Group #26021786 — subgroups

Subgroups — #26021786

Order 11392.

290 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 91 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×#15667943	C₂	1	2	yes	maximal, complemented
2	C₂×#15667932	C₂	1	2	yes	maximal
3	#19151500	C₂	1	1	yes	maximal, characteristic
4	#19151353	C₂	1	1	yes	maximal, characteristic, complemented
5	#384×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
7	C₂²×#10623494	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
9	C₂×#13083551	—	2	2	no	—
10	#15667943	—	2	4	no	—
11	#15667932	—	2	4	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	1	no	—
16	C₂×D₃₅₆	—	2	2	no	—
17	C₂×D₃₅₆	—	2	2	no	—
18	C₂×D₃₅₆	—	2	1	no	—
19	C₂×D₃₅₆	Q₈	1	2	yes	—
20	C₂²×Dic₈₉	—	2	1	no	—
21	C₂×#10623494	—	2	2	no	—
22	C₂×#10623494	—	2	2	no	—
23	C₂×#10623494	D₄	1	4	yes	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	#13083551	—	4	4	no	—
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	2	no	—
30	C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₄⋊C₄×C₈₉	—	4	4	no	—
32	C₄²×C₈₉	—	2	2	no	—
33	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₈₉	—	2	2	no	—
35	C₂²×D₈₉	—	4	1	no	—
36	C₂²×D₈₉	—	4	1	no	—
37	C₂²×D₈₉	—	4	2	no	—
38	C₂²×D₈₉	—	4	2	no	—
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	4	2	no	—
42	C₂×C₄×C₈₉	—	4	4	no	—
43	C₂×C₄×C₈₉	SD₁₆	1	4	yes	—
44	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#41	1	2	yes	—
47	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
48	C₂×Dic₈₉	—	4	1	no	—
49	C₂×Dic₈₉	—	4	2	no	—
50	D₃₅₆	—	2	4	no	—
51	D₃₅₆	—	4	2	no	—
52	#10623494	—	2	4	no	—
53	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₈₉	#123	1	1	yes	characteristic
55	C₂²×C₈₉	#127	1	1	yes	characteristic
56	C₂²×C₈₉	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
58	C₂×D₈₉	—	4	2	no	—
59	C₂×D₈₉	—	4	4	no	—
60	C₂×D₈₉	—	8	2	no	—
61	C₂×D₈₉	—	8	1	no	—
62	C₂×D₈₉	—	8	1	no	—
63	C₄×C₈₉	—	2	4	no	—
64	C₄×C₈₉	—	8	2	no	—
65	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₈₉	#123	1	2	yes	—
67	Dic₈₉	—	8	1	no	—
68	C₂×C₈₉	#399	1	1	yes	characteristic
69	C₂×C₈₉	#476	1	2	yes	—
70	C₂×C₈₉	#481	1	2	yes	—
71	C₂×C₈₉	#492	1	1	yes	characteristic
72	C₂×C₈₉	#496	1	1	yes	characteristic
73	D₈₉	—	8	2	no	—
74	#2094	—	89	1	no	maximal, Sylow
75	C₈₉	#2094	1	1	yes	characteristic, complemented, Sylow
76	C₂×#106	—	89	2	no	—
77	C₂×#102	—	89	2	no	—
78	#390	—	89	1	no	—
79	#384	D₈₉	1	1	yes	characteristic, complemented
80	#336	—	89	1	no	—
81	C₂²×D₄	—	89	1	no	—
82	C₂²×C₈	—	89	2	no	—
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	—	89	1	no	—
85	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
86	C₂×C₂²⋊C₄	—	178	2	no	—
87	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
88	#106	—	178	4	no	—
89	#102	—	178	4	no	—
90	C₂⁴	—	178	1	no	—
91	C₂×D₄	—	89	2	no	—
92	C₂×D₄	—	178	2	no	—
93	C₂×D₄	—	178	2	no	—
94	C₂×D₄	—	178	1	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	178	1	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	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	178	2	no	—
106	C₂²⋊C₄	—	356	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	356	1	no	—
109	C₂³	—	356	1	no	—
110	C₂³	—	356	2	no	—
111	C₂³	—	356	2	no	—
112	C₂³	—	178	2	no	—
113	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
114	D₄	—	356	2	no	—
115	D₄	—	178	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	356	1	no	—
121	C₂×C₄	—	356	2	no	—
122	C₂×C₄	#13083561	1	4	yes	—
123	C₂×C₄	D₄×D₈₉	1	2	yes	—
124	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
125	C₂×C₄	#13083574	1	2	yes	—
126	C₂×C₄	#13083575	1	1	yes	characteristic
127	C₈	—	178	4	no	—
128	C₂²	—	356	4	no	—
129	C₂²	—	356	2	no	—
130	C₂²	—	712	2	no	—
131	C₂²	—	712	1	no	—
132	C₂²	—	712	1	no	—
133	C₂²	#15667988	1	1	yes	characteristic
134	C₂²	#15667995	1	1	yes	characteristic
135	C₂²	#15668008	1	1	yes	characteristic
136	C₂²	C₂×#13083561	1	2	yes	—
137	C₂²	#15668038	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	8	2	no	—
140	C₄	—	712	1	no	—
141	C₄	#15667990	1	2	yes	—
142	C₄	#15667997	1	2	yes	—
143	C₂	—	712	2	no	—
144	C₂	#19151533	1	1	yes	characteristic
145	C₂	#19151885	1	2	yes	—
146	C₂	#19151916	1	2	yes	—
147	C₂	#19151972	1	1	yes	characteristic
148	C₂	#19151985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667943

C₂

yes

maximal, complemented

C₂×#15667932

C₂

yes

maximal

#19151500

C₂

yes

maximal, characteristic

#19151353

C₂

yes

maximal, characteristic, complemented

#384×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

—

—

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

Q₈

yes

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

D₄

yes

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

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

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

#10623494

—

C₂²×C₈₉

C₂²⋊Q₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×SD₁₆

yes

—

C₂²×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#123

yes

—

Dic₈₉

—

C₂×C₈₉

#399

yes

characteristic

C₂×C₈₉

#476

yes

—

C₂×C₈₉

#481

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2094

—

maximal, Sylow

C₈₉

#2094

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic, complemented

#336

—

C₂²×D₄

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#106

—

178

—

#102

—

178

—

C₂⁴

—

178

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

—

178

—

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

C₄⋊C₄

—

105

C₄⋊C₄

—

178

—

106

C₂²⋊C₄

—

356

—

107

C₄²

—

108

C₂³

—

356

—

109

C₂³

—

356

—

110

C₂³

—

356

—

111

C₂³

—

356

—

112

C₂³

—

178

—

113

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

114

D₄

—

356

—

115

D₄

—

178

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

356

—

121

C₂×C₄

—

356

—

122

C₂×C₄

#13083561

yes

—

123

C₂×C₄

D₄×D₈₉

yes

—

124

C₂×C₄

Q₈×D₈₉

yes

characteristic

125

C₂×C₄

#13083574

yes

—

126

C₂×C₄

#13083575

yes

characteristic

127

C₈

—

178

—

128

C₂²

—

356

—

129

C₂²

—

356

—

130

C₂²

—

712

—

131

C₂²

—

712

—

132

C₂²

—

712

—

133

C₂²

#15667988

yes

characteristic

134

C₂²

#15667995

yes

characteristic

135

C₂²

#15668008

yes

characteristic

136

C₂²

C₂×#13083561

yes

—

137

C₂²

#15668038

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

712

—

141

C₄

#15667990

yes

—

142

C₄

#15667997

yes

—

143

C₂

—

712

—

144

C₂

#19151533

yes

characteristic

145

C₂

#19151885

yes

—

146

C₂

#19151916

yes

—

147

C₂

#19151972

yes

characteristic

148

C₂

#19151985

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#384×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	#384	D₈₉
hypocenter	C₈₉	#2094
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#384×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

prime radical

C₂²×D₃₅₆

C₂²

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
1. C₂×#15667943	C₂	semidirect
4. #19151353	C₂	semidirect
5. #384×C₈₉	C₂	semidirect
75. C₈₉	#2094	semidirect
79. #384	D₈₉	semidirect

H ≅ G/N

type

1. C₂×#15667943

C₂

semidirect

4. #19151353

C₂

semidirect

5. #384×C₈₉

C₂

semidirect

75. C₈₉

#2094

semidirect

79. #384

D₈₉

semidirect

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

subgroup	index
C₂×#15667943	2
C₂×#15667932	2
#19151500	2
#19151353	2
#384×C₈₉	2
#2094	89

subgroup

index

#384×C₈₉

Subgroups — #26021786

Conjugacy classes of subgroups

Complements

Maximal subgroups