Group #26021057 — subgroups

Subgroups — #26021057

Order 11392.

436 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15668009	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667932	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
4	#384×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₅₆	—	2	2	no	—
6	C₂²×D₃₅₆	C₂²	1	2	yes	—
7	C₂²×#10623494	C₂²	1	2	yes	—
8	#15668009	—	2	2	no	—
9	#15668009	C₄	1	4	yes	complemented
10	#15667932	—	2	8	no	—
11	C₄×#10623494	—	2	2	no	—
12	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×D₈₉	—	2	2	no	—
16	C₂×D₃₅₆	—	2	4	no	—
17	C₂×D₃₅₆	—	2	4	no	—
18	C₂×D₃₅₆	—	2	8	no	—
19	C₂×D₃₅₆	—	2	2	no	—
20	C₂×D₃₅₆	—	4	4	no	—
21	C₂×D₃₅₆	—	4	4	no	—
22	C₂×D₃₅₆	C₂×C₄	1	4	yes	—
23	C₂×#10623494	—	2	2	no	—
24	C₂×#10623494	—	2	2	no	—
25	C₂×#10623494	D₄	1	4	yes	complemented
26	C₂²×C₄×C₈₉	—	2	2	no	—
27	C₂²×C₄×C₈₉	D₄	1	2	yes	—
28	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₈₉	—	2	4	no	—
30	C₄⋊C₄×C₈₉	—	4	4	no	—
31	C₄²×C₈₉	C₂×C₄	1	2	yes	—
32	C₄²×C₈₉	D₄	1	2	yes	—
33	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₈₉	—	2	4	no	—
35	C₂²×D₈₉	—	4	2	no	—
36	C₂²×D₈₉	—	4	2	no	—
37	C₂²×D₈₉	—	4	4	no	—
38	C₂²×D₈₉	—	4	4	no	—
39	C₂×C₄×C₈₉	—	4	2	no	—
40	C₂×C₄×C₈₉	—	4	4	no	—
41	C₂×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₈₉	D₈	1	4	yes	—
43	C₂×C₄×C₈₉	SD₁₆	1	4	yes	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	#41	1	2	yes	—
48	D₃₅₆	—	2	8	no	—
49	D₃₅₆	—	4	16	no	—
50	D₃₅₆	—	4	4	no	—
51	#10623494	—	2	4	no	—
52	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₈₉	#124	1	1	yes	characteristic
54	C₂²×C₈₉	#127	1	1	yes	characteristic
55	C₂²×C₈₉	C₂×D₈	1	2	yes	—
56	C₂²×C₈₉	C₂×SD₁₆	1	2	yes	—
57	C₂×D₈₉	—	4	4	no	—
58	C₂×D₈₉	—	4	8	no	—
59	C₂×D₈₉	—	8	4	no	—
60	C₂×D₈₉	—	8	2	no	—
61	C₂×D₈₉	—	8	2	no	—
62	C₄×C₈₉	—	8	2	no	—
63	C₄×C₈₉	#102	1	8	yes	complemented
64	C₄×C₈₉	C₄×D₄	1	2	yes	—
65	C₄×C₈₉	#121	1	2	yes	—
66	C₂×C₈₉	#390	1	1	yes	characteristic
67	C₂×C₈₉	C₂×#102	1	2	yes	—
68	C₂×C₈₉	#486	1	2	yes	—
69	C₂×C₈₉	#492	1	1	yes	characteristic
70	C₂×C₈₉	#493	1	1	yes	characteristic
71	D₈₉	—	8	4	no	—
72	#1957	—	89	1	no	maximal, Sylow
73	C₈₉	#1957	1	1	yes	characteristic, complemented, Sylow
74	C₂×#127	—	89	1	no	—
75	C₂×#102	—	89	4	no	—
76	C₂×C₄×C₈	—	89	1	no	—
77	#384	D₈₉	1	1	yes	characteristic, complemented
78	C₂²×D₄	—	89	2	no	—
79	C₂²×D₄	—	178	2	no	—
80	C₂²×C₈	—	89	2	no	—
81	#127	—	89	4	no	—
82	#127	—	178	2	no	—
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
85	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
86	#102	—	178	8	no	—
87	C₄×C₈	—	178	2	no	—
88	C₂⁴	—	178	2	no	—
89	C₂×D₄	—	89	4	no	—
90	C₂×D₄	—	356	4	no	—
91	C₂×D₄	—	356	4	no	—
92	C₂×D₄	—	178	8	no	—
93	C₂×D₄	—	178	4	no	—
94	C₂×D₄	—	178	2	no	—
95	C₂×D₄	—	178	4	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
98	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
99	C₂×C₈	—	89	4	no	—
100	C₂×C₈	—	178	2	no	—
101	C₂×C₈	—	178	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	4	4	no	—
104	C₄²	C₄×D₈₉	1	2	yes	—
105	C₄²	D₃₅₆	1	2	yes	—
106	C₂³	—	356	2	no	—
107	C₂³	—	356	2	no	—
108	C₂³	—	356	4	no	—
109	C₂³	—	356	4	no	—
110	C₂³	—	178	4	no	—
111	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
112	D₄	—	356	4	no	—
113	D₄	—	356	16	no	—
114	D₄	—	178	8	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	#13083551	1	4	yes	—
118	C₂×C₄	#13083559	1	4	yes	—
119	C₂×C₄	#13083561	1	4	yes	—
120	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
121	C₂×C₄	D₄×D₈₉	1	2	yes	—
122	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
123	C₂×C₄	#13083575	1	2	yes	—
124	C₈	—	178	4	no	—
125	C₂²	—	356	4	no	—
126	C₂²	—	356	8	no	—
127	C₂²	—	712	4	no	—
128	C₂²	—	712	2	no	—
129	C₂²	—	712	2	no	—
130	C₂²	C₂×#13083551	1	1	yes	characteristic
131	C₂²	#15668001	1	1	yes	characteristic
132	C₂²	#15668008	1	1	yes	characteristic
133	C₂²	C₂×#13083559	1	2	yes	—
134	C₂²	C₂×#13083561	1	2	yes	—
135	C₄	—	8	2	no	—
136	C₄	#15667932	1	8	yes	complemented
137	C₄	#15667974	1	2	yes	—
138	C₄	#15667987	1	2	yes	—
139	C₂	—	712	4	no	—
140	C₂	#19151499	1	1	yes	characteristic
141	C₂	C₂×#15667932	1	2	yes	—
142	C₂	#19151946	1	2	yes	—
143	C₂	#19151972	1	1	yes	characteristic
144	C₂	#19151974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668009

C₂

yes

maximal, characteristic

C₂×#15667932

C₂

yes

maximal, complemented

C₂×C₄×#10623494

C₂

yes

maximal, characteristic, complemented

#384×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

—

C₂²×D₃₅₆

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

#15668009

—

#15668009

C₄

yes

complemented

#15667932

—

C₄×#10623494

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂×C₄

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

D₄

yes

complemented

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

D₈

yes

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

#10623494

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×D₈

yes

—

C₂²×C₈₉

C₂×SD₁₆

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#102

yes

complemented

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

#121

yes

—

C₂×C₈₉

#390

yes

characteristic

C₂×C₈₉

C₂×#102

yes

—

C₂×C₈₉

#486

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#1957

—

maximal, Sylow

C₈₉

#1957

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#102

—

C₂×C₄×C₈

—

#384

D₈₉

yes

characteristic, complemented

C₂²×D₄

—

C₂²×D₄

—

178

—

C₂²×C₈

—

#127

—

#127

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#102

—

178

—

C₄×C₈

—

178

—

C₂⁴

—

178

—

C₂×D₄

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

178

—

101

C₂×C₈

—

178

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₄×D₈₉

yes

—

105

C₄²

D₃₅₆

yes

—

106

C₂³

—

356

—

107

C₂³

—

356

—

108

C₂³

—

356

—

109

C₂³

—

356

—

110

C₂³

—

178

—

111

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

112

D₄

—

356

—

113

D₄

—

356

—

114

D₄

—

178

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#13083551

yes

—

118

C₂×C₄

#13083559

yes

—

119

C₂×C₄

#13083561

yes

—

120

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

121

C₂×C₄

D₄×D₈₉

yes

—

122

C₂×C₄

C₂×D₃₅₆

yes

characteristic

123

C₂×C₄

#13083575

yes

—

124

C₈

—

178

—

125

C₂²

—

356

—

126

C₂²

—

356

—

127

C₂²

—

712

—

128

C₂²

—

712

—

129

C₂²

—

712

—

130

C₂²

C₂×#13083551

yes

characteristic

131

C₂²

#15668001

yes

characteristic

132

C₂²

#15668008

yes

characteristic

133

C₂²

C₂×#13083559

yes

—

134

C₂²

C₂×#13083561

yes

—

135

C₄

—

136

C₄

#15667932

yes

complemented

137

C₄

#15667974

yes

—

138

C₄

#15667987

yes

—

139

C₂

—

712

—

140

C₂

#19151499

yes

characteristic

141

C₂

C₂×#15667932

yes

—

142

C₂

#19151946

yes

—

143

C₂

#19151972

yes

characteristic

144

C₂

#19151974

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#384×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂×#15668009	C₂
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#384	D₈₉
hypocenter	C₈₉	#1957
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#384×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

prime radical

C₂×#15668009

C₂

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
2. C₂×#15667932	C₂	semidirect
3. C₂×C₄×#10623494	C₂	semidirect
4. #384×C₈₉	C₂	semidirect
9. #15668009	C₄	semidirect
25. C₂×#10623494	D₄	semidirect
63. C₄×C₈₉	#102	semidirect
73. C₈₉	#1957	semidirect
77. #384	D₈₉	semidirect
136. C₄	#15667932	semidirect

H ≅ G/N

type

2. C₂×#15667932

C₂

semidirect

3. C₂×C₄×#10623494

C₂

semidirect

4. #384×C₈₉

C₂

semidirect

9. #15668009

C₄

semidirect

25. C₂×#10623494

D₄

semidirect

63. C₄×C₈₉

#102

semidirect

73. C₈₉

#1957

semidirect

77. #384

D₈₉

semidirect

136. C₄

#15667932

semidirect

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

subgroup	index
C₂×#15668009	2
C₂×#15667932	2
C₂×C₄×#10623494	2
#384×C₈₉	2
#1957	89

subgroup

index

#384×C₈₉

Subgroups — #26021057

Conjugacy classes of subgroups

Complements

Maximal subgroups