Group #26020149 — subgroups

Subgroups — #26020149

Order 11392.

272 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15667946	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667944	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₈₉	C₂	1	2	yes	maximal
4	#19151361	C₂	1	2	yes	maximal
5	C₂×#97×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
8	C₂×C₄×Dic₈₉	—	2	2	no	—
9	#15667946	C₄	1	4	yes	complemented
10	#15667944	C₄	1	4	yes	complemented
11	#15667919	—	2	2	no	—
12	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	#97×C₈₉	C₄	1	4	yes	complemented
16	C₂²×Dic₈₉	—	2	2	no	—
17	C₂²×Dic₈₉	—	2	2	no	—
18	C₄⋊Dic₈₉	—	2	2	no	—
19	C₄⋊Dic₈₉	—	2	2	no	—
20	C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
21	C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
22	C₄×Dic₈₉	—	4	4	no	—
23	C₂²×C₄×C₈₉	D₄	1	2	yes	—
24	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₈₉	—	2	1	no	—
26	C₂×C₈×C₈₉	—	2	1	no	—
27	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₄²×C₈₉	C₂×C₄	1	2	yes	—
32	C₄²×C₈₉	Q₈	1	2	yes	—
33	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	—	2	2	no	—
35	C₂×C₄×C₈₉	—	2	2	no	—
36	C₂×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
43	C₂×Dic₈₉	—	2	4	no	—
44	C₂×Dic₈₉	—	2	4	no	—
45	C₂×Dic₈₉	—	4	2	no	—
46	C₂×Dic₈₉	—	4	2	no	—
47	C₂×Dic₈₉	—	4	2	no	—
48	C₂×Dic₈₉	—	4	2	no	—
49	C₈×C₈₉	—	2	2	no	—
50	C₈×C₈₉	C₄²	1	4	yes	complemented
51	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₈₉	#137	1	2	yes	—
56	C₄×C₈₉	—	2	4	no	—
57	C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
58	C₄×C₈₉	#117	1	1	yes	characteristic
59	C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
61	Dic₈₉	—	4	4	no	—
62	Dic₈₉	—	4	4	no	—
63	C₂×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈₉	#428	1	2	yes	—
65	C₂×C₈₉	#440	1	2	yes	—
66	C₂×C₈₉	#441	1	1	yes	characteristic
67	C₂×C₈₉	#442	1	1	yes	characteristic
68	#1776	—	89	1	no	maximal, Sylow
69	C₈₉	#1776	1	1	yes	characteristic, complemented, Sylow
70	C₂×#107	—	89	1	no	—
71	C₂×#106	—	89	1	no	—
72	C₂×#97	D₈₉	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	89	2	no	—
74	#340	—	89	2	no	—
75	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	89	2	no	—
78	C₂×C₄⋊C₄	—	89	2	no	—
79	C₂×C₄²	—	178	2	no	—
80	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
81	#107	—	89	4	no	—
82	#106	—	89	4	no	—
83	#97	Dic₈₉	1	4	yes	complemented
84	#95	—	178	2	no	—
85	C₂²×C₄	—	178	2	no	—
86	C₂²×C₄	—	178	2	no	—
87	C₂²×C₄	D₃₅₆	1	2	yes	—
88	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	C₄×D₈₉	1	2	yes	—
92	C₂×C₈	C₄×D₈₉	1	2	yes	—
93	C₂×C₈	C₂×Dic₈₉	1	2	yes	—
94	C₂×C₈	C₂×Dic₈₉	1	2	yes	—
95	C₄⋊C₄	—	89	4	no	—
96	C₄⋊C₄	—	89	4	no	—
97	C₄⋊C₄	—	178	2	no	—
98	C₄⋊C₄	—	178	2	no	—
99	C₄²	—	356	4	no	—
100	C₄²	Dic₁₇₈	1	2	yes	—
101	C₄²	C₂×Dic₈₉	1	2	yes	—
102	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	356	2	no	—
106	C₂×C₄	—	356	2	no	—
107	C₂×C₄	—	356	2	no	—
108	C₂×C₄	—	356	2	no	—
109	C₂×C₄	—	178	4	no	—
110	C₂×C₄	—	178	4	no	—
111	C₂×C₄	C₄⋊Dic₈₉	1	4	yes	—
112	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
114	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
116	C₂×C₄	#13083574	1	1	yes	characteristic
117	C₂×C₄	#13083574	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	C₄×Dic₈₉	1	4	yes	complemented
120	C₂²	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
121	C₂²	C₄×D₃₅₆	1	1	yes	characteristic
122	C₂²	C₄×D₃₅₆	1	1	yes	characteristic
123	C₂²	#15668040	1	2	yes	—
124	C₂²	#15668045	1	2	yes	—
125	C₄	—	2	4	no	—
126	C₄	—	356	4	no	—
127	C₄	—	356	4	no	—
128	C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
129	C₄	#15667971	1	1	yes	characteristic
130	C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
131	C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
132	C₂	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
133	C₂	#19151640	1	2	yes	—
134	C₂	#19151700	1	2	yes	—
135	C₂	#19151706	1	1	yes	characteristic
136	C₂	#19151711	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667946

C₂

yes

maximal, characteristic

C₂×#15667944

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₈₉

C₂

yes

maximal

#19151361

C₂

yes

maximal

C₂×#97×C₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

—

#15667946

C₄

yes

complemented

#15667944

C₄

yes

complemented

#15667919

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#97×C₈₉

C₄

yes

complemented

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

Q₈

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₄²

yes

complemented

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

C₄×D₄

yes

characteristic

C₂²×C₈₉

C₄×D₄

yes

characteristic

C₂²×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²×C₈₉

#137

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄²

yes

characteristic

C₄×C₈₉

#117

yes

characteristic

C₄×C₈₉

C₄×Q₈

yes

characteristic

C₄×C₈₉

C₄×Q₈

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₈₉

#428

yes

—

C₂×C₈₉

#440

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#106

—

C₂×#97

D₈₉

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

178

—

C₂×C₄²

C₂×D₈₉

yes

—

—

Dic₈₉

yes

complemented

#95

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₈₉

yes

—

C₂×C₈

C₄×D₈₉

yes

—

C₂×C₈

C₂×Dic₈₉

yes

—

C₂×C₈

C₂×Dic₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

356

—

100

C₄²

Dic₁₇₈

yes

—

101

C₄²

C₂×Dic₈₉

yes

—

102

C₂³

C₂×D₃₅₆

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

356

—

106

C₂×C₄

—

356

—

107

C₂×C₄

—

356

—

108

C₂×C₄

—

356

—

109

C₂×C₄

—

178

—

110

C₂×C₄

—

178

—

111

C₂×C₄

C₄⋊Dic₈₉

yes

—

112

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

113

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

114

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

115

C₂×C₄

C₂×Dic₁₇₈

yes

characteristic

116

C₂×C₄

#13083574

yes

characteristic

117

C₂×C₄

#13083574

yes

characteristic

118

C₈

—

119

C₈

C₄×Dic₈₉

yes

complemented

120

C₂²

C₂×C₄⋊Dic₈₉

yes

characteristic

121

C₂²

C₄×D₃₅₆

yes

characteristic

122

C₂²

C₄×D₃₅₆

yes

characteristic

123

C₂²

#15668040

yes

—

124

C₂²

#15668045

yes

—

125

C₄

—

126

C₄

—

356

—

127

C₄

—

356

—

128

C₄

C₂×C₄×Dic₈₉

yes

characteristic

129

C₄

#15667971

yes

characteristic

130

C₄

C₄×Dic₁₇₈

yes

characteristic

131

C₄

C₄×Dic₁₇₈

yes

characteristic

132

C₂

C₄×C₄⋊Dic₈₉

yes

characteristic

133

C₂

#19151640

yes

—

134

C₂

#19151700

yes

—

135

C₂

#19151706

yes

characteristic

136

C₂

#19151711

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator	C₄×C₈₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×#97×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×#97	D₈₉
hypocenter	C₈₉	#1776
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator

C₄×C₈₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×#97×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂×#97

D₈₉

hypocenter

C₈₉

#1776

LCS

G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
9. #15667946	C₄	semidirect
10. #15667944	C₄	semidirect
15. #97×C₈₉	C₄	semidirect
50. C₈×C₈₉	C₄²	semidirect
69. C₈₉	#1776	semidirect
83. #97	Dic₈₉	semidirect
119. C₈	C₄×Dic₈₉	semidirect

H ≅ G/N

type

9. #15667946

C₄

semidirect

10. #15667944

C₄

semidirect

15. #97×C₈₉

C₄

semidirect

50. C₈×C₈₉

C₄²

semidirect

69. C₈₉

#1776

semidirect

83. #97

Dic₈₉

semidirect

119. C₈

C₄×Dic₈₉

semidirect

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

subgroup	index
C₂×#15667946	2
C₂×#15667944	2
C₄×C₄⋊Dic₈₉	2
#19151361	2
C₂×#97×C₈₉	2
#1776	89

subgroup

index

C₄×C₄⋊Dic₈₉

C₂×#97×C₈₉

Subgroups — #26020149

Conjugacy classes of subgroups

Complements

Maximal subgroups