Group #36442212 — subgroups

Subgroups — #36442212

Order 22784.

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

normal subgroups only

This group has 286 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	#26021937	C₂	1	2	yes	maximal
2	C₈₉×#1962	C₂	1	1	yes	maximal, characteristic
3	#19151582	—	2	2	no	—
4	#19151397	—	2	2	no	—
5	#19151395	C₄	1	4	yes	complemented
6	#532×C₈₉	C₂²	1	1	yes	characteristic
7	#423×C₈₉	C₄	1	1	yes	characteristic
8	C₂×#104×C₈₉	—	2	2	no	—
9	C₄³×C₈₉	C₄	1	1	yes	characteristic
10	C₂×#13083558	—	4	2	no	—
11	#15667931	—	2	4	no	—
12	#15667929	—	4	2	no	—
13	C₂×#41×C₈₉	D₄	1	2	yes	—
14	C₂×M₁₆×C₈₉	—	2	1	no	—
15	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
16	#127×C₈₉	Q₈	1	1	yes	characteristic
17	#124×C₈₉	C₂×C₄	1	2	yes	—
18	#121×C₈₉	—	4	1	no	—
19	C₄×D₄×C₈₉	—	4	1	no	—
20	C₂×C₄²×C₈₉	—	2	1	no	—
21	C₂×C₄²×C₈₉	—	4	1	no	—
22	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₈×C₈₉	—	4	1	no	—
24	#104×C₈₉	—	4	4	no	—
25	#13083558	—	4	4	no	—
26	#13083558	—	8	2	no	—
27	C₂×#10623494	—	8	2	no	—
28	#41×C₈₉	—	4	2	no	—
29	#41×C₈₉	—	4	2	no	—
30	C₂×Q₈×C₈₉	C₂²⋊C₄	1	2	yes	—
31	C₂×D₄×C₈₉	—	2	2	no	—
32	C₂×D₄×C₈₉	—	4	1	no	—
33	C₂×D₄×C₈₉	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₈₉	—	2	1	no	—
35	C₂²×C₄×C₈₉	—	2	2	no	—
36	C₂²×C₄×C₈₉	—	4	1	no	—
37	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
38	M₁₆×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	—	4	1	no	—
40	C₄⋊C₄×C₈₉	—	4	1	no	—
41	C₂²⋊C₄×C₈₉	—	4	2	no	—
42	C₄²×C₈₉	—	2	1	no	—
43	C₄²×C₈₉	—	2	2	no	—
44	C₄²×C₈₉	—	4	1	no	—
45	C₄²×C₈₉	—	4	4	no	—
46	C₄²×C₈₉	C₄²	1	1	yes	characteristic
47	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂³×C₈₉	—	2	2	no	—
49	C₂³×C₈₉	#99	1	1	yes	characteristic
50	Q₈×C₈₉	—	4	2	no	—
51	D₄×C₈₉	—	4	2	no	—
52	D₄×C₈₉	—	8	2	no	—
53	D₄×C₈₉	—	8	2	no	—
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	—	2	1	no	—
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	4	2	no	—
60	C₂×C₄×C₈₉	—	4	2	no	—
61	C₂×C₄×C₈₉	—	4	4	no	—
62	C₂×C₄×C₈₉	—	4	2	no	—
63	C₂×C₄×C₈₉	—	4	2	no	—
64	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
66	C₈×C₈₉	—	8	1	no	—
67	#10623494	—	8	4	no	—
68	C₂²×C₈₉	—	2	1	no	—
69	C₂²×C₈₉	—	4	1	no	—
70	C₂²×C₈₉	—	4	2	no	—
71	C₂²×C₈₉	—	8	2	no	—
72	C₂²×C₈₉	#342	1	1	yes	characteristic
73	C₄×C₈₉	—	2	2	no	—
74	C₄×C₈₉	—	2	1	no	—
75	C₄×C₈₉	—	4	1	no	—
76	C₄×C₈₉	—	4	4	no	—
77	C₄×C₈₉	—	4	2	no	—
78	C₄×C₈₉	#353	1	1	yes	characteristic
79	C₄×C₈₉	#354	1	1	yes	characteristic
80	#7501	—	89	1	no	maximal, Sylow
81	C₂×C₈₉	—	2	1	no	—
82	C₂×C₈₉	—	4	1	no	—
83	C₂×C₈₉	—	8	2	no	—
84	C₂×C₈₉	#1393	1	1	yes	characteristic
85	#2132	—	89	2	no	—
86	#1962	D₈₉	1	1	yes	characteristic
87	C₈₉	#7501	1	1	yes	characteristic, complemented, Sylow
88	#532	C₂×D₈₉	1	1	yes	characteristic
89	#423	Dic₈₉	1	1	yes	characteristic
90	C₂×#104	—	2	2	no	—
91	#413	—	178	2	no	—
92	C₄³	Dic₈₉	1	1	yes	characteristic
93	#356	—	178	2	no	—
94	#355	—	89	4	no	—
95	C₂×#41	C₂²⋊D₈₉	1	2	yes	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	356	2	no	—
98	C₄⋊Q₈	D₃₅₆	1	1	yes	characteristic
99	#127	Dic₁₇₈	1	1	yes	characteristic
100	#124	C₄×D₈₉	1	2	yes	—
101	#121	—	4	1	no	—
102	C₄×D₄	—	4	1	no	—
103	C₂×C₄²	—	2	1	no	—
104	C₂×C₄²	—	4	1	no	—
105	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
106	C₄⋊C₈	—	4	1	no	—
107	#104	—	4	4	no	—
108	#101	—	178	4	no	—
109	#100	—	356	2	no	—
110	#41	—	4	2	no	—
111	#41	—	4	2	no	—
112	C₂×Q₈	#13083551	1	2	yes	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	4	1	no	—
115	C₂×D₄	#13083553	1	2	yes	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	4	1	no	—
119	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
120	M₁₆	—	4	2	no	—
121	M₁₆	—	356	4	no	—
122	M₁₆	—	712	2	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	712	2	no	—
125	C₄⋊C₄	—	4	1	no	—
126	C₂²⋊C₄	—	4	2	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	4	1	no	—
130	C₄²	—	4	4	no	—
131	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
132	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
133	C₂³	—	2	2	no	—
134	C₂³	#15667927	1	1	yes	characteristic
135	Q₈	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	8	2	no	—
138	D₄	—	8	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	4	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	#15667919	1	1	yes	characteristic
150	C₂×C₄	#15667927	1	1	yes	characteristic
151	C₈	—	8	1	no	—
152	C₈	—	712	4	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	#19151365	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	4	no	—
162	C₄	—	4	2	no	—
163	C₄	#19151391	1	1	yes	characteristic
164	C₄	#19151393	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	4	1	no	—
167	C₂	—	8	2	no	—
168	C₂	#26018377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021937

C₂

yes

maximal

C₈₉×#1962

C₂

yes

maximal, characteristic

#19151582

—

#19151397

—

#19151395

C₄

yes

complemented

#532×C₈₉

C₂²

yes

characteristic

#423×C₈₉

C₄

yes

characteristic

C₂×#104×C₈₉

—

C₄³×C₈₉

C₄

yes

—

—

—

C₂×#41×C₈₉

D₄

yes

—

C₂×M₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

D₄

yes

characteristic

#127×C₈₉

Q₈

yes

characteristic

#124×C₈₉

C₂×C₄

yes

—

#121×C₈₉

—

C₄×D₄×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₈₉

—

#104×C₈₉

—

#13083558

—

#13083558

—

C₂×#10623494

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂²⋊C₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₄²

yes

characteristic

C₄²×C₈₉

C₄⋊C₄

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

#99

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#342

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

C₈₉

#7501

yes

characteristic, complemented, Sylow

#532

C₂×D₈₉

yes

characteristic

#423

Dic₈₉

yes

characteristic

C₂×#104

—

#413

—

178

—

C₄³

Dic₈₉

yes

characteristic

#356

—

178

—

#355

—

C₂×#41

C₂²⋊D₈₉

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

356

—

C₄⋊Q₈

D₃₅₆

yes

characteristic

#127

Dic₁₇₈

yes

characteristic

100

#124

C₄×D₈₉

yes

—

101

#121

—

102

C₄×D₄

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

106

C₄⋊C₈

—

107

#104

—

108

#101

—

178

—

109

#100

—

356

—

110

#41

—

111

#41

—

112

C₂×Q₈

#13083551

yes

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

#13083553

yes

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊Dic₈₉

yes

characteristic

120

M₁₆

—

121

M₁₆

—

356

—

122

M₁₆

—

712

—

123

C₂×C₈

—

124

C₂×C₈

—

712

—

125

C₄⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₄×Dic₈₉

yes

characteristic

132

C₄²

C₄⋊Dic₈₉

yes

characteristic

133

C₂³

—

134

C₂³

#15667927

yes

characteristic

135

Q₈

—

136

D₄

—

137

D₄

—

138

D₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

#15667919

yes

characteristic

150

C₂×C₄

#15667927

yes

characteristic

151

C₈

—

152

C₈

—

712

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#19151365

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#19151391

yes

characteristic

164

C₄

#19151393

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

168

C₂

#26018377

yes

characteristic

	subgroup	G/H
center	C₄	#19151393
commutator	C₄²×C₈₉	C₄²
Frattini	#532	C₂×D₈₉
Fitting	C₈₉×#1962	C₂
socle	C₂×C₈₉	#1393
prime radical, cosocle	#532×C₈₉	C₂²
hypercenter	#1962	D₈₉
hypocenter	C₈₉	#7501
LCS	G ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ #1962

subgroup

G/H

center

C₄

#19151393

commutator

C₄²×C₈₉

C₄²

Frattini

#532

C₂×D₈₉

Fitting

C₈₉×#1962

C₂

socle

C₂×C₈₉

#1393

prime radical, cosocle

#532×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ #1962

N	H ≅ G/N	type
5. #19151395	C₄	semidirect
87. C₈₉	#7501	semidirect

H ≅ G/N

type

5. #19151395

C₄

semidirect

87. C₈₉

#7501

semidirect

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

subgroup	index
#26021937	2
C₈₉×#1962	2
#7501	89

subgroup

index

#26021937

C₈₉×#1962

#7501

Subgroups — #36442212

Conjugacy classes of subgroups

Complements

Maximal subgroups