Group #26020321 — subgroups

Subgroups — #26020321

Order 11392.

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

normal subgroups only

This group has 292 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	C₂×#15667971	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	#19151352	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10623494	C₂²	1	2	yes	—
6	#15667971	—	2	2	no	—
7	#15667971	—	2	2	no	—
8	C₂×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₈₉	C₄	1	2	yes	—
11	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
12	C₂²⋊#10623494	—	2	2	no	—
13	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₈₉	—	2	2	no	—
16	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
17	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
18	C₂×#10623494	—	2	2	no	—
19	C₂×#10623494	—	2	4	no	—
20	C₄⋊Dic₈₉	—	2	2	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₂²⋊Dic₈₉	—	2	4	no	—
23	C₄×Dic₈₉	—	2	2	no	—
24	C₄×Dic₈₉	—	2	2	no	—
25	C₂⁴×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	—	2	4	no	—
27	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	D₄	1	2	yes	—
29	C₂²×C₄×C₈₉	D₄	1	2	yes	—
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₈₉	—	2	2	no	—
32	C₂×C₈×C₈₉	—	2	4	no	—
33	C₂³×C₈₉	—	2	2	no	—
34	C₂³×C₈₉	—	2	2	no	—
35	C₂³×C₈₉	—	2	2	no	—
36	C₂³×C₈₉	C₂²⋊C₄	1	2	yes	—
37	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	—	2	2	no	—
39	C₂×C₄×C₈₉	—	2	4	no	—
40	C₂×C₄×C₈₉	—	2	4	no	—
41	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	#41	1	4	yes	—
45	C₂×Dic₈₉	—	2	2	no	—
46	C₂×Dic₈₉	—	2	2	no	—
47	C₂×Dic₈₉	—	2	4	no	—
48	C₂×Dic₈₉	—	2	4	no	—
49	C₈×C₈₉	—	4	2	no	—
50	#10623494	—	4	2	no	—
51	C₂²×C₈₉	—	2	2	no	—
52	C₂²×C₈₉	—	2	2	no	—
53	C₂²×C₈₉	—	2	2	no	—
54	C₂²×C₈₉	—	4	4	no	—
55	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₈₉	#117	1	2	yes	—
57	C₂²×C₈₉	#131	1	2	yes	—
58	C₂²×C₈₉	#131	1	2	yes	—
59	C₄×C₈₉	—	2	2	no	—
60	C₄×C₈₉	#123	1	2	yes	—
61	C₄×C₈₉	#123	1	2	yes	—
62	Dic₈₉	—	4	2	no	—
63	Dic₈₉	—	4	2	no	—
64	C₂×C₈₉	—	4	2	no	—
65	C₂×C₈₉	#381	1	1	yes	characteristic
66	C₂×C₈₉	#408	1	1	yes	characteristic
67	C₂×C₈₉	#408	1	1	yes	characteristic
68	C₂×C₈₉	#433	1	4	yes	—
69	#1818	—	89	1	no	maximal, Sylow
70	C₈₉	#1818	1	1	yes	characteristic, complemented, Sylow
71	C₂×#117	—	89	1	no	—
72	C₂×C₂²⋊C₈	—	89	1	no	—
73	C₂×C₂²⋊C₈	D₈₉	1	1	yes	characteristic
74	#336	—	89	4	no	—
75	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
76	C₂²×C₈	—	89	2	no	—
77	C₂²×C₈	C₂×D₈₉	1	2	yes	—
78	#117	—	178	2	no	—
79	#117	—	178	2	no	—
80	C₂×C₄⋊C₄	—	89	1	no	—
81	C₂×C₄⋊C₄	—	89	1	no	—
82	C₂×C₂²⋊C₄	—	89	2	no	—
83	C₂×C₄²	—	89	2	no	—
84	C₂²⋊C₈	—	2	2	no	—
85	C₂²⋊C₈	—	178	2	no	—
86	C₂⁴	C₄×D₈₉	1	1	yes	characteristic
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	89	2	no	—
89	C₂²×C₄	—	89	2	no	—
90	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
91	C₂²×C₄	D₃₅₆	1	2	yes	—
92	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
93	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	178	2	no	—
97	C₂×C₈	—	178	4	no	—
98	C₄⋊C₄	—	178	2	no	—
99	C₄⋊C₄	—	178	2	no	—
100	C₂²⋊C₄	—	178	4	no	—
101	C₄²	—	178	2	no	—
102	C₄²	—	178	2	no	—
103	C₂³	—	2	2	no	—
104	C₂³	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	#13083551	1	2	yes	—
107	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	178	4	no	—
112	C₂×C₄	—	178	4	no	—
113	C₂×C₄	—	178	2	no	—
114	C₂×C₄	—	178	2	no	—
115	C₂×C₄	#13083551	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
117	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
118	C₂×C₄	#13083573	1	4	yes	—
119	C₈	—	4	2	no	—
120	C₈	—	356	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	4	4	no	—
125	C₂²	C₂×#13083551	1	1	yes	characteristic
126	C₂²	#15667968	1	2	yes	—
127	C₂²	#15668018	1	2	yes	—
128	C₂²	#15668020	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	356	2	no	—
131	C₄	—	356	2	no	—
132	C₄	#15667993	1	2	yes	—
133	C₄	#15667996	1	2	yes	—
134	C₂	—	4	2	no	—
135	C₂	#19151459	1	1	yes	characteristic
136	C₂	#19151564	1	1	yes	characteristic
137	C₂	#19151566	1	1	yes	characteristic
138	C₂	#19151656	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667971

C₂

yes

maximal, characteristic

C₂×C₂²⋊#10623494

C₂

yes

maximal, characteristic

#19151352

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

#15667971

—

#15667971

—

C₂×C₄⋊Dic₈₉

C₄

yes

characteristic

C₂×C₄⋊Dic₈₉

C₄

yes

characteristic

C₂×C₂²⋊Dic₈₉

C₄

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

C₂²⋊#10623494

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

—

C₂²⋊C₈×C₈₉

—

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²⋊C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

#117

yes

—

C₂²×C₈₉

#131

yes

—

C₂²×C₈₉

#131

yes

—

C₄×C₈₉

—

C₄×C₈₉

#123

yes

—

C₄×C₈₉

#123

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#1818

—

maximal, Sylow

C₈₉

#1818

yes

characteristic, complemented, Sylow

C₂×#117

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

D₈₉

yes

characteristic

#336

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

—

#117

—

178

—

#117

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

178

—

C₂⁴

C₄×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

100

C₂²⋊C₄

—

178

—

101

C₄²

—

178

—

102

C₄²

—

178

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

#13083551

yes

—

107

C₂³

C₂×C₄×D₈₉

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

178

—

112

C₂×C₄

—

178

—

113

C₂×C₄

—

178

—

114

C₂×C₄

—

178

—

115

C₂×C₄

#13083551

yes

—

116

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

117

C₂×C₄

C₂×D₃₅₆

yes

characteristic

118

C₂×C₄

#13083573

yes

—

119

C₈

—

120

C₈

—

356

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×#13083551

yes

characteristic

126

C₂²

#15667968

yes

—

127

C₂²

#15668018

yes

—

128

C₂²

#15668020

yes

—

129

C₄

—

130

C₄

—

356

—

131

C₄

—

356

—

132

C₄

#15667993

yes

—

133

C₄

#15667996

yes

—

134

C₂

—

135

C₂

#19151459

yes

characteristic

136

C₂

#19151564

yes

characteristic

137

C₂

#19151566

yes

characteristic

138

C₂

#19151656

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₉
commutator, socle	C₂³×C₈₉	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂×C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×C₂²⋊C₈	D₈₉
hypocenter	C₈₉	#1818
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₉

commutator, socle

C₂³×C₈₉

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂×C₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂×C₂²⋊C₈

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #19151352	C₂	semidirect
70. C₈₉	#1818	semidirect

H ≅ G/N

type

3. #19151352

C₂

semidirect

70. C₈₉

#1818

semidirect

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

subgroup	index
C₂×#15667971	2
C₂×C₂²⋊#10623494	2
#19151352	2
C₂×C₂²⋊C₈×C₈₉	2
#1818	89

subgroup

index

C₂×#15667971

C₂×C₂²⋊#10623494

#19151352

C₂×C₂²⋊C₈×C₈₉

#1818

Subgroups — #26020321

Conjugacy classes of subgroups

Complements

Maximal subgroups