Group #26031633 — subgroups

Subgroups — #26031633

Order 11392.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19152148	C₂	1	2	yes	maximal, complemented
2	#19151889	C₂	1	4	yes	maximal, complemented
3	#19151851	C₂	1	4	yes	maximal, complemented
4	#19151794	C₂	1	4	yes	maximal, complemented
5	#423×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	—	2	2	no	—
7	C₂×D₇₁₂	C₂²	1	4	yes	—
8	#15668012	C₂²	1	2	yes	—
9	#15668009	C₂²	1	2	yes	—
10	#15668000	—	2	2	no	—
11	#15667986	—	2	2	no	—
12	#15667986	C₂²	1	4	yes	—
13	C₄×D₃₅₆	—	2	2	no	—
14	C₄×D₃₅₆	C₂²	1	4	yes	—
15	#15667944	C₂²	1	4	yes	—
16	#15667934	C₂²	1	8	yes	—
17	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
20	#13083574	—	4	8	no	—
21	C₂×Dic₁₇₈	—	2	2	no	—
22	C₂×D₃₅₆	—	2	2	no	—
23	C₂×D₃₅₆	—	2	2	no	—
24	C₂×D₃₅₆	C₂³	1	4	yes	—
25	C₂×C₂²⋊D₈₉	—	2	4	no	—
26	C₂×C₄×D₈₉	—	2	4	no	—
27	D₇₁₂	—	4	4	no	—
28	C₄⋊Dic₈₉	—	2	2	no	—
29	C₄⋊Dic₈₉	C₂³	1	4	yes	—
30	#13083551	—	2	4	no	—
31	#13083551	—	2	4	no	—
32	C₂²×C₄×C₈₉	D₄	1	2	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	M₁₆×C₈₉	—	4	2	no	—
35	C₂×C₈×C₈₉	C₂³	1	4	yes	—
36	C₄²×C₈₉	D₄	1	1	yes	characteristic
37	C₄²×C₈₉	D₄	1	1	yes	characteristic
38	C₄²×C₈₉	C₂³	1	2	yes	—
39	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×D₈₉	—	2	4	no	—
41	C₂×C₄×C₈₉	—	2	4	no	—
42	C₂×C₄×C₈₉	—	2	1	no	—
43	C₂×C₄×C₈₉	—	2	1	no	—
44	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
48	C₂²⋊D₈₉	—	4	8	no	—
49	C₂×Dic₈₉	—	2	4	no	—
50	D₃₅₆	—	2	4	no	—
51	D₃₅₆	—	2	4	no	—
52	D₃₅₆	—	4	4	no	—
53	D₃₅₆	—	4	4	no	—
54	C₄×D₈₉	—	4	8	no	—
55	Dic₁₇₈	—	4	4	no	—
56	C₈×C₈₉	—	2	4	no	—
57	C₂²×C₈₉	—	2	1	no	—
58	C₂²×C₈₉	—	2	2	no	—
59	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
60	C₂×D₈₉	—	4	4	no	—
61	C₂×D₈₉	—	4	4	no	—
62	C₂×D₈₉	—	4	4	no	—
63	C₄×C₈₉	—	2	1	no	—
64	C₄×C₈₉	—	2	2	no	—
65	C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
66	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
67	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₈₉	—	4	4	no	—
69	C₂×C₈₉	—	4	1	no	—
70	C₂×C₈₉	#537	1	1	yes	characteristic
71	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
72	D₈₉	—	8	4	no	—
73	#3253	—	89	1	no	maximal, Sylow
74	C₈₉	#3253	1	1	yes	characteristic, complemented, Sylow
75	#532	—	89	2	no	—
76	#476	—	89	4	no	—
77	#469	—	89	4	no	—
78	#459	—	89	4	no	—
79	#423	D₈₉	1	1	yes	characteristic, complemented
80	C₂×#41	—	178	2	no	—
81	C₂×D₈	—	89	4	no	—
82	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
83	C₄⋊Q₈	—	89	2	no	—
84	#127	—	89	2	no	—
85	#124	—	178	2	no	—
86	#121	—	89	4	no	—
87	#121	—	178	2	no	—
88	C₄×D₄	—	89	4	no	—
89	C₄×D₄	—	178	2	no	—
90	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
91	#106	—	89	4	no	—
92	C₄⋊C₈	C₂×D₈₉	1	4	yes	—
93	#102	—	89	8	no	—
94	#41	—	356	8	no	—
95	C₂×Q₈	—	178	2	no	—
96	C₂×D₄	—	89	4	no	—
97	C₂×D₄	—	178	4	no	—
98	C₂×D₄	—	178	2	no	—
99	C₂×D₄	—	178	2	no	—
100	C₂²×C₄	—	178	4	no	—
101	C₂²×C₄	D₃₅₆	1	2	yes	—
102	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
103	D₈	—	356	4	no	—
104	M₁₆	—	4	2	no	—
105	C₂×C₈	C₂²×D₈₉	1	4	yes	—
106	C₄⋊C₄	—	89	4	no	—
107	C₄⋊C₄	—	178	2	no	—
108	C₂²⋊C₄	—	178	4	no	—
109	C₂²⋊C₄	—	178	4	no	—
110	C₄²	D₃₅₆	1	1	yes	characteristic
111	C₄²	D₃₅₆	1	1	yes	characteristic
112	C₄²	C₂²×D₈₉	1	2	yes	—
113	C₂³	—	178	4	no	—
114	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
115	Q₈	—	356	4	no	—
116	D₄	—	356	4	no	—
117	D₄	—	356	8	no	—
118	D₄	—	356	4	no	—
119	D₄	—	178	4	no	—
120	D₄	—	178	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	356	8	no	—
125	C₂×C₄	—	178	4	no	—
126	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
127	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
128	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
129	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
130	C₈	—	2	4	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	356	4	no	—
134	C₂²	—	356	4	no	—
135	C₂²	—	356	4	no	—
136	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	356	4	no	—
140	C₄	#15668040	1	4	yes	—
141	C₄	#15668057	1	1	yes	characteristic
142	C₄	#15668060	1	1	yes	characteristic
143	C₂	—	4	1	no	—
144	C₂	—	712	4	no	—
145	C₂	#19152179	1	1	yes	characteristic
146	C₂	C₂×#15668040	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152148

C₂

yes

maximal, complemented

#19151889

C₂

yes

maximal, complemented

#19151851

C₂

yes

maximal, complemented

#19151794

C₂

yes

maximal, complemented

#423×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

—

C₂×D₇₁₂

C₂²

yes

—

#15668012

C₂²

yes

—

#15668009

C₂²

yes

—

#15668000

—

#15667986

—

#15667986

C₂²

yes

—

C₄×D₃₅₆

—

C₄×D₃₅₆

C₂²

yes

—

#15667944

C₂²

yes

—

#15667934

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

#13083574

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083551

—

#13083551

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₄×C₈₉

H(5,𝔽₂)

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#537

yes

characteristic

C₂×C₈₉

C₂×AGL(1,ℤ/8)

yes

—

D₈₉

—

#3253

—

maximal, Sylow

C₈₉

#3253

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂×#41

—

178

—

C₂×D₈

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₄⋊Q₈

—

#127

—

#124

—

178

—

#121

—

#121

—

178

—

C₄×D₄

—

C₄×D₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#106

—

C₄⋊C₈

C₂×D₈₉

yes

—

#102

—

#41

—

356

—

C₂×Q₈

—

178

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

100

C₂²×C₄

—

178

—

101

C₂²×C₄

D₃₅₆

yes

—

102

C₂²×C₄

C₂²×D₈₉

yes

characteristic

103

D₈

—

356

—

104

M₁₆

—

105

C₂×C₈

C₂²×D₈₉

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

178

—

108

C₂²⋊C₄

—

178

—

109

C₂²⋊C₄

—

178

—

110

C₄²

D₃₅₆

yes

characteristic

111

C₄²

D₃₅₆

yes

characteristic

112

C₄²

C₂²×D₈₉

yes

—

113

C₂³

—

178

—

114

C₂³

C₂×D₃₅₆

yes

characteristic

115

Q₈

—

356

—

116

D₄

—

356

—

117

D₄

—

356

—

118

D₄

—

356

—

119

D₄

—

178

—

120

D₄

—

178

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

356

—

125

C₂×C₄

—

178

—

126

C₂×C₄

C₂×D₃₅₆

yes

characteristic

127

C₂×C₄

C₂×D₃₅₆

yes

—

128

C₂×C₄

C₂×D₃₅₆

yes

—

129

C₂×C₄

C₂³×D₈₉

yes

characteristic

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

356

—

134

C₂²

—

356

—

135

C₂²

—

356

—

136

C₂²

C₂²×D₃₅₆

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

356

—

140

C₄

#15668040

yes

—

141

C₄

#15668057

yes

characteristic

142

C₄

#15668060

yes

characteristic

143

C₂

—

144

C₂

—

712

—

145

C₂

#19152179

yes

characteristic

146

C₂

C₂×#15668040

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#423×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#423	D₈₉
hypocenter	C₈₉	#3253
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#423×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #19152148	C₂	semidirect
2. #19151889	C₂	semidirect
3. #19151851	C₂	semidirect
4. #19151794	C₂	semidirect
5. #423×C₈₉	C₂	semidirect
74. C₈₉	#3253	semidirect
79. #423	D₈₉	semidirect

H ≅ G/N

type

1. #19152148

C₂

semidirect

2. #19151889

C₂

semidirect

3. #19151851

C₂

semidirect

4. #19151794

C₂

semidirect

5. #423×C₈₉

C₂

semidirect

74. C₈₉

#3253

semidirect

79. #423

D₈₉

semidirect

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

subgroup	index
#19152148	2
#19151889	2
#19151851	2
#19151794	2
#423×C₈₉	2
#3253	89

subgroup

index

#423×C₈₉

Subgroups — #26031633

Conjugacy classes of subgroups

Complements

Maximal subgroups