Group #26019294 — subgroups

Subgroups — #26019294

Order 11392.

202 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19151931	C₂	1	2	yes	maximal
2	#19151537	C₂	1	1	yes	maximal, characteristic
3	#19151335	C₂	1	2	yes	maximal
4	#19151324	C₂	1	1	yes	maximal, characteristic
5	#544×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
7	#15667945	—	2	2	no	—
8	#15667936	—	2	2	no	—
9	C₂²⋊#10623494	C₂²	1	2	yes	—
10	#15667919	—	2	1	no	—
11	#15667919	—	2	1	no	—
12	#15667919	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₈₉	—	2	2	no	—
14	#125×C₈₉	—	2	2	no	—
15	C₂²⋊Q₈×C₈₉	—	2	1	no	—
16	C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
17	#117×C₈₉	—	2	1	no	—
18	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×Dic₈₉	—	2	1	no	—
20	C₂²×Dic₈₉	—	2	1	no	—
21	C₂×#10623494	—	2	2	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₂×Q₈×C₈₉	—	2	1	no	—
24	C₂×Q₈×C₈₉	D₄	1	2	yes	—
25	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	—	2	1	no	—
29	C₄⋊C₄×C₈₉	—	2	1	no	—
30	C₄⋊C₄×C₈₉	—	2	1	no	—
31	C₄⋊C₄×C₈₉	—	2	2	no	—
32	C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
35	C₂²⋊C₄×C₈₉	—	2	2	no	—
36	C₄²×C₈₉	—	2	2	no	—
37	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₈₉	—	4	2	no	—
39	Q₈×C₈₉	—	8	1	no	—
40	C₂×C₄×C₈₉	—	2	1	no	—
41	C₂×C₄×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	1	no	—
45	C₂×C₄×C₈₉	—	4	1	no	—
46	C₂×C₄×C₈₉	—	4	1	no	—
47	C₂×C₄×C₈₉	Q₁₆	1	2	yes	—
48	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
49	C₂×Dic₈₉	—	4	1	no	—
50	C₂×Dic₈₉	—	4	1	no	—
51	C₂×Dic₈₉	—	4	2	no	—
52	C₂×Dic₈₉	—	4	1	no	—
53	C₂×Dic₈₉	—	4	1	no	—
54	#10623494	—	4	2	no	—
55	C₂²×C₈₉	—	2	1	no	—
56	C₂²×C₈₉	—	2	1	no	—
57	C₂²×C₈₉	#120	1	1	yes	characteristic
58	C₂²×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂²×C₈₉	#137	1	1	yes	characteristic
60	C₄×C₈₉	—	2	2	no	—
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	—	4	2	no	—
63	C₄×C₈₉	—	4	1	no	—
64	C₄×C₈₉	—	4	2	no	—
65	Dic₈₉	—	8	1	no	—
66	Dic₈₉	—	8	1	no	—
67	C₂×C₈₉	—	2	1	no	—
68	C₂×C₈₉	—	2	1	no	—
69	C₂×C₈₉	#451	1	1	yes	characteristic
70	C₂×C₈₉	#454	1	1	yes	characteristic
71	C₂×C₈₉	#458	1	1	yes	characteristic
72	#1617	—	89	1	no	maximal, Sylow
73	C₈₉	#1617	1	1	yes	characteristic, complemented, Sylow
74	#544	D₈₉	1	1	yes	characteristic
75	#484	—	89	2	no	—
76	#400	—	89	1	no	—
77	#328	—	89	2	no	—
78	#324	—	89	1	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	#125	—	2	2	no	—
81	C₂²⋊Q₈	—	2	1	no	—
82	C₂²⋊Q₈	C₂×D₈₉	1	2	yes	—
83	#117	—	2	1	no	—
84	C₂×C₄⋊C₄	—	89	1	no	—
85	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
86	#107	—	178	2	no	—
87	#103	—	178	2	no	—
88	C₂²⋊C₈	—	89	2	no	—
89	#95	—	89	1	no	—
90	#95	—	178	1	no	—
91	#95	—	178	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	C₂²⋊D₈₉	1	2	yes	—
94	C₂²×C₄	—	178	1	no	—
95	C₂²×C₄	—	178	1	no	—
96	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
99	C₂×C₈	—	178	2	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	178	2	no	—
107	C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
111	Q₈	—	4	2	no	—
112	Q₈	—	8	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	356	1	no	—
121	C₂×C₄	—	356	1	no	—
122	C₂×C₄	—	356	2	no	—
123	C₂×C₄	—	356	1	no	—
124	C₂×C₄	—	356	1	no	—
125	C₂×C₄	#13083564	1	2	yes	—
126	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
127	C₈	—	356	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#15667981	1	1	yes	characteristic
131	C₂²	C₂×#13083564	1	1	yes	characteristic
132	C₂²	#15668043	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	2	no	—
135	C₄	—	4	2	no	—
136	C₄	—	4	1	no	—
137	C₄	—	4	2	no	—
138	C₄	—	712	1	no	—
139	C₄	—	712	1	no	—
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#19151754	1	1	yes	characteristic
143	C₂	#19151771	1	1	yes	characteristic
144	C₂	#19151788	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151931

C₂

yes

maximal

#19151537

C₂

yes

maximal, characteristic

#19151335

C₂

yes

maximal

#19151324

C₂

yes

maximal, characteristic

#544×C₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

—

C₂²

yes

—

#15667919

—

#15667919

—

#15667919

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

—

#125×C₈₉

—

C₂²⋊Q₈×C₈₉

—

C₂²⋊Q₈×C₈₉

C₂²

yes

—

#117×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂²×C₈₉

C₂×Q₁₆

yes

characteristic

C₂²×C₈₉

#137

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#544

D₈₉

yes

—

—

—

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₈₉

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

#107

—

178

—

#103

—

178

—

C₂²⋊C₈

—

#95

—

#95

—

178

—

#95

—

178

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₈₉

yes

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

178

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

C₂²⋊D₈₉

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

356

—

121

C₂×C₄

—

356

—

122

C₂×C₄

—

356

—

123

C₂×C₄

—

356

—

124

C₂×C₄

—

356

—

125

C₂×C₄

#13083564

yes

—

126

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

127

C₈

—

356

—

128

C₂²

—

129

C₂²

—

130

C₂²

#15667981

yes

characteristic

131

C₂²

C₂×#13083564

yes

characteristic

132

C₂²

#15668043

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

712

—

139

C₄

—

712

—

140

C₂

—

141

C₂

—

142

C₂

#19151754

yes

characteristic

143

C₂

#19151771

yes

characteristic

144

C₂

#19151788

yes

characteristic

	subgroup	G/H
center	C₂²	#15667981
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#544×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#544	D₈₉
hypocenter	C₈₉	#1617
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#15667981

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#544×C₈₉

C₂

socle

C₂²×C₈₉

#120

prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
73. C₈₉	#1617	semidirect

H ≅ G/N

type

73. C₈₉

#1617

semidirect

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

subgroup	index
#19151931	2
#19151537	2
#19151335	2
#19151324	2
#544×C₈₉	2
#1617	89

subgroup

index

#544×C₈₉

Subgroups — #26019294

Conjugacy classes of subgroups

Complements

Maximal subgroups