Group #36470844 — subgroups

Subgroups — #36470844

Order 22784.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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×#19151418	C₂	1	4	yes	maximal
2	#26022008	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	C₂²	1	2	yes	—
6	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
8	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
9	C₄²×Dic₈₉	C₄	1	1	yes	characteristic
10	#19151418	—	2	8	no	—
11	C₄⋊#13083549	—	2	2	no	—
12	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₈₉	—	2	2	no	—
15	C₂²×#10623494	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₈₉	—	2	2	no	—
17	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
18	#15667952	—	2	4	no	—
19	C₂×#13083549	—	2	2	no	—
20	C₄×#10623494	C₈	1	2	yes	—
21	C₄×#10623494	C₈	1	2	yes	—
22	C₈×Dic₈₉	—	2	4	no	—
23	C₈×Dic₈₉	D₄	1	4	yes	—
24	C₈×Dic₈₉	Q₈	1	4	yes	—
25	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₈₉	—	2	4	no	—
29	C₂×C₁₆×C₈₉	—	2	2	no	—
30	C₄×C₈×C₈₉	D₄	1	2	yes	—
31	C₄×C₈×C₈₉	Q₈	1	2	yes	—
32	C₂²×Dic₈₉	—	2	2	no	—
33	C₂×#10623494	C₄⋊C₄	1	4	yes	—
34	C₂×#10623494	C₂×C₈	1	4	yes	—
35	C₂×#10623494	M₁₆	1	4	yes	—
36	C₄×Dic₈₉	—	2	8	no	—
37	C₄×Dic₈₉	—	2	2	no	—
38	C₄×Dic₈₉	C₄⋊C₄	1	4	yes	—
39	#13083549	—	4	2	no	—
40	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
41	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	—	2	2	no	—
43	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
49	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
51	C₁₆×C₈₉	—	4	2	no	—
52	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₄×C₈₉	—	2	4	no	—
54	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₈₉	—	2	8	no	—
61	C₂×Dic₈₉	—	2	4	no	—
62	C₈×C₈₉	—	2	2	no	—
63	C₈×C₈₉	#127	1	1	yes	characteristic
64	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
66	#10623494	C₄⋊C₈	1	8	yes	—
67	C₂²×C₈₉	—	2	2	no	—
68	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₈₉	#423	1	2	yes	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	#364	1	4	yes	complemented
72	C₄×C₈₉	#384	1	1	yes	characteristic
73	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
74	C₄×C₈₉	#423	1	1	yes	characteristic
75	Dic₈₉	—	2	8	no	—
76	#12900	—	89	1	no	maximal, Sylow
77	C₂×C₈₉	—	2	2	no	—
78	C₂×C₈₉	#1842	1	1	yes	characteristic
79	C₂×C₈₉	C₂×#364	1	2	yes	—
80	C₂×#364	—	89	4	no	—
81	#2150	—	89	1	no	—
82	#2150	D₈₉	1	1	yes	characteristic
83	C₄²×C₈	—	89	1	no	—
84	C₈₉	#12900	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	89	2	no	—
86	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
87	C₂×C₄×C₈	—	89	2	no	—
88	C₂×C₄×C₈	—	89	1	no	—
89	C₂×C₄×C₈	—	89	2	no	—
90	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
91	C₄³	—	89	1	no	—
92	#364	—	178	8	no	—
93	C₄⋊C₁₆	—	2	2	no	—
94	C₄⋊C₁₆	—	178	2	no	—
95	C₂²×C₈	—	89	2	no	—
96	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
98	C₂×C₄²	—	89	2	no	—
99	C₂×C₄²	—	178	2	no	—
100	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	M₃₂	—	178	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	178	2	no	—
105	C₄×C₈	—	89	4	no	—
106	C₄×C₈	—	89	4	no	—
107	C₄×C₈	—	89	2	no	—
108	C₄×C₈	—	89	2	no	—
109	C₄×C₈	—	178	4	no	—
110	C₄×C₈	Dic₁₇₈	1	2	yes	—
111	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
112	C₂²×C₄	—	178	2	no	—
113	C₂²×C₄	#13083557	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	89	4	no	—
117	C₂×C₈	—	89	4	no	—
118	C₂×C₈	—	89	4	no	—
119	C₂×C₈	#13083553	1	2	yes	—
120	C₂×C₈	D₄×D₈₉	1	2	yes	—
121	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
122	C₂×C₈	Q₈×D₈₉	1	2	yes	—
123	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
124	C₄²	—	89	4	no	—
125	C₄²	—	178	8	no	—
126	C₄²	—	178	2	no	—
127	C₄²	#13083553	1	2	yes	—
128	C₄²	C₈×D₈₉	1	1	yes	characteristic
129	C₄²	C₈×D₈₉	1	1	yes	characteristic
130	C₁₆	—	4	2	no	—
131	C₁₆	—	356	2	no	—
132	C₂³	C₂×#13083557	1	1	yes	characteristic
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	178	8	no	—
135	C₂×C₄	—	178	4	no	—
136	C₂×C₄	#15667941	1	2	yes	—
137	C₂×C₄	#15667941	1	2	yes	—
138	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
139	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
141	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	89	8	no	—
144	C₈	#15668008	1	1	yes	characteristic
145	C₈	#15668010	1	1	yes	characteristic
146	C₈	#15668011	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×#15667941	1	1	yes	characteristic
149	C₂²	#19151620	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	178	8	no	—
152	C₄	#19151418	1	4	yes	complemented
153	C₄	#19151467	1	1	yes	characteristic
154	C₄	C₄⋊C₈×D₈₉	1	2	yes	—
155	C₄	#19151621	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#26020403	1	1	yes	characteristic
158	C₂	C₂×#19151418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151418

C₂

yes

maximal

#26022008

C₂

yes

maximal, characteristic

C₄×C₈×Dic₈₉

C₂

yes

maximal, characteristic

C₈₉×#2150

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

C₂×C₄×#10623494

C₄

yes

characteristic

C₂×C₈×Dic₈₉

C₄

yes

—

C₂×C₈×Dic₈₉

C₂²

yes

—

C₄²×Dic₈₉

C₄

yes

characteristic

#19151418

—

C₄⋊#13083549

—

C₂×M₃₂×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₂²×#10623494

C₂×C₄

yes

—

C₂×C₄×Dic₈₉

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

#15667952

—

C₂×#13083549

—

C₄×#10623494

C₈

yes

—

C₄×#10623494

C₈

yes

—

C₈×Dic₈₉

—

C₈×Dic₈₉

D₄

yes

—

C₈×Dic₈₉

Q₈

yes

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

C₂²×Dic₈₉

—

C₂×#10623494

C₄⋊C₄

yes

—

C₂×#10623494

C₂×C₈

yes

—

C₂×#10623494

M₁₆

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₄⋊C₄

yes

—

#13083549

—

C₂²×C₄×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₈

yes

—

C₂×C₄×C₈₉

C₄⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#127

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

—

#10623494

C₄⋊C₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₈₉

#423

yes

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×C₄⋊C₈

yes

—

C₄×C₈₉

#423

yes

characteristic

Dic₈₉

—

#12900

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1842

yes

characteristic

C₂×C₈₉

C₂×#364

yes

—

C₂×#364

—

#2150

—

#2150

D₈₉

yes

characteristic

C₄²×C₈

—

C₈₉

#12900

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₈₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄³

—

#364

—

178

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

178

—

C₂²×C₈

—

C₂²×C₈

C₄×D₈₉

yes

characteristic

C₂²×C₈

C₂²×D₈₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

178

—

100

C₂×C₄²

C₄×D₈₉

yes

characteristic

101

M₃₂

—

102

M₃₂

—

178

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

178

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

178

—

110

C₄×C₈

Dic₁₇₈

yes

—

111

C₄×C₈

C₂²⋊D₈₉

yes

—

112

C₂²×C₄

—

178

—

113

C₂²×C₄

#13083557

yes

—

114

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

#13083553

yes

—

120

C₂×C₈

D₄×D₈₉

yes

—

121

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

122

C₂×C₈

Q₈×D₈₉

yes

—

123

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

124

C₄²

—

125

C₄²

—

178

—

126

C₄²

—

178

—

127

C₄²

#13083553

yes

—

128

C₄²

C₈×D₈₉

yes

characteristic

129

C₄²

C₈×D₈₉

yes

characteristic

130

C₁₆

—

131

C₁₆

—

356

—

132

C₂³

C₂×#13083557

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

178

—

135

C₂×C₄

—

178

—

136

C₂×C₄

#15667941

yes

—

137

C₂×C₄

#15667941

yes

—

138

C₂×C₄

C₄⋊C₄×D₈₉

yes

—

139

C₂×C₄

C₂×#13083553

yes

characteristic

140

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

141

C₂×C₄

M₁₆×D₈₉

yes

—

142

C₈

—

143

C₈

—

144

C₈

#15668008

yes

characteristic

145

C₈

#15668010

yes

characteristic

146

C₈

#15668011

yes

—

147

C₂²

—

148

C₂²

C₂×#15667941

yes

characteristic

149

C₂²

#19151620

yes

—

150

C₄

—

151

C₄

—

178

—

152

C₄

#19151418

yes

complemented

153

C₄

#19151467

yes

characteristic

154

C₄

C₄⋊C₈×D₈₉

yes

—

155

C₄

#19151621

yes

characteristic

156

C₂

—

157

C₂

#26020403

yes

characteristic

158

C₂

C₂×#19151418

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2150	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#12900
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#2150

C₂

socle

C₂²×C₈₉

C₂×C₄⋊C₈

prime radical

C₂³×C₈₉

C₂×M₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
71. C₄×C₈₉	#364	semidirect
84. C₈₉	#12900	semidirect
152. C₄	#19151418	semidirect

H ≅ G/N

type

71. C₄×C₈₉

#364

semidirect

84. C₈₉

#12900

semidirect

152. C₄

#19151418

semidirect

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

subgroup	index
C₂×#19151418	2
#26022008	2
C₄×C₈×Dic₈₉	2
C₈₉×#2150	2
#12900	89

subgroup

index

C₂×#19151418

#26022008

C₄×C₈×Dic₈₉

C₈₉×#2150

#12900

Subgroups — #36470844

Conjugacy classes of subgroups

Complements

Maximal subgroups