Group #26028814 — subgroups

Subgroups — #26028814

Order 11392.

484 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 213 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×M₁₆×D₈₉	C₂	1	2	yes	maximal
2	C₂×#15667969	C₂	1	1	yes	maximal, characteristic
3	#19151623	C₂	1	8	yes	maximal, complemented
4	#19151622	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151621	C₂	1	2	yes	maximal, complemented
6	#423×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
8	C₂×#13083558	C₂²	1	2	yes	—
9	C₂×#13083557	C₂²	1	4	yes	—
10	M₁₆×D₈₉	—	2	8	no	—
11	C₂×C₈×D₈₉	C₂²	1	4	yes	—
12	#15667969	C₄	1	4	yes	—
13	#15667969	C₂²	1	4	yes	—
14	C₂×#13083553	C₄	1	2	yes	—
15	C₂×#13083551	C₄	1	2	yes	—
16	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
17	C₄⋊#10623494	C₂²	1	4	yes	complemented
18	#15667941	C₂²	1	8	yes	complemented
19	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₈₉	C₂²	1	4	yes	complemented
22	C₂³×D₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₈₉	D₄	1	4	yes	—
27	C₂×C₄×D₈₉	Q₈	1	4	yes	—
28	C₂×C₄×D₈₉	C₂³	1	2	yes	—
29	#13083558	—	2	4	no	—
30	#13083557	—	2	8	no	—
31	C₂×#10623494	C₂³	1	4	yes	—
32	C₈×D₈₉	—	2	8	no	—
33	#13083553	C₂×C₄	1	8	yes	—
34	#13083551	C₂×C₄	1	8	yes	—
35	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
36	C₄×Dic₈₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	M₁₆×C₈₉	—	2	4	no	—
40	C₂×C₈×C₈₉	C₂³	1	4	yes	—
41	C₄²×C₈₉	C₂×C₄	1	2	yes	—
42	C₄²×C₈₉	C₂³	1	2	yes	—
43	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×D₈₉	—	4	2	no	—
45	C₂²×D₈₉	C₄⋊C₄	1	4	yes	—
46	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₈₉	—	4	2	no	—
48	C₂×C₄×C₈₉	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₈₉	—	4	2	no	—
54	C₂×Dic₈₉	C₄⋊C₄	1	4	yes	—
55	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
56	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
57	C₄×D₈₉	C₄⋊C₄	1	8	yes	—
58	C₄×D₈₉	C₂×D₄	1	4	yes	—
59	C₄×D₈₉	C₂×Q₈	1	4	yes	—
60	C₈×C₈₉	—	2	4	no	—
61	#10623494	—	2	4	no	—
62	C₂²×C₈₉	—	2	2	no	—
63	C₂²×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
64	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₈₉	—	2	4	no	—
66	C₂×D₈₉	—	4	4	no	—
67	C₂×D₈₉	C₂×C₄⋊C₄	1	4	yes	—
68	C₄×C₈₉	—	2	4	no	—
69	C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
70	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
72	Dic₈₉	—	2	4	no	—
73	Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
74	C₂×C₈₉	—	2	2	no	—
75	C₂×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₈₉	#568	1	2	yes	—
77	D₈₉	—	2	4	no	—
78	#2905	—	89	1	no	maximal, Sylow
79	C₈₉	#2905	1	1	yes	characteristic, complemented, Sylow
80	C₂²×M₁₆	—	89	2	no	—
81	C₂×#117	—	89	1	no	—
82	#424	—	89	8	no	—
83	#423	—	89	2	no	—
84	#423	—	89	1	no	—
85	#423	D₈₉	1	1	yes	characteristic, complemented
86	C₂³×C₄	—	89	1	no	—
87	C₂×M₁₆	—	89	2	no	—
88	C₂×M₁₆	—	89	4	no	—
89	C₂×M₁₆	—	178	8	no	—
90	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
91	C₂²×C₈	—	89	4	no	—
92	#117	—	89	4	no	—
93	#117	—	89	4	no	—
94	C₂×C₄⋊C₄	—	89	2	no	—
95	C₂×C₂²⋊C₄	—	89	2	no	—
96	C₂×C₄²	—	89	1	no	—
97	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
98	C₄⋊C₈	—	89	8	no	—
99	C₄⋊C₈	—	89	4	no	—
100	C₄⋊C₈	C₂×D₈₉	1	4	yes	complemented
101	C₂⁴	—	89	1	no	—
102	C₂²×C₄	—	89	1	no	—
103	C₂²×C₄	—	89	2	no	—
104	C₂²×C₄	—	89	4	no	—
105	C₂²×C₄	—	89	4	no	—
106	C₂²×C₄	—	89	2	no	—
107	C₂²×C₄	—	89	2	no	—
108	C₂²×C₄	C₄×D₈₉	1	2	yes	—
109	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	M₁₆	—	178	8	no	—
112	M₁₆	—	178	4	no	—
113	C₂×C₈	—	89	4	no	—
114	C₂×C₈	—	178	8	no	—
115	C₂×C₈	C₂²×D₈₉	1	4	yes	—
116	C₄⋊C₄	—	89	8	no	—
117	C₂²⋊C₄	—	89	8	no	—
118	C₄²	—	89	2	no	—
119	C₄²	—	89	2	no	—
120	C₄²	C₄×D₈₉	1	2	yes	—
121	C₄²	C₂²×D₈₉	1	2	yes	—
122	C₂³	—	89	4	no	—
123	C₂³	—	89	2	no	—
124	C₂³	—	356	2	no	—
125	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	89	2	no	—
128	C₂×C₄	—	89	4	no	—
129	C₂×C₄	—	89	4	no	—
130	C₂×C₄	—	89	4	no	—
131	C₂×C₄	—	89	8	no	—
132	C₂×C₄	—	89	4	no	—
133	C₂×C₄	—	356	2	no	—
134	C₂×C₄	C₂×C₄×D₈₉	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
136	C₂×C₄	D₄×D₈₉	1	2	yes	—
137	C₂×C₄	Q₈×D₈₉	1	2	yes	—
138	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	178	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	89	4	no	—
143	C₂²	—	356	4	no	—
144	C₂²	—	178	4	no	—
145	C₂²	C₄⋊C₄×D₈₉	1	2	yes	—
146	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	89	4	no	—
149	C₄	—	178	4	no	—
150	C₄	C₄⋊C₄×D₈₉	1	2	yes	—
151	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
152	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	178	4	no	—
155	C₂	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
156	C₂	#19152364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×D₈₉

C₂

yes

maximal

C₂×#15667969

C₂

yes

maximal, characteristic

#19151623

C₂

yes

maximal, complemented

#19151622

C₂

yes

maximal, characteristic, complemented

#19151621

C₂

yes

maximal, complemented

#423×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₈₉

C₂²

yes

characteristic

C₂×#13083558

C₂²

yes

—

C₂×#13083557

C₂²

yes

—

M₁₆×D₈₉

—

C₂×C₈×D₈₉

C₂²

yes

—

#15667969

C₄

yes

—

#15667969

C₂²

yes

—

C₂×#13083553

C₄

yes

—

C₂×#13083551

C₄

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

complemented

#15667941

C₂²

yes

complemented

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

complemented

C₂³×D₈₉

C₂×C₄

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

characteristic

C₂×C₄×D₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

D₄

yes

—

C₂×C₄×D₈₉

Q₈

yes

—

C₂×C₄×D₈₉

C₂³

yes

—

#13083558

—

#13083557

—

C₂×#10623494

C₂³

yes

—

C₈×D₈₉

—

#13083553

C₂×C₄

yes

—

#13083551

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

C₄⋊C₄

yes

—

C₂²×D₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

C₂×Dic₈₉

C₂²×C₄

yes

—

C₂×Dic₈₉

C₂²×C₄

yes

—

C₄×D₈₉

C₄⋊C₄

yes

—

C₄×D₈₉

C₂×D₄

yes

—

C₄×D₈₉

C₂×Q₈

yes

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₂×C₄⋊C₄

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₄×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

C₂²×Q₈

yes

characteristic

Dic₈₉

—

Dic₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₈₉

#568

yes

—

D₈₉

—

#2905

—

maximal, Sylow

C₈₉

#2905

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

178

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

C₂×D₈₉

yes

complemented

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₈₉

yes

—

109

C₂²×C₄

C₂²×D₈₉

yes

characteristic

110

M₁₆

—

111

M₁₆

—

178

—

112

M₁₆

—

178

—

113

C₂×C₈

—

114

C₂×C₈

—

178

—

115

C₂×C₈

C₂²×D₈₉

yes

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₈₉

yes

—

121

C₄²

C₂²×D₈₉

yes

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

356

—

125

C₂³

C₂×C₄×D₈₉

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

356

—

134

C₂×C₄

C₂×C₄×D₈₉

yes

—

135

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

136

C₂×C₄

D₄×D₈₉

yes

—

137

C₂×C₄

Q₈×D₈₉

yes

—

138

C₂×C₄

C₂³×D₈₉

yes

characteristic

139

C₈

—

140

C₈

—

178

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

356

—

144

C₂²

—

178

—

145

C₂²

C₄⋊C₄×D₈₉

yes

—

146

C₂²

C₂²×C₄×D₈₉

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

178

—

150

C₄

C₄⋊C₄×D₈₉

yes

—

151

C₄

C₂×D₄×D₈₉

yes

characteristic

152

C₄

C₂×Q₈×D₈₉

yes

characteristic

153

C₂

—

154

C₂

—

178

—

155

C₂

C₂×C₄⋊C₄×D₈₉

yes

characteristic

156

C₂

#19152364

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#423×C₈₉	C₂
prime radical	C₂³×D₈₉	C₂×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#423	D₈₉
hypocenter	C₈₉	#2905
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

#423×C₈₉

C₂

prime radical

C₂³×D₈₉

C₂×C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
3. #19151623	C₂	semidirect
4. #19151622	C₂	semidirect
5. #19151621	C₂	semidirect
6. #423×C₈₉	C₂	semidirect
17. C₄⋊#10623494	C₂²	semidirect
18. #15667941	C₂²	semidirect
21. C₄⋊C₈×C₈₉	C₂²	semidirect
79. C₈₉	#2905	semidirect
85. #423	D₈₉	semidirect
100. C₄⋊C₈	C₂×D₈₉	semidirect

H ≅ G/N

type

3. #19151623

C₂

semidirect

4. #19151622

C₂

semidirect

5. #19151621

C₂

semidirect

6. #423×C₈₉

C₂

semidirect

17. C₄⋊#10623494

C₂²

semidirect

18. #15667941

C₂²

semidirect

21. C₄⋊C₈×C₈₉

C₂²

semidirect

79. C₈₉

#2905

semidirect

85. #423

D₈₉

semidirect

100. C₄⋊C₈

C₂×D₈₉

semidirect

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

subgroup	index
C₂×M₁₆×D₈₉	2
C₂×#15667969	2
#19151623	2
#19151622	2
#19151621	2
#423×C₈₉	2
#2905	89

subgroup

index

C₂×M₁₆×D₈₉

#423×C₈₉

Subgroups — #26028814

Conjugacy classes of subgroups

Complements

Maximal subgroups