Group #26021829 — subgroups

Subgroups — #26021829

Order 11392.

158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19151646	C₂	1	1	yes	maximal, characteristic
2	#19151621	C₂	1	1	yes	maximal, characteristic
3	#19151370	C₂	1	1	yes	maximal, characteristic
4	#19151359	C₂	1	1	yes	maximal, characteristic
5	#19151357	C₂	1	1	yes	maximal, characteristic
6	#19151354	C₂	1	1	yes	maximal, characteristic
7	#344×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13083558	C₂²	1	1	yes	characteristic
9	C₂×#13083558	C₂²	1	1	yes	characteristic
10	#15667971	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
12	#15667947	—	2	2	no	—
13	#15667941	—	2	2	no	—
14	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	#117×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₈₉	—	2	1	no	—
18	#13083558	—	2	1	no	—
19	#13083558	—	2	1	no	—
20	#13083558	—	2	1	no	—
21	#13083558	—	2	1	no	—
22	C₂×#10623494	—	2	1	no	—
23	C₂×#10623494	D₄	1	2	yes	—
24	C₄⋊Dic₈₉	—	2	1	no	—
25	C₂²⋊Dic₈₉	—	2	1	no	—
26	C₄×Dic₈₉	—	2	1	no	—
27	C₄×Dic₈₉	—	2	1	no	—
28	C₄×Dic₈₉	Q₈	1	1	yes	characteristic
29	C₄×Dic₈₉	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	M₁₆×C₈₉	—	2	1	no	—
32	M₁₆×C₈₉	—	2	1	no	—
33	M₁₆×C₈₉	—	2	1	no	—
34	M₁₆×C₈₉	—	2	1	no	—
35	C₂×C₈×C₈₉	—	2	1	no	—
36	C₂×C₈×C₈₉	—	2	1	no	—
37	C₄⋊C₄×C₈₉	—	2	1	no	—
38	C₂²⋊C₄×C₈₉	—	2	1	no	—
39	C₄²×C₈₉	—	2	1	no	—
40	C₂³×C₈₉	#41	1	1	yes	characteristic
41	C₂×C₄×C₈₉	—	4	1	no	—
42	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
47	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
48	C₂×Dic₈₉	—	2	1	no	—
49	C₂×Dic₈₉	—	2	1	no	—
50	C₂×Dic₈₉	—	4	2	no	—
51	C₂×Dic₈₉	—	4	1	no	—
52	C₈×C₈₉	—	4	1	no	—
53	C₈×C₈₉	—	4	1	no	—
54	#10623494	—	2	2	no	—
55	#10623494	—	4	1	no	—
56	C₂²×C₈₉	—	4	1	no	—
57	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂²×C₈₉	#126	1	1	yes	characteristic
59	C₂²×C₈₉	#126	1	1	yes	characteristic
60	C₄×C₈₉	—	8	1	no	—
61	C₄×C₈₉	#123	1	1	yes	characteristic
62	C₄×C₈₉	#124	1	1	yes	characteristic
63	C₄×C₈₉	#125	1	1	yes	characteristic
64	C₄×C₈₉	#125	1	1	yes	characteristic
65	Dic₈₉	—	2	2	no	—
66	Dic₈₉	—	4	1	no	—
67	Dic₈₉	—	8	1	no	—
68	C₂×C₈₉	—	2	1	no	—
69	C₂×C₈₉	—	2	1	no	—
70	C₂×C₈₉	—	2	1	no	—
71	C₂×C₈₉	#400	1	1	yes	characteristic
72	#2102	—	89	1	no	maximal, Sylow
73	C₈₉	#2102	1	1	yes	characteristic, complemented, Sylow
74	#430	—	89	1	no	—
75	#423	—	89	1	no	—
76	#344	—	89	1	no	—
77	#344	D₈₉	1	1	yes	characteristic
78	#339	—	89	1	no	—
79	#339	—	89	1	no	—
80	H(3,ℤ/4)	—	89	1	no	—
81	C₂×M₁₆	—	89	1	no	—
82	C₂×M₁₆	—	89	1	no	—
83	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
84	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
85	#117	—	89	1	no	—
86	#117	C₂×D₈₉	1	1	yes	characteristic
87	C₂×C₄²	—	89	1	no	—
88	#108	—	178	2	no	—
89	C₄⋊C₈	—	178	2	no	—
90	C₂²×C₄	—	178	1	no	—
91	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
92	M₁₆	—	2	1	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	178	1	no	—
97	M₁₆	—	178	1	no	—
98	M₁₆	—	178	1	no	—
99	M₁₆	—	178	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	89	2	no	—
103	C₂×C₈	—	178	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	178	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	178	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	89	1	no	—
110	C₄²	—	89	1	no	—
111	C₄²	—	178	1	no	—
112	C₄²	—	178	1	no	—
113	C₂³	#13083574	1	1	yes	characteristic
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	356	2	no	—
116	C₂×C₄	—	356	1	no	—
117	C₂×C₄	—	178	1	no	—
118	C₂×C₄	—	178	1	no	—
119	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
121	C₂×C₄	#13083573	1	1	yes	characteristic
122	C₂×C₄	#13083573	1	1	yes	characteristic
123	C₂×C₄	#13083574	1	1	yes	characteristic
124	C₂×C₄	#13083574	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	356	1	no	—
128	C₈	—	178	2	no	—
129	C₂²	—	4	1	no	—
130	C₂²	#15667990	1	1	yes	characteristic
131	C₂²	#15668005	1	1	yes	characteristic
132	C₂²	#15668006	1	1	yes	characteristic
133	C₄	—	8	1	no	—
134	C₄	—	356	1	no	—
135	C₄	—	178	2	no	—
136	C₄	—	712	1	no	—
137	C₄	#15667994	1	1	yes	characteristic
138	C₄	#15667998	1	1	yes	characteristic
139	C₄	#15668002	1	1	yes	characteristic
140	C₄	#15668002	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#19151539	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151646

C₂

yes

maximal, characteristic

#19151621

C₂

yes

maximal, characteristic

#19151370

C₂

yes

maximal, characteristic

#19151359

C₂

yes

maximal, characteristic

#19151357

C₂

yes

maximal, characteristic

#19151354

C₂

yes

maximal, characteristic

#344×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

characteristic

#15667971

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667947

—

#15667941

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

—

—

—

—

—

—

D₄

yes

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

Q₈

yes

characteristic

C₄×Dic₈₉

Q₈

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²⋊Q₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂×M₁₆

C₂×D₈₉

yes

characteristic

#117

—

#117

C₂×D₈₉

yes

characteristic

C₂×C₄²

—

#108

—

178

—

C₄⋊C₈

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

M₁₆

—

178

—

M₁₆

—

178

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

178

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

178

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

178

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

178

—

112

C₄²

—

178

—

113

C₂³

#13083574

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

356

—

116

C₂×C₄

—

356

—

117

C₂×C₄

—

178

—

118

C₂×C₄

—

178

—

119

C₂×C₄

D₄×D₈₉

yes

characteristic

120

C₂×C₄

Q₈×D₈₉

yes

characteristic

121

C₂×C₄

#13083573

yes

characteristic

122

C₂×C₄

#13083573

yes

characteristic

123

C₂×C₄

#13083574

yes

characteristic

124

C₂×C₄

#13083574

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

356

—

128

C₈

—

178

—

129

C₂²

—

130

C₂²

#15667990

yes

characteristic

131

C₂²

#15668005

yes

characteristic

132

C₂²

#15668006

yes

characteristic

133

C₄

—

134

C₄

—

356

—

135

C₄

—

178

—

136

C₄

—

712

—

137

C₄

#15667994

yes

characteristic

138

C₄

#15667998

yes

characteristic

139

C₄

#15668002

yes

characteristic

140

C₄

#15668002

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#19151539

yes

characteristic

	subgroup	G/H
center	C₄	#15667998
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#344×C₈₉	C₂
socle	C₂×C₈₉	#400
prime radical	C₂³×C₈₉	#41
hypercenter	#344	D₈₉
hypocenter	C₈₉	#2102
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#15667998

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#344×C₈₉

C₂

C₂×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
73. C₈₉	#2102	semidirect

H ≅ G/N

type

73. C₈₉

#2102

semidirect

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

subgroup	index
#19151646	2
#19151621	2
#19151370	2
#19151359	2
#19151357	2
#19151354	2
#344×C₈₉	2
#2102	89

subgroup

index

#344×C₈₉

Subgroups — #26021829

Conjugacy classes of subgroups

Complements

Maximal subgroups