Group #36474607 — subgroups

Subgroups — #36474607

Order 22784.

238 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#19151432	C₂	1	1	yes	maximal, characteristic
2	#26021848	C₂	1	1	yes	maximal, characteristic
3	#26018385	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₈₉	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₄	1	1	yes	characteristic
7	#19151553	C₄	1	1	yes	characteristic
8	#19151432	C₄	1	4	yes	complemented
9	C₂×#15667918	C₂²	1	2	yes	—
10	#19151373	C₄	1	2	yes	—
11	C₁₆×Dic₈₉	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₈₉	C₄	1	4	yes	complemented
14	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#13083558	C₈	1	1	yes	characteristic
17	#15667971	C₈	1	1	yes	characteristic
18	#15667952	C₈	1	4	yes	—
19	C₂×#13083549	C₂×C₄	1	2	yes	—
20	#15667922	C₈	1	2	yes	—
21	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
22	#15667918	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
28	C₃₂×C₈₉	C₂×C₄	1	2	yes	—
29	C₃₂×C₈₉	C₂×C₄	1	2	yes	—
30	#13083558	C₁₆	1	4	yes	—
31	C₂×#10623494	C₂×C₈	1	2	yes	—
32	C₄⋊Dic₈₉	C₁₆	1	2	yes	—
33	C₂²⋊Dic₈₉	C₁₆	1	2	yes	—
34	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
35	#13083549	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₈₉	C₄²	1	2	yes	—
38	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₁₆×C₈₉	C₄²	1	2	yes	—
42	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂³×C₈₉	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₈₉	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	4	yes	—
50	C₈×C₈₉	C₄×C₈	1	2	yes	—
51	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
52	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
53	#10623494	C₂×C₁₆	1	4	yes	—
54	C₂²×C₈₉	—	2	2	no	—
55	C₂²×C₈₉	C₄×C₁₆	1	2	yes	—
56	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
57	C₄×C₈₉	C₄×C₁₆	1	2	yes	—
58	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
59	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
60	Dic₈₉	—	2	4	no	—
61	#13549	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	—	2	2	no	—
63	C₂×C₈₉	C₂×C₄×C₁₆	1	1	yes	characteristic
64	C₂×C₈₉	#2258	1	2	yes	—
65	C₂×M₆₄	—	89	1	no	—
66	C₂²×C₃₂	D₈₉	1	1	yes	characteristic
67	#2108	—	89	1	no	—
68	#1397	—	89	2	no	—
69	C₄×C₃₂	—	89	2	no	—
70	C₈₉	#13549	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₃₂	—	89	1	no	—
72	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
73	#405	—	89	1	no	—
74	M₆₄	—	89	4	no	—
75	C₂×C₃₂	—	89	2	no	—
76	C₂×C₃₂	Dic₈₉	1	4	yes	complemented
77	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
78	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
79	#346	—	89	2	no	—
80	C₄×C₁₆	—	89	2	no	—
81	C₂×M₁₆	—	89	1	no	—
82	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
83	#117	—	89	1	no	—
84	M₃₂	—	89	4	no	—
85	C₂×C₁₆	—	89	2	no	—
86	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
87	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
88	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
89	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
90	#97	—	89	2	no	—
91	C₄×C₈	—	89	2	no	—
92	C₃₂	—	89	4	no	—
93	C₃₂	C₂×Dic₈₉	1	2	yes	—
94	C₃₂	C₂×Dic₈₉	1	2	yes	—
95	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
96	M₁₆	—	89	4	no	—
97	C₂×C₈	—	89	2	no	—
98	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
99	C₂×C₈	C₈×D₈₉	1	2	yes	—
100	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
101	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
102	C₄⋊C₄	—	89	2	no	—
103	C₂²⋊C₄	—	89	2	no	—
104	C₄²	—	89	2	no	—
105	C₁₆	—	89	4	no	—
106	C₁₆	C₄×Dic₈₉	1	2	yes	—
107	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
108	C₁₆	C₂²×Dic₈₉	1	1	yes	characteristic
109	C₂³	C₁₆×D₈₉	1	1	yes	characteristic
110	C₂×C₄	—	89	4	no	—
111	C₂×C₄	C₈×Dic₈₉	1	2	yes	—
112	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
113	C₂×C₄	C₁₆×D₈₉	1	2	yes	—
114	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
115	C₈	—	89	4	no	—
116	C₈	C₈×Dic₈₉	1	2	yes	—
117	C₈	C₂×C₄×Dic₈₉	1	1	yes	characteristic
118	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₁₆×Dic₈₉	1	2	yes	—
121	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
122	C₄	—	178	4	no	—
123	C₄	C₁₆×Dic₈₉	1	2	yes	—
124	C₄	C₂×C₈×Dic₈₉	1	1	yes	characteristic
125	C₄	C₂×C₁₆×D₈₉	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₁₆×Dic₈₉	1	1	yes	characteristic
128	C₂	#26022545	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151432

C₂

yes

maximal, characteristic

#26021848

C₂

yes

maximal, characteristic

#26018385

C₂

yes

maximal, complemented

C₃₂×Dic₈₉

C₂

yes

maximal, complemented

C₂²×C₃₂×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667952

C₄

yes

characteristic

#19151553

C₄

yes

characteristic

#19151432

C₄

yes

complemented

C₂×#15667918

C₂²

yes

—

#19151373

C₄

yes

—

C₁₆×Dic₈₉

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₄

yes

complemented

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₂×#13083558

C₈

yes

characteristic

#15667971

C₈

yes

characteristic

#15667952

C₈

yes

—

C₂×#13083549

C₂×C₄

yes

—

#15667922

C₈

yes

—

C₈×Dic₈₉

C₂×C₄

yes

—

#15667918

C₂×C₄

yes

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₃₂×C₈₉

C₂×C₄

yes

—

C₃₂×C₈₉

C₂×C₄

yes

—

#13083558

C₁₆

yes

—

C₂×#10623494

C₂×C₈

yes

—

C₄⋊Dic₈₉

C₁₆

yes

—

C₂²⋊Dic₈₉

C₁₆

yes

—

C₄×Dic₈₉

C₂×C₈

yes

—

#13083549

C₂×C₈

yes

—

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₄²

yes

—

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂×C₈

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

C₄²

yes

—

C₁₆×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

C₂²×C₄

yes

characteristic

C₂³×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₄×C₈

yes

—

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×C₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×Dic₈₉

C₂×C₁₆

yes

—

C₈×C₈₉

C₄×C₈

yes

—

C₈×C₈₉

C₂×C₄²

yes

characteristic

C₈×C₈₉

C₂²×C₈

yes

characteristic

#10623494

C₂×C₁₆

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×C₁₆

yes

—

C₂²×C₈₉

C₂²×C₁₆

yes

characteristic

C₄×C₈₉

C₄×C₁₆

yes

—

C₄×C₈₉

C₂×C₄×C₈

yes

characteristic

C₄×C₈₉

C₂²×C₁₆

yes

characteristic

Dic₈₉

—

#13549

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄×C₁₆

yes

characteristic

C₂×C₈₉

#2258

yes

—

C₂×M₆₄

—

C₂²×C₃₂

D₈₉

yes

characteristic

#2108

—

#1397

—

C₄×C₃₂

—

C₈₉

#13549

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

Dic₈₉

yes

complemented

C₂×C₃₂

C₂×D₈₉

yes

characteristic

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#346

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₄×D₈₉

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₄×D₈₉

yes

—

C₂×C₁₆

C₂×Dic₈₉

yes

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

C₂×Dic₈₉

yes

—

C₃₂

C₂×Dic₈₉

yes

—

C₂²×C₄

C₈×D₈₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄×Dic₈₉

yes

—

C₂×C₈

C₈×D₈₉

yes

—

100

C₂×C₈

C₈×D₈₉

yes

characteristic

101

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

C₄×Dic₈₉

yes

—

107

C₁₆

C₂×C₄×D₈₉

yes

characteristic

108

C₁₆

C₂²×Dic₈₉

yes

characteristic

109

C₂³

C₁₆×D₈₉

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

C₈×Dic₈₉

yes

—

112

C₂×C₄

C₁₆×D₈₉

yes

characteristic

113

C₂×C₄

C₁₆×D₈₉

yes

—

114

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

115

C₈

—

116

C₈

C₈×Dic₈₉

yes

—

117

C₈

C₂×C₄×Dic₈₉

yes

characteristic

118

C₈

C₂×C₈×D₈₉

yes

characteristic

119

C₂²

—

120

C₂²

C₁₆×Dic₈₉

yes

—

121

C₂²

C₂×C₁₆×D₈₉

yes

characteristic

122

C₄

—

178

—

123

C₄

C₁₆×Dic₈₉

yes

—

124

C₄

C₂×C₈×Dic₈₉

yes

characteristic

125

C₄

C₂×C₁₆×D₈₉

yes

characteristic

126

C₂

—

127

C₂

C₂×C₁₆×Dic₈₉

yes

characteristic

128

C₂

#26022545

yes

—

	subgroup	G/H
center	C₂×C₃₂	C₂×D₈₉
commutator	C₂×C₈₉	C₂×C₄×C₁₆
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₂²×C₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₁₆
prime radical	C₂³×C₈₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	C₂²×C₃₂	D₈₉
hypocenter	C₈₉	#13549
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂

subgroup

G/H

center

C₂×C₃₂

C₂×D₈₉

commutator

C₂×C₈₉

C₂×C₄×C₁₆

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₂²×C₃₂×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×C₁₆

prime radical

C₂³×C₈₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

hypercenter

C₂²×C₃₂

D₈₉

C₈₉

G ▷ C₂×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂

N	H ≅ G/N	type
3. #26018385	C₂	semidirect
4. C₃₂×Dic₈₉	C₂	semidirect
8. #19151432	C₄	semidirect
13. C₂×C₃₂×C₈₉	C₄	semidirect
70. C₈₉	#13549	semidirect
76. C₂×C₃₂	Dic₈₉	semidirect

H ≅ G/N

type

3. #26018385

C₂

semidirect

4. C₃₂×Dic₈₉

C₂

semidirect

8. #19151432

C₄

semidirect

13. C₂×C₃₂×C₈₉

C₄

semidirect

70. C₈₉

#13549

semidirect

76. C₂×C₃₂

Dic₈₉

semidirect

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

subgroup	index
C₂×#19151432	2
#26021848	2
#26018385	2
C₃₂×Dic₈₉	2
C₂²×C₃₂×C₈₉	2
#13549	89

subgroup

index

C₂×#19151432

#26021848

#26018385

C₃₂×Dic₈₉

C₂²×C₃₂×C₈₉

#13549

Subgroups — #36474607

Conjugacy classes of subgroups

Complements

Maximal subgroups