Group #26020050 — subgroups

Subgroups — #101×Dic₈₉ #26020050

Order 11392.

296 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂×Q₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667931	C₂	1	1	yes	maximal, characteristic
3	M₁₆×Dic₈₉	C₂	1	2	yes	maximal
4	#19151366	C₂	1	2	yes	maximal
5	C₂×#101×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	Q₈×Dic₈₉	—	2	2	no	—
8	Q₈×Dic₈₉	—	2	2	no	—
9	C₂×C₄⋊Dic₈₉	C₄	1	2	yes	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₉	C₄	1	2	yes	—
12	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
13	#15667931	C₄	1	4	yes	complemented
14	#15667922	—	2	2	no	—
15	C₈×Dic₈₉	—	2	2	no	—
16	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
18	#101×C₈₉	C₄	1	4	yes	complemented
19	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
21	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
22	#13083558	—	2	2	no	—
23	#13083558	C₂×C₄	1	4	yes	—
24	C₂×#10623494	—	2	2	no	—
25	C₄⋊Dic₈₉	—	2	2	no	—
26	C₄⋊Dic₈₉	—	2	4	no	—
27	C₄×Dic₈₉	—	2	2	no	—
28	C₄×Dic₈₉	—	2	2	no	—
29	C₄×Dic₈₉	D₄	1	2	yes	—
30	C₄×Dic₈₉	D₄	1	2	yes	—
31	C₂×Q₈×C₈₉	—	2	2	no	—
32	C₂×Q₈×C₈₉	—	2	2	no	—
33	C₂×Q₈×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×Q₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	M₁₆×C₈₉	—	2	2	no	—
38	M₁₆×C₈₉	C₂×C₄	1	4	yes	—
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	Q₈×C₈₉	—	2	4	no	—
42	Q₈×C₈₉	—	2	4	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	C₄²	1	4	yes	—
46	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₈₉	#41	1	2	yes	—
50	C₂×Dic₈₉	—	2	2	no	—
51	C₂×Dic₈₉	—	2	2	no	—
52	C₂×Dic₈₉	—	4	1	no	—
53	C₂×Dic₈₉	—	4	2	no	—
54	C₂×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₈₉	C₂²⋊C₄	1	2	yes	—
56	C₂×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₈×C₈₉	—	2	4	no	—
58	#10623494	—	2	4	no	—
59	C₂²×C₈₉	—	2	1	no	—
60	C₂²×C₈₉	—	2	1	no	—
61	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
62	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₈₉	#117	1	1	yes	characteristic
64	C₄×C₈₉	—	2	4	no	—
65	C₄×C₈₉	C₄×D₄	1	2	yes	—
66	C₄×C₈₉	C₄×D₄	1	2	yes	—
67	Dic₈₉	—	2	1	no	—
68	Dic₈₉	—	2	2	no	—
69	Dic₈₉	—	4	2	no	—
70	Dic₈₉	#101	1	2	yes	direct product factor
71	C₂×C₈₉	—	2	1	no	—
72	C₂×C₈₉	—	2	1	no	—
73	C₂×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈₉	C₂×#101	1	1	yes	characteristic
75	C₂×C₈₉	#413	1	1	yes	characteristic
76	C₄×#101	—	89	1	no	maximal, Sylow
77	C₈₉	C₄×#101	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₄×Q₈	—	89	1	no	—
79	C₂×#101	—	89	1	no	—
80	C₂×#101	D₈₉	1	1	yes	characteristic
81	C₄×M₁₆	—	89	2	no	—
82	#343	—	89	2	no	—
83	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
84	C₂×M₁₆	—	89	2	no	—
85	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
86	C₄×Q₈	—	178	2	no	—
87	C₄×Q₈	—	178	2	no	—
88	C₂×C₄⋊C₄	—	89	2	no	—
89	C₂×C₄⋊C₄	—	89	1	no	—
90	C₂×C₄²	—	89	1	no	—
91	C₂×C₄²	—	89	2	no	—
92	#101	—	89	4	no	—
93	#101	Dic₈₉	1	4	yes	direct product factor
94	#97	—	178	2	no	—
95	C₄×C₈	—	178	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₄×D₈₉	1	2	yes	—
99	C₂×Q₈	C₂×Dic₈₉	1	2	yes	—
100	C₂²×C₄	—	89	1	no	—
101	C₂²×C₄	—	89	1	no	—
102	C₂²×C₄	—	89	2	no	—
103	C₂²×C₄	C₄×D₈₉	1	2	yes	—
104	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	89	4	no	—
107	M₁₆	—	178	2	no	—
108	M₁₆	C₂×Dic₈₉	1	4	yes	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	178	2	no	—
111	C₄⋊C₄	—	178	4	no	—
112	C₄⋊C₄	—	178	2	no	—
113	C₄²	—	89	2	no	—
114	C₄²	—	89	2	no	—
115	C₄²	—	178	2	no	—
116	C₄²	—	178	2	no	—
117	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	89	1	no	—
123	C₂×C₄	—	89	1	no	—
124	C₂×C₄	—	89	2	no	—
125	C₂×C₄	—	356	2	no	—
126	C₂×C₄	—	356	1	no	—
127	C₂×C₄	—	178	2	no	—
128	C₂×C₄	—	178	2	no	—
129	C₂×C₄	C₄×Dic₈₉	1	4	yes	—
130	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
131	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
132	C₂×C₄	D₄×D₈₉	1	2	yes	—
133	C₂×C₄	#13083573	1	2	yes	—
134	C₈	—	2	4	no	—
135	C₈	—	178	4	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	C₂×C₄×Dic₈₉	1	1	yes	characteristic
139	C₂²	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
140	C₂²	#15667968	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	89	2	no	—
143	C₄	—	356	2	no	—
144	C₄	—	178	1	no	—
145	C₄	—	178	2	no	—
146	C₄	#15667972	1	2	yes	—
147	C₄	D₄×Dic₈₉	1	2	yes	—
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	C₂²⋊C₄×Dic₈₉	1	1	yes	characteristic
151	C₂	#101×D₈₉	1	1	yes	characteristic
152	C₂	#19151581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₈₉

C₂

yes

maximal, characteristic

C₂×#15667931

C₂

yes

maximal, characteristic

M₁₆×Dic₈₉

C₂

yes

maximal

#19151366

C₂

yes

maximal

C₂×#101×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

Q₈×Dic₈₉

—

Q₈×Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₄

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₄

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667931

C₄

yes

complemented

#15667922

—

C₈×Dic₈₉

—

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

—

#101×C₈₉

C₄

yes

complemented

C₂²×Dic₈₉

C₂×C₄

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

characteristic

#13083558

—

#13083558

C₂×C₄

yes

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

—

C₄×Dic₈₉

D₄

yes

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₄

yes

—

C₂×Q₈×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄²

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₈₉

C₂²⋊C₄

yes

—

C₂×Dic₈₉

C₂²⋊C₄

yes

characteristic

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄²

yes

characteristic

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

#117

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

C₄×D₄

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

#101

yes

direct product factor

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₄×C₂²⋊C₄

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₂×#101

—

C₂×#101

D₈₉

yes

characteristic

C₄×M₁₆

—

#343

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₄×Q₈

—

178

—

C₄×Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

#101

—

#101

Dic₈₉

yes

direct product factor

#97

—

178

—

C₄×C₈

—

178

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄×D₈₉

yes

—

C₂×Q₈

C₂×Dic₈₉

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₄×D₈₉

yes

—

104

C₂²×C₄

C₂²×D₈₉

yes

characteristic

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

178

—

108

M₁₆

C₂×Dic₈₉

yes

—

109

C₂×C₈

—

110

C₂×C₈

—

178

—

111

C₄⋊C₄

—

178

—

112

C₄⋊C₄

—

178

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

178

—

116

C₄²

—

178

—

117

C₂³

C₂×C₄×D₈₉

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

356

—

126

C₂×C₄

—

356

—

127

C₂×C₄

—

178

—

128

C₂×C₄

—

178

—

129

C₂×C₄

C₄×Dic₈₉

yes

—

130

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

131

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

132

C₂×C₄

D₄×D₈₉

yes

—

133

C₂×C₄

#13083573

yes

—

134

C₈

—

135

C₈

—

178

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄×Dic₈₉

yes

characteristic

139

C₂²

C₂²⋊C₄×D₈₉

yes

characteristic

140

C₂²

#15667968

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

356

—

144

C₄

—

178

—

145

C₄

—

178

—

146

C₄

#15667972

yes

—

147

C₄

D₄×Dic₈₉

yes

—

148

C₂

—

149

C₂

—

150

C₂

C₂²⋊C₄×Dic₈₉

yes

characteristic

151

C₂

#101×D₈₉

yes

characteristic

152

C₂

#19151581

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₈₉
commutator	C₂²×C₈₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×#101×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×#101	D₈₉
hypocenter	C₈₉	C₄×#101
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₈₉

commutator

C₂²×C₈₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×#101×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
13. #15667931	C₄	semidirect
18. #101×C₈₉	C₄	semidirect
70. Dic₈₉	#101	direct (N × H)
77. C₈₉	C₄×#101	semidirect
93. #101	Dic₈₉	direct (N × H)

H ≅ G/N

type

13. #15667931

C₄

semidirect

18. #101×C₈₉

C₄

semidirect

70. Dic₈₉

#101

direct (N × H)

77. C₈₉

C₄×#101

semidirect

93. #101

Dic₈₉

direct (N × H)

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

subgroup	index
C₂×Q₈×Dic₈₉	2
C₂×#15667931	2
M₁₆×Dic₈₉	2
#19151366	2
C₂×#101×C₈₉	2
C₄×#101	89

subgroup

index

C₂×Q₈×Dic₈₉

C₂×#15667931

M₁₆×Dic₈₉

#19151366

C₂×#101×C₈₉

C₄×#101

Subgroups — #101×Dic₈₉ #26020050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #101×Dic89 #26020050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #101×Dic₈₉ #26020050