Group #36470851 — subgroups

Subgroups — #36470851

Order 22784.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#26022017	C₂	1	2	yes	maximal
2	#26022012	C₂	1	2	yes	maximal
3	#26022011	C₂	1	1	yes	maximal, characteristic
4	#26020007	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2153	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	1	yes	characteristic
7	C₂²×#13083549	C₂²	1	1	yes	characteristic
8	#19151553	C₂²	1	2	yes	—
9	#19151418	—	2	4	no	—
10	C₄⋊#13083549	—	2	2	no	—
11	#19151416	—	2	4	no	—
12	#19151356	C₄	1	1	yes	characteristic
13	#19151355	C₄	1	1	yes	characteristic
14	#19151354	C₄	1	2	yes	—
15	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
16	#405×C₈₉	C₂²	1	1	yes	characteristic
17	#364×C₈₉	—	2	2	no	—
18	C₂×#13083558	C₂×C₄	1	2	yes	—
19	#15667971	C₂×C₄	1	2	yes	—
20	#15667952	—	2	2	no	—
21	C₂×#13083549	—	2	2	no	—
22	C₂×#13083549	—	2	1	no	—
23	C₂×#13083549	—	2	1	no	—
24	#15667922	Q₈	1	4	yes	—
25	C₈×Dic₈₉	D₄	1	4	yes	—
26	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
29	M₃₂×C₈₉	—	2	2	no	—
30	M₃₂×C₈₉	—	2	2	no	—
31	C₂×C₁₆×C₈₉	—	2	2	no	—
32	#97×C₈₉	Q₈	1	2	yes	—
33	C₄×C₈×C₈₉	D₄	1	2	yes	—
34	#13083558	—	2	2	no	—
35	#13083558	—	2	2	no	—
36	C₂×#10623494	C₄⋊C₄	1	4	yes	—
37	C₄⋊Dic₈₉	—	2	2	no	—
38	C₂²⋊Dic₈₉	—	2	2	no	—
39	C₄×Dic₈₉	C₄⋊C₄	1	4	yes	—
40	#13083549	—	4	1	no	—
41	#13083549	—	4	1	no	—
42	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	M₁₆×C₈₉	—	2	2	no	—
44	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
49	C₄⋊C₄×C₈₉	—	2	1	no	—
50	C₂²⋊C₄×C₈₉	—	2	1	no	—
51	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
52	C₁₆×C₈₉	—	4	2	no	—
53	C₂³×C₈₉	#131	1	1	yes	characteristic
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#131	1	2	yes	—
58	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
59	C₂×Dic₈₉	—	2	4	no	—
60	C₈×C₈₉	—	2	2	no	—
61	C₈×C₈₉	#127	1	1	yes	characteristic
62	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
64	#10623494	—	2	4	no	—
65	C₂²×C₈₉	—	4	1	no	—
66	C₂²×C₈₉	#424	1	2	yes	—
67	C₂²×C₈₉	#424	1	1	yes	characteristic
68	C₄×C₈₉	—	4	2	no	—
69	C₄×C₈₉	#384	1	1	yes	characteristic
70	C₄×C₈₉	#424	1	2	yes	—
71	C₄×C₈₉	#424	1	1	yes	characteristic
72	Dic₈₉	—	4	4	no	—
73	#12902	—	89	1	no	maximal, Sylow
74	C₂×C₈₉	—	2	2	no	—
75	C₂×C₈₉	—	2	1	no	—
76	C₂×C₈₉	#1843	1	1	yes	characteristic
77	#2153	—	89	2	no	—
78	#2153	D₈₉	1	1	yes	characteristic
79	#2151	—	89	2	no	—
80	#2151	—	89	1	no	—
81	#1747	—	89	1	no	—
82	C₈₉	#12902	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	89	1	no	—
84	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
85	C₂²×C₁₆	—	89	1	no	—
86	#405	—	89	2	no	—
87	#405	C₂×D₈₉	1	1	yes	characteristic
88	#364	—	2	2	no	—
89	#364	—	178	4	no	—
90	C₄⋊C₁₆	—	178	4	no	—
91	C₄⋊C₁₆	—	178	2	no	—
92	#338	—	89	1	no	—
93	H(3,ℤ/4)	—	89	1	no	—
94	H(3,ℤ/4)	—	89	2	no	—
95	C₂×M₁₆	—	89	2	no	—
96	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
98	#117	—	89	2	no	—
99	#117	C₄×D₈₉	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	2	2	no	—
102	M₃₂	—	178	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	178	2	no	—
105	C₂×C₁₆	—	178	1	no	—
106	C₂×C₁₆	—	178	1	no	—
107	#97	—	89	4	no	—
108	#97	Dic₁₇₈	1	2	yes	—
109	C₄×C₈	—	89	4	no	—
110	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	178	2	no	—
114	M₁₆	—	178	2	no	—
115	C₂×C₈	—	89	4	no	—
116	C₂×C₈	#13083553	1	2	yes	—
117	C₂×C₈	D₄×D₈₉	1	2	yes	—
118	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
119	C₂×C₈	Q₈×D₈₉	1	2	yes	—
120	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	178	2	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	178	2	no	—
125	C₄²	—	89	4	no	—
126	C₄²	#13083553	1	2	yes	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	356	1	no	—
129	C₁₆	—	356	1	no	—
130	C₂³	#15668020	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	178	4	no	—
133	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
134	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
135	C₂×C₄	#15668018	1	2	yes	—
136	C₂×C₄	#15668020	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	178	4	no	—
139	C₈	#15668008	1	1	yes	characteristic
140	C₈	#15668010	1	1	yes	characteristic
141	C₈	#15668011	1	2	yes	—
142	C₂²	—	4	1	no	—
143	C₂²	#19151623	1	2	yes	—
144	C₂²	#19151626	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	356	4	no	—
147	C₄	#19151467	1	1	yes	characteristic
148	C₄	#19151623	1	2	yes	—
149	C₄	#19151624	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#26020405	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022017

C₂

yes

maximal

#26022012

C₂

yes

maximal

#26022011

C₂

yes

maximal, characteristic

#26020007

C₂

yes

maximal, characteristic

C₈₉×#2153

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

characteristic

C₂²×#13083549

C₂²

yes

characteristic

#19151553

C₂²

yes

—

—

—

—

C₄

yes

characteristic

#19151355

C₄

yes

characteristic

#19151354

C₄

yes

—

C₂×M₃₂×C₈₉

C₂²

yes

—

#405×C₈₉

C₂²

yes

characteristic

#364×C₈₉

—

C₂×#13083558

C₂×C₄

yes

—

#15667971

C₂×C₄

yes

—

—

—

—

—

Q₈

yes

—

C₈×Dic₈₉

D₄

yes

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

#117×C₈₉

C₂×C₄

yes

characteristic

M₃₂×C₈₉

—

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

#97×C₈₉

Q₈

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#13083558

—

#13083558

—

C₂×#10623494

C₄⋊C₄

yes

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

C₄⋊C₄

yes

—

#13083549

—

#13083549

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

M₁₆×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

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

#131

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

#131

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#127

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#424

yes

—

C₂²×C₈₉

#424

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#384

yes

characteristic

C₄×C₈₉

#424

yes

—

C₄×C₈₉

#424

yes

characteristic

Dic₈₉

—

#12902

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

D₈₉

yes

—

—

—

C₈₉

#12902

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₈₉

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₈₉

yes

characteristic

#364

—

#364

—

178

—

C₄⋊C₁₆

—

178

—

C₄⋊C₁₆

—

178

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₈₉

yes

characteristic

C₂²×C₈

C₂²×D₈₉

yes

characteristic

#117

—

#117

C₄×D₈₉

yes

characteristic

100

M₃₂

—

101

M₃₂

—

102

M₃₂

—

178

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

178

—

105

C₂×C₁₆

—

178

—

106

C₂×C₁₆

—

178

—

107

#97

—

108

#97

Dic₁₇₈

yes

—

109

C₄×C₈

—

110

C₄×C₈

C₂²⋊D₈₉

yes

—

111

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

112

M₁₆

—

113

M₁₆

—

178

—

114

M₁₆

—

178

—

115

C₂×C₈

—

116

C₂×C₈

#13083553

yes

—

117

C₂×C₈

D₄×D₈₉

yes

—

118

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

119

C₂×C₈

Q₈×D₈₉

yes

—

120

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

178

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

178

—

125

C₄²

—

126

C₄²

#13083553

yes

—

127

C₁₆

—

128

C₁₆

—

356

—

129

C₁₆

—

356

—

130

C₂³

#15668020

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

178

—

133

C₂×C₄

C₄⋊C₄×D₈₉

yes

—

134

C₂×C₄

C₂×#13083553

yes

characteristic

135

C₂×C₄

#15668018

yes

—

136

C₂×C₄

#15668020

yes

characteristic

137

C₈

—

138

C₈

—

178

—

139

C₈

#15668008

yes

characteristic

140

C₈

#15668010

yes

characteristic

141

C₈

#15668011

yes

—

142

C₂²

—

143

C₂²

#19151623

yes

—

144

C₂²

#19151626

yes

characteristic

145

C₄

—

146

C₄

—

356

—

147

C₄

#19151467

yes

characteristic

148

C₄

#19151623

yes

—

149

C₄

#19151624

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#26020405

yes

characteristic

	subgroup	G/H
center	C₈	#15668008
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2153	C₂
socle	C₂×C₈₉	#1843
prime radical	C₂³×C₈₉	#131
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2153	D₈₉
hypocenter	C₈₉	#12902
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#15668008

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#2153

C₂

C₂×C₈₉

C₂³×C₈₉

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

N	H ≅ G/N	type
82. C₈₉	#12902	semidirect

H ≅ G/N

type

82. C₈₉

#12902

semidirect

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

subgroup	index
#26022017	2
#26022012	2
#26022011	2
#26020007	2
C₈₉×#2153	2
#12902	89

subgroup

index

C₈₉×#2153

Subgroups — #36470851

Conjugacy classes of subgroups

Complements

Maximal subgroups