Group #36474963 — subgroups

Subgroups — #36474963

Order 22784.

180 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 91 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	#26021847	C₂	1	1	yes	maximal, characteristic
2	#26018447	C₂	1	2	yes	maximal, complemented
3	#26018389	C₂	1	2	yes	maximal
4	C₄×#15667918	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15667951	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
8	#19151552	C₄	1	1	yes	characteristic
9	C₂×#15667918	C₂²	1	2	yes	—
10	#19151374	C₄	1	1	yes	characteristic
11	#19151373	C₄	1	1	yes	characteristic
12	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083557	C₈	1	1	yes	characteristic
16	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
17	#15667969	C₈	1	1	yes	characteristic
18	#15667951	—	2	2	no	—
19	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
20	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
21	C₁₆×D₈₉	—	2	2	no	—
22	#15667923	C₈	1	1	yes	characteristic
23	#15667922	C₈	1	1	yes	characteristic
24	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
25	#15667918	—	2	2	no	—
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₃₂×C₈₉	—	2	2	no	—
30	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
31	#13083557	—	2	2	no	—
32	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
33	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
34	C₈×D₈₉	—	2	1	no	—
35	C₈×D₈₉	—	2	1	no	—
36	#13083553	C₁₆	1	2	yes	—
37	#13083551	C₁₆	1	2	yes	—
38	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
39	#13083549	—	2	1	no	—
40	#13083549	—	2	1	no	—
41	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
44	C₁₆×C₈₉	M₁₆	1	2	yes	—
45	C₁₆×C₈₉	#41	1	2	yes	—
46	C₂²×D₈₉	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
50	C₂×Dic₈₉	C₂×C₁₆	1	1	yes	characteristic
51	C₄×D₈₉	—	2	1	no	—
52	C₄×D₈₉	—	2	1	no	—
53	C₈×C₈₉	M₃₂	1	2	yes	—
54	C₈×C₈₉	#117	1	1	yes	characteristic
55	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
56	#10623494	—	2	1	no	—
57	#10623494	—	2	1	no	—
58	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
59	C₂×D₈₉	—	2	1	no	—
60	C₂×D₈₉	—	2	2	no	—
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	#431	1	1	yes	characteristic
63	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
64	Dic₈₉	—	2	1	no	—
65	Dic₈₉	—	2	2	no	—
66	#13641	—	89	1	no	maximal, Sylow
67	C₂×C₈₉	#2162	1	1	yes	characteristic
68	C₂×C₈₉	#2258	1	2	yes	—
69	D₈₉	—	4	1	no	—
70	#2108	—	89	1	no	—
71	C₄⋊C₃₂	—	89	2	no	—
72	C₂²⋊C₃₂	—	89	2	no	—
73	#1397	D₈₉	1	1	yes	characteristic, complemented
74	C₄×C₃₂	—	89	1	no	—
75	C₈₉	#13641	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₃₂	—	89	1	no	—
77	C₂²×C₁₆	—	89	1	no	—
78	#405	—	89	1	no	—
79	C₂×C₃₂	—	89	2	no	—
80	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
81	#346	—	89	1	no	—
82	#346	—	89	1	no	—
83	C₄×C₁₆	—	89	1	no	—
84	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
85	C₂×M₁₆	—	89	1	no	—
86	C₂²×C₈	—	89	1	no	—
87	#117	—	89	1	no	—
88	M₃₂	—	178	2	no	—
89	C₂×C₁₆	—	89	1	no	—
90	C₂×C₁₆	—	89	1	no	—
91	C₂×C₁₆	—	178	2	no	—
92	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
93	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
94	#97	—	89	1	no	—
95	#97	—	89	1	no	—
96	C₄×C₈	—	89	1	no	—
97	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
98	C₃₂	—	2	2	no	—
99	C₃₂	—	178	2	no	—
100	C₂²×C₄	—	89	1	no	—
101	M₁₆	—	178	2	no	—
102	C₂×C₈	—	89	1	no	—
103	C₂×C₈	—	89	1	no	—
104	C₂×C₈	—	178	1	no	—
105	C₂×C₈	—	178	1	no	—
106	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
107	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
108	C₄⋊C₄	—	89	2	no	—
109	C₂²⋊C₄	—	89	2	no	—
110	C₄²	—	89	1	no	—
111	C₄²	C₈×D₈₉	1	1	yes	characteristic
112	C₁₆	—	178	1	no	—
113	C₁₆	—	178	1	no	—
114	C₁₆	#13083557	1	2	yes	—
115	C₁₆	#13083574	1	2	yes	—
116	C₂³	—	89	1	no	—
117	C₂×C₄	—	89	2	no	—
118	C₂×C₄	—	89	1	no	—
119	C₂×C₄	—	178	1	no	—
120	C₂×C₄	—	178	1	no	—
121	C₂×C₄	C₁₆×D₈₉	1	2	yes	—
122	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
123	C₈	—	178	1	no	—
124	C₈	—	178	1	no	—
125	C₈	#15667951	1	2	yes	—
126	C₈	#15667969	1	1	yes	characteristic
127	C₈	C₂×#13083557	1	1	yes	characteristic
128	C₂²	—	178	2	no	—
129	C₂²	—	178	1	no	—
130	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	178	2	no	—
133	C₄	—	178	1	no	—
134	C₄	#19151649	1	1	yes	characteristic
135	C₄	C₂×#15667951	1	1	yes	characteristic
136	C₂	—	356	1	no	—
137	C₂	#26022049	1	1	yes	characteristic
138	C₂	#26022547	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021847

C₂

yes

maximal, characteristic

#26018447

C₂

yes

maximal, complemented

#26018389

C₂

yes

maximal

C₄×#15667918

C₂

yes

maximal, characteristic, complemented

C₈₉×#1397

C₂

yes

maximal, characteristic, complemented

C₂×#15667951

C₄

yes

characteristic

C₂×C₁₆×D₈₉

C₂²

yes

characteristic

#19151552

C₄

yes

characteristic

C₂×#15667918

C₂²

yes

—

#19151374

C₄

yes

characteristic

#19151373

C₄

yes

characteristic

C₁₆×Dic₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#13083557

C₈

yes

characteristic

C₂×C₈×D₈₉

C₂×C₄

yes

characteristic

#15667969

C₈

yes

characteristic

#15667951

—

C₂×#13083549

C₂×C₄

yes

characteristic

C₂×#13083549

C₂×C₄

yes

characteristic

C₁₆×D₈₉

—

#15667923

C₈

yes

characteristic

#15667922

C₈

yes

characteristic

C₈×Dic₈₉

C₂×C₄

yes

characteristic

#15667918

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₃₂×C₈₉

—

C₂×C₄×D₈₉

C₂×C₈

yes

characteristic

#13083557

—

C₂×#10623494

C₂×C₈

yes

characteristic

C₂×#10623494

C₂×C₈

yes

characteristic

C₈×D₈₉

—

C₈×D₈₉

—

#13083553

C₁₆

yes

—

#13083551

C₁₆

yes

—

C₄×Dic₈₉

C₂×C₈

yes

characteristic

#13083549

—

#13083549

—

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

M₁₆

yes

—

C₁₆×C₈₉

#41

yes

—

C₂²×D₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×C₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×Dic₈₉

C₂×C₁₆

yes

—

C₂×Dic₈₉

C₂×C₁₆

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

M₃₂

yes

—

C₈×C₈₉

#117

yes

characteristic

C₈×C₈₉

C₂×M₁₆

yes

characteristic

#10623494

—

#10623494

—

C₂²×C₈₉

C₂²×C₁₆

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#431

yes

characteristic

C₄×C₈₉

C₂×M₃₂

yes

characteristic

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

#2108

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

#1397

D₈₉

yes

characteristic, complemented

C₄×C₃₂

—

C₈₉

#13641

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

178

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₄×D₈₉

yes

characteristic

C₃₂

—

C₃₂

—

178

—

100

C₂²×C₄

—

101

M₁₆

—

178

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

178

—

105

C₂×C₈

—

178

—

106

C₂×C₈

C₈×D₈₉

yes

characteristic

107

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₈×D₈₉

yes

characteristic

112

C₁₆

—

178

—

113

C₁₆

—

178

—

114

C₁₆

#13083557

yes

—

115

C₁₆

#13083574

yes

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

178

—

120

C₂×C₄

—

178

—

121

C₂×C₄

C₁₆×D₈₉

yes

—

122

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

123

C₈

—

178

—

124

C₈

—

178

—

125

C₈

#15667951

yes

—

126

C₈

#15667969

yes

characteristic

127

C₈

C₂×#13083557

yes

characteristic

128

C₂²

—

178

—

129

C₂²

—

178

—

130

C₂²

C₂×C₁₆×D₈₉

yes

characteristic

131

C₄

—

132

C₄

—

178

—

133

C₄

—

178

—

134

C₄

#19151649

yes

characteristic

135

C₄

C₂×#15667951

yes

characteristic

136

C₂

—

356

—

137

C₂

#26022049

yes

characteristic

138

C₂

#26022547

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₁₆
Fitting	C₈₉×#1397	C₂
prime radical	C₂²×D₈₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1397	D₈₉
hypocenter	C₈₉	#13641
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ #1397

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₈₉

commutator, socle

C₂²×C₈₉

C₂²×C₁₆

Fitting

C₈₉×#1397

C₂

prime radical

C₂²×D₈₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₁₆ ◁ #1397

N	H ≅ G/N	type
2. #26018447	C₂	semidirect
4. C₄×#15667918	C₂	semidirect
5. C₈₉×#1397	C₂	semidirect
73. #1397	D₈₉	semidirect
75. C₈₉	#13641	semidirect

H ≅ G/N

type

2. #26018447

C₂

semidirect

4. C₄×#15667918

C₂

semidirect

5. C₈₉×#1397

C₂

semidirect

73. #1397

D₈₉

semidirect

75. C₈₉

#13641

semidirect

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

subgroup	index
#26021847	2
#26018447	2
#26018389	2
C₄×#15667918	2
C₈₉×#1397	2
#13641	89

subgroup

index

C₈₉×#1397

Subgroups — #36474963

Conjugacy classes of subgroups

Complements

Maximal subgroups