Group #26021558 — subgroups

Subgroups — #26021558

Order 11392.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19152398	C₂	1	1	yes	maximal, characteristic
2	#19151595	C₂	1	2	yes	maximal, complemented
3	#19151571	C₂	1	2	yes	maximal, complemented
4	#19151355	C₂	1	1	yes	maximal, characteristic, complemented
5	#424×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	C₂²	1	2	yes	—
7	#15668045	—	2	2	no	—
8	#15668040	—	2	2	no	—
9	#15668034	—	2	2	no	—
10	#15667971	C₂²	1	2	yes	—
11	#15667937	—	2	2	no	—
12	#15667934	—	2	2	no	—
13	#15667926	—	2	4	no	—
14	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#117×C₈₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₈₉	—	2	2	no	—
18	#13083574	—	2	2	no	—
19	#13083574	—	2	2	no	—
20	#13083574	—	2	2	no	—
21	#13083574	—	2	2	no	—
22	C₂×Dic₁₇₈	D₄	1	2	yes	—
23	C₂×D₃₅₆	D₄	1	2	yes	—
24	C₂×C₂²⋊D₈₉	—	2	2	no	—
25	C₂×C₄×D₈₉	—	2	2	no	—
26	Dic₃₅₆	—	2	2	no	—
27	#13083563	—	2	4	no	—
28	D₇₁₂	—	2	2	no	—
29	C₄⋊Dic₈₉	—	2	2	no	—
30	C₂²⋊Dic₈₉	—	2	2	no	—
31	C₄×Dic₈₉	—	2	2	no	—
32	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₈₉	—	2	2	no	—
34	C₂×C₈×C₈₉	—	2	1	no	—
35	C₂×C₈×C₈₉	—	2	2	no	—
36	C₂×C₈×C₈₉	D₄	1	2	yes	—
37	C₄⋊C₄×C₈₉	—	2	1	no	—
38	C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
39	C₄²×C₈₉	—	2	1	no	—
40	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#41	1	2	yes	—
47	C₂²⋊D₈₉	—	4	2	no	—
48	C₂²⋊D₈₉	—	4	2	no	—
49	C₂×Dic₈₉	—	2	2	no	—
50	C₂×Dic₈₉	—	4	2	no	—
51	D₃₅₆	—	2	2	no	—
52	D₃₅₆	—	2	2	no	—
53	C₄×D₈₉	—	4	2	no	—
54	C₄×D₈₉	—	4	2	no	—
55	Dic₁₇₈	—	2	2	no	—
56	Dic₁₇₈	—	2	2	no	—
57	C₈×C₈₉	—	2	2	no	—
58	C₈×C₈₉	—	4	1	no	—
59	C₂²×C₈₉	—	4	1	no	—
60	C₂²×C₈₉	#121	1	1	yes	characteristic
61	C₂²×C₈₉	#121	1	2	yes	—
62	C₂×D₈₉	—	4	2	no	—
63	C₂×D₈₉	—	8	2	no	—
64	C₄×C₈₉	—	4	2	no	—
65	C₄×C₈₉	#120	1	1	yes	characteristic
66	C₄×C₈₉	#123	1	2	yes	—
67	C₄×C₈₉	#124	1	1	yes	characteristic
68	Dic₈₉	—	4	2	no	—
69	Dic₈₉	—	8	2	no	—
70	C₂×C₈₉	—	2	2	no	—
71	C₂×C₈₉	—	2	1	no	—
72	C₂×C₈₉	#394	1	1	yes	characteristic
73	D₈₉	—	8	2	no	—
74	#2053	—	89	1	no	maximal, Sylow
75	C₈₉	#2053	1	1	yes	characteristic, complemented, Sylow
76	#575	—	89	1	no	—
77	#424	D₈₉	1	1	yes	characteristic, complemented
78	#416	—	89	2	no	—
79	#410	—	89	2	no	—
80	H(3,ℤ/4)	—	89	1	no	—
81	C₂×#41	—	89	2	no	—
82	#137	—	178	2	no	—
83	AGL(1,ℤ/8)	—	178	2	no	—
84	#135	—	178	2	no	—
85	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
87	#117	—	89	2	no	—
88	#117	C₂×D₈₉	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	#103	—	178	2	no	—
91	#102	—	178	2	no	—
92	#99	—	178	4	no	—
93	#41	—	178	2	no	—
94	#41	—	178	2	no	—
95	#41	—	178	2	no	—
96	#41	—	178	2	no	—
97	C₂×Q₈	—	89	2	no	—
98	C₂×D₄	—	89	2	no	—
99	C₂×D₄	—	178	2	no	—
100	C₂²×C₄	—	178	2	no	—
101	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
102	Q₁₆	—	178	2	no	—
103	SD₁₆	—	178	4	no	—
104	D₈	—	178	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	178	2	no	—
111	C₂²⋊C₄	—	178	2	no	—
112	C₂²⋊C₄	D₃₅₆	1	2	yes	—
113	C₄²	—	2	1	no	—
114	C₄²	—	178	2	no	—
115	C₂³	—	178	2	no	—
116	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
117	Q₈	—	178	2	no	—
118	Q₈	—	178	2	no	—
119	D₄	—	356	2	no	—
120	D₄	—	356	2	no	—
121	D₄	—	178	2	no	—
122	D₄	—	178	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	356	2	no	—
125	C₂×C₄	—	356	2	no	—
126	C₂×C₄	—	356	2	no	—
127	C₂×C₄	—	178	2	no	—
128	C₂×C₄	D₄×D₈₉	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₂×C₄	#13083574	1	1	yes	characteristic
131	C₂×C₄	#13083575	1	2	yes	—
132	C₈	—	2	2	no	—
133	C₈	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	356	2	no	—
136	C₂²	—	712	2	no	—
137	C₂²	#15667986	1	1	yes	characteristic
138	C₂²	#15667987	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	356	2	no	—
141	C₄	—	712	2	no	—
142	C₄	#15667980	1	1	yes	characteristic
143	C₄	#15667997	1	2	yes	—
144	C₄	#15668001	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	2	no	—
147	C₂	—	712	2	no	—
148	C₂	#19151515	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152398

C₂

yes

maximal, characteristic

#19151595

C₂

yes

maximal, complemented

#19151571

C₂

yes

maximal, complemented

#19151355

C₂

yes

maximal, characteristic, complemented

#424×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

—

—

—

—

C₂²

yes

—

#15667937

—

#15667934

—

#15667926

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

—

—

—

—

—

C₂×Dic₁₇₈

D₄

yes

—

C₂×D₃₅₆

D₄

yes

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

Dic₃₅₆

—

#13083563

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

#121

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#120

yes

characteristic

C₄×C₈₉

#123

yes

—

C₄×C₈₉

#124

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#394

yes

characteristic

D₈₉

—

#2053

—

maximal, Sylow

C₈₉

#2053

yes

characteristic, complemented, Sylow

#575

—

#424

D₈₉

yes

characteristic, complemented

#416

—

#410

—

H(3,ℤ/4)

—

C₂×#41

—

#137

—

178

—

AGL(1,ℤ/8)

—

178

—

#135

—

178

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

#117

—

#117

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

#103

—

178

—

#102

—

178

—

#99

—

178

—

#41

—

178

—

#41

—

178

—

#41

—

178

—

#41

—

178

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

178

—

100

C₂²×C₄

—

178

—

101

C₂²×C₄

C₂²×D₈₉

yes

characteristic

102

Q₁₆

—

178

—

103

SD₁₆

—

178

—

104

D₈

—

178

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₈₉

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

178

—

111

C₂²⋊C₄

—

178

—

112

C₂²⋊C₄

D₃₅₆

yes

—

113

C₄²

—

114

C₄²

—

178

—

115

C₂³

—

178

—

116

C₂³

C₂×D₃₅₆

yes

characteristic

117

Q₈

—

178

—

118

Q₈

—

178

—

119

D₄

—

356

—

120

D₄

—

356

—

121

D₄

—

178

—

122

D₄

—

178

—

123

C₂×C₄

—

124

C₂×C₄

—

356

—

125

C₂×C₄

—

356

—

126

C₂×C₄

—

356

—

127

C₂×C₄

—

178

—

128

C₂×C₄

D₄×D₈₉

yes

—

129

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

130

C₂×C₄

#13083574

yes

characteristic

131

C₂×C₄

#13083575

yes

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

356

—

136

C₂²

—

712

—

137

C₂²

#15667986

yes

characteristic

138

C₂²

#15667987

yes

—

139

C₄

—

140

C₄

—

356

—

141

C₄

—

712

—

142

C₄

#15667980

yes

characteristic

143

C₄

#15667997

yes

—

144

C₄

#15668001

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

712

—

148

C₂

#19151515

yes

characteristic

	subgroup	G/H
center	C₄	#15668001
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#424×C₈₉	C₂
socle	C₂×C₈₉	#394
prime radical	#19152398	C₂
hypercenter	#424	D₈₉
hypocenter	C₈₉	#2053
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₄

#15668001

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#424×C₈₉

C₂

C₂×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. #19151595	C₂	semidirect
3. #19151571	C₂	semidirect
4. #19151355	C₂	semidirect
5. #424×C₈₉	C₂	semidirect
75. C₈₉	#2053	semidirect
77. #424	D₈₉	semidirect

H ≅ G/N

type

2. #19151595

C₂

semidirect

3. #19151571

C₂

semidirect

4. #19151355

C₂

semidirect

5. #424×C₈₉

C₂

semidirect

75. C₈₉

#2053

semidirect

77. #424

D₈₉

semidirect

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

subgroup	index
#19152398	2
#19151595	2
#19151571	2
#19151355	2
#424×C₈₉	2
#2053	89

subgroup

index

#424×C₈₉

Subgroups — #26021558

Conjugacy classes of subgroups

Complements

Maximal subgroups