Group #26021581 — subgroups

Subgroups — #26021581

Order 11392.

282 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 99 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×#15668012	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667946	C₂	1	2	yes	maximal
3	#19151496	C₂	1	1	yes	maximal, characteristic
4	#19151361	C₂	1	2	yes	maximal
5	#336×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	—	2	1	no	—
7	#15668012	—	2	4	no	—
8	C₂×C₄⋊Dic₈₉	—	2	1	no	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
12	C₂×#13083553	—	2	1	no	—
13	#15667946	—	2	4	no	—
14	#15667919	—	2	1	no	—
15	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	2	4	no	—
18	C₂×Dic₁₇₈	—	4	2	no	—
19	C₂×Dic₁₇₈	—	4	2	no	—
20	C₂²×Dic₈₉	—	2	2	no	—
21	C₂²×Dic₈₉	—	2	1	no	—
22	C₂²×Dic₈₉	—	2	1	no	—
23	C₄⋊Dic₈₉	—	2	4	no	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	C₄⋊Dic₈₉	—	2	1	no	—
26	C₄⋊Dic₈₉	—	2	2	no	—
27	C₄⋊Dic₈₉	D₄	1	4	yes	—
28	C₄⋊Dic₈₉	Q₈	1	2	yes	complemented
29	#13083553	—	4	2	no	—
30	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₂×C₈×C₈₉	—	2	2	no	—
35	C₂×C₈×C₈₉	Q₈	1	4	yes	complemented
36	C₄²×C₈₉	—	2	2	no	—
37	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈₉	—	2	1	no	—
41	C₂×C₄×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈₉	D₈	1	2	yes	—
43	C₂×C₄×C₈₉	Q₁₆	1	2	yes	—
44	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
48	C₂×Dic₈₉	—	2	4	no	—
49	C₂×Dic₈₉	—	2	2	no	—
50	C₂×Dic₈₉	—	4	1	no	—
51	C₂×Dic₈₉	—	4	2	no	—
52	C₂×Dic₈₉	—	4	2	no	—
53	C₂×Dic₈₉	—	4	1	no	—
54	C₂×Dic₈₉	—	4	1	no	—
55	C₂×Dic₈₉	—	4	1	no	—
56	C₂×Dic₈₉	—	4	1	no	—
57	Dic₁₇₈	—	4	8	no	—
58	C₈×C₈₉	—	2	4	no	—
59	C₂²×C₈₉	#120	1	1	yes	characteristic
60	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₈₉	#137	1	1	yes	characteristic
66	C₄×C₈₉	—	2	2	no	—
67	C₄×C₈₉	—	2	2	no	—
68	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
69	C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
70	Dic₈₉	—	4	4	no	—
71	Dic₈₉	—	4	2	no	—
72	Dic₈₉	—	8	1	no	—
73	C₂×C₈₉	#395	1	1	yes	characteristic
74	C₂×C₈₉	#447	1	1	yes	characteristic
75	C₂×C₈₉	#451	1	1	yes	characteristic
76	C₂×C₈₉	#474	1	1	yes	characteristic
77	C₂×C₈₉	#477	1	1	yes	characteristic
78	C₂×C₈₉	#500	1	1	yes	characteristic
79	C₂×C₈₉	#501	1	1	yes	characteristic
80	#2057	—	89	1	no	maximal, Sylow
81	C₈₉	#2057	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	—	89	1	no	—
83	C₂×#107	—	89	2	no	—
84	#389	—	89	1	no	—
85	#340	—	89	2	no	—
86	#336	D₈₉	1	1	yes	characteristic
87	C₂²×Q₈	—	178	1	no	—
88	C₂²×C₈	C₂×D₈₉	1	2	yes	—
89	C₄⋊Q₈	—	178	4	no	—
90	C₂×C₄⋊C₄	—	89	2	no	—
91	C₂×C₄⋊C₄	—	89	1	no	—
92	C₂×C₄⋊C₄	—	89	1	no	—
93	C₂×C₄⋊C₄	—	178	1	no	—
94	C₂×C₄⋊C₄	—	178	1	no	—
95	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
96	#107	—	178	4	no	—
97	#95	—	178	1	no	—
98	C₂×Q₈	—	356	2	no	—
99	C₂×Q₈	—	356	2	no	—
100	C₂×Q₈	—	178	4	no	—
101	C₂²×C₄	—	178	1	no	—
102	C₂²×C₄	—	178	1	no	—
103	C₂²×C₄	—	178	2	no	—
104	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
105	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	Dic₁₇₈	1	4	yes	complemented
110	C₄⋊C₄	—	89	4	no	—
111	C₄⋊C₄	—	89	2	no	—
112	C₄⋊C₄	—	356	2	no	—
113	C₄⋊C₄	—	178	4	no	—
114	C₄⋊C₄	—	178	2	no	—
115	C₄⋊C₄	—	178	2	no	—
116	C₄⋊C₄	—	178	1	no	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
119	Q₈	—	356	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	356	1	no	—
125	C₂×C₄	—	356	2	no	—
126	C₂×C₄	—	356	2	no	—
127	C₂×C₄	—	356	1	no	—
128	C₂×C₄	—	356	1	no	—
129	C₂×C₄	—	356	1	no	—
130	C₂×C₄	—	356	1	no	—
131	C₂×C₄	—	178	4	no	—
132	C₂×C₄	—	178	2	no	—
133	C₂×C₄	D₇₁₂	1	2	yes	—
134	C₂×C₄	Dic₃₅₆	1	2	yes	—
135	C₂×C₄	D₄×D₈₉	1	2	yes	—
136	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
137	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
138	C₂×C₄	#13083573	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₂²	#15667982	1	1	yes	characteristic
141	C₂²	#15667992	1	1	yes	characteristic
142	C₂²	#15668012	1	1	yes	characteristic
143	C₂²	C₂×D₇₁₂	1	1	yes	characteristic
144	C₂²	C₂×Dic₃₅₆	1	1	yes	characteristic
145	C₂²	#15668040	1	1	yes	characteristic
146	C₂²	#15668045	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	356	2	no	—
150	C₄	—	356	4	no	—
151	C₄	—	712	1	no	—
152	C₄	#15667989	1	2	yes	—
153	C₄	#15668011	1	2	yes	—
154	C₂	#19151519	1	1	yes	characteristic
155	C₂	#19151734	1	1	yes	characteristic
156	C₂	#19151756	1	1	yes	characteristic
157	C₂	#19151879	1	1	yes	characteristic
158	C₂	#19151894	1	1	yes	characteristic
159	C₂	#19152003	1	1	yes	characteristic
160	C₂	#19152007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic

C₂×#15667946

C₂

yes

maximal

#19151496

C₂

yes

maximal, characteristic

#19151361

C₂

yes

maximal

#336×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

—

#15668012

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

—

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

C₄⋊Dic₈₉

Q₈

yes

complemented

#13083553

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

Q₈

yes

complemented

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₈

yes

—

C₂×C₄×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₉

C₄⋊Q₈

yes

characteristic

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₂²×C₈₉

C₂×Q₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

C₄⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

178

—

C₂²×C₈

C₂×D₈₉

yes

—

C₄⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#107

—

178

—

#95

—

178

—

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

100

C₂×Q₈

—

178

—

101

C₂²×C₄

—

178

—

102

C₂²×C₄

—

178

—

103

C₂²×C₄

—

178

—

104

C₂²×C₄

D₃₅₆

yes

characteristic

105

C₂²×C₄

D₃₅₆

yes

characteristic

106

C₂²×C₄

C₂²×D₈₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

Dic₁₇₈

yes

complemented

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

356

—

113

C₄⋊C₄

—

178

—

114

C₄⋊C₄

—

178

—

115

C₄⋊C₄

—

178

—

116

C₄⋊C₄

—

178

—

117

C₄²

—

118

C₂³

C₂×D₃₅₆

yes

characteristic

119

Q₈

—

356

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

356

—

125

C₂×C₄

—

356

—

126

C₂×C₄

—

356

—

127

C₂×C₄

—

356

—

128

C₂×C₄

—

356

—

129

C₂×C₄

—

356

—

130

C₂×C₄

—

356

—

131

C₂×C₄

—

178

—

132

C₂×C₄

—

178

—

133

C₂×C₄

D₇₁₂

yes

—

134

C₂×C₄

Dic₃₅₆

yes

—

135

C₂×C₄

D₄×D₈₉

yes

—

136

C₂×C₄

Q₈×D₈₉

yes

characteristic

137

C₂×C₄

C₂×Dic₁₇₈

yes

—

138

C₂×C₄

#13083573

yes

characteristic

139

C₈

—

140

C₂²

#15667982

yes

characteristic

141

C₂²

#15667992

yes

characteristic

142

C₂²

#15668012

yes

characteristic

143

C₂²

C₂×D₇₁₂

yes

characteristic

144

C₂²

C₂×Dic₃₅₆

yes

characteristic

145

C₂²

#15668040

yes

characteristic

146

C₂²

#15668045

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

356

—

150

C₄

—

356

—

151

C₄

—

712

—

152

C₄

#15667989

yes

—

153

C₄

#15668011

yes

—

154

C₂

#19151519

yes

characteristic

155

C₂

#19151734

yes

characteristic

156

C₂

#19151756

yes

characteristic

157

C₂

#19151879

yes

characteristic

158

C₂

#19151894

yes

characteristic

159

C₂

#19152003

yes

characteristic

160

C₂

#19152007

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#336×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#336	D₈₉
hypocenter	C₈₉	#2057
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#336×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
28. C₄⋊Dic₈₉	Q₈	semidirect
35. C₂×C₈×C₈₉	Q₈	semidirect
81. C₈₉	#2057	semidirect
109. C₂×C₈	Dic₁₇₈	semidirect

H ≅ G/N

type

28. C₄⋊Dic₈₉

Q₈

semidirect

35. C₂×C₈×C₈₉

Q₈

semidirect

81. C₈₉

#2057

semidirect

109. C₂×C₈

Dic₁₇₈

semidirect

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

subgroup	index
C₂×#15668012	2
C₂×#15667946	2
#19151496	2
#19151361	2
#336×C₈₉	2
#2057	89

subgroup

index

#336×C₈₉

Subgroups — #26021581

Conjugacy classes of subgroups

Complements

Maximal subgroups