Group #26019293 — subgroups

Subgroups — #26019293

Order 11392.

202 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19152229	C₂	1	1	yes	maximal, characteristic
2	#19151933	C₂	1	2	yes	maximal
3	#19151334	C₂	1	2	yes	maximal
4	#19151324	C₂	1	1	yes	maximal, characteristic
5	#400×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668011	—	2	2	no	—
7	#15668003	—	2	2	no	—
8	#15667991	—	2	1	no	—
9	#15667991	C₂²	1	2	yes	—
10	#15667971	—	2	1	no	—
11	#15667945	—	2	2	no	—
12	#15667935	—	2	2	no	—
13	C₂²⋊#10623494	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
16	#95×C₈₉	—	2	1	no	—
17	#95×C₈₉	—	2	1	no	—
18	#95×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₇₈	—	2	1	no	—
20	C₂×Dic₁₇₈	D₄	1	2	yes	—
21	C₂×#10623494	—	2	2	no	—
22	C₄⋊Dic₈₉	—	2	1	no	—
23	C₄⋊Dic₈₉	—	2	1	no	—
24	C₄⋊Dic₈₉	—	2	1	no	—
25	C₄⋊Dic₈₉	D₄	1	2	yes	—
26	#13083553	—	2	2	no	—
27	#13083553	—	2	2	no	—
28	C₂²⋊Dic₈₉	—	2	2	no	—
29	C₄×Dic₈₉	—	2	2	no	—
30	C₂²×C₄×C₈₉	—	2	1	no	—
31	C₂²×C₄×C₈₉	—	2	1	no	—
32	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈₉	—	2	1	no	—
41	C₂×C₄×C₈₉	—	4	1	no	—
42	C₂×C₄×C₈₉	—	4	1	no	—
43	C₂×C₄×C₈₉	—	4	1	no	—
44	C₂×C₄×C₈₉	—	4	1	no	—
45	C₂×C₄×C₈₉	—	4	2	no	—
46	C₂×C₄×C₈₉	—	4	1	no	—
47	C₂×C₄×C₈₉	—	4	1	no	—
48	C₂×C₄×C₈₉	Q₁₆	1	2	yes	—
49	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
50	C₂×Dic₈₉	—	2	2	no	—
51	C₂×Dic₈₉	—	2	2	no	—
52	Dic₁₇₈	—	4	2	no	—
53	Dic₁₇₈	—	8	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	1	no	—
63	C₄×C₈₉	—	8	1	no	—
64	C₄×C₈₉	—	8	1	no	—
65	Dic₈₉	—	4	2	no	—
66	Dic₈₉	—	4	2	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	—	89	1	no	—
75	#484	—	89	2	no	—
76	#400	D₈₉	1	1	yes	characteristic
77	#328	—	89	2	no	—
78	#324	—	89	1	no	—
79	C₄⋊Q₈	—	178	2	no	—
80	#125	—	178	2	no	—
81	C₂²⋊Q₈	—	89	2	no	—
82	C₂²⋊Q₈	—	178	1	no	—
83	#117	—	178	1	no	—
84	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
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	—	2	1	no	—
90	#95	—	2	1	no	—
91	#95	C₂×D₈₉	1	1	yes	characteristic
92	C₂×Q₈	—	89	2	no	—
93	C₂×Q₈	—	178	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	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	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	89	2	no	—
103	C₄⋊C₄	—	178	1	no	—
104	C₄⋊C₄	—	178	1	no	—
105	C₄⋊C₄	—	178	1	no	—
106	C₄⋊C₄	—	178	2	no	—
107	C₄⋊C₄	—	178	2	no	—
108	C₂²⋊C₄	—	178	2	no	—
109	C₄²	—	178	2	no	—
110	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
111	Q₈	—	356	2	no	—
112	Q₈	—	712	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	178	2	no	—
124	C₂×C₄	—	178	2	no	—
125	C₂×C₄	#13083564	1	2	yes	—
126	C₂×C₄	D₄×D₈₉	1	2	yes	—
127	C₈	—	356	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#15667980	1	1	yes	characteristic
131	C₂²	C₂×#13083564	1	1	yes	characteristic
132	C₂²	#15668042	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	2	no	—
135	C₄	—	4	1	no	—
136	C₄	—	8	1	no	—
137	C₄	—	8	1	no	—
138	C₄	—	356	2	no	—
139	C₄	—	356	2	no	—
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#19151753	1	1	yes	characteristic
143	C₂	#19151770	1	1	yes	characteristic
144	C₂	#19151787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152229

C₂

yes

maximal, characteristic

#19151933

C₂

yes

maximal

#19151334

C₂

yes

maximal

#19151324

C₂

yes

maximal, characteristic

#400×C₈₉

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

#95×C₈₉

—

#95×C₈₉

—

#95×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

#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

—

#484

—

#400

D₈₉

yes

characteristic

#328

—

#324

—

C₄⋊Q₈

—

178

—

#125

—

178

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

178

—

#117

—

178

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

#107

—

178

—

#103

—

178

—

C₂²⋊C₈

—

#95

—

#95

—

#95

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

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

—

178

—

104

C₄⋊C₄

—

178

—

105

C₄⋊C₄

—

178

—

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

—

178

—

108

C₂²⋊C₄

—

178

—

109

C₄²

—

178

—

110

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

111

Q₈

—

356

—

112

Q₈

—

712

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

178

—

124

C₂×C₄

—

178

—

125

C₂×C₄

#13083564

yes

—

126

C₂×C₄

D₄×D₈₉

yes

—

127

C₈

—

356

—

128

C₂²

—

129

C₂²

—

130

C₂²

#15667980

yes

characteristic

131

C₂²

C₂×#13083564

yes

characteristic

132

C₂²

#15668042

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

356

—

139

C₄

—

356

—

140

C₂

—

141

C₂

—

142

C₂

#19151753

yes

characteristic

143

C₂

#19151770

yes

characteristic

144

C₂

#19151787

yes

characteristic

	subgroup	G/H
center	C₂²	#15667980
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#400×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#400	D₈₉
hypocenter	C₈₉	#1617
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #400

subgroup

G/H

center

C₂²

#15667980

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#400×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 ◁ #400

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
#19152229	2
#19151933	2
#19151334	2
#19151324	2
#400×C₈₉	2
#1617	89

subgroup

index

#400×C₈₉

Subgroups — #26019293

Conjugacy classes of subgroups

Complements

Maximal subgroups