Group #26030626 — subgroups

Subgroups — #26030626

Order 11392.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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	#19152155	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151962	C₂	1	4	yes	maximal, complemented
3	#19151944	C₂	1	4	yes	maximal, complemented
4	#19151603	C₂	1	4	yes	maximal, complemented
5	C₄×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#127×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#13083558	C₂²	1	2	yes	—
8	#15668012	C₂²	1	2	yes	complemented
9	#15668004	C₂²	1	2	yes	complemented
10	#15667991	—	2	2	no	—
11	C₄×Dic₁₇₈	—	2	2	no	—
12	#15667971	C₂²	1	2	yes	—
13	#15667933	C₂²	1	16	yes	complemented
14	#15667923	C₂²	1	2	yes	complemented
15	C₄×#10623494	C₂²	1	2	yes	complemented
16	C₂²×D₄×C₈₉	—	2	2	no	—
17	C₂²×D₄×C₈₉	C₂²	1	2	yes	—
18	#127×C₈₉	—	2	2	no	—
19	#127×C₈₉	C₂²	1	2	yes	—
20	#127×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₇₈	—	2	2	no	—
23	#13083558	—	2	4	no	—
24	C₂×#10623494	C₂³	1	4	yes	—
25	C₄⋊Dic₈₉	—	2	2	no	—
26	C₄⋊Dic₈₉	C₂³	1	4	yes	—
27	#13083553	—	2	4	no	—
28	C₂²⋊Dic₈₉	—	2	2	no	—
29	C₄×Dic₈₉	—	2	2	no	—
30	C₂⁴×C₈₉	—	2	2	no	—
31	C₂×D₄×C₈₉	—	2	2	no	—
32	C₂×D₄×C₈₉	—	2	4	no	—
33	C₂×D₄×C₈₉	—	2	4	no	—
34	C₂×D₄×C₈₉	—	2	8	no	—
35	C₂×D₄×C₈₉	—	4	8	no	—
36	C₂×D₄×C₈₉	C₂³	1	4	yes	—
37	C₂²×C₄×C₈₉	D₄	1	2	yes	—
38	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₄²×C₈₉	D₄	1	2	yes	—
40	C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	C₄²×C₈₉	C₂³	1	1	yes	characteristic
42	C₂³×C₈₉	—	2	4	no	—
43	C₂³×C₈₉	—	4	8	no	—
44	C₂³×C₈₉	—	4	4	no	—
45	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
46	D₄×C₈₉	—	2	8	no	—
47	D₄×C₈₉	—	4	8	no	—
48	D₄×C₈₉	—	4	4	no	—
49	D₄×C₈₉	—	4	8	no	—
50	C₂×C₄×C₈₉	—	2	4	no	—
51	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₈₉	#41	1	4	yes	—
55	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₈₉	—	2	4	no	—
57	Dic₁₇₈	—	4	4	no	—
58	#10623494	—	2	4	no	—
59	C₂²×C₈₉	—	2	2	no	—
60	C₂²×C₈₉	—	4	4	no	—
61	C₂²×C₈₉	—	4	8	no	—
62	C₂²×C₈₉	—	8	8	no	—
63	C₂²×C₈₉	#124	1	2	yes	—
64	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₈₉	—	2	2	no	—
66	C₄×C₈₉	#124	1	2	yes	—
67	C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
68	C₄×C₈₉	C₂×#41	1	2	yes	—
69	Dic₈₉	—	4	4	no	—
70	C₂×C₈₉	—	2	2	no	—
71	C₂×C₈₉	—	8	4	no	—
72	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
73	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
74	#3138	—	89	1	no	maximal, Sylow
75	C₈₉	#3138	1	1	yes	characteristic, complemented, Sylow
76	#533	—	89	1	no	—
77	C₂×#127	D₈₉	1	1	yes	characteristic
78	#490	—	89	4	no	—
79	#486	—	89	4	no	—
80	#418	—	89	4	no	—
81	C₄×M₁₆	—	89	1	no	—
82	C₂²×D₄	—	2	2	no	—
83	C₂²×D₄	C₂×D₈₉	1	2	yes	—
84	C₂×M₁₆	—	89	2	no	—
85	C₄⋊Q₈	—	89	2	no	—
86	#127	—	2	2	no	—
87	#127	C₂×D₈₉	1	2	yes	—
88	#127	C₂×D₈₉	1	2	yes	—
89	#125	—	89	2	no	—
90	C₂²⋊Q₈	—	178	2	no	—
91	C₄×Q₈	—	178	2	no	—
92	#117	—	89	2	no	—
93	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
94	#102	—	89	16	no	—
95	#97	—	89	2	no	—
96	C₄×C₈	—	89	2	no	—
97	C₂⁴	—	2	2	no	—
98	C₂×Q₈	—	178	2	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	—	2	4	no	—
101	C₂×D₄	—	2	4	no	—
102	C₂×D₄	—	2	8	no	—
103	C₂×D₄	—	4	8	no	—
104	C₂×D₄	C₂²×D₈₉	1	4	yes	—
105	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
106	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
107	M₁₆	—	178	4	no	—
108	C₂×C₈	—	89	4	no	—
109	C₄⋊C₄	—	89	4	no	—
110	C₄⋊C₄	—	178	4	no	—
111	C₄⋊C₄	—	178	2	no	—
112	C₂²⋊C₄	—	178	2	no	—
113	C₄²	—	178	2	no	—
114	C₄²	C₂²⋊D₈₉	1	2	yes	—
115	C₄²	C₂²×D₈₉	1	1	yes	characteristic
116	C₄²	C₂²×D₈₉	1	1	yes	characteristic
117	C₂³	—	2	4	no	—
118	C₂³	—	4	8	no	—
119	C₂³	—	4	4	no	—
120	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
121	Q₈	—	356	4	no	—
122	D₄	—	2	8	no	—
123	D₄	—	4	8	no	—
124	D₄	—	4	4	no	—
125	D₄	—	4	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	178	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
131	C₂×C₄	#13083573	1	4	yes	—
132	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
133	C₈	—	178	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	8	no	—
136	C₂²	—	4	4	no	—
137	C₂²	—	8	8	no	—
138	C₂²	#15667999	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	356	4	no	—
142	C₄	#15667999	1	2	yes	—
143	C₄	#15668038	1	4	yes	—
144	C₄	C₂×#13083573	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	8	4	no	—
147	C₂	C₂×#15667999	1	1	yes	characteristic
148	C₂	C₂×#15668038	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152155

C₂

yes

maximal, characteristic, complemented

#19151962

C₂

yes

maximal, complemented

#19151944

C₂

yes

maximal, complemented

#19151603

C₂

yes

maximal, complemented

C₄×#13083558

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

#15668012

C₂²

yes

complemented

#15668004

C₂²

yes

complemented

#15667991

—

C₄×Dic₁₇₈

—

#15667971

C₂²

yes

—

#15667933

C₂²

yes

complemented

#15667923

C₂²

yes

complemented

C₄×#10623494

C₂²

yes

complemented

C₂²×D₄×C₈₉

—

C₂²×D₄×C₈₉

C₂²

yes

—

#127×C₈₉

—

#127×C₈₉

C₂²

yes

—

#127×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

#13083558

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#124

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#124

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₄×C₈₉

C₂×#41

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#124

yes

characteristic

C₂×C₈₉

C₂×AGL(1,ℤ/8)

yes

—

#3138

—

maximal, Sylow

C₈₉

#3138

yes

characteristic, complemented, Sylow

#533

—

C₂×#127

D₈₉

yes

—

—

—

C₄×M₁₆

—

C₂²×D₄

—

C₂²×D₄

C₂×D₈₉

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#127

C₂×D₈₉

yes

—

#127

C₂×D₈₉

yes

—

#125

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

178

—

#117

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₂⁴

—

C₂×Q₈

—

178

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

C₂²×D₈₉

yes

—

105

C₂²×C₄

C₂²⋊D₈₉

yes

—

106

C₂²×C₄

C₂²×D₈₉

yes

characteristic

107

M₁₆

—

178

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

178

—

111

C₄⋊C₄

—

178

—

112

C₂²⋊C₄

—

178

—

113

C₄²

—

178

—

114

C₄²

C₂²⋊D₈₉

yes

—

115

C₄²

C₂²×D₈₉

yes

characteristic

116

C₄²

C₂²×D₈₉

yes

characteristic

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

121

Q₈

—

356

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

178

—

128

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

129

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

131

C₂×C₄

#13083573

yes

—

132

C₂×C₄

C₂³×D₈₉

yes

characteristic

133

C₈

—

178

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#15667999

yes

—

139

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

140

C₄

—

141

C₄

—

356

—

142

C₄

#15667999

yes

—

143

C₄

#15668038

yes

—

144

C₄

C₂×#13083573

yes

—

145

C₂

—

146

C₂

—

147

C₂

C₂×#15667999

yes

characteristic

148

C₂

C₂×#15668038

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting, prime radical	C₂×#127×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	C₂×#127	D₈₉
hypocenter	C₈₉	#3138
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting, prime radical

C₂×#127×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #19152155	C₂	semidirect
2. #19151962	C₂	semidirect
3. #19151944	C₂	semidirect
4. #19151603	C₂	semidirect
5. C₄×#13083558	C₂	semidirect
8. #15668012	C₂²	semidirect
9. #15668004	C₂²	semidirect
13. #15667933	C₂²	semidirect
14. #15667923	C₂²	semidirect
15. C₄×#10623494	C₂²	semidirect
75. C₈₉	#3138	semidirect

H ≅ G/N

type

1. #19152155

C₂

semidirect

2. #19151962

C₂

semidirect

3. #19151944

C₂

semidirect

4. #19151603

C₂

semidirect

5. C₄×#13083558

C₂

semidirect

8. #15668012

C₂²

semidirect

9. #15668004

C₂²

semidirect

13. #15667933

C₂²

semidirect

14. #15667923

C₂²

semidirect

15. C₄×#10623494

C₂²

semidirect

75. C₈₉

#3138

semidirect

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

subgroup	index
#19152155	2
#19151962	2
#19151944	2
#19151603	2
C₄×#13083558	2
C₂×#127×C₈₉	2
#3138	89

subgroup

index

C₂×#127×C₈₉

Subgroups — #26030626

Conjugacy classes of subgroups

Complements

Maximal subgroups