Group #26020469 — subgroups

Subgroups — #26020469

Order 11392.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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	#19152076	C₂	1	1	yes	maximal, characteristic
2	#19151640	C₂	1	2	yes	maximal
3	#19151505	C₂	1	1	yes	maximal, characteristic
4	#19151359	C₂	1	2	yes	maximal
5	C₂×#101×C₈₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₈₉	—	2	2	no	—
7	Q₈×Dic₈₉	—	2	2	no	—
8	#15667971	—	2	2	no	—
9	#15667971	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
12	#15667946	—	2	2	no	—
13	#15667944	—	2	2	no	—
14	#15667919	—	2	2	no	—
15	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
17	#101×C₈₉	C₄	1	4	yes	complemented
18	C₂²×Dic₈₉	—	2	1	no	—
19	C₂²×Dic₈₉	—	2	1	no	—
20	C₄⋊Dic₈₉	—	2	2	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₄⋊Dic₈₉	D₄	1	2	yes	—
24	C₄⋊Dic₈₉	D₄	1	2	yes	—
25	C₂²⋊Dic₈₉	—	2	2	no	—
26	C₄×Dic₈₉	—	2	4	no	—
27	C₄×Dic₈₉	—	2	2	no	—
28	C₄×Dic₈₉	—	2	2	no	—
29	C₂×Q₈×C₈₉	—	2	2	no	—
30	C₂×Q₈×C₈₉	—	2	2	no	—
31	C₂×Q₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×Q₈×C₈₉	Q₈	1	2	yes	—
33	C₂²×C₄×C₈₉	D₄	1	2	yes	—
34	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
35	M₁₆×C₈₉	—	2	2	no	—
36	M₁₆×C₈₉	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₈₉	—	2	2	no	—
38	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₈₉	—	2	4	no	—
40	Q₈×C₈₉	—	2	4	no	—
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	C₄⋊C₄	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×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₈₉	#41	1	2	yes	—
48	C₂×Dic₈₉	—	2	2	no	—
49	C₂×Dic₈₉	—	2	2	no	—
50	C₂×Dic₈₉	—	2	2	no	—
51	C₂×Dic₈₉	—	4	1	no	—
52	C₂×Dic₈₉	—	4	1	no	—
53	C₂×Dic₈₉	—	4	2	no	—
54	C₂×Dic₈₉	—	4	1	no	—
55	C₈×C₈₉	—	2	4	no	—
56	C₂²×C₈₉	—	2	1	no	—
57	C₂²×C₈₉	—	2	1	no	—
58	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₈₉	#120	1	1	yes	characteristic
60	C₂²×C₈₉	#123	1	1	yes	characteristic
61	C₄×C₈₉	—	2	4	no	—
62	C₄×C₈₉	C₄×D₄	1	2	yes	—
63	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
64	Dic₈₉	—	4	2	no	—
65	Dic₈₉	—	4	2	no	—
66	Dic₈₉	—	4	2	no	—
67	Dic₈₉	—	8	1	no	—
68	C₂×C₈₉	—	2	1	no	—
69	C₂×C₈₉	—	2	1	no	—
70	C₂×C₈₉	#385	1	1	yes	characteristic
71	C₂×C₈₉	#454	1	1	yes	characteristic
72	C₂×C₈₉	#456	1	1	yes	characteristic
73	#1857	—	89	1	no	maximal, Sylow
74	C₈₉	#1857	1	1	yes	characteristic, complemented, Sylow
75	#519	—	89	1	no	—
76	#428	—	89	2	no	—
77	C₂×#101	D₈₉	1	1	yes	characteristic
78	#391	—	89	1	no	—
79	#339	—	89	2	no	—
80	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
81	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
82	C₄×Q₈	—	178	2	no	—
83	C₄×Q₈	—	178	2	no	—
84	#117	—	89	2	no	—
85	#117	—	178	2	no	—
86	C₂×C₄⋊C₄	—	89	1	no	—
87	C₂×C₄²	—	89	1	no	—
88	#107	—	178	2	no	—
89	#106	—	178	2	no	—
90	#101	Dic₈₉	1	4	yes	complemented
91	#95	—	178	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	Dic₁₇₈	1	2	yes	—
95	C₂×Q₈	C₂×Dic₈₉	1	2	yes	—
96	C₂²×C₄	—	178	1	no	—
97	C₂²×C₄	—	178	1	no	—
98	C₂²×C₄	D₃₅₆	1	2	yes	—
99	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	M₁₆	C₂×Dic₈₉	1	4	yes	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	89	2	no	—
104	C₄⋊C₄	—	89	2	no	—
105	C₄⋊C₄	—	178	2	no	—
106	C₄⋊C₄	—	178	2	no	—
107	C₄⋊C₄	—	178	2	no	—
108	C₂²⋊C₄	—	178	2	no	—
109	C₄²	—	178	4	no	—
110	C₄²	—	178	2	no	—
111	C₄²	—	178	2	no	—
112	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	356	1	no	—
118	C₂×C₄	—	356	1	no	—
119	C₂×C₄	—	356	2	no	—
120	C₂×C₄	—	356	1	no	—
121	C₂×C₄	—	178	2	no	—
122	C₂×C₄	—	178	2	no	—
123	C₂×C₄	—	178	2	no	—
124	C₂×C₄	C₄⋊Dic₈₉	1	4	yes	—
125	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
126	C₂×C₄	D₄×D₈₉	1	2	yes	—
127	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
128	C₂×C₄	#13083573	1	2	yes	—
129	C₈	—	2	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
133	C₂²	#15667982	1	1	yes	characteristic
134	C₂²	#15667996	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	356	2	no	—
137	C₄	—	356	2	no	—
138	C₄	—	356	2	no	—
139	C₄	—	712	1	no	—
140	C₄	D₄×Dic₈₉	1	2	yes	—
141	C₄	#15667989	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#19151473	1	1	yes	characteristic
145	C₂	#19151773	1	1	yes	characteristic
146	C₂	#19151783	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152076

C₂

yes

maximal, characteristic

#19151640

C₂

yes

maximal

#19151505

C₂

yes

maximal, characteristic

#19151359

C₂

yes

maximal

C₂×#101×C₈₉

C₂

yes

maximal, characteristic

Q₈×Dic₈₉

—

Q₈×Dic₈₉

—

#15667971

—

#15667971

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

—

—

—

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

—

#101×C₈₉

C₄

yes

complemented

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

C₄⋊Dic₈₉

D₄

yes

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₄

yes

—

C₂×Q₈×C₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#519

—

#428

—

C₂×#101

D₈₉

yes

characteristic

#391

—

#339

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic

C₂×M₁₆

C₂×D₈₉

yes

—

C₄×Q₈

—

178

—

C₄×Q₈

—

178

—

#117

—

#117

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

178

—

#106

—

178

—

#101

Dic₈₉

yes

complemented

#95

—

178

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₇₈

yes

—

C₂×Q₈

C₂×Dic₈₉

yes

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

100

M₁₆

—

101

M₁₆

C₂×Dic₈₉

yes

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

178

—

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

—

178

—

108

C₂²⋊C₄

—

178

—

109

C₄²

—

178

—

110

C₄²

—

178

—

111

C₄²

—

178

—

112

C₂³

C₂×D₃₅₆

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

356

—

118

C₂×C₄

—

356

—

119

C₂×C₄

—

356

—

120

C₂×C₄

—

356

—

121

C₂×C₄

—

178

—

122

C₂×C₄

—

178

—

123

C₂×C₄

—

178

—

124

C₂×C₄

C₄⋊Dic₈₉

yes

—

125

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

126

C₂×C₄

D₄×D₈₉

yes

—

127

C₂×C₄

C₂×Dic₁₇₈

yes

characteristic

128

C₂×C₄

#13083573

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₄⋊Dic₈₉

yes

characteristic

133

C₂²

#15667982

yes

characteristic

134

C₂²

#15667996

yes

characteristic

135

C₄

—

136

C₄

—

356

—

137

C₄

—

356

—

138

C₄

—

356

—

139

C₄

—

712

—

140

C₄

D₄×Dic₈₉

yes

—

141

C₄

#15667989

yes

—

142

C₂

—

143

C₂

—

144

C₂

#19151473

yes

characteristic

145

C₂

#19151773

yes

characteristic

146

C₂

#19151783

yes

characteristic

	subgroup	G/H
center	C₂²	#15667982
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×#101×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	C₂³×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×#101	D₈₉
hypocenter	C₈₉	#1857
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#15667982

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×#101×C₈₉

C₂

socle

C₂²×C₈₉

#120

prime radical

C₂³×C₈₉

C₂×D₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
17. #101×C₈₉	C₄	semidirect
74. C₈₉	#1857	semidirect
90. #101	Dic₈₉	semidirect

H ≅ G/N

type

17. #101×C₈₉

C₄

semidirect

74. C₈₉

#1857

semidirect

90. #101

Dic₈₉

semidirect

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

subgroup	index
#19152076	2
#19151640	2
#19151505	2
#19151359	2
C₂×#101×C₈₉	2
#1857	89

subgroup

index

C₂×#101×C₈₉

Subgroups — #26020469

Conjugacy classes of subgroups

Complements

Maximal subgroups