Group #26030908 — subgroups

Subgroups — #26030908

Order 11392.

338 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19152209	C₂	1	2	yes	maximal
2	#19152000	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151966	C₂	1	2	yes	maximal, complemented
4	#19151853	C₂	1	2	yes	maximal
5	#19151836	C₂	1	2	yes	maximal
6	#19151705	C₂	1	2	yes	maximal, complemented
7	Q₁₆×Dic₈₉	C₂	1	2	yes	maximal, complemented
8	#19151552	C₂	1	1	yes	maximal, characteristic
9	#494×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13083557	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
12	#15668007	—	2	4	no	—
13	#15668004	C₂²	1	2	yes	—
14	#15668002	—	2	4	no	—
15	#15667988	C₂²	1	4	yes	—
16	Q₈×Dic₈₉	C₂²	1	4	yes	—
17	#15667969	C₂²	1	1	yes	characteristic
18	#15667946	C₂²	1	2	yes	—
19	#15667944	C₂²	1	2	yes	—
20	#15667936	C₂²	1	4	yes	—
21	#15667936	C₂²	1	4	yes	—
22	#15667923	C₂²	1	1	yes	characteristic
23	#15667922	C₂²	1	1	yes	characteristic
24	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
25	C₂×Q₁₆×C₈₉	C₂²	1	2	yes	—
26	C₂×Q₁₆×C₈₉	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
28	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
30	#13083557	—	2	2	no	—
31	C₂×#10623494	C₂³	1	1	yes	characteristic
32	C₂×#10623494	C₂³	1	1	yes	characteristic
33	C₈×D₈₉	—	2	2	no	—
34	C₄⋊Dic₈₉	—	2	4	no	—
35	C₄⋊Dic₈₉	C₂³	1	4	yes	—
36	#13083553	—	2	4	no	—
37	#13083553	—	2	4	no	—
38	#13083553	D₄	1	2	yes	—
39	#13083551	—	2	4	no	—
40	#13083551	D₄	1	2	yes	—
41	C₄×Dic₈₉	—	2	4	no	—
42	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₈₉	C₂³	1	4	yes	—
44	Q₁₆×C₈₉	—	2	4	no	—
45	Q₁₆×C₈₉	—	2	4	no	—
46	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₈₉	—	2	4	no	—
49	C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₈₉	—	2	4	no	—
52	Q₈×C₈₉	—	2	4	no	—
53	C₂×C₄×C₈₉	—	2	4	no	—
54	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₈₉	—	2	4	no	—
57	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
58	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
59	C₄×D₈₉	—	2	1	no	—
60	C₄×D₈₉	—	2	1	no	—
61	C₈×C₈₉	#41	1	2	yes	—
62	C₈×C₈₉	#41	1	2	yes	—
63	#10623494	—	2	1	no	—
64	#10623494	—	2	1	no	—
65	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₈₉	—	2	1	no	—
67	C₂×D₈₉	—	2	2	no	—
68	C₄×C₈₉	—	2	2	no	—
69	C₄×C₈₉	—	4	4	no	—
70	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
71	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
72	Dic₈₉	—	2	1	no	—
73	Dic₈₉	—	2	2	no	—
74	Dic₈₉	—	4	4	no	—
75	C₂×C₈₉	#532	1	1	yes	characteristic
76	C₂×C₈₉	#578	1	2	yes	—
77	D₈₉	—	4	1	no	—
78	#3177	—	89	1	no	maximal, Sylow
79	C₈₉	#3177	1	1	yes	characteristic, complemented, Sylow
80	#541	—	89	2	no	—
81	#499	—	89	1	no	—
82	#494	D₈₉	1	1	yes	characteristic, complemented
83	#491	—	89	2	no	—
84	#470	—	89	2	no	—
85	#467	—	89	2	no	—
86	#441	—	89	2	no	—
87	C₄×Q₁₆	—	89	2	no	—
88	#405	—	89	1	no	—
89	C₂×Q₁₆	C₂×D₈₉	1	2	yes	—
90	C₂×Q₁₆	C₂×D₈₉	1	2	yes	—
91	C₂×M₁₆	—	89	1	no	—
92	C₂²×C₈	—	89	1	no	—
93	C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
94	#126	—	178	4	no	—
95	#125	—	89	2	no	—
96	#125	—	178	4	no	—
97	C₂²⋊Q₈	—	89	4	no	—
98	C₄×Q₈	—	89	4	no	—
99	#117	—	89	1	no	—
100	#107	—	89	2	no	—
101	#106	—	89	2	no	—
102	#103	—	89	4	no	—
103	#103	—	89	4	no	—
104	#97	—	89	1	no	—
105	#97	—	89	1	no	—
106	C₄×C₈	—	89	1	no	—
107	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
108	C₂×Q₈	C₂²×D₈₉	1	4	yes	—
109	C₂²×C₄	—	89	1	no	—
110	Q₁₆	—	2	4	no	—
111	Q₁₆	—	2	4	no	—
112	M₁₆	—	178	2	no	—
113	C₂×C₈	—	89	1	no	—
114	C₂×C₈	—	89	1	no	—
115	C₂×C₈	—	178	2	no	—
116	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	89	2	no	—
120	C₄⋊C₄	—	89	4	no	—
121	C₄⋊C₄	—	178	4	no	—
122	C₄⋊C₄	—	178	4	no	—
123	C₄⋊C₄	—	178	4	no	—
124	C₂²⋊C₄	—	89	2	no	—
125	C₂²⋊C₄	—	178	4	no	—
126	C₄²	—	89	1	no	—
127	C₄²	—	178	4	no	—
128	C₄²	C₂²×D₈₉	1	1	yes	characteristic
129	C₂³	—	89	1	no	—
130	Q₈	—	2	4	no	—
131	Q₈	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	89	2	no	—
134	C₂×C₄	—	89	1	no	—
135	C₂×C₄	—	178	4	no	—
136	C₂×C₄	—	178	1	no	—
137	C₂×C₄	—	178	1	no	—
138	C₂×C₄	D₄×D₈₉	1	2	yes	—
139	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
140	C₈	—	178	1	no	—
141	C₈	—	178	1	no	—
142	C₈	#13083573	1	2	yes	—
143	C₈	#13083573	1	2	yes	—
144	C₂²	—	178	1	no	—
145	C₂²	—	178	2	no	—
146	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	356	4	no	—
150	C₄	—	178	1	no	—
151	C₄	—	178	2	no	—
152	C₄	C₂×#13083573	1	1	yes	characteristic
153	C₄	C₂×#13083573	1	1	yes	characteristic
154	C₂	—	356	1	no	—
155	C₂	#19152146	1	1	yes	characteristic
156	C₂	#19152411	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152209

C₂

yes

maximal

#19152000

C₂

yes

maximal, characteristic, complemented

#19151966

C₂

yes

maximal, complemented

#19151853

C₂

yes

maximal

#19151836

C₂

yes

maximal

#19151705

C₂

yes

maximal, complemented

Q₁₆×Dic₈₉

C₂

yes

maximal, complemented

#19151552

C₂

yes

maximal, characteristic

#494×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083557

C₂²

yes

characteristic

C₂×C₈×D₈₉

C₂²

yes

characteristic

#15668007

—

#15668004

C₂²

yes

—

#15668002

—

#15667988

C₂²

yes

—

Q₈×Dic₈₉

C₂²

yes

—

#15667969

C₂²

yes

characteristic

#15667946

C₂²

yes

—

#15667944

C₂²

yes

—

#15667936

C₂²

yes

—

#15667936

C₂²

yes

—

#15667923

C₂²

yes

characteristic

#15667922

C₂²

yes

characteristic

C₈×Dic₈₉

C₂²

yes

characteristic

C₂×Q₁₆×C₈₉

C₂²

yes

—

C₂×Q₁₆×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂³

yes

characteristic

#13083557

—

C₂×#10623494

C₂³

yes

characteristic

C₂×#10623494

C₂³

yes

characteristic

C₈×D₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

#13083553

D₄

yes

—

#13083551

—

#13083551

D₄

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂³

yes

characteristic

C₂×Q₈×C₈₉

C₂³

yes

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

#41

yes

—

C₈×C₈₉

#41

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#41

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#532

yes

characteristic

C₂×C₈₉

#578

yes

—

D₈₉

—

#3177

—

maximal, Sylow

C₈₉

#3177

yes

characteristic, complemented, Sylow

#541

—

#499

—

#494

D₈₉

yes

characteristic, complemented

—

—

—

—

C₄×Q₁₆

—

#405

—

C₂×Q₁₆

C₂×D₈₉

yes

—

C₂×Q₁₆

C₂×D₈₉

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₈₉

yes

—

#126

—

178

—

#125

—

#125

—

178

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

100

#107

—

101

#106

—

102

#103

—

103

#103

—

104

#97

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

C₂×D₈₉

yes

characteristic

108

C₂×Q₈

C₂²×D₈₉

yes

—

109

C₂²×C₄

—

110

Q₁₆

—

111

Q₁₆

—

112

M₁₆

—

178

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

178

—

116

C₂×C₈

C₂²×D₈₉

yes

characteristic

117

C₂×C₈

C₂²×D₈₉

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

178

—

122

C₄⋊C₄

—

178

—

123

C₄⋊C₄

—

178

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

178

—

126

C₄²

—

127

C₄²

—

178

—

128

C₄²

C₂²×D₈₉

yes

characteristic

129

C₂³

—

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

178

—

136

C₂×C₄

—

178

—

137

C₂×C₄

—

178

—

138

C₂×C₄

D₄×D₈₉

yes

—

139

C₂×C₄

C₂³×D₈₉

yes

characteristic

140

C₈

—

178

—

141

C₈

—

178

—

142

C₈

#13083573

yes

—

143

C₈

#13083573

yes

—

144

C₂²

—

178

—

145

C₂²

—

178

—

146

C₂²

C₂×D₄×D₈₉

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

356

—

150

C₄

—

178

—

151

C₄

—

178

—

152

C₄

C₂×#13083573

yes

characteristic

153

C₄

C₂×#13083573

yes

characteristic

154

C₂

—

356

—

155

C₂

#19152146

yes

characteristic

156

C₂

#19152411

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#494×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂²×D₈₉	C₂×D₄
hypercenter	#494	D₈₉
hypocenter	C₈₉	#3177
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#494×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂²×D₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #19152000	C₂	semidirect
3. #19151966	C₂	semidirect
6. #19151705	C₂	semidirect
7. Q₁₆×Dic₈₉	C₂	semidirect
9. #494×C₈₉	C₂	semidirect
79. C₈₉	#3177	semidirect
82. #494	D₈₉	semidirect

H ≅ G/N

type

2. #19152000

C₂

semidirect

3. #19151966

C₂

semidirect

6. #19151705

C₂

semidirect

7. Q₁₆×Dic₈₉

C₂

semidirect

9. #494×C₈₉

C₂

semidirect

79. C₈₉

#3177

semidirect

82. #494

D₈₉

semidirect

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

subgroup	index
#19152209	2
#19152000	2
#19151966	2
#19151853	2
#19151836	2
#19151705	2
Q₁₆×Dic₈₉	2
#19151552	2
#494×C₈₉	2
#3177	89

subgroup

index

Q₁₆×Dic₈₉

#494×C₈₉

Subgroups — #26030908

Conjugacy classes of subgroups

Complements

Maximal subgroups