Group #26023550 — subgroups

Subgroups — #26023550

Order 11392.

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

normal subgroups only

This group has 432 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	C₈₉⋊#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151539	C₂	1	3	yes	maximal, complemented
3	#19151527	C₂	1	3	yes	maximal
4	#19151522	C₂	1	1	yes	maximal, characteristic
5	C₄×#13083553	C₂	1	3	yes	maximal, complemented
6	C₄×#13083551	C₂	1	3	yes	maximal
7	#376×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
9	C₂×#13083553	C₂²	1	3	yes	—
10	C₂×#13083553	C₂²	1	3	yes	—
11	C₂×#13083551	C₂²	1	3	yes	—
12	C₂×#13083551	C₂²	1	3	yes	—
13	C₂×C₄×Dic₈₉	C₂²	1	3	yes	—
14	#15667919	C₂²	1	3	yes	—
15	#15667919	C₂²	1	3	yes	—
16	#15667919	C₂²	1	3	yes	—
17	#15667919	C₂²	1	3	yes	—
18	C₂×C₄²×C₈₉	C₂²	1	3	yes	—
19	#95×C₈₉	C₂²	1	1	yes	characteristic
20	#95×C₈₉	C₂²	1	3	yes	—
21	C₂³×D₈₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₈₉	C₂³	1	3	yes	—
24	C₂²×Dic₈₉	C₂³	1	3	yes	—
25	C₂×C₄×D₈₉	—	2	6	no	—
26	#13083553	—	2	6	no	—
27	#13083553	—	2	6	no	—
28	#13083551	—	2	6	no	—
29	#13083551	—	2	6	no	—
30	C₄×Dic₈₉	—	2	6	no	—
31	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₄²×C₈₉	—	2	6	no	—
35	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
36	C₂²×D₈₉	—	2	3	no	—
37	C₂²×D₈₉	—	2	3	no	—
38	C₂²×D₈₉	#41	1	2	yes	—
39	C₂×C₄×C₈₉	—	2	3	no	—
40	C₂×C₄×C₈₉	—	2	3	no	—
41	C₂×C₄×C₈₉	—	2	3	no	—
42	C₂×C₄×C₈₉	—	2	3	no	—
43	C₂×C₄×C₈₉	—	2	3	no	—
44	C₂×C₄×C₈₉	—	2	3	no	—
45	C₂×C₄×C₈₉	#41	1	6	yes	—
46	C₂×Dic₈₉	—	2	3	no	—
47	C₂×Dic₈₉	—	2	3	no	—
48	C₂×Dic₈₉	—	2	3	no	—
49	C₂×Dic₈₉	—	2	3	no	—
50	C₂×Dic₈₉	—	2	3	no	—
51	C₂×Dic₈₉	—	2	3	no	—
52	C₂×Dic₈₉	#41	1	6	yes	—
53	C₄×D₈₉	—	4	4	no	—
54	C₂²×C₈₉	C₂×#41	1	3	yes	—
55	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
56	C₂²×C₈₉	C₂×#41	1	3	yes	—
57	C₂×D₈₉	—	2	6	no	—
58	C₂×D₈₉	—	4	1	no	—
59	C₂×D₈₉	—	4	3	no	—
60	C₄×C₈₉	—	4	3	no	—
61	C₄×C₈₉	—	4	3	no	—
62	C₄×C₈₉	#126	1	4	yes	—
63	Dic₈₉	—	2	6	no	—
64	Dic₈₉	—	4	1	no	—
65	Dic₈₉	—	4	3	no	—
66	C₂×C₈₉	C₂×#126	1	1	yes	characteristic
67	C₂×C₈₉	#529	1	3	yes	—
68	C₂×C₈₉	#529	1	3	yes	—
69	D₈₉	—	4	2	no	—
70	#2401	—	89	1	no	maximal, Sylow
71	C₈₉	#2401	1	1	yes	characteristic, complemented, Sylow
72	#401	—	89	1	no	—
73	#400	—	89	3	no	—
74	#397	—	89	3	no	—
75	#396	—	89	1	no	—
76	C₄×C₄⋊C₄	—	89	3	no	—
77	C₄×C₂²⋊C₄	—	89	3	no	—
78	#376	D₈₉	1	1	yes	characteristic, complemented
79	C₂³×C₄	—	89	1	no	—
80	C₂×C₄⋊C₄	—	89	3	no	—
81	C₂×C₄⋊C₄	—	89	3	no	—
82	C₂×C₂²⋊C₄	—	89	3	no	—
83	C₂×C₂²⋊C₄	—	89	3	no	—
84	C₂×C₄²	—	89	3	no	—
85	C₂×C₄²	C₂×D₈₉	1	3	yes	—
86	#95	—	89	3	no	—
87	#95	—	89	3	no	—
88	#95	—	89	3	no	—
89	#95	—	89	3	no	—
90	#95	C₂×D₈₉	1	1	yes	characteristic
91	#95	C₂×D₈₉	1	3	yes	—
92	C₂⁴	—	89	1	no	—
93	C₂²×C₄	—	89	3	no	—
94	C₂²×C₄	—	89	1	no	—
95	C₂²×C₄	—	89	3	no	—
96	C₂²×C₄	—	178	6	no	—
97	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
98	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
99	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
100	C₄⋊C₄	—	178	6	no	—
101	C₄⋊C₄	—	178	6	no	—
102	C₂²⋊C₄	—	178	6	no	—
103	C₂²⋊C₄	—	178	6	no	—
104	C₄²	—	2	6	no	—
105	C₄²	—	178	6	no	—
106	C₂³	—	89	2	no	—
107	C₂³	—	178	3	no	—
108	C₂³	—	178	3	no	—
109	C₂³	C₂³×D₈₉	1	1	yes	characteristic
110	C₂×C₄	—	2	3	no	—
111	C₂×C₄	—	2	3	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	3	no	—
115	C₂×C₄	—	2	3	no	—
116	C₂×C₄	—	89	6	no	—
117	C₂×C₄	—	356	4	no	—
118	C₂×C₄	—	178	3	no	—
119	C₂×C₄	—	178	3	no	—
120	C₂×C₄	—	178	3	no	—
121	C₂×C₄	—	178	3	no	—
122	C₂×C₄	—	178	3	no	—
123	C₂×C₄	—	178	3	no	—
124	C₂×C₄	#13083574	1	6	yes	—
125	C₂²	—	356	1	no	—
126	C₂²	—	356	3	no	—
127	C₂²	—	178	6	no	—
128	C₂²	#41×D₈₉	1	3	yes	—
129	C₂²	#41×D₈₉	1	1	yes	characteristic
130	C₂²	C₂×#13083574	1	3	yes	—
131	C₄	—	4	3	no	—
132	C₄	—	4	3	no	—
133	C₄	—	356	1	no	—
134	C₄	—	356	3	no	—
135	C₄	—	178	6	no	—
136	C₄	#15668006	1	4	yes	—
137	C₂	—	356	2	no	—
138	C₂	C₂×#15668006	1	1	yes	characteristic
139	C₂	#19152126	1	3	yes	—
140	C₂	#19152126	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉⋊#401

C₂

yes

maximal, characteristic, complemented

#19151539

C₂

yes

maximal, complemented

#19151527

C₂

yes

maximal

#19151522

C₂

yes

maximal, characteristic

C₄×#13083553

C₂

yes

maximal, complemented

C₄×#13083551

C₂

yes

maximal

#376×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₈₉

C₂²

yes

characteristic

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂³×D₈₉

C₂³

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂×C₄×D₈₉

—

—

—

—

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

#41

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#41

yes

—

C₄×D₈₉

—

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#126

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×#126

yes

characteristic

C₂×C₈₉

#529

yes

—

C₂×C₈₉

#529

yes

—

D₈₉

—

#2401

—

maximal, Sylow

C₈₉

#2401

yes

characteristic, complemented, Sylow

—

—

—

—

C₄×C₄⋊C₄

—

C₄×C₂²⋊C₄

—

#376

D₈₉

yes

characteristic, complemented

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

—

—

—

—

—

C₂×D₈₉

yes

characteristic

#95

C₂×D₈₉

yes

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

100

C₄⋊C₄

—

178

—

101

C₄⋊C₄

—

178

—

102

C₂²⋊C₄

—

178

—

103

C₂²⋊C₄

—

178

—

104

C₄²

—

105

C₄²

—

178

—

106

C₂³

—

107

C₂³

—

178

—

108

C₂³

—

178

—

109

C₂³

C₂³×D₈₉

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

356

—

118

C₂×C₄

—

178

—

119

C₂×C₄

—

178

—

120

C₂×C₄

—

178

—

121

C₂×C₄

—

178

—

122

C₂×C₄

—

178

—

123

C₂×C₄

—

178

—

124

C₂×C₄

#13083574

yes

—

125

C₂²

—

356

—

126

C₂²

—

356

—

127

C₂²

—

178

—

128

C₂²

#41×D₈₉

yes

—

129

C₂²

#41×D₈₉

yes

characteristic

130

C₂²

C₂×#13083574

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

356

—

134

C₄

—

356

—

135

C₄

—

178

—

136

C₄

#15668006

yes

—

137

C₂

—

356

—

138

C₂

C₂×#15668006

yes

characteristic

139

C₂

#19152126

yes

—

140

C₂

#19152126

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Frattini	C₂³	C₂³×D₈₉
Fitting	#376×C₈₉	C₂
prime radical	C₂³×D₈₉	C₂³
hypercenter	#376	D₈₉
hypocenter	C₈₉	#2401
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center

C₂²×C₄

C₂²×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Frattini

C₂³

C₂³×D₈₉

Fitting

#376×C₈₉

C₂

prime radical

C₂³×D₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
1. C₈₉⋊#401	C₂	semidirect
2. #19151539	C₂	semidirect
5. C₄×#13083553	C₂	semidirect
7. #376×C₈₉	C₂	semidirect
71. C₈₉	#2401	semidirect
78. #376	D₈₉	semidirect

H ≅ G/N

type

1. C₈₉⋊#401

C₂

semidirect

2. #19151539

C₂

semidirect

5. C₄×#13083553

C₂

semidirect

7. #376×C₈₉

C₂

semidirect

71. C₈₉

#2401

semidirect

78. #376

D₈₉

semidirect

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

subgroup	index
C₈₉⋊#401	2
#19151539	2
#19151527	2
#19151522	2
C₄×#13083553	2
C₄×#13083551	2
#376×C₈₉	2
#2401	89

subgroup

index

C₈₉⋊#401

#376×C₈₉

Subgroups — #26023550

Conjugacy classes of subgroups

Complements

Maximal subgroups