no	—
134	Q₁₆	—	4	2	no	—
135	Q₁₆	—	356	2	no	—
136	Q₁₆	—	356	2	no	—
137	Q₁₆	#13083559	1	2	yes	—
138	SD₁₆	—	2	2	no	—
139	SD₁₆	—	4	4	no	—
140	D₈	#13083559	1	2	yes	—
141	M₁₆	—	2	2	no	—
142	M₁₆	—	4	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	178	2	no	—
145	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	356	2	no	—
148	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	C₁₆	—	356	2	no	—
150	Q₈	—	2	2	no	—
151	Q₈	—	4	4	no	—
152	Q₈	—	8	2	no	—
153	Q₈	—	712	2	no	—
154	D₄	—	2	2	no	—
155	D₄	—	4	2	no	—
156	D₄	—	8	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	712	1	no	—
162	C₂×C₄	#15667981	1	1	yes	characteristic
163	C₈	—	4	2	no	—
164	C₈	—	178	4	no	—
165	C₈	C₂×#13083559	1	2	yes	—
166	C₈	#15668038	1	2	yes	—
167	C₂²	—	4	2	no	—
168	C₂²	#19151766	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	2	no	—
172	C₄	—	1424	1	no	—
173	C₄	#19151732	1	2	yes	—
174	C₂	—	2	1	no	—
175	C₂	—	8	2	no	—
176	C₂	#26019306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022333

C₂

yes

maximal, complemented

#26019868

C₂

yes

maximal, characteristic, complemented

#26018291

C₂

yes

maximal, characteristic, complemented

#26018225

C₂

yes

maximal, complemented

C₈₉×#3293

C₂

yes

maximal, characteristic

#19152038

—

#19152001

—

#19151978

C₂²

yes

characteristic

#19151419

C₂²

yes

—

#19151413

—

C₈×#10623494

C₂²

yes

characteristic

#578×C₈₉

—

#494×C₈₉

C₂²

yes

characteristic

#473×C₈₉

—

#456×C₈₉

—

#443×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

—

—

—

—

—

—

—

2₋¹⁺⁴×C₈₉

—

#137×C₈₉

—

#137×C₈₉

—

#135×C₈₉

—

#135×C₈₉

D₄

yes

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

#131×C₈₉

—

C₄⋊Q₈×C₈₉

D₄

yes

—

#108×C₈₉

D₄

yes

—

#104×C₈₉

—

#101×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083549

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

D₈

yes

—

SD₁₆×C₈₉

—

SD₁₆×C₈₉

—

D₈×C₈₉

D₈

yes

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#453

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#447

yes

—

Dic₈₉

—

#12313

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1619

yes

characteristic

#3293

D₈₉

yes

—

—

—

—

C₈₉

#12313

yes

characteristic, complemented, Sylow

#578

—

#510

—

178

—

#500

—

178

—

#494

—

#494

C₂×D₈₉

yes

characteristic

100

#473

—

101

#456

—

102

#443

C₂×D₈₉

yes

—

103

#364

—

104

#362

—

178

—

105

C₈²

—

106

2₋¹⁺⁴

—

107

#137

—

108

#137

—

109

#135

—

110

#135

C₂²⋊D₈₉

yes

—

111

C₂×Q₁₆

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

178

—

114

#131

—

115

C₄⋊Q₈

—

178

—

116

C₄⋊Q₈

C₂²⋊D₈₉

yes

—

117

Q₃₂

—

356

—

118

SD₃₂

—

356

—

119

M₃₂

—

178

—

120

#108

C₂²⋊D₈₉

yes

—

121

#107

—

356

—

122

#104

—

123

#101

—

124

C₄×C₈

—

178

—

125

C₄×C₈

C₂²×D₈₉

yes

characteristic

126

#41

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

356

—

132

Q₁₆

—

133

Q₁₆

—

134

Q₁₆

—

135

Q₁₆

—

356

—

136

Q₁₆

—

356

—

137

Q₁₆

#13083559

yes

—

138

SD₁₆

—

139

SD₁₆

—

140

D₈

#13083559

yes

—

141

M₁₆

—

142

M₁₆

—

143

C₂×C₈

—

144

C₂×C₈

—

178

—

145

C₂×C₈

C₂×C₂²⋊D₈₉

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

356

—

148

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

149

C₁₆

—

356

—

150

Q₈

—

151

Q₈

—

152

Q₈

—

153

Q₈

—

712

—

154

D₄

—

155

D₄

—

156

D₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

712

—

162

C₂×C₄

#15667981

yes

characteristic

163

C₈

—

164

C₈

—

178

—

165

C₈

C₂×#13083559

yes

—

166

C₈

#15668038

yes

—

167

C₂²

—

168

C₂²

#19151766

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

1424

—

173

C₄

#19151732

yes

—

174

C₂

—

175

C₂

—

176

C₂

#26019306

yes

characteristic

	subgroup	G/H
center	C₂	#26019306
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting, prime radical	C₈₉×#3293	C₂
socle	C₂×C₈₉	#1619
hypercenter	#3293	D₈₉
hypocenter	C₈₉	#12313
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3293

subgroup

G/H

center

C₂

#26019306

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting, prime radical

C₈₉×#3293

C₂

C₂×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3293

N	H ≅ G/N	type
1. #26022333	C₂	semidirect
2. #26019868	C₂	semidirect
3. #26018291	C₂	semidirect
4. #26018225	C₂	semidirect
94. C₈₉	#12313	semidirect

H ≅ G/N

type

1. #26022333

C₂

semidirect

2. #26019868

C₂

semidirect

3. #26018291

C₂

semidirect

4. #26018225

C₂

semidirect

94. C₈₉

#12313

semidirect

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

subgroup	index
#26022333	2
#26019868	2
#26018291	2
#26018225	2
C₈₉×#3293	2
#12313	89

subgroup

index

C₈₉×#3293

Subgroups — #36467836

Conjugacy classes of subgroups

Complements

Maximal subgroups