Group #26028161 — subgroups

Subgroups — #26028161

Order 11392.

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

normal subgroups only

This group has 420 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	C₈₉⋊#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151535	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151514	C₂	1	3	yes	maximal
4	#19151497	C₂	1	3	yes	maximal, complemented
5	#19151491	C₂	1	3	yes	maximal, complemented
6	#19151488	C₂	1	3	yes	maximal, complemented
7	#396×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₈₉	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	3	yes	—
11	C₂×C₂²⋊Dic₈₉	C₂²	1	3	yes	—
12	C₂×#13083551	C₂²	1	3	yes	—
13	C₂×#13083551	C₂²	1	3	yes	—
14	C₂×C₄×Dic₈₉	C₂²	1	3	yes	—
15	#15667919	C₂²	1	3	yes	—
16	#15667919	C₂²	1	3	yes	—
17	#15667919	C₂²	1	3	yes	—
18	C₂×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
19	C₂×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
20	#95×C₈₉	C₂²	1	1	yes	characteristic
21	C₂³×D₈₉	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₈₉	—	4	3	no	—
23	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₈₉	C₂³	1	3	yes	—
25	C₂²×Dic₈₉	C₂³	1	3	yes	—
26	C₄⋊Dic₈₉	—	4	3	no	—
27	C₄⋊Dic₈₉	—	4	3	no	—
28	C₂²⋊Dic₈₉	—	4	3	no	—
29	#13083551	—	2	6	no	—
30	#13083551	—	4	3	no	—
31	C₄×Dic₈₉	—	4	3	no	—
32	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
34	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
35	C₄⋊C₄×C₈₉	—	2	6	no	—
36	C₂²⋊C₄×C₈₉	—	2	6	no	—
37	C₂³×C₈₉	—	2	3	no	—
38	C₂³×C₈₉	—	2	3	no	—
39	C₂³×C₈₉	—	2	1	no	—
40	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
41	C₂²×D₈₉	—	2	3	no	—
42	C₂²×D₈₉	—	2	3	no	—
43	C₂²×D₈₉	—	2	1	no	—
44	C₂×C₄×C₈₉	—	2	3	no	—
45	C₂×C₄×C₈₉	—	2	3	no	—
46	C₂×C₄×C₈₉	—	2	3	no	—
47	C₂×C₄×C₈₉	—	2	3	no	—
48	C₂×C₄×C₈₉	—	2	3	no	—
49	C₂×C₄×C₈₉	#41	1	6	yes	—
50	C₂²⋊D₈₉	—	4	4	no	—
51	C₂×Dic₈₉	—	2	3	no	—
52	C₂×Dic₈₉	—	2	3	no	—
53	C₂×Dic₈₉	—	2	3	no	—
54	C₂×Dic₈₉	—	2	3	no	—
55	C₂×Dic₈₉	—	2	3	no	—
56	C₂×Dic₈₉	—	2	3	no	—
57	C₂×Dic₈₉	—	2	3	no	—
58	C₂²×C₈₉	—	4	3	no	—
59	C₂²×C₈₉	—	4	3	no	—
60	C₂²×C₈₉	—	4	1	no	—
61	C₂²×C₈₉	C₂×#41	1	3	yes	—
62	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
63	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
64	C₂×D₈₉	—	4	3	no	—
65	C₂×D₈₉	—	4	3	no	—
66	C₂×D₈₉	—	4	1	no	—
67	C₄×C₈₉	—	2	6	no	—
68	C₄×C₈₉	—	4	3	no	—
69	Dic₈₉	—	4	3	no	—
70	Dic₈₉	—	4	3	no	—
71	Dic₈₉	—	4	1	no	—
72	C₂×C₈₉	—	8	1	no	—
73	C₂×C₈₉	#542	1	3	yes	—
74	C₂×C₈₉	#553	1	3	yes	—
75	C₂×C₈₉	#560	1	1	yes	characteristic
76	D₈₉	—	8	1	no	—
77	#2801	—	89	1	no	maximal, Sylow
78	C₈₉	#2801	1	1	yes	characteristic, complemented, Sylow
79	#401	—	89	1	no	—
80	#399	—	89	1	no	—
81	#396	D₈₉	1	1	yes	characteristic, complemented
82	#394	—	89	3	no	—
83	#389	—	89	3	no	—
84	#388	—	89	3	no	—
85	#388	—	89	3	no	—
86	C₂²×D₄	—	89	1	no	—
87	C₂×C₄⋊C₄	—	89	3	no	—
88	C₂×C₄⋊C₄	—	89	3	no	—
89	C₂×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
90	C₂×C₂²⋊C₄	—	89	3	no	—
91	C₂×C₂²⋊C₄	—	89	3	no	—
92	C₂×C₂²⋊C₄	—	89	3	no	—
93	C₂×C₂²⋊C₄	C₂×D₈₉	1	3	yes	—
94	C₂×C₄²	—	89	3	no	—
95	#95	—	89	3	no	—
96	#95	—	89	3	no	—
97	#95	—	89	3	no	—
98	#95	C₂×D₈₉	1	1	yes	characteristic
99	C₂⁴	—	89	1	no	—
100	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
101	C₂×D₄	—	356	3	no	—
102	C₂²×C₄	—	89	3	no	—
103	C₂²×C₄	—	89	1	no	—
104	C₂²×C₄	—	89	3	no	—
105	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
106	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
107	C₄⋊C₄	—	2	6	no	—
108	C₄⋊C₄	—	356	3	no	—
109	C₄⋊C₄	—	356	3	no	—
110	C₂²⋊C₄	—	2	6	no	—
111	C₂²⋊C₄	—	356	3	no	—
112	C₂²⋊C₄	—	356	3	no	—
113	C₂²⋊C₄	—	178	6	no	—
114	C₄²	—	356	3	no	—
115	C₂³	—	2	3	no	—
116	C₂³	—	2	3	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	178	3	no	—
119	C₂³	—	178	3	no	—
120	C₂³	—	178	1	no	—
121	C₂³	C₂³×D₈₉	1	1	yes	characteristic
122	D₄	—	356	4	no	—
123	C₂×C₄	—	2	3	no	—
124	C₂×C₄	—	2	3	no	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	178	3	no	—
129	C₂×C₄	—	178	3	no	—
130	C₂×C₄	—	178	3	no	—
131	C₂×C₄	—	178	3	no	—
132	C₂×C₄	—	178	3	no	—
133	C₂×C₄	—	178	3	no	—
134	C₂×C₄	—	178	3	no	—
135	C₂×C₄	#13083573	1	6	yes	—
136	C₂²	—	4	3	no	—
137	C₂²	—	4	3	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	356	3	no	—
140	C₂²	—	356	3	no	—
141	C₂²	—	356	1	no	—
142	C₂²	C₂×#13083573	1	3	yes	—
143	C₂²	#15668058	1	3	yes	—
144	C₂²	#15668060	1	1	yes	characteristic
145	C₄	—	2	6	no	—
146	C₄	—	4	3	no	—
147	C₄	—	356	3	no	—
148	C₄	—	356	1	no	—
149	C₄	—	356	3	no	—
150	C₂	—	8	1	no	—
151	C₂	—	712	1	no	—
152	C₂	#19152221	1	3	yes	—
153	C₂	#19152300	1	3	yes	—
154	C₂	#19152339	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉⋊#401

C₂

yes

maximal, characteristic, complemented

#19151535

C₂

yes

maximal, characteristic, complemented

#19151514

C₂

yes

maximal

#19151497

C₂

yes

maximal, complemented

#19151491

C₂

yes

maximal, complemented

#19151488

C₂

yes

maximal, complemented

#396×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

C₂³

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

C₂³

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

#13083551

—

#13083551

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#41

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#542

yes

—

C₂×C₈₉

#553

yes

—

C₂×C₈₉

#560

yes

characteristic

D₈₉

—

#2801

—

maximal, Sylow

C₈₉

#2801

yes

characteristic, complemented, Sylow

#401

—

#399

—

#396

D₈₉

yes

characteristic, complemented

—

—

—

—

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

—

—

—

C₂×D₈₉

yes

characteristic

C₂⁴

—

100

C₂⁴

C₂²×D₈₉

yes

characteristic

101

C₂×D₄

—

356

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₈₉

yes

—

106

C₂²×C₄

C₂²×D₈₉

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

356

—

109

C₄⋊C₄

—

356

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

356

—

112

C₂²⋊C₄

—

356

—

113

C₂²⋊C₄

—

178

—

114

C₄²

—

356

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

178

—

119

C₂³

—

178

—

120

C₂³

—

178

—

121

C₂³

C₂³×D₈₉

yes

characteristic

122

D₄

—

356

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

178

—

129

C₂×C₄

—

178

—

130

C₂×C₄

—

178

—

131

C₂×C₄

—

178

—

132

C₂×C₄

—

178

—

133

C₂×C₄

—

178

—

134

C₂×C₄

—

178

—

135

C₂×C₄

#13083573

yes

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

356

—

140

C₂²

—

356

—

141

C₂²

—

356

—

142

C₂²

C₂×#13083573

yes

—

143

C₂²

#15668058

yes

—

144

C₂²

#15668060

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

356

—

148

C₄

—

356

—

149

C₄

—

356

—

150

C₂

—

151

C₂

—

712

—

152

C₂

#19152221

yes

—

153

C₂

#19152300

yes

—

154

C₂

#19152339

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#396×C₈₉	C₂
prime radical	C₂²×C₂²⋊D₈₉	C₂²
hypercenter	#396	D₈₉
hypocenter	C₈₉	#2801
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #396

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#396×C₈₉

C₂

prime radical

C₂²×C₂²⋊D₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #396

N	H ≅ G/N	type
1. C₈₉⋊#401	C₂	semidirect
2. #19151535	C₂	semidirect
4. #19151497	C₂	semidirect
5. #19151491	C₂	semidirect
6. #19151488	C₂	semidirect
7. #396×C₈₉	C₂	semidirect
78. C₈₉	#2801	semidirect
81. #396	D₈₉	semidirect

H ≅ G/N

type

1. C₈₉⋊#401

C₂

semidirect

2. #19151535

C₂

semidirect

4. #19151497

C₂

semidirect

5. #19151491

C₂

semidirect

6. #19151488

C₂

semidirect

7. #396×C₈₉

C₂

semidirect

78. C₈₉

#2801

semidirect

81. #396

D₈₉

semidirect

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

subgroup	index
C₈₉⋊#401	2
#19151535	2
#19151514	2
#19151497	2
#19151491	2
#19151488	2
#396×C₈₉	2
#2801	89

subgroup

index

C₈₉⋊#401

#396×C₈₉

Subgroups — #26028161

Conjugacy classes of subgroups

Complements

Maximal subgroups