Group #36453890 — subgroups

Subgroups — #36453890

Order 22784.

478 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 151 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×#19151978	C₂	1	1	yes	maximal, characteristic
2	#26018925	C₂	1	4	yes	maximal
3	#26018535	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₅₆	—	2	2	no	—
6	C₂×#15668012	C₂²	1	2	yes	—
7	#19151978	—	2	1	no	—
8	#19151978	—	2	1	no	—
9	#19151978	C₄	1	4	yes	—
10	#19151651	C₂²	1	2	yes	—
11	#19151345	—	2	8	no	—
12	#19151319	—	2	1	no	—
13	C₈×#10623494	—	2	1	no	—
14	C₂×#108×C₈₉	—	2	2	no	—
15	#423×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₃₅₆	—	2	4	no	—
19	C₂×Dic₃₅₆	—	2	4	no	—
20	C₂×Dic₃₅₆	—	4	4	no	—
21	C₂×Dic₃₅₆	—	4	4	no	—
22	#15668012	—	2	4	no	—
23	#15668012	—	2	2	no	—
24	#15668012	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₈₉	—	2	4	no	—
26	C₄⋊#10623494	D₄	1	4	yes	—
27	C₂²⋊#10623494	—	2	2	no	—
28	C₄×#10623494	—	2	2	no	—
29	C₂×M₁₆×C₈₉	—	2	2	no	—
30	C₂²×C₈×C₈₉	D₄	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	#108×C₈₉	—	4	4	no	—
33	C₄⋊C₈×C₈₉	—	2	4	no	—
34	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
36	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₂×Dic₁₇₈	—	2	4	no	—
38	C₂×Dic₁₇₈	—	4	4	no	—
39	C₂×Dic₁₇₈	—	4	4	no	—
40	C₂×Dic₁₇₈	—	4	2	no	—
41	C₂²×Dic₈₉	—	4	2	no	—
42	Dic₃₅₆	—	4	8	no	—
43	Dic₃₅₆	—	4	8	no	—
44	C₂×#10623494	—	2	4	no	—
45	C₄⋊Dic₈₉	—	2	8	no	—
46	C₄⋊Dic₈₉	—	4	4	no	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	#41	1	2	yes	—
49	M₁₆×C₈₉	—	4	4	no	—
50	C₂×C₈×C₈₉	—	4	2	no	—
51	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₈₉	D₈	1	4	yes	—
54	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
55	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂×D₄	1	2	yes	—
58	C₂³×C₈₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	2	4	no	—
60	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₈₉	#121	1	2	yes	—
63	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
66	C₂×Dic₈₉	—	4	4	no	—
67	C₂×Dic₈₉	—	8	4	no	—
68	Dic₁₇₈	—	4	8	no	—
69	Dic₁₇₈	—	8	2	no	—
70	Dic₁₇₈	—	8	2	no	—
71	C₈×C₈₉	—	8	2	no	—
72	C₈×C₈₉	#102	1	4	yes	—
73	C₈×C₈₉	#102	1	4	yes	—
74	#10623494	—	4	4	no	—
75	C₂²×C₈₉	—	2	2	no	—
76	C₂²×C₈₉	#390	1	1	yes	characteristic
77	C₂²×C₈₉	#486	1	2	yes	—
78	C₄×C₈₉	—	2	2	no	—
79	C₄×C₈₉	C₂×#102	1	2	yes	—
80	C₄×C₈₉	#473	1	4	yes	—
81	C₄×C₈₉	#492	1	1	yes	characteristic
82	C₄×C₈₉	#493	1	1	yes	characteristic
83	Dic₈₉	—	8	4	no	—
84	#9772	—	89	1	no	maximal, Sylow
85	C₂×C₈₉	—	2	2	no	—
86	C₂×C₈₉	#1957	1	1	yes	characteristic
87	C₂×C₈₉	#1979	1	2	yes	—
88	C₂×#494	—	89	1	no	—
89	#1848	D₈₉	1	1	yes	characteristic
90	#1546	—	89	4	no	—
91	#1450	—	89	1	no	—
92	C₈₉	#9772	1	1	yes	characteristic, complemented, Sylow
93	C₂²×Q₁₆	—	178	2	no	—
94	C₂×C₄⋊Q₈	—	89	2	no	—
95	#494	—	89	4	no	—
96	#494	—	178	1	no	—
97	#494	—	178	1	no	—
98	#431	—	89	2	no	—
99	C₂×#108	—	2	2	no	—
100	#423	C₂×D₈₉	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
102	#332	—	178	8	no	—
103	#322	—	178	1	no	—
104	C₈²	—	178	1	no	—
105	C₂²×Q₈	—	178	2	no	—
106	C₂×Q₁₆	—	356	4	no	—
107	C₂×Q₁₆	—	356	4	no	—
108	C₂×Q₁₆	—	178	4	no	—
109	C₂×Q₁₆	—	178	4	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
112	C₄⋊Q₈	—	89	4	no	—
113	C₄⋊Q₈	—	178	4	no	—
114	C₄⋊Q₈	—	178	2	no	—
115	C₂×C₄⋊C₄	—	178	4	no	—
116	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
117	#108	—	4	4	no	—
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	89	4	no	—
120	C₂²⋊C₈	—	178	2	no	—
121	C₄×C₈	—	178	2	no	—
122	C₄×C₈	C₄×D₈₉	1	2	yes	—
123	C₄×C₈	D₃₅₆	1	1	yes	characteristic
124	C₄×C₈	D₃₅₆	1	1	yes	characteristic
125	C₂×Q₈	—	356	2	no	—
126	C₂×Q₈	—	356	4	no	—
127	C₂×Q₈	—	356	4	no	—
128	C₂×Q₈	—	178	4	no	—
129	C₂²×C₄	—	356	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
131	C₂²×C₄	#13083575	1	2	yes	—
132	Q₁₆	—	356	8	no	—
133	Q₁₆	—	356	8	no	—
134	M₁₆	—	4	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	178	4	no	—
137	C₂×C₈	#13083551	1	2	yes	—
138	C₂×C₈	#13083551	1	2	yes	—
139	C₂×C₈	#13083559	1	4	yes	—
140	C₂×C₈	#13083561	1	4	yes	—
141	C₄⋊C₄	—	356	4	no	—
142	C₄⋊C₄	—	178	8	no	—
143	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
144	C₄²	D₄×D₈₉	1	2	yes	—
145	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
146	C₂³	#15668001	1	1	yes	characteristic
147	Q₈	—	356	8	no	—
148	Q₈	—	712	2	no	—
149	Q₈	—	712	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	356	4	no	—
152	C₂×C₄	—	712	4	no	—
153	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
154	C₂×C₄	#15667974	1	2	yes	—
155	C₂×C₄	#15667987	1	2	yes	—
156	C₂×C₄	#15668008	1	1	yes	characteristic
157	C₂×C₄	C₂×#13083559	1	2	yes	—
158	C₂×C₄	C₂×#13083561	1	2	yes	—
159	C₈	—	8	2	no	—
160	C₈	—	356	4	no	—
161	C₈	#15667932	1	4	yes	—
162	C₈	#15667932	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	#19151499	1	1	yes	characteristic
165	C₂²	#19151946	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	712	4	no	—
168	C₄	C₂×#15667932	1	2	yes	—
169	C₄	#19151874	1	4	yes	—
170	C₄	#19151972	1	1	yes	characteristic
171	C₄	#19151974	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	#26021057	1	1	yes	characteristic
174	C₂	#26021165	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151978

C₂

yes

maximal, characteristic

#26018925

C₂

yes

maximal

#26018535

C₂

yes

maximal, characteristic

C₈₉×#1848

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₆

—

C₂×#15668012

C₂²

yes

—

#19151978

—

#19151978

—

#19151978

C₄

yes

—

#19151651

C₂²

yes

—

#19151345

—

#19151319

—

C₈×#10623494

—

C₂×#108×C₈₉

—

#423×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂²×Dic₁₇₈

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

#15668012

—

#15668012

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₄⋊#10623494

D₄

yes

—

C₂²⋊#10623494

—

C₄×#10623494

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#108×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

#102

yes

—

C₈×C₈₉

#102

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#486

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#102

yes

—

C₄×C₈₉

#473

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#9772

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1957

yes

characteristic

C₂×C₈₉

#1979

yes

—

C₂×#494

—

#1848

D₈₉

yes

characteristic

#1546

—

#1450

—

C₈₉

#9772

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

178

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

178

—

#494

—

178

—

#431

—

C₂×#108

—

100

#423

C₂×D₈₉

yes

—

101

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

102

#332

—

178

—

103

#322

—

178

—

104

C₈²

—

178

—

105

C₂²×Q₈

—

178

—

106

C₂×Q₁₆

—

356

—

107

C₂×Q₁₆

—

356

—

108

C₂×Q₁₆

—

178

—

109

C₂×Q₁₆

—

178

—

110

C₂×M₁₆

—

111

C₂²×C₈

C₂²⋊D₈₉

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

178

—

114

C₄⋊Q₈

—

178

—

115

C₂×C₄⋊C₄

—

178

—

116

C₂×C₄²

C₂²×D₈₉

yes

characteristic

117

#108

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₂²⋊C₈

—

178

—

121

C₄×C₈

—

178

—

122

C₄×C₈

C₄×D₈₉

yes

—

123

C₄×C₈

D₃₅₆

yes

characteristic

124

C₄×C₈

D₃₅₆

yes

characteristic

125

C₂×Q₈

—

356

—

126

C₂×Q₈

—

356

—

127

C₂×Q₈

—

356

—

128

C₂×Q₈

—

178

—

129

C₂²×C₄

—

356

—

130

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

131

C₂²×C₄

#13083575

yes

—

132

Q₁₆

—

356

—

133

Q₁₆

—

356

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

178

—

137

C₂×C₈

#13083551

yes

—

138

C₂×C₈

#13083551

yes

—

139

C₂×C₈

#13083559

yes

—

140

C₂×C₈

#13083561

yes

—

141

C₄⋊C₄

—

356

—

142

C₄⋊C₄

—

178

—

143

C₄²

C₂×C₄×D₈₉

yes

characteristic

144

C₄²

D₄×D₈₉

yes

—

145

C₄²

C₂×D₃₅₆

yes

characteristic

146

C₂³

#15668001

yes

characteristic

147

Q₈

—

356

—

148

Q₈

—

712

—

149

Q₈

—

712

—

150

C₂×C₄

—

151

C₂×C₄

—

356

—

152

C₂×C₄

—

712

—

153

C₂×C₄

C₂×#13083551

yes

characteristic

154

C₂×C₄

#15667974

yes

—

155

C₂×C₄

#15667987

yes

—

156

C₂×C₄

#15668008

yes

characteristic

157

C₂×C₄

C₂×#13083559

yes

—

158

C₂×C₄

C₂×#13083561

yes

—

159

C₈

—

160

C₈

—

356

—

161

C₈

#15667932

yes

—

162

C₈

#15667932

yes

—

163

C₂²

—

164

C₂²

#19151499

yes

characteristic

165

C₂²

#19151946

yes

—

166

C₄

—

167

C₄

—

712

—

168

C₄

C₂×#15667932

yes

—

169

C₄

#19151874

yes

—

170

C₄

#19151972

yes

characteristic

171

C₄

#19151974

yes

characteristic

172

C₂

—

173

C₂

#26021057

yes

characteristic

174

C₂

#26021165

yes

—

	subgroup	G/H
center	C₂²	#19151499
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂³×C₈₉	#124
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#9772
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂²

#19151499

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1848

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
92. C₈₉	#9772	semidirect

H ≅ G/N

type

92. C₈₉

#9772

semidirect

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

subgroup	index
C₂×#19151978	2
#26018925	2
#26018535	2
C₈₉×#1848	2
#9772	89

subgroup

index

C₈₉×#1848

Subgroups — #36453890

Conjugacy classes of subgroups

Complements

Maximal subgroups