Group #36459213 — subgroups

Subgroups — #36459213

Order 22784.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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	#26021233	C₂	1	1	yes	maximal, characteristic
2	#26020372	C₂	1	1	yes	maximal, characteristic
3	#26018907	C₂	1	2	yes	maximal
4	#26018148	C₂	1	2	yes	maximal
5	C₈₉×#1274	C₂	1	1	yes	maximal, characteristic
6	#19152155	C₂²	1	1	yes	characteristic
7	#19151622	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	#19151358	—	2	2	no	—
10	#19151353	—	2	2	no	—
11	#19151346	—	2	2	no	—
12	#19151345	—	2	2	no	—
13	#431×C₈₉	C₂²	1	2	yes	—
14	#376×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083558	—	2	1	no	—
16	C₂²×#10623494	—	2	2	no	—
17	#15668012	D₄	1	2	yes	—
18	#15668004	—	2	1	no	—
19	#15667991	—	2	2	no	—
20	C₄×Dic₁₇₈	—	2	2	no	—
21	#15667971	—	2	1	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₂×C₄²×C₈₉	D₄	1	2	yes	—
26	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	—	2	2	no	—
28	C₂²⋊C₈×C₈₉	—	2	2	no	—
29	C₄×C₈×C₈₉	—	2	2	no	—
30	#95×C₈₉	—	2	2	no	—
31	C₂×Dic₁₇₈	—	2	2	no	—
32	#13083558	—	4	2	no	—
33	C₂×#10623494	—	4	2	no	—
34	C₂×#10623494	—	4	1	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₂×#10623494	—	4	2	no	—
37	C₄⋊Dic₈₉	—	2	2	no	—
38	C₄⋊Dic₈₉	—	2	2	no	—
39	#13083553	—	4	2	no	—
40	C₂²⋊Dic₈₉	—	4	1	no	—
41	C₄×Dic₈₉	—	4	1	no	—
42	C₂²×C₄×C₈₉	—	2	2	no	—
43	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	#41	1	2	yes	—
45	C₂×C₈×C₈₉	—	4	2	no	—
46	C₄²×C₈₉	—	2	2	no	—
47	C₄²×C₈₉	—	2	2	no	—
48	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₈₉	#41	1	1	yes	characteristic
50	C₄²×C₈₉	#41	1	2	yes	—
51	C₂³×C₈₉	#123	1	1	yes	characteristic
52	C₂×C₄×C₈₉	—	4	2	no	—
53	C₂×C₄×C₈₉	—	4	4	no	—
54	C₂×C₄×C₈₉	—	4	2	no	—
55	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#123	1	2	yes	—
57	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#126	1	2	yes	—
59	C₂×C₄×C₈₉	#135	1	2	yes	—
60	C₂×C₄×C₈₉	#137	1	2	yes	—
61	C₂×Dic₈₉	—	4	2	no	—
62	Dic₁₇₈	—	8	2	no	—
63	C₈×C₈₉	—	8	2	no	—
64	#10623494	—	8	2	no	—
65	#10623494	—	8	1	no	—
66	C₂²×C₈₉	—	2	2	no	—
67	C₂²×C₈₉	#397	1	1	yes	characteristic
68	C₂²×C₈₉	#485	1	2	yes	—
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	—	8	2	no	—
72	C₄×C₈₉	#464	1	2	yes	—
73	C₄×C₈₉	#488	1	1	yes	characteristic
74	C₄×C₈₉	#491	1	1	yes	characteristic
75	Dic₈₉	—	8	2	no	—
76	#10630	—	89	1	no	maximal, Sylow
77	C₂×C₈₉	—	2	2	no	—
78	C₂×C₈₉	#2074	1	1	yes	characteristic
79	C₂×C₈₉	#2080	1	2	yes	—
80	#1995	—	89	1	no	—
81	#1832	—	89	1	no	—
82	#1542	—	89	2	no	—
83	#1301	—	89	2	no	—
84	#1274	D₈₉	1	1	yes	characteristic
85	C₈₉	#10630	1	1	yes	characteristic, complemented, Sylow
86	#533	—	89	1	no	—
87	#431	C₂×D₈₉	1	2	yes	—
88	#423	—	89	1	no	—
89	C₂×C₄⋊C₈	—	89	2	no	—
90	#376	C₂×D₈₉	1	1	yes	characteristic
91	#339	—	178	2	no	—
92	#336	—	178	2	no	—
93	#333	—	178	2	no	—
94	#332	—	178	2	no	—
95	C₂×M₁₆	—	178	1	no	—
96	C₂²×C₈	—	178	2	no	—
97	C₄⋊Q₈	—	89	2	no	—
98	#125	—	178	1	no	—
99	C₂²⋊Q₈	—	178	2	no	—
100	C₄×Q₈	—	178	2	no	—
101	#117	—	178	1	no	—
102	C₂×C₄²	D₃₅₆	1	2	yes	—
103	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	178	2	no	—
106	C₄⋊C₈	—	178	2	no	—
107	C₄⋊C₈	—	178	2	no	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₄×C₈	—	2	2	no	—
110	#95	—	2	2	no	—
111	C₂×Q₈	—	178	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
114	C₂²×C₄	#13083573	1	2	yes	—
115	M₁₆	—	356	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	356	2	no	—
118	C₂×C₈	—	356	1	no	—
119	C₂×C₈	—	356	2	no	—
120	C₂×C₈	—	356	2	no	—
121	C₄⋊C₄	—	356	2	no	—
122	C₄⋊C₄	—	178	2	no	—
123	C₄⋊C₄	—	178	2	no	—
124	C₂²⋊C₄	—	356	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	356	1	no	—
128	C₄²	D₄×D₈₉	1	1	yes	characteristic
129	C₄²	#13083574	1	2	yes	—
130	C₄²	#13083575	1	1	yes	characteristic
131	C₂³	#15667996	1	1	yes	characteristic
132	Q₈	—	712	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	356	2	no	—
137	C₂×C₄	#15667987	1	1	yes	characteristic
138	C₂×C₄	#15667994	1	2	yes	—
139	C₂×C₄	#15668000	1	1	yes	characteristic
140	C₂×C₄	#15668007	1	2	yes	—
141	C₂×C₄	#15668034	1	2	yes	—
142	C₂×C₄	#15668045	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	712	1	no	—
145	C₈	—	712	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#19151525	1	1	yes	characteristic
148	C₂²	#19151940	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	712	2	no	—
153	C₄	#19151821	1	2	yes	—
154	C₄	#19151953	1	1	yes	characteristic
155	C₄	#19151967	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#26021672	1	1	yes	characteristic
158	C₂	#26021704	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021233

C₂

yes

maximal, characteristic

#26020372

C₂

yes

maximal, characteristic

#26018907

C₂

yes

maximal

#26018148

C₂

yes

maximal

C₈₉×#1274

C₂

yes

maximal, characteristic

#19152155

C₂²

yes

characteristic

#19151622

C₂²

yes

characteristic

C₂×C₄⋊#10623494

C₂²

yes

—

—

—

—

—

#431×C₈₉

C₂²

yes

—

#376×C₈₉

C₂²

yes

—

—

D₄

yes

—

#15668004

—

#15667991

—

C₄×Dic₁₇₈

—

—

—

—

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

#95×C₈₉

—

C₂×Dic₁₇₈

—

—

—

—

—

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

C₄²×C₈₉

yes

C₄²×C₈₉

yes

—

C₂³×C₈₉

#123

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

#126

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#397

yes

characteristic

C₂²×C₈₉

#485

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#464

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#10630

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2074

yes

characteristic

C₂×C₈₉

#2080

yes

—

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#10630

yes

characteristic, complemented, Sylow

#533

—

#431

C₂×D₈₉

yes

—

#423

—

C₂×C₄⋊C₈

—

#376

C₂×D₈₉

yes

characteristic

#339

—

178

—

#336

—

178

—

#333

—

178

—

#332

—

178

—

C₂×M₁₆

—

178

—

C₂²×C₈

—

178

—

C₄⋊Q₈

—

#125

—

178

—

C₂²⋊Q₈

—

178

—

100

C₄×Q₈

—

178

—

101

#117

—

178

—

102

C₂×C₄²

D₃₅₆

yes

—

103

C₂×C₄²

C₂²×D₈₉

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

178

—

106

C₄⋊C₈

—

178

—

107

C₄⋊C₈

—

178

—

108

C₂²⋊C₈

—

109

C₄×C₈

—

110

#95

—

111

C₂×Q₈

—

178

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

114

C₂²×C₄

#13083573

yes

—

115

M₁₆

—

356

—

116

C₂×C₈

—

117

C₂×C₈

—

356

—

118

C₂×C₈

—

356

—

119

C₂×C₈

—

356

—

120

C₂×C₈

—

356

—

121

C₄⋊C₄

—

356

—

122

C₄⋊C₄

—

178

—

123

C₄⋊C₄

—

178

—

124

C₂²⋊C₄

—

356

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

356

—

128

C₄²

D₄×D₈₉

yes

characteristic

129

C₄²

#13083574

yes

—

130

C₄²

#13083575

yes

characteristic

131

C₂³

#15667996

yes

characteristic

132

Q₈

—

712

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

356

—

137

C₂×C₄

#15667987

yes

characteristic

138

C₂×C₄

#15667994

yes

—

139

C₂×C₄

#15668000

yes

characteristic

140

C₂×C₄

#15668007

yes

—

141

C₂×C₄

#15668034

yes

—

142

C₂×C₄

#15668045

yes

—

143

C₈

—

144

C₈

—

712

—

145

C₈

—

712

—

146

C₂²

—

147

C₂²

#19151525

yes

characteristic

148

C₂²

#19151940

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

712

—

153

C₄

#19151821

yes

—

154

C₄

#19151953

yes

characteristic

155

C₄

#19151967

yes

characteristic

156

C₂

—

157

C₂

#26021672

yes

characteristic

158

C₂

#26021704

yes

—

	subgroup	G/H
center	C₂×C₄	#15668000
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1274	C₂
socle	C₂²×C₈₉	#397
prime radical	C₂³×C₈₉	#123
hypercenter	#1274	D₈₉
hypocenter	C₈₉	#10630
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#15668000

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1274

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
85. C₈₉	#10630	semidirect

H ≅ G/N

type

85. C₈₉

#10630

semidirect

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

subgroup	index
#26021233	2
#26020372	2
#26018907	2
#26018148	2
C₈₉×#1274	2
#10630	89

subgroup

index

C₈₉×#1274

Subgroups — #36459213

Conjugacy classes of subgroups

Complements

Maximal subgroups