Group #36466022 — subgroups

Subgroups — #36466022

Order 22784.

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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	#364×D₈₉	C₂	1	2	yes	maximal
2	C₄×#15667951	C₂	1	1	yes	maximal, characteristic
3	#26018328	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018327	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15667951	C₄	1	1	yes	characteristic
7	C₂×#15667951	C₄	1	1	yes	characteristic
8	M₃₂×D₈₉	—	2	2	no	—
9	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	#19151419	C₂²	1	2	yes	—
11	#19151418	—	2	2	no	—
12	#19151373	C₄	1	1	yes	characteristic
13	#19151373	C₄	1	1	yes	characteristic
14	C₄×#13083549	C₂²	1	1	yes	characteristic
15	#364×C₈₉	C₂²	1	2	yes	—
16	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
19	C₄²×D₈₉	Q₈	1	1	yes	characteristic
20	#15667952	—	2	2	no	—
21	#15667951	—	2	2	no	—
22	#15667951	—	4	2	no	—
23	#15667951	C₈	1	4	yes	—
24	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
25	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
26	C₁₆×D₈₉	—	4	2	no	—
27	C₄×#10623494	Q₈	1	1	yes	characteristic
28	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₈₉	D₄	1	1	yes	characteristic
30	M₃₂×C₈₉	—	2	2	no	—
31	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₈₉	—	2	1	no	—
35	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
37	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
38	C₈×D₈₉	C₂×C₈	1	2	yes	—
39	C₈×D₈₉	M₁₆	1	2	yes	—
40	C₈×D₈₉	D₈	1	2	yes	—
41	C₈×D₈₉	Q₁₆	1	2	yes	—
42	C₄×Dic₈₉	—	2	1	no	—
43	C₄×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
44	#13083549	—	2	1	no	—
45	#13083549	—	4	2	no	—
46	#13083549	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₁₆×C₈₉	—	2	1	no	—
51	C₁₆×C₈₉	—	4	2	no	—
52	C₁₆×C₈₉	C₂×C₈	1	2	yes	—
53	C₂²×D₈₉	#108	1	1	yes	characteristic
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×Dic₈₉	—	2	1	no	—
57	C₂×Dic₈₉	#108	1	1	yes	characteristic
58	C₄×D₈₉	—	4	2	no	—
59	C₄×D₈₉	C₄⋊C₈	1	2	yes	—
60	C₄×D₈₉	#107	1	2	yes	—
61	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
62	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
64	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
65	#10623494	C₄⋊C₈	1	2	yes	—
66	#10623494	#107	1	2	yes	—
67	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
68	C₂×D₈₉	—	4	1	no	—
69	C₂×D₈₉	#335	1	2	yes	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
73	Dic₈₉	—	4	1	no	—
74	Dic₈₉	#335	1	2	yes	—
75	#12011	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	—	2	1	no	—
77	C₂×C₈₉	C₂×#335	1	1	yes	characteristic
78	D₈₉	—	2	2	no	—
79	C₂×#364	—	89	2	no	—
80	C₄×M₃₂	—	89	1	no	—
81	#1373	—	89	2	no	—
82	#1373	—	89	1	no	—
83	#1373	D₈₉	1	1	yes	characteristic, complemented
84	C₈₉	#12011	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	89	1	no	—
86	C₂×M₃₂	—	89	1	no	—
87	C₂×M₃₂	—	178	2	no	—
88	C₂×C₄×C₈	—	89	1	no	—
89	#364	—	89	2	no	—
90	#364	—	178	2	no	—
91	#364	C₂×D₈₉	1	2	yes	—
92	#346	—	89	1	no	—
93	#346	—	89	1	no	—
94	C₄×C₁₆	—	89	1	no	—
95	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
96	C₂²×C₈	—	89	1	no	—
97	C₂²×C₈	—	89	1	no	—
98	C₂×C₄²	—	89	1	no	—
99	M₃₂	—	2	2	no	—
100	M₃₂	—	89	4	no	—
101	M₃₂	—	356	2	no	—
102	M₃₂	—	178	2	no	—
103	M₃₂	—	178	2	no	—
104	C₂×C₁₆	—	89	1	no	—
105	C₂×C₁₆	—	89	1	no	—
106	C₂×C₁₆	—	356	2	no	—
107	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
108	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
109	C₄×C₈	—	89	1	no	—
110	C₄×C₈	—	89	1	no	—
111	C₄×C₈	—	89	1	no	—
112	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
113	C₂²×C₄	—	89	1	no	—
114	C₂²×C₄	—	178	1	no	—
115	C₂×C₈	—	89	1	no	—
116	C₂×C₈	—	89	1	no	—
117	C₂×C₈	—	89	2	no	—
118	C₂×C₈	—	89	2	no	—
119	C₂×C₈	—	89	2	no	—
120	C₂×C₈	—	89	2	no	—
121	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
122	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
123	C₄²	—	89	1	no	—
124	C₄²	—	178	1	no	—
125	C₄²	Q₈×D₈₉	1	1	yes	characteristic
126	C₁₆	—	2	1	no	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	89	2	no	—
129	C₁₆	—	356	2	no	—
130	C₁₆	—	178	1	no	—
131	C₁₆	C₈×D₈₉	1	2	yes	—
132	C₂³	—	89	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	89	1	no	—
135	C₂×C₄	—	89	2	no	—
136	C₂×C₄	—	89	2	no	—
137	C₂×C₄	—	356	2	no	—
138	C₂×C₄	—	178	1	no	—
139	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
140	C₈	—	89	2	no	—
141	C₈	—	89	2	no	—
142	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
143	C₈	M₁₆×D₈₉	1	1	yes	characteristic
144	C₈	D₈×D₈₉	1	1	yes	characteristic
145	C₈	Q₁₆×D₈₉	1	1	yes	characteristic
146	C₂²	—	89	2	no	—
147	C₂²	—	356	1	no	—
148	C₂²	#108×D₈₉	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	89	2	no	—
151	C₄	—	356	1	no	—
152	C₄	C₄⋊C₈×D₈₉	1	1	yes	characteristic
153	C₄	#107×D₈₉	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	178	2	no	—
156	C₂	#335×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#364×D₈₉

C₂

yes

maximal

C₄×#15667951

C₂

yes

maximal, characteristic

#26018328

C₂

yes

maximal, characteristic, complemented

#26018327

C₂

yes

maximal, complemented

C₈₉×#1373

C₂

yes

maximal, characteristic, complemented

C₂×#15667951

C₄

yes

characteristic

C₂×#15667951

C₄

yes

characteristic

M₃₂×D₈₉

—

C₄×C₈×D₈₉

C₂²

yes

characteristic

#19151419

C₂²

yes

—

#19151418

—

#19151373

C₄

yes

characteristic

#19151373

C₄

yes

characteristic

C₄×#13083549

C₂²

yes

characteristic

#364×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₈×D₈₉

C₂×C₄

yes

characteristic

C₂×C₈×D₈₉

D₄

yes

characteristic

C₄²×D₈₉

Q₈

yes

—

—

—

C₈

yes

—

C₂×#13083549

C₂×C₄

yes

characteristic

C₂×#13083549

C₂×C₄

yes

characteristic

C₁₆×D₈₉

—

C₄×#10623494

Q₈

yes

characteristic

C₈×Dic₈₉

C₂×C₄

yes

characteristic

C₈×Dic₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

C₄⋊C₄

yes

characteristic

C₂×#10623494

C₄⋊C₄

yes

characteristic

C₂×#10623494

C₄⋊C₄

yes

characteristic

C₈×D₈₉

C₂×C₈

yes

—

C₈×D₈₉

M₁₆

yes

—

C₈×D₈₉

D₈

yes

—

C₈×D₈₉

Q₁₆

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₄⋊C₄

yes

—

—

C₂×C₈

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×C₈

yes

—

C₂²×D₈₉

#108

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

#108

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

C₄⋊C₈

yes

—

C₄×D₈₉

#107

yes

—

C₈×C₈₉

C₂²×C₈

yes

characteristic

C₈×C₈₉

C₂×M₁₆

yes

characteristic

C₈×C₈₉

C₂×D₈

yes

characteristic

C₈×C₈₉

C₂×Q₁₆

yes

characteristic

#10623494

C₄⋊C₈

yes

—

#10623494

#107

yes

—

C₂²×C₈₉

C₂×#108

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

#335

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄⋊C₈

yes

characteristic

C₄×C₈₉

C₂×#107

yes

characteristic

Dic₈₉

—

Dic₈₉

#335

yes

—

#12011

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×#335

yes

characteristic

D₈₉

—

C₂×#364

—

C₄×M₃₂

—

#1373

—

#1373

—

#1373

D₈₉

yes

characteristic, complemented

C₈₉

#12011

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

178

—

C₂×C₄×C₈

—

#364

—

#364

—

178

—

#364

C₂×D₈₉

yes

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

100

M₃₂

—

101

M₃₂

—

356

—

102

M₃₂

—

178

—

103

M₃₂

—

178

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

356

—

107

C₂×C₁₆

C₄×D₈₉

yes

characteristic

108

C₂×C₁₆

C₄×D₈₉

yes

characteristic

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₂²×D₈₉

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

178

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

122

C₂×C₈

D₄×D₈₉

yes

characteristic

123

C₄²

—

124

C₄²

—

178

—

125

C₄²

Q₈×D₈₉

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

356

—

130

C₁₆

—

178

—

131

C₁₆

C₈×D₈₉

yes

—

132

C₂³

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

356

—

138

C₂×C₄

—

178

—

139

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

C₂×C₈×D₈₉

yes

characteristic

143

C₈

M₁₆×D₈₉

yes

characteristic

144

C₈

D₈×D₈₉

yes

characteristic

145

C₈

Q₁₆×D₈₉

yes

characteristic

146

C₂²

—

147

C₂²

—

356

—

148

C₂²

#108×D₈₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

356

—

152

C₄

C₄⋊C₈×D₈₉

yes

characteristic

153

C₄

#107×D₈₉

yes

characteristic

154

C₂

—

155

C₂

—

178

—

156

C₂

#335×D₈₉

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₈₉
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1373	C₂
socle	C₂×C₈₉	C₂×#335
prime radical	C₂²×D₈₉	#108
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1373	D₈₉
hypocenter	C₈₉	#12011
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₈₉

commutator

C₈×C₈₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1373

C₂

C₂×C₈₉

C₂²×D₈₉

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
3. #26018328	C₂	semidirect
4. #26018327	C₂	semidirect
5. C₈₉×#1373	C₂	semidirect
83. #1373	D₈₉	semidirect
84. C₈₉	#12011	semidirect

H ≅ G/N

type

3. #26018328

C₂

semidirect

4. #26018327

C₂

semidirect

5. C₈₉×#1373

C₂

semidirect

83. #1373

D₈₉

semidirect

84. C₈₉

#12011

semidirect

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

subgroup	index
#364×D₈₉	2
C₄×#15667951	2
#26018328	2
#26018327	2
C₈₉×#1373	2
#12011	89

subgroup

index

#364×D₈₉

C₈₉×#1373

Subgroups — #36466022

Conjugacy classes of subgroups

Complements

Maximal subgroups