Group #26018552 — subgroups

Subgroups — #26018552

Order 11392.

178 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 101 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	#19151651	C₂	1	1	yes	maximal, characteristic
2	#19151651	C₂	1	1	yes	maximal, characteristic
3	#19151320	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151320	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151319	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
7	#431×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#10623494	C₄	1	2	yes	—
9	C₄⋊#10623494	C₄	1	2	yes	—
10	C₂²⋊#10623494	C₄	1	2	yes	—
11	C₂²⋊#10623494	C₄	1	2	yes	—
12	C₄×#10623494	C₂²	1	1	yes	characteristic
13	C₄×#10623494	C₂²	1	1	yes	characteristic
14	C₄×#10623494	C₂²	1	1	yes	characteristic
15	C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₈₉	C₄	1	2	yes	—
18	C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
19	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#10623494	C₂×C₄	1	2	yes	—
22	C₂×#10623494	C₂×C₄	1	2	yes	—
23	C₂×#10623494	C₂×C₄	1	2	yes	—
24	C₂×#10623494	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₂³×C₈₉	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₈₉	—	2	2	no	—
36	C₂×C₄×C₈₉	—	2	2	no	—
37	C₂×C₄×C₈₉	—	2	2	no	—
38	C₂×C₄×C₈₉	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₄²	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₈×C₈₉	—	2	2	no	—
45	C₈×C₈₉	—	2	2	no	—
46	#10623494	—	2	2	no	—
47	#10623494	—	2	2	no	—
48	#10623494	—	2	2	no	—
49	#10623494	—	2	2	no	—
50	C₂²×C₈₉	—	2	1	no	—
51	C₂²×C₈₉	—	2	1	no	—
52	C₂²×C₈₉	—	2	1	no	—
53	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
54	C₄×C₈₉	—	2	1	no	—
55	C₄×C₈₉	—	2	1	no	—
56	C₄×C₈₉	—	2	1	no	—
57	C₄×C₈₉	#131	1	2	yes	—
58	C₄×C₈₉	#131	1	2	yes	—
59	C₄×C₈₉	#131	1	2	yes	—
60	C₂×C₈₉	—	4	1	no	—
61	C₂×C₈₉	#405	1	1	yes	characteristic
62	C₂×C₈₉	#405	1	1	yes	characteristic
63	C₂×C₈₉	#405	1	1	yes	characteristic
64	#1454	—	89	1	no	maximal, Sylow
65	C₈₉	#1454	1	1	yes	characteristic, complemented, Sylow
66	#431	—	89	1	no	—
67	#431	—	89	1	no	—
68	#431	D₈₉	1	1	yes	characteristic
69	#322	—	89	1	no	—
70	#322	—	89	1	no	—
71	#322	—	89	1	no	—
72	C₈²	—	89	1	no	—
73	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
74	C₄⋊C₈	—	89	2	no	—
75	C₄⋊C₈	—	89	2	no	—
76	C₄⋊C₈	Dic₈₉	1	2	yes	—
77	C₂²⋊C₈	—	89	2	no	—
78	C₂²⋊C₈	—	89	2	no	—
79	C₂²⋊C₈	Dic₈₉	1	2	yes	—
80	C₄×C₈	—	89	1	no	—
81	C₄×C₈	—	89	1	no	—
82	C₄×C₈	—	89	1	no	—
83	C₄×C₈	—	89	1	no	—
84	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
85	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
89	C₂×C₈	—	89	2	no	—
90	C₂×C₈	—	89	2	no	—
91	C₂×C₈	—	89	2	no	—
92	C₂×C₈	—	89	2	no	—
93	C₂×C₈	C₂×Dic₈₉	1	2	yes	—
94	C₂×C₈	C₂×Dic₈₉	1	2	yes	—
95	C₄²	C₄×D₈₉	1	1	yes	characteristic
96	C₄²	C₄×D₈₉	1	1	yes	characteristic
97	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
98	C₄²	C₂²×D₈₉	1	1	yes	characteristic
99	C₂³	C₄×Dic₈₉	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₄×Dic₈₉	1	1	yes	characteristic
104	C₂×C₄	C₄×Dic₈₉	1	1	yes	characteristic
105	C₂×C₄	C₄×Dic₈₉	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
108	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	178	2	no	—
112	C₈	—	178	2	no	—
113	C₈	—	178	2	no	—
114	C₈	—	178	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	C₂×C₄×Dic₈₉	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	#15668018	1	2	yes	—
123	C₄	#15668019	1	2	yes	—
124	C₄	#15668020	1	2	yes	—
125	C₂	—	4	1	no	—
126	C₂	#19151553	1	1	yes	characteristic
127	C₂	#19151554	1	1	yes	characteristic
128	C₂	#19151556	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151651

C₂

yes

maximal, characteristic

#19151651

C₂

yes

maximal, characteristic

#19151320

C₂

yes

maximal, characteristic, complemented

#19151320

C₂

yes

maximal, characteristic, complemented

#19151319

C₂

yes

maximal, characteristic, complemented

C₈×#10623494

C₂

yes

maximal, characteristic, complemented

#431×C₈₉

C₂

yes

maximal, characteristic

C₄⋊#10623494

C₄

yes

—

C₄⋊#10623494

C₄

yes

—

C₂²⋊#10623494

C₄

yes

—

C₂²⋊#10623494

C₄

yes

—

C₄×#10623494

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₄

yes

—

C₂²⋊C₈×C₈₉

C₄

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

—

—

—

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄²

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#131

yes

—

C₄×C₈₉

#131

yes

—

C₄×C₈₉

#131

yes

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#431

—

#431

—

#431

D₈₉

yes

—

—

—

C₈²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₈₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₈₉

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

C₂×Dic₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₈₉

yes

—

C₂×C₈

C₂×Dic₈₉

yes

—

C₄²

C₄×D₈₉

yes

characteristic

C₄²

C₄×D₈₉

yes

characteristic

C₄²

C₂×Dic₈₉

yes

characteristic

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

C₄×Dic₈₉

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₄×Dic₈₉

yes

characteristic

104

C₂×C₄

C₄×Dic₈₉

yes

characteristic

105

C₂×C₄

C₄×Dic₈₉

yes

characteristic

106

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

107

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

108

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

178

—

112

C₈

—

178

—

113

C₈

—

178

—

114

C₈

—

178

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×C₄×Dic₈₉

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#15668018

yes

—

123

C₄

#15668019

yes

—

124

C₄

#15668020

yes

—

125

C₂

—

126

C₂

#19151553

yes

characteristic

127

C₂

#19151554

yes

characteristic

128

C₂

#19151556

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂×C₄²
Fitting	#431×C₈₉	C₂
prime radical	C₂³×C₈₉	C₄²
cosocle	C₄²×C₈₉	C₂³
hypercenter	#431	D₈₉
hypocenter	C₈₉	#1454
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₈₉

commutator, socle

C₂²×C₈₉

C₂×C₄²

Fitting

#431×C₈₉

C₂

prime radical

C₂³×C₈₉

C₄²

cosocle

C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #19151320	C₂	semidirect
4. #19151320	C₂	semidirect
5. #19151319	C₂	semidirect
6. C₈×#10623494	C₂	semidirect
65. C₈₉	#1454	semidirect

H ≅ G/N

type

3. #19151320

C₂

semidirect

4. #19151320

C₂

semidirect

5. #19151319

C₂

semidirect

6. C₈×#10623494

C₂

semidirect

65. C₈₉

#1454

semidirect

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

subgroup	index
#19151651	2
#19151320	2
#19151319	2
C₈×#10623494	2
#431×C₈₉	2
#1454	89

subgroup

index

#431×C₈₉

Subgroups — #26018552

Conjugacy classes of subgroups

Complements

Maximal subgroups