Group #26020997 — subgroups

Subgroups — #26020997

Order 11392.

208 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15667947	C₂	1	1	yes	maximal, characteristic
2	#19151634	C₂	1	1	yes	maximal, characteristic
3	#19151624	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151359	C₂	1	1	yes	maximal, characteristic
6	#19151357	C₂	1	1	yes	maximal, characteristic
7	#424×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13083558	C₂²	1	1	yes	characteristic
9	C₂²×#10623494	C₂²	1	1	yes	characteristic
10	#15667971	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
12	#15667947	—	2	2	no	—
13	#15667945	C₄	1	2	yes	complemented
14	#15667943	C₄	1	2	yes	complemented
15	#15667941	—	2	2	no	—
16	C₈×Dic₈₉	—	2	2	no	—
17	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#117×C₈₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₈₉	—	2	2	no	—
21	C₂²×Dic₈₉	—	2	1	no	—
22	#13083558	—	2	1	no	—
23	#13083558	—	2	1	no	—
24	C₂×#10623494	—	2	1	no	—
25	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
26	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
27	C₂×#10623494	D₄	1	2	yes	—
28	C₂×#10623494	Q₈	1	2	yes	—
29	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
30	C₂²⋊Dic₈₉	—	2	1	no	—
31	C₄×Dic₈₉	—	2	1	no	—
32	C₄×Dic₈₉	—	2	1	no	—
33	C₄×Dic₈₉	Q₈	1	2	yes	—
34	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
35	M₁₆×C₈₉	—	2	1	no	—
36	M₁₆×C₈₉	—	2	1	no	—
37	C₂×C₈×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	—	2	1	no	—
39	C₂×C₈×C₈₉	—	2	1	no	—
40	C₂×C₈×C₈₉	D₄	1	2	yes	—
41	C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
42	C₂²⋊C₄×C₈₉	—	2	1	no	—
43	C₄²×C₈₉	—	2	1	no	—
44	C₂³×C₈₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
52	C₂×Dic₈₉	—	2	4	no	—
53	C₂×Dic₈₉	—	2	2	no	—
54	C₂×Dic₈₉	—	2	2	no	—
55	C₈×C₈₉	—	2	2	no	—
56	C₈×C₈₉	—	4	1	no	—
57	#10623494	—	4	1	no	—
58	#10623494	C₄⋊C₄	1	2	yes	—
59	#10623494	C₄⋊C₄	1	2	yes	—
60	C₂²×C₈₉	—	2	2	no	—
61	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₈₉	#125	1	1	yes	characteristic
64	C₄×C₈₉	—	4	2	no	—
65	C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
67	C₄×C₈₉	#121	1	1	yes	characteristic
68	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
69	Dic₈₉	—	2	4	no	—
70	Dic₈₉	—	4	2	no	—
71	C₂×C₈₉	—	2	1	no	—
72	C₂×C₈₉	—	2	1	no	—
73	C₂×C₈₉	#389	1	1	yes	characteristic
74	C₂×C₈₉	#443	1	2	yes	—
75	#1945	—	89	1	no	maximal, Sylow
76	C₈₉	#1945	1	1	yes	characteristic, complemented, Sylow
77	C₂×#108	—	89	1	no	—
78	#427	—	89	1	no	—
79	#424	—	89	1	no	—
80	#424	D₈₉	1	1	yes	characteristic
81	C₂×C₄×C₈	—	89	1	no	—
82	#339	—	89	1	no	—
83	#339	—	89	1	no	—
84	C₂×M₁₆	—	89	1	no	—
85	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
86	C₂²×C₈	—	89	1	no	—
87	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
88	#117	—	89	1	no	—
89	#117	C₂×D₈₉	1	1	yes	characteristic
90	C₂×C₄²	—	89	1	no	—
91	#108	—	178	2	no	—
92	#107	—	89	2	no	—
93	#106	—	89	2	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	178	2	no	—
96	C₄×C₈	—	178	2	no	—
97	C₂²×C₄	—	178	1	no	—
98	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	178	1	no	—
102	M₁₆	—	178	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	89	1	no	—
107	C₂×C₈	—	89	1	no	—
108	C₂×C₈	—	89	2	no	—
109	C₂×C₈	—	89	2	no	—
110	C₂×C₈	—	178	1	no	—
111	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
112	C₄⋊C₄	—	89	2	no	—
113	C₄⋊C₄	C₄×D₈₉	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	178	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	89	2	no	—
118	C₄²	—	178	1	no	—
119	C₄²	—	178	1	no	—
120	C₂³	#13083574	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	178	2	no	—
123	C₂×C₄	—	178	4	no	—
124	C₂×C₄	—	178	2	no	—
125	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
126	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
129	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
130	C₂×C₄	#13083573	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	4	1	no	—
133	C₈	—	89	2	no	—
134	C₈	—	89	2	no	—
135	C₈	—	356	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
138	C₂²	#15667990	1	1	yes	characteristic
139	C₂²	#15668002	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	356	2	no	—
142	C₄	—	178	4	no	—
143	C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
144	C₄	#15667977	1	1	yes	characteristic
145	C₄	#15667984	1	1	yes	characteristic
146	C₄	#15668010	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#19151493	1	1	yes	characteristic
150	C₂	#19151713	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667947

C₂

yes

maximal, characteristic

#19151634

C₂

yes

maximal, characteristic

#19151624

C₂

yes

maximal, characteristic

C₂×C₈×Dic₈₉

C₂

yes

maximal, characteristic

#19151359

C₂

yes

maximal, characteristic

#19151357

C₂

yes

maximal, characteristic

#424×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

#15667971

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667947

—

#15667945

C₄

yes

complemented

#15667943

C₄

yes

complemented

#15667941

—

C₈×Dic₈₉

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²×Dic₈₉

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#10623494

C₂×C₄

yes

characteristic

C₂×#10623494

D₄

yes

—

C₂×#10623494

Q₈

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

C₄⋊C₄

yes

—

#10623494

C₄⋊C₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×D₄

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₉

#125

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₉

C₄×Q₈

yes

characteristic

C₄×C₈₉

#121

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#389

yes

characteristic

C₂×C₈₉

#443

yes

—

#1945

—

maximal, Sylow

C₈₉

#1945

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic

C₂×C₄×C₈

—

#339

—

#339

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

#117

—

#117

C₂×D₈₉

yes

characteristic

C₂×C₄²

—

#108

—

178

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄×C₈

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

178

—

102

M₁₆

—

178

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

178

—

111

C₂×C₈

C₂²⋊D₈₉

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₈₉

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

178

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

178

—

119

C₄²

—

178

—

120

C₂³

#13083574

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

178

—

123

C₂×C₄

—

178

—

124

C₂×C₄

—

178

—

125

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

126

C₂×C₄

D₄×D₈₉

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

128

C₂×C₄

Q₈×D₈₉

yes

characteristic

129

C₂×C₄

Q₈×D₈₉

yes

characteristic

130

C₂×C₄

#13083573

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

356

—

136

C₂²

—

137

C₂²

C₄×C₂²⋊D₈₉

yes

characteristic

138

C₂²

#15667990

yes

characteristic

139

C₂²

#15668002

yes

characteristic

140

C₄

—

141

C₄

—

356

—

142

C₄

—

178

—

143

C₄

C₄⋊C₄×D₈₉

yes

characteristic

144

C₄

#15667977

yes

characteristic

145

C₄

#15667984

yes

characteristic

146

C₄

#15668010

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#19151493

yes

characteristic

150

C₂

#19151713

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#424×C₈₉	C₂
socle	C₂²×C₈₉	C₄×D₄
prime radical	C₂³×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#424	D₈₉
hypocenter	C₈₉	#1945
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#424×C₈₉

C₂

socle

C₂²×C₈₉

C₄×D₄

prime radical

C₂³×C₈₉

#41

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

N	H ≅ G/N	type
13. #15667945	C₄	semidirect
14. #15667943	C₄	semidirect
76. C₈₉	#1945	semidirect

H ≅ G/N

type

13. #15667945

C₄

semidirect

14. #15667943

C₄

semidirect

76. C₈₉

#1945

semidirect

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

subgroup	index
C₂×#15667947	2
#19151634	2
#19151624	2
C₂×C₈×Dic₈₉	2
#19151359	2
#19151357	2
#424×C₈₉	2
#1945	89

subgroup

index

C₂×C₈×Dic₈₉

#424×C₈₉

Subgroups — #26020997

Conjugacy classes of subgroups

Complements

Maximal subgroups