Group #36448238 — subgroups

Subgroups — #36448238

Order 22784.

338 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 147 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	#26020423	C₂	1	2	yes	maximal
2	#26019008	C₂	1	1	yes	maximal, characteristic
3	#26019007	C₂	1	2	yes	maximal
4	C₂×#19151320	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667947	—	2	4	no	—
7	#19151622	C₂²	1	2	yes	—
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	#19151349	—	2	2	no	—
11	#19151348	—	2	4	no	—
12	#19151320	C₄	1	4	yes	—
13	#423×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	#334×C₈₉	—	2	2	no	—
16	C₂×#13083558	—	2	2	no	—
17	C₂²×#10623494	D₄	1	2	yes	—
18	C₂²×#10623494	Q₈	1	2	yes	—
19	#15667947	—	4	8	no	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	—	2	2	no	—
22	C₄×#10623494	C₂×C₄	1	2	yes	—
23	C₄×#10623494	C₂×C₄	1	2	yes	—
24	C₄×#10623494	D₄	1	2	yes	—
25	C₄×#10623494	Q₈	1	2	yes	—
26	C₂×M₁₆×C₈₉	—	2	2	no	—
27	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
28	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₈₉	—	2	2	no	—
30	C₄⋊C₈×C₈₉	—	2	2	no	—
31	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₉	D₄	1	2	yes	—
33	#13083558	—	4	4	no	—
34	C₂×#10623494	—	4	2	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₂×#10623494	C₄⋊C₄	1	4	yes	—
37	C₂×#10623494	C₄⋊C₄	1	4	yes	—
38	C₂×#10623494	SD₁₆	1	8	yes	—
39	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₈₉	—	4	4	no	—
43	C₂×C₈×C₈₉	—	4	2	no	—
44	C₂×C₈×C₈₉	—	4	2	no	—
45	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
46	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
60	C₈×C₈₉	—	2	4	no	—
61	C₈×C₈₉	—	8	2	no	—
62	#10623494	—	2	4	no	—
63	#10623494	—	8	2	no	—
64	#10623494	#106	1	8	yes	—
65	C₂²×C₈₉	—	2	1	no	—
66	C₂²×C₈₉	—	2	1	no	—
67	C₂²×C₈₉	#384	1	1	yes	characteristic
68	C₂²×C₈₉	#498	1	2	yes	—
69	C₄×C₈₉	—	2	1	no	—
70	C₄×C₈₉	—	2	1	no	—
71	C₄×C₈₉	C₂×#106	1	2	yes	—
72	C₄×C₈₉	#430	1	2	yes	—
73	C₄×C₈₉	#430	1	2	yes	—
74	C₄×C₈₉	#492	1	2	yes	—
75	#8825	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	—	2	2	no	—
77	C₂×C₈₉	#1844	1	1	yes	characteristic
78	C₂×C₈₉	#1849	1	1	yes	characteristic
79	C₂×C₈₉	#1849	1	1	yes	characteristic
80	#1848	—	89	2	no	—
81	#1566	—	89	2	no	—
82	#1566	—	89	1	no	—
83	#1566	D₈₉	1	1	yes	characteristic
84	C₂×#322	—	89	1	no	—
85	C₈₉	#8825	1	1	yes	characteristic, complemented, Sylow
86	C₂×#108	—	178	4	no	—
87	#423	—	89	2	no	—
88	#423	C₂×D₈₉	1	2	yes	—
89	C₂×C₄×C₈	—	89	1	no	—
90	C₂×C₄×C₈	—	89	1	no	—
91	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
92	#334	—	2	2	no	—
93	#334	—	178	4	no	—
94	#334	—	178	2	no	—
95	#322	—	89	4	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	178	2	no	—
98	C₂²×C₈	—	89	2	no	—
99	C₂²×C₈	—	89	2	no	—
100	C₂²×C₈	Dic₁₇₈	1	2	yes	—
101	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
102	#108	—	356	8	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	178	2	no	—
106	C₄⋊C₈	—	178	2	no	—
107	C₄×C₈	—	89	2	no	—
108	C₄×C₈	—	89	2	no	—
109	C₄×C₈	—	89	2	no	—
110	C₄×C₈	—	89	2	no	—
111	C₄×C₈	C₄×D₈₉	1	2	yes	—
112	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
113	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
114	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
116	M₁₆	—	4	4	no	—
117	M₁₆	—	356	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	89	8	no	—
121	C₂×C₈	—	89	4	no	—
122	C₂×C₈	—	89	4	no	—
123	C₂×C₈	—	356	2	no	—
124	C₂×C₈	—	356	2	no	—
125	C₂×C₈	#13083553	1	4	yes	—
126	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
127	C₄²	D₄×D₈₉	1	1	yes	characteristic
128	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
129	C₄²	Q₈×D₈₉	1	1	yes	characteristic
130	C₂³	#15668011	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
134	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
135	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
136	C₂×C₄	#15668008	1	1	yes	characteristic
137	C₂×C₄	#15668010	1	1	yes	characteristic
138	C₂×C₄	#15668011	1	1	yes	characteristic
139	C₂×C₄	SD₁₆×D₈₉	1	4	yes	—
140	C₈	—	2	4	no	—
141	C₈	—	8	2	no	—
142	C₈	—	89	8	no	—
143	C₈	—	178	4	no	—
144	C₈	—	712	2	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#19151467	1	1	yes	characteristic
148	C₂²	#19151995	1	2	yes	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	#106×D₈₉	1	2	yes	—
152	C₄	#19151645	1	2	yes	—
153	C₄	#19151646	1	2	yes	—
154	C₄	#19151969	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	#26020407	1	1	yes	characteristic
157	C₂	#26020425	1	1	yes	characteristic
158	C₂	#26020426	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020423

C₂

yes

maximal

#26019008

C₂

yes

maximal, characteristic

#26019007

C₂

yes

maximal

C₂×#19151320

C₂

yes

maximal, characteristic

C₈₉×#1566

C₂

yes

maximal, characteristic

C₂×#15667947

—

#19151622

C₂²

yes

—

C₂×C₄×#10623494

C₂²

yes

characteristic

C₂×C₄×#10623494

C₂²

yes

—

—

C₄

yes

—

#423×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#334×C₈₉

—

C₂×#13083558

—

C₂²×#10623494

D₄

yes

—

C₂²×#10623494

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

C₄×#10623494

C₂×C₄

yes

—

C₄×#10623494

D₄

yes

—

C₄×#10623494

Q₈

yes

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

Q₈

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10623494

C₄⋊C₄

yes

—

C₂×#10623494

SD₁₆

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

—

—

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#384

yes

characteristic

C₂²×C₈₉

#498

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#106

yes

—

C₄×C₈₉

#430

yes

—

C₄×C₈₉

#430

yes

—

C₄×C₈₉

#492

yes

—

#8825

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

D₈₉

yes

characteristic

C₂×#322

—

C₈₉

#8825

yes

characteristic, complemented, Sylow

C₂×#108

—

178

—

#423

—

#423

C₂×D₈₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#334

—

#334

—

178

—

#334

—

178

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

178

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

Dic₁₇₈

yes

—

101

C₂×C₄²

C₂²×D₈₉

yes

characteristic

102

#108

—

356

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

178

—

106

C₄⋊C₈

—

178

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

C₄×D₈₉

yes

—

112

C₄×C₈

C₂²⋊D₈₉

yes

—

113

C₂²×C₄

D₄×D₈₉

yes

characteristic

114

C₂²×C₄

Q₈×D₈₉

yes

characteristic

115

C₂²×C₄

C₂×Dic₁₇₈

yes

characteristic

116

M₁₆

—

117

M₁₆

—

356

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

356

—

124

C₂×C₈

—

356

—

125

C₂×C₈

#13083553

yes

—

126

C₄²

C₂×C₄×D₈₉

yes

characteristic

127

C₄²

D₄×D₈₉

yes

characteristic

128

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

129

C₄²

Q₈×D₈₉

yes

characteristic

130

C₂³

#15668011

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

134

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

135

C₂×C₄

C₂×#13083553

yes

characteristic

136

C₂×C₄

#15668008

yes

characteristic

137

C₂×C₄

#15668010

yes

characteristic

138

C₂×C₄

#15668011

yes

characteristic

139

C₂×C₄

SD₁₆×D₈₉

yes

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

178

—

144

C₈

—

712

—

145

C₂²

—

146

C₂²

—

147

C₂²

#19151467

yes

characteristic

148

C₂²

#19151995

yes

—

149

C₄

—

150

C₄

—

151

C₄

#106×D₈₉

yes

—

152

C₄

#19151645

yes

—

153

C₄

#19151646

yes

—

154

C₄

#19151969

yes

—

155

C₂

—

156

C₂

#26020407

yes

characteristic

157

C₂

#26020425

yes

characteristic

158

C₂

#26020426

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	#384
prime radical	C₂³×C₈₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#8825
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#15668008

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1566

C₂

socle

C₂²×C₈₉

#384

prime radical

C₂³×C₈₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
85. C₈₉	#8825	semidirect

H ≅ G/N

type

85. C₈₉

#8825

semidirect

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

subgroup	index
#26020423	2
#26019008	2
#26019007	2
C₂×#19151320	2
C₈₉×#1566	2
#8825	89

subgroup

index

C₈₉×#1566

Subgroups — #36448238

Conjugacy classes of subgroups

Complements

Maximal subgroups