Group #26022813 — subgroups

Subgroups — #125×Dic₈₉ #26022813

Order 11392.

484 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 249 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	#19151497	C₂	1	2	yes	maximal
2	#19151484	C₂	1	4	yes	maximal
3	#19151464	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₈₉	C₂	1	4	yes	maximal
5	C₄⋊C₄×Dic₈₉	C₂	1	2	yes	maximal
6	C₄²×Dic₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
10	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
11	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
12	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₈₉	C₂²	1	4	yes	—
14	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
15	#15667919	C₂²	1	4	yes	—
16	#15667919	C₂²	1	4	yes	—
17	#15667919	C₂²	1	4	yes	—
18	#125×C₈₉	C₄	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	C₂²×Dic₈₉	C₂³	1	4	yes	—
23	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₈₉	C₂³	1	2	yes	—
25	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₈₉	—	2	4	no	—
27	C₄⋊Dic₈₉	—	2	8	no	—
28	C₄×Dic₈₉	—	2	2	no	—
29	C₄×Dic₈₉	—	2	4	no	—
30	C₄×Dic₈₉	—	2	4	no	—
31	C₄×Dic₈₉	—	2	8	no	—
32	C₄×Dic₈₉	Q₈	1	4	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
34	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
35	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₈₉	C₂×C₄	1	16	yes	—
37	C₄⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
38	C₄²×C₈₉	C₂×C₄	1	4	yes	—
39	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	2	4	no	—
42	C₂×C₄×C₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	—	2	4	no	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₈₉	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₈₉	#41	1	2	yes	—
49	C₂×Dic₈₉	—	2	2	no	—
50	C₂×Dic₈₉	—	2	2	no	—
51	C₂×Dic₈₉	—	2	2	no	—
52	C₂×Dic₈₉	—	2	4	no	—
53	C₂×Dic₈₉	—	2	4	no	—
54	C₂×Dic₈₉	—	2	4	no	—
55	C₂×Dic₈₉	C₂×Q₈	1	2	yes	—
56	C₂×Dic₈₉	C₂×Q₈	1	2	yes	—
57	C₂×Dic₈₉	#41	1	4	yes	—
58	C₂×Dic₈₉	#41	1	4	yes	—
59	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
62	C₂²×C₈₉	C₂×#41	1	2	yes	—
63	C₂²×C₈₉	C₂×#41	1	2	yes	—
64	C₄×C₈₉	—	2	4	no	—
65	C₄×C₈₉	—	2	8	no	—
66	C₄×C₈₉	C₄×Q₈	1	4	yes	—
67	Dic₈₉	—	2	4	no	—
68	Dic₈₉	—	2	2	no	—
69	Dic₈₉	—	4	4	no	—
70	Dic₈₉	#125	1	4	yes	direct product factor
71	C₂×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
72	C₂×C₈₉	C₄×#41	1	2	yes	—
73	C₂×C₈₉	C₂×#125	1	1	yes	characteristic
74	C₂×C₈₉	#529	1	2	yes	—
75	C₂×C₈₉	#533	1	1	yes	characteristic
76	C₄×#125	—	89	1	no	maximal, Sylow
77	C₈₉	C₄×#125	1	1	yes	characteristic, complemented, Sylow
78	C₂×#125	D₈₉	1	1	yes	characteristic
79	#389	—	89	2	no	—
80	#387	—	89	4	no	—
81	#382	—	89	1	no	—
82	C₄×C₄⋊C₄	—	89	4	no	—
83	C₄×C₄⋊C₄	—	89	2	no	—
84	C₄³	—	89	1	no	—
85	#125	Dic₈₉	1	8	yes	direct product factor
86	C₂×C₄⋊C₄	—	89	2	no	—
87	C₂×C₄⋊C₄	—	89	4	no	—
88	C₂×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
89	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
90	C₂×C₄²	—	89	1	no	—
91	C₂×C₄²	—	89	2	no	—
92	C₂×C₄²	—	89	2	no	—
93	C₂×C₄²	—	89	1	no	—
94	C₂×C₄²	—	89	4	no	—
95	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
96	#95	—	89	4	no	—
97	#95	—	89	4	no	—
98	#95	—	89	4	no	—
99	C₂²×C₄	—	89	1	no	—
100	C₂²×C₄	—	89	1	no	—
101	C₂²×C₄	—	89	2	no	—
102	C₂²×C₄	—	89	4	no	—
103	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
104	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
105	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
106	C₄⋊C₄	—	178	8	no	—
107	C₄⋊C₄	—	178	4	no	—
108	C₄⋊C₄	C₂×Dic₈₉	1	16	yes	—
109	C₄⋊C₄	C₂×Dic₈₉	1	8	yes	—
110	C₄²	—	89	4	no	—
111	C₄²	—	178	8	no	—
112	C₄²	—	178	4	no	—
113	C₄²	—	178	4	no	—
114	C₄²	—	178	2	no	—
115	C₄²	C₂×Dic₈₉	1	4	yes	—
116	C₂³	C₂³×D₈₉	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	89	2	no	—
122	C₂×C₄	—	89	4	no	—
123	C₂×C₄	—	89	2	no	—
124	C₂×C₄	—	89	4	no	—
125	C₂×C₄	—	178	4	no	—
126	C₂×C₄	—	178	4	no	—
127	C₂×C₄	—	178	4	no	—
128	C₂×C₄	—	178	2	no	—
129	C₂×C₄	—	178	2	no	—
130	C₂×C₄	—	178	2	no	—
131	C₂×C₄	C₂²×Dic₈₉	1	4	yes	—
132	C₂×C₄	C₂²×Dic₈₉	1	8	yes	—
133	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
134	C₂×C₄	Q₈×D₈₉	1	2	yes	—
135	C₂×C₄	#13083575	1	2	yes	—
136	C₂²	C₂³×Dic₈₉	1	1	yes	characteristic
137	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
138	C₂²	#41×D₈₉	1	2	yes	—
139	C₂²	#41×D₈₉	1	2	yes	—
140	C₂²	C₂×#13083575	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	8	no	—
143	C₄	—	89	4	no	—
144	C₄	—	356	4	no	—
145	C₄	—	178	4	no	—
146	C₄	—	178	2	no	—
147	C₄	Q₈×Dic₈₉	1	4	yes	—
148	C₂	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
149	C₂	#41×Dic₈₉	1	2	yes	—
150	C₂	#125×D₈₉	1	1	yes	characteristic
151	C₂	#19152133	1	2	yes	—
152	C₂	#19152154	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151497

C₂

yes

maximal

#19151484

C₂

yes

maximal

#19151464

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₈₉

C₂

yes

maximal

C₄⋊C₄×Dic₈₉

C₂

yes

maximal

C₄²×Dic₈₉

C₂

yes

maximal, characteristic

C₂×#125×C₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#125×C₈₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×Q₈

yes

—

C₂×Dic₈₉

C₂×Q₈

yes

—

C₂×Dic₈₉

#41

yes

—

C₂×Dic₈₉

#41

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

C₂×#41

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

#125

yes

direct product factor

C₂×C₈₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₈₉

C₄×#41

yes

—

C₂×C₈₉

C₂×#125

yes

characteristic

C₂×C₈₉

#529

yes

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×#125

D₈₉

yes

—

—

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

—

#125

Dic₈₉

yes

direct product factor

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

—

—

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×D₈₉

yes

—

104

C₂²×C₄

C₂²×D₈₉

yes

—

105

C₂²×C₄

C₂²×D₈₉

yes

characteristic

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

—

178

—

108

C₄⋊C₄

C₂×Dic₈₉

yes

—

109

C₄⋊C₄

C₂×Dic₈₉

yes

—

110

C₄²

—

111

C₄²

—

178

—

112

C₄²

—

178

—

113

C₄²

—

178

—

114

C₄²

—

178

—

115

C₄²

C₂×Dic₈₉

yes

—

116

C₂³

C₂³×D₈₉

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

178

—

126

C₂×C₄

—

178

—

127

C₂×C₄

—

178

—

128

C₂×C₄

—

178

—

129

C₂×C₄

—

178

—

130

C₂×C₄

—

178

—

131

C₂×C₄

C₂²×Dic₈₉

yes

—

132

C₂×C₄

C₂²×Dic₈₉

yes

—

133

C₂×C₄

C₂²×Dic₈₉

yes

—

134

C₂×C₄

Q₈×D₈₉

yes

—

135

C₂×C₄

#13083575

yes

—

136

C₂²

C₂³×Dic₈₉

yes

characteristic

137

C₂²

C₂×Q₈×D₈₉

yes

characteristic

138

C₂²

#41×D₈₉

yes

—

139

C₂²

#41×D₈₉

yes

—

140

C₂²

C₂×#13083575

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

356

—

145

C₄

—

178

—

146

C₄

—

178

—

147

C₄

Q₈×Dic₈₉

yes

—

148

C₂

C₂×Q₈×Dic₈₉

yes

characteristic

149

C₂

#41×Dic₈₉

yes

—

150

C₂

#125×D₈₉

yes

characteristic

151

C₂

#19152133

yes

—

152

C₂

#19152154

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂³×C₄
Fitting	C₂×#125×C₈₉	C₂
socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
hypercenter	C₂×#125	D₈₉
hypocenter	C₈₉	C₄×#125
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator

C₂²×C₈₉

C₂³×C₄

Fitting

C₂×#125×C₈₉

C₂

socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
18. #125×C₈₉	C₄	semidirect
70. Dic₈₉	#125	direct (N × H)
77. C₈₉	C₄×#125	semidirect
85. #125	Dic₈₉	direct (N × H)

H ≅ G/N

type

18. #125×C₈₉

C₄

semidirect

70. Dic₈₉

#125

direct (N × H)

77. C₈₉

C₄×#125

semidirect

85. #125

Dic₈₉

direct (N × H)

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

subgroup	index
#19151497	2
#19151484	2
#19151464	2
C₄⋊C₄×Dic₈₉	2
C₄²×Dic₈₉	2
C₂×#125×C₈₉	2
C₄×#125	89

subgroup

index

#19151497

#19151484

#19151464

C₄⋊C₄×Dic₈₉

C₄²×Dic₈₉

C₂×#125×C₈₉

C₄×#125

Subgroups — #125×Dic₈₉ #26022813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #125×Dic89 #26022813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #125×Dic₈₉ #26022813