Group #26030652 — subgroups

Subgroups — #26030652

Order 11392.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15668004	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151966	C₂	1	2	yes	maximal, complemented
3	#19151962	C₂	1	2	yes	maximal, complemented
4	#19151952	C₂	1	4	yes	maximal, complemented
5	#19151599	C₂	1	4	yes	maximal, complemented
6	C₄×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13083558	C₂²	1	2	yes	—
9	#15668004	—	2	2	no	—
10	#15668004	C₂²	1	2	yes	—
11	#15668004	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	C₂×#13083553	—	2	2	no	—
14	#15667936	C₂²	1	8	yes	—
15	#15667933	C₂²	1	8	yes	—
16	#15667923	C₂²	1	2	yes	—
17	C₄×#10623494	C₂²	1	2	yes	—
18	C₂×#41×C₈₉	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	#127×C₈₉	C₂²	1	1	yes	characteristic
21	#124×C₈₉	C₂²	1	2	yes	—
22	#121×C₈₉	—	2	2	no	—
23	C₄×D₄×C₈₉	—	2	2	no	—
24	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×Dic₈₉	—	2	2	no	—
26	#13083558	—	2	4	no	—
27	C₂×#10623494	C₂³	1	4	yes	—
28	C₄⋊Dic₈₉	—	2	2	no	—
29	C₄⋊Dic₈₉	C₂³	1	4	yes	—
30	#13083553	—	2	4	no	—
31	#13083553	—	2	4	no	—
32	#41×C₈₉	—	2	8	no	—
33	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
34	C₂×D₄×C₈₉	—	2	2	no	—
35	C₂×D₄×C₈₉	—	2	2	no	—
36	C₂×D₄×C₈₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₈₉	—	2	2	no	—
38	C₂²×C₄×C₈₉	D₄	1	2	yes	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₈₉	—	2	2	no	—
41	C₂²⋊C₄×C₈₉	—	2	4	no	—
42	C₄²×C₈₉	D₄	1	2	yes	—
43	C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	C₂³×C₈₉	—	2	2	no	—
46	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₈₉	—	2	4	no	—
48	D₄×C₈₉	—	2	4	no	—
49	D₄×C₈₉	—	4	4	no	—
50	D₄×C₈₉	—	4	4	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	4	2	no	—
53	C₂×C₄×C₈₉	—	4	4	no	—
54	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#41	1	4	yes	—
58	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₈₉	—	2	4	no	—
60	C₂×Dic₈₉	—	4	4	no	—
61	#10623494	—	2	4	no	—
62	C₂²×C₈₉	—	2	2	no	—
63	C₂²×C₈₉	—	4	2	no	—
64	C₂²×C₈₉	—	4	4	no	—
65	C₂²×C₈₉	#124	1	2	yes	—
66	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
67	C₄×C₈₉	—	2	2	no	—
68	C₄×C₈₉	—	2	2	no	—
69	C₄×C₈₉	—	4	2	no	—
70	C₄×C₈₉	#124	1	2	yes	—
71	C₄×C₈₉	C₂×#41	1	2	yes	—
72	Dic₈₉	—	4	4	no	—
73	C₂×C₈₉	—	2	2	no	—
74	C₂×C₈₉	—	8	2	no	—
75	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
76	C₂×C₈₉	#575	1	2	yes	—
77	#3142	—	89	1	no	maximal, Sylow
78	C₈₉	#3142	1	1	yes	characteristic, complemented, Sylow
79	#532	D₈₉	1	1	yes	characteristic
80	C₂×#125	—	89	1	no	—
81	#491	—	89	2	no	—
82	#490	—	89	2	no	—
83	#488	—	89	4	no	—
84	#417	—	89	4	no	—
85	C₄×M₁₆	—	89	1	no	—
86	C₂×#41	C₂×D₈₉	1	2	yes	—
87	C₂×M₁₆	—	89	2	no	—
88	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
89	#127	C₂×D₈₉	1	1	yes	characteristic
90	#125	—	89	2	no	—
91	#125	—	89	2	no	—
92	#125	—	178	2	no	—
93	#124	C₂×D₈₉	1	2	yes	—
94	#121	—	2	2	no	—
95	C₄×D₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	89	2	no	—
97	C₂×C₄⋊C₄	—	178	2	no	—
98	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
99	#103	—	89	8	no	—
100	#102	—	89	8	no	—
101	#97	—	89	2	no	—
102	C₄×C₈	—	89	2	no	—
103	#41	—	2	8	no	—
104	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	C₂²×D₈₉	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	178	2	no	—
110	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
112	M₁₆	—	178	4	no	—
113	C₂×C₈	—	89	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	89	4	no	—
116	C₄⋊C₄	—	178	2	no	—
117	C₄⋊C₄	—	178	4	no	—
118	C₄⋊C₄	—	178	4	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₄²	C₂²⋊D₈₉	1	2	yes	—
121	C₄²	C₂²×D₈₉	1	1	yes	characteristic
122	C₄²	C₂²×D₈₉	1	1	yes	characteristic
123	C₂³	—	2	2	no	—
124	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	D₄	—	2	4	no	—
127	D₄	—	4	4	no	—
128	D₄	—	4	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	356	4	no	—
133	C₂×C₄	—	178	4	no	—
134	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
137	C₂×C₄	#13083573	1	4	yes	—
138	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
139	C₈	—	178	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	2	no	—
143	C₂²	#15667999	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	356	4	no	—
149	C₄	#15667999	1	2	yes	—
150	C₄	C₂×#13083573	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	8	2	no	—
153	C₂	C₂×#15667999	1	1	yes	characteristic
154	C₂	#19152396	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668004

C₂

yes

maximal, characteristic, complemented

#19151966

C₂

yes

maximal, complemented

#19151962

C₂

yes

maximal, complemented

#19151952

C₂

yes

maximal, complemented

#19151599

C₂

yes

maximal, complemented

C₄×#13083558

C₂

yes

maximal, characteristic, complemented

#532×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

#15668004

—

#15668004

C₂²

yes

—

#15668004

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

—

#15667936

C₂²

yes

—

#15667933

C₂²

yes

—

#15667923

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₂×#41×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#127×C₈₉

C₂²

yes

characteristic

#124×C₈₉

C₂²

yes

—

#121×C₈₉

—

C₄×D₄×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

—

#13083558

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂³

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#124

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#124

yes

—

C₄×C₈₉

C₂×#41

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#124

yes

characteristic

C₂×C₈₉

#575

yes

—

#3142

—

maximal, Sylow

C₈₉

#3142

yes

characteristic, complemented, Sylow

#532

D₈₉

yes

—

—

—

—

—

C₄×M₁₆

—

C₂×#41

C₂×D₈₉

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

#127

C₂×D₈₉

yes

—

—

—

178

—

#124

C₂×D₈₉

yes

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#103

—

100

#102

—

101

#97

—

102

C₄×C₈

—

103

#41

—

104

C₂×Q₈

C₂²×D₈₉

yes

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₈₉

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

178

—

110

C₂²×C₄

C₂²⋊D₈₉

yes

—

111

C₂²×C₄

C₂²×D₈₉

yes

characteristic

112

M₁₆

—

178

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

178

—

117

C₄⋊C₄

—

178

—

118

C₄⋊C₄

—

178

—

119

C₂²⋊C₄

—

120

C₄²

C₂²⋊D₈₉

yes

—

121

C₄²

C₂²×D₈₉

yes

characteristic

122

C₄²

C₂²×D₈₉

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

356

—

133

C₂×C₄

—

178

—

134

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

135

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

136

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

137

C₂×C₄

#13083573

yes

—

138

C₂×C₄

C₂³×D₈₉

yes

characteristic

139

C₈

—

178

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#15667999

yes

—

144

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

356

—

149

C₄

#15667999

yes

—

150

C₄

C₂×#13083573

yes

—

151

C₂

—

152

C₂

—

153

C₂

C₂×#15667999

yes

characteristic

154

C₂

#19152396

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting, prime radical	#532×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#532	D₈₉
hypocenter	C₈₉	#3142
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting, prime radical

#532×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. C₂×#15668004	C₂	semidirect
2. #19151966	C₂	semidirect
3. #19151962	C₂	semidirect
4. #19151952	C₂	semidirect
5. #19151599	C₂	semidirect
6. C₄×#13083558	C₂	semidirect
78. C₈₉	#3142	semidirect

H ≅ G/N

type

1. C₂×#15668004

C₂

semidirect

2. #19151966

C₂

semidirect

3. #19151962

C₂

semidirect

4. #19151952

C₂

semidirect

5. #19151599

C₂

semidirect

6. C₄×#13083558

C₂

semidirect

78. C₈₉

#3142

semidirect

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

subgroup	index
C₂×#15668004	2
#19151966	2
#19151962	2
#19151952	2
#19151599	2
C₄×#13083558	2
#532×C₈₉	2
#3142	89

subgroup

index

#532×C₈₉

Subgroups — #26030652

Conjugacy classes of subgroups

Complements

Maximal subgroups