Group #26021039 — subgroups

Subgroups — #26021039

Order 11392.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₂×#15667947	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667947	C₂	1	1	yes	maximal, characteristic
4	#19151556	C₂	1	1	yes	maximal, characteristic
5	#19151370	C₂	1	1	yes	maximal, characteristic
6	#19151369	C₂	1	1	yes	maximal, characteristic
7	#430×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13083558	C₂²	1	1	yes	characteristic
9	C₂×#13083558	C₂²	1	1	yes	characteristic
10	C₂²×#10623494	C₂²	1	1	yes	characteristic
11	#15667971	C₂²	1	1	yes	characteristic
12	#15667947	—	2	1	no	—
13	#15667947	—	2	1	no	—
14	#15667947	—	2	2	no	—
15	#15667947	C₄	1	4	yes	complemented
16	#15667922	—	2	1	no	—
17	C₈×Dic₈₉	—	2	1	no	—
18	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	#108×C₈₉	—	2	2	no	—
22	#13083558	—	2	1	no	—
23	#13083558	—	2	1	no	—
24	#13083558	—	2	1	no	—
25	#13083558	C₂×C₄	1	2	yes	—
26	C₂×#10623494	—	2	1	no	—
27	C₂×#10623494	—	2	1	no	—
28	C₂×#10623494	C₂×C₄	1	2	yes	—
29	C₂×#10623494	D₄	1	1	yes	characteristic
30	C₂×#10623494	D₄	1	1	yes	characteristic
31	C₂×#10623494	Q₈	1	2	yes	—
32	C₄⋊Dic₈₉	—	2	1	no	—
33	C₂²⋊Dic₈₉	Q₈	1	2	yes	—
34	C₄×Dic₈₉	—	2	1	no	—
35	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₈₉	—	2	1	no	—
37	M₁₆×C₈₉	—	2	1	no	—
38	M₁₆×C₈₉	—	2	1	no	—
39	M₁₆×C₈₉	—	2	1	no	—
40	M₁₆×C₈₉	C₂×C₄	1	2	yes	—
41	M₁₆×C₈₉	D₄	1	2	yes	—
42	C₂×C₈×C₈₉	—	2	1	no	—
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₂×C₈×C₈₉	—	2	1	no	—
45	C₂³×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
52	C₂×Dic₈₉	—	2	2	no	—
53	C₈×C₈₉	—	2	2	no	—
54	C₈×C₈₉	—	2	2	no	—
55	C₈×C₈₉	—	4	1	no	—
56	#10623494	—	2	2	no	—
57	#10623494	—	4	1	no	—
58	#10623494	C₄⋊C₄	1	2	yes	—
59	#10623494	C₄⋊C₄	1	2	yes	—
60	C₂²×C₈₉	—	4	1	no	—
61	C₂²×C₈₉	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
66	C₄×C₈₉	#121	1	1	yes	characteristic
67	C₄×C₈₉	#125	1	1	yes	characteristic
68	Dic₈₉	—	4	2	no	—
69	C₂×C₈₉	—	2	1	no	—
70	C₂×C₈₉	—	2	1	no	—
71	C₂×C₈₉	—	2	1	no	—
72	C₂×C₈₉	#389	1	1	yes	characteristic
73	#1953	—	89	1	no	maximal, Sylow
74	C₈₉	#1953	1	1	yes	characteristic, complemented, Sylow
75	#430	—	89	1	no	—
76	#430	D₈₉	1	1	yes	characteristic
77	C₂×#108	—	89	1	no	—
78	C₂×#108	—	89	1	no	—
79	#405	—	89	1	no	—
80	#344	—	89	1	no	—
81	#344	—	89	1	no	—
82	C₂×M₁₆	—	89	1	no	—
83	C₂×M₁₆	—	89	1	no	—
84	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
85	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
87	C₂²×C₈	—	89	1	no	—
88	#117	—	89	1	no	—
89	#108	—	2	2	no	—
90	#108	—	89	4	no	—
91	#108	—	178	1	no	—
92	#108	—	178	1	no	—
93	#108	—	178	2	no	—
94	#97	—	178	1	no	—
95	C₄×C₈	—	178	1	no	—
96	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	89	2	no	—
102	M₁₆	—	178	1	no	—
103	M₁₆	—	178	1	no	—
104	M₁₆	—	178	1	no	—
105	M₁₆	C₄×D₈₉	1	2	yes	—
106	M₁₆	C₂²⋊D₈₉	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	89	2	no	—
111	C₂×C₈	—	89	1	no	—
112	C₂×C₈	—	89	1	no	—
113	C₂×C₈	—	89	2	no	—
114	C₂×C₈	—	178	1	no	—
115	C₂×C₈	—	178	1	no	—
116	C₄⋊C₄	—	178	1	no	—
117	C₂²⋊C₄	—	89	2	no	—
118	C₄²	—	178	1	no	—
119	C₂³	Q₈×D₈₉	1	1	yes	characteristic
120	C₂×C₄	—	178	2	no	—
121	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
122	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
125	C₂×C₄	#13083573	1	1	yes	characteristic
126	C₂×C₄	#13083574	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	2	2	no	—
129	C₈	—	4	1	no	—
130	C₈	—	89	2	no	—
131	C₈	—	89	2	no	—
132	C₈	—	356	1	no	—
133	C₈	—	178	2	no	—
134	C₂²	—	4	1	no	—
135	C₂²	#15667977	1	1	yes	characteristic
136	C₂²	#15667990	1	1	yes	characteristic
137	C₂²	#15668010	1	1	yes	characteristic
138	C₄	—	356	2	no	—
139	C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
140	C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
141	C₄	#15667984	1	1	yes	characteristic
142	C₄	#15668002	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#19151493	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151646

C₂

yes

maximal, characteristic

C₂×#15667947

C₂

yes

maximal, characteristic

C₂×#15667947

C₂

yes

maximal, characteristic

#19151556

C₂

yes

maximal, characteristic

#19151370

C₂

yes

maximal, characteristic

#19151369

C₂

yes

maximal, characteristic

#430×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

#15667971

C₂²

yes

—

—

—

C₄

yes

complemented

#15667922

—

C₈×Dic₈₉

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

#108×C₈₉

—

—

—

—

C₂×C₄

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

D₄

yes

characteristic

C₂×#10623494

D₄

yes

characteristic

C₂×#10623494

Q₈

yes

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

Q₈

yes

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

C₂×C₄

yes

—

M₁₆×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

#10623494

C₄⋊C₄

yes

—

#10623494

C₄⋊C₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×Q₈

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₉

C₄⋊Q₈

yes

characteristic

C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₉

C₄×D₄

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#430

—

#430

D₈₉

yes

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

—

—

—

—

178

—

#108

—

178

—

#108

—

178

—

#97

—

178

—

C₄×C₈

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

178

—

103

M₁₆

—

178

—

104

M₁₆

—

178

—

105

M₁₆

C₄×D₈₉

yes

—

106

M₁₆

C₂²⋊D₈₉

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

178

—

115

C₂×C₈

—

178

—

116

C₄⋊C₄

—

178

—

117

C₂²⋊C₄

—

118

C₄²

—

178

—

119

C₂³

Q₈×D₈₉

yes

characteristic

120

C₂×C₄

—

178

—

121

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

122

C₂×C₄

D₄×D₈₉

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

124

C₂×C₄

Q₈×D₈₉

yes

characteristic

125

C₂×C₄

#13083573

yes

characteristic

126

C₂×C₄

#13083574

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

356

—

133

C₈

—

178

—

134

C₂²

—

135

C₂²

#15667977

yes

characteristic

136

C₂²

#15667990

yes

characteristic

137

C₂²

#15668010

yes

characteristic

138

C₄

—

356

—

139

C₄

C₄⋊C₄×D₈₉

yes

characteristic

140

C₄

C₄×C₂²⋊D₈₉

yes

characteristic

141

C₄

#15667984

yes

characteristic

142

C₄

#15668002

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#19151493

yes

characteristic

	subgroup	G/H
center	C₄	#15667984
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#430×C₈₉	C₂
socle	C₂×C₈₉	#389
prime radical	C₂³×C₈₉	C₂×Q₈
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#430	D₈₉
hypocenter	C₈₉	#1953
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

#15667984

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#430×C₈₉

C₂

socle

C₂×C₈₉

#389

prime radical

C₂³×C₈₉

C₂×Q₈

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
15. #15667947	C₄	semidirect
74. C₈₉	#1953	semidirect

H ≅ G/N

type

15. #15667947

C₄

semidirect

74. C₈₉

#1953

semidirect

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

subgroup	index
#19151646	2
C₂×#15667947	2
#19151556	2
#19151370	2
#19151369	2
#430×C₈₉	2
#1953	89

subgroup

index

#430×C₈₉

Subgroups — #26021039

Conjugacy classes of subgroups

Complements

Maximal subgroups