Group #26019882 — subgroups

Subgroups — #26019882

Order 11392.

220 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 61 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	#19151994	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151974	C₂	1	1	yes	maximal, characteristic
3	#19151972	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151971	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151944	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151320	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083563	—	2	2	no	—
9	C₂×#13083561	—	2	2	no	—
10	C₂×#13083559	—	2	2	no	—
11	#15668012	C₂²	1	1	yes	characteristic
12	#15668009	C₂²	1	1	yes	characteristic
13	#15667943	—	2	2	no	—
14	#15667933	—	2	2	no	—
15	C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	#127×C₈₉	C₂²	1	1	yes	characteristic
19	#102×C₈₉	—	2	2	no	—
20	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₇₈	—	2	1	no	—
22	C₂×D₃₅₆	—	2	1	no	—
23	C₂×D₃₅₆	—	2	1	no	—
24	C₂×D₃₅₆	—	2	1	no	—
25	#13083563	—	4	4	no	—
26	#13083561	—	4	4	no	—
27	#13083559	—	4	4	no	—
28	C₂×#10623494	D₄	1	2	yes	—
29	C₂×#10623494	D₄	1	2	yes	—
30	C₄⋊Dic₈₉	—	2	1	no	—
31	C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×Q₈×C₈₉	—	2	1	no	—
33	C₂×D₄×C₈₉	—	2	1	no	—
34	C₂×D₄×C₈₉	—	2	1	no	—
35	C₂×D₄×C₈₉	—	2	1	no	—
36	C₂×C₈×C₈₉	D₄	1	2	yes	—
37	C₄⋊C₄×C₈₉	—	2	1	no	—
38	C₄⋊C₄×C₈₉	—	2	1	no	—
39	C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₂³×C₈₉	—	4	1	no	—
41	C₂²×D₈₉	—	4	1	no	—
42	Q₈×C₈₉	—	4	2	no	—
43	D₄×C₈₉	—	4	2	no	—
44	D₄×C₈₉	—	4	2	no	—
45	D₄×C₈₉	—	4	2	no	—
46	C₂×C₄×C₈₉	—	4	1	no	—
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×D₄	1	1	yes	characteristic
50	C₂×Dic₈₉	—	4	1	no	—
51	D₃₅₆	—	4	2	no	—
52	D₃₅₆	—	4	2	no	—
53	D₃₅₆	—	4	2	no	—
54	Dic₁₇₈	—	4	2	no	—
55	C₈×C₈₉	—	2	2	no	—
56	#10623494	—	2	2	no	—
57	#10623494	SD₁₆	1	4	yes	complemented
58	C₂²×C₈₉	—	8	1	no	—
59	C₂²×C₈₉	—	8	1	no	—
60	C₂²×C₈₉	—	8	1	no	—
61	C₂²×C₈₉	#127	1	1	yes	characteristic
62	C₂×D₈₉	—	8	1	no	—
63	C₂×D₈₉	—	8	1	no	—
64	C₂×D₈₉	—	8	1	no	—
65	C₄×C₈₉	—	8	1	no	—
66	C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
67	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₈₉	—	8	1	no	—
70	C₂×C₈₉	—	16	1	no	—
71	C₂×C₈₉	#492	1	1	yes	characteristic
72	C₂×C₈₉	#496	1	1	yes	characteristic
73	C₂×C₈₉	#496	1	1	yes	characteristic
74	D₈₉	—	16	1	no	—
75	#1715	—	89	1	no	maximal, Sylow
76	C₈₉	#1715	1	1	yes	characteristic, complemented, Sylow
77	#498	—	89	1	no	—
78	#493	—	89	1	no	—
79	#492	—	89	1	no	—
80	#492	—	89	1	no	—
81	#486	—	89	1	no	—
82	#486	D₈₉	1	1	yes	characteristic, complemented
83	#322	—	89	1	no	—
84	C₂×SD₁₆	—	178	2	no	—
85	C₂×SD₁₆	—	178	2	no	—
86	C₂×D₈	—	178	2	no	—
87	C₄⋊Q₈	—	89	1	no	—
88	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
89	#127	—	89	1	no	—
90	#127	C₂×D₈₉	1	1	yes	characteristic
91	#106	—	178	2	no	—
92	#102	—	2	2	no	—
93	#102	—	178	2	no	—
94	C₄×C₈	—	89	1	no	—
95	C₄×C₈	—	89	1	no	—
96	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	178	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	178	1	no	—
103	C₂×D₄	—	178	1	no	—
104	C₂×D₄	—	178	1	no	—
105	SD₁₆	—	356	4	no	—
106	SD₁₆	—	356	4	no	—
107	D₈	—	356	4	no	—
108	C₂×C₈	—	89	2	no	—
109	C₂×C₈	—	89	2	no	—
110	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	178	1	no	—
114	C₄⋊C₄	—	178	1	no	—
115	C₄²	C₂²×D₈₉	1	1	yes	characteristic
116	C₂³	—	4	1	no	—
117	C₂³	—	356	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	356	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	356	2	no	—
124	D₄	—	356	2	no	—
125	D₄	—	356	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	356	1	no	—
128	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
129	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	89	4	no	—
133	C₈	—	178	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	712	1	no	—
138	C₂²	—	712	1	no	—
139	C₂²	—	712	1	no	—
140	C₂²	#15668008	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	712	1	no	—
143	C₄	SD₁₆×D₈₉	1	2	yes	—
144	C₄	#15668036	1	2	yes	—
145	C₄	#15668037	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	1424	1	no	—
148	C₂	#19151969	1	1	yes	characteristic
149	C₂	#19151986	1	1	yes	characteristic
150	C₂	#19151988	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151994

C₂

yes

maximal, characteristic, complemented

#19151974

C₂

yes

maximal, characteristic

#19151972

C₂

yes

maximal, characteristic, complemented

#19151971

C₂

yes

maximal, characteristic, complemented

#19151944

C₂

yes

maximal, characteristic, complemented

#19151320

C₂

yes

maximal, characteristic, complemented

#486×C₈₉

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#15668009

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#127×C₈₉

C₂²

yes

characteristic

#102×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

—

—

—

D₄

yes

—

C₂×#10623494

D₄

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

SD₁₆

yes

complemented

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#127

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#1715

—

maximal, Sylow

C₈₉

#1715

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₈₉

yes

characteristic, complemented

#322

—

C₂×SD₁₆

—

178

—

C₂×SD₁₆

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

#127

—

#127

C₂×D₈₉

yes

characteristic

#106

—

178

—

#102

—

#102

—

178

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

178

—

103

C₂×D₄

—

178

—

104

C₂×D₄

—

178

—

105

SD₁₆

—

356

—

106

SD₁₆

—

356

—

107

D₈

—

356

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂²⋊D₈₉

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

178

—

114

C₄⋊C₄

—

178

—

115

C₄²

C₂²×D₈₉

yes

characteristic

116

C₂³

—

117

C₂³

—

356

—

118

Q₈

—

119

Q₈

—

356

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

356

—

124

D₄

—

356

—

125

D₄

—

356

—

126

C₂×C₄

—

127

C₂×C₄

—

356

—

128

C₂×C₄

D₄×D₈₉

yes

characteristic

129

C₂×C₄

D₄×D₈₉

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

178

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

712

—

138

C₂²

—

712

—

139

C₂²

—

712

—

140

C₂²

#15668008

yes

characteristic

141

C₄

—

142

C₄

—

712

—

143

C₄

SD₁₆×D₈₉

yes

—

144

C₄

#15668036

yes

—

145

C₄

#15668037

yes

—

146

C₂

—

147

C₂

—

1424

—

148

C₂

#19151969

yes

characteristic

149

C₂

#19151986

yes

characteristic

150

C₂

#19151988

yes

characteristic

	subgroup	G/H
center	C₂²	#15668008
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#486×C₈₉	C₂
socle	C₂²×C₈₉	#127
prime radical	#19151974	C₂
hypercenter	#486	D₈₉
hypocenter	C₈₉	#1715
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#15668008

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#486×C₈₉

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #19151994	C₂	semidirect
3. #19151972	C₂	semidirect
4. #19151971	C₂	semidirect
5. #19151944	C₂	semidirect
6. #19151320	C₂	semidirect
7. #486×C₈₉	C₂	semidirect
57. #10623494	SD₁₆	semidirect
76. C₈₉	#1715	semidirect
82. #486	D₈₉	semidirect

H ≅ G/N

type

1. #19151994

C₂

semidirect

3. #19151972

C₂

semidirect

4. #19151971

C₂

semidirect

5. #19151944

C₂

semidirect

6. #19151320

C₂

semidirect

7. #486×C₈₉

C₂

semidirect

57. #10623494

SD₁₆

semidirect

76. C₈₉

#1715

semidirect

82. #486

D₈₉

semidirect

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

subgroup	index
#19151994	2
#19151974	2
#19151972	2
#19151971	2
#19151944	2
#19151320	2
#486×C₈₉	2
#1715	89

subgroup

index

#486×C₈₉

Subgroups — #26019882

Conjugacy classes of subgroups

Complements

Maximal subgroups