Group #26021662 — subgroups

Subgroups — #26021662

Order 11392.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#15667937	C₂	1	2	yes	maximal
2	#19151504	C₂	1	1	yes	maximal, characteristic
3	#19151468	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	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₈₉	—	2	1	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	#15667937	—	2	4	no	—
12	#15667919	—	2	1	no	—
13	#15667919	—	2	1	no	—
14	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₈	—	2	2	no	—
17	C₂×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₁₇₈	—	2	1	no	—
19	C₂×Dic₁₇₈	D₄	1	2	yes	—
20	C₂²×Dic₈₉	—	2	1	no	—
21	C₂²×Dic₈₉	—	2	1	no	—
22	C₂²×Dic₈₉	—	2	2	no	—
23	C₄⋊Dic₈₉	—	2	2	no	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	C₄⋊Dic₈₉	—	2	2	no	—
26	C₄⋊Dic₈₉	—	4	2	no	—
27	C₄⋊Dic₈₉	—	4	2	no	—
28	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₈₉	—	2	2	no	—
32	C₂×C₈×C₈₉	—	2	2	no	—
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₄²×C₈₉	—	2	2	no	—
35	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	—	2	1	no	—
37	C₂×C₄×C₈₉	—	2	1	no	—
38	C₂×C₄×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈₉	SD₁₆	1	2	yes	—
41	C₂×C₄×C₈₉	Q₁₆	1	2	yes	—
42	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	#41	1	2	yes	—
44	C₂×C₄×C₈₉	#41	1	2	yes	—
45	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
46	C₂×Dic₈₉	—	2	2	no	—
47	C₂×Dic₈₉	—	4	1	no	—
48	C₂×Dic₈₉	—	4	1	no	—
49	C₂×Dic₈₉	—	4	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	Dic₁₇₈	—	2	4	no	—
56	Dic₁₇₈	—	4	2	no	—
57	C₈×C₈₉	—	4	2	no	—
58	C₂²×C₈₉	#121	1	1	yes	characteristic
59	C₂²×C₈₉	#123	1	1	yes	characteristic
60	C₂²×C₈₉	#124	1	1	yes	characteristic
61	C₂²×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₈₉	#137	1	1	yes	characteristic
65	C₄×C₈₉	—	2	2	no	—
66	C₄×C₈₉	—	2	2	no	—
67	C₄×C₈₉	#123	1	2	yes	—
68	C₄×C₈₉	#126	1	2	yes	—
69	Dic₈₉	—	4	2	no	—
70	Dic₈₉	—	8	2	no	—
71	Dic₈₉	—	8	1	no	—
72	C₂×C₈₉	#397	1	1	yes	characteristic
73	C₂×C₈₉	#460	1	1	yes	characteristic
74	C₂×C₈₉	#462	1	1	yes	characteristic
75	C₂×C₈₉	#483	1	1	yes	characteristic
76	C₂×C₈₉	#484	1	1	yes	characteristic
77	C₂×C₈₉	#487	1	1	yes	characteristic
78	C₂×C₈₉	#489	1	1	yes	characteristic
79	#2072	—	89	1	no	maximal, Sylow
80	C₈₉	#2072	1	1	yes	characteristic, complemented, Sylow
81	C₂×#103	—	89	2	no	—
82	#391	—	89	1	no	—
83	#384	—	89	1	no	—
84	#340	—	89	2	no	—
85	#336	D₈₉	1	1	yes	characteristic
86	C₂²×Q₈	—	89	1	no	—
87	C₂²×C₈	C₂×D₈₉	1	2	yes	—
88	C₂×C₄⋊C₄	—	89	1	no	—
89	C₂×C₄⋊C₄	—	89	2	no	—
90	C₂×C₄⋊C₄	—	178	1	no	—
91	C₂×C₄⋊C₄	—	178	1	no	—
92	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
93	#103	—	178	4	no	—
94	#95	—	178	1	no	—
95	#95	—	178	1	no	—
96	C₂×Q₈	—	89	2	no	—
97	C₂×Q₈	—	178	2	no	—
98	C₂×Q₈	—	178	2	no	—
99	C₂×Q₈	—	178	1	no	—
100	C₂²×C₄	—	178	1	no	—
101	C₂²×C₄	—	178	2	no	—
102	C₂²×C₄	—	178	1	no	—
103	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
104	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₄⋊C₄	—	356	2	no	—
110	C₄⋊C₄	—	356	2	no	—
111	C₄⋊C₄	—	178	2	no	—
112	C₄⋊C₄	—	178	2	no	—
113	C₄⋊C₄	—	178	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
116	Q₈	—	356	2	no	—
117	Q₈	—	178	4	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	356	1	no	—
123	C₂×C₄	—	356	1	no	—
124	C₂×C₄	—	356	2	no	—
125	C₂×C₄	—	356	1	no	—
126	C₂×C₄	—	356	2	no	—
127	C₂×C₄	—	356	2	no	—
128	C₂×C₄	—	356	1	no	—
129	C₂×C₄	—	356	1	no	—
130	C₂×C₄	—	178	2	no	—
131	C₂×C₄	#13083563	1	2	yes	—
132	C₂×C₄	Dic₃₅₆	1	2	yes	—
133	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
134	C₂×C₄	#13083573	1	2	yes	—
135	C₂×C₄	#13083574	1	2	yes	—
136	C₂×C₄	#13083575	1	1	yes	characteristic
137	C₈	—	4	2	no	—
138	C₂²	#15667987	1	1	yes	characteristic
139	C₂²	#15667996	1	1	yes	characteristic
140	C₂²	#15668000	1	1	yes	characteristic
141	C₂²	C₂×#13083563	1	1	yes	characteristic
142	C₂²	C₂×Dic₃₅₆	1	1	yes	characteristic
143	C₂²	#15668040	1	1	yes	characteristic
144	C₂²	#15668045	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	356	2	no	—
148	C₄	—	712	1	no	—
149	C₄	—	712	2	no	—
150	C₄	#15667994	1	2	yes	—
151	C₄	#15668007	1	2	yes	—
152	C₂	#19151525	1	1	yes	characteristic
153	C₂	#19151799	1	1	yes	characteristic
154	C₂	#19151809	1	1	yes	characteristic
155	C₂	#19151929	1	1	yes	characteristic
156	C₂	#19151934	1	1	yes	characteristic
157	C₂	#19151949	1	1	yes	characteristic
158	C₂	#19151958	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667937

C₂

yes

maximal

#19151504

C₂

yes

maximal, characteristic

#19151468

C₂

yes

maximal, characteristic

#19151361

C₂

yes

maximal

#336×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667937

—

#15667919

—

#15667919

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

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

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

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

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

C₂×Q₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#123

yes

—

C₄×C₈₉

#126

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

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#103

—

178

—

#95

—

178

—

#95

—

178

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

100

C₂²×C₄

—

178

—

101

C₂²×C₄

—

178

—

102

C₂²×C₄

—

178

—

103

C₂²×C₄

D₃₅₆

yes

characteristic

104

C₂²×C₄

D₃₅₆

yes

characteristic

105

C₂²×C₄

C₂²×D₈₉

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

356

—

110

C₄⋊C₄

—

356

—

111

C₄⋊C₄

—

178

—

112

C₄⋊C₄

—

178

—

113

C₄⋊C₄

—

178

—

114

C₄²

—

115

C₂³

C₂×D₃₅₆

yes

characteristic

116

Q₈

—

356

—

117

Q₈

—

178

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

356

—

123

C₂×C₄

—

356

—

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

—

178

—

131

C₂×C₄

#13083563

yes

—

132

C₂×C₄

Dic₃₅₆

yes

—

133

C₂×C₄

D₄×D₈₉

yes

characteristic

134

C₂×C₄

#13083573

yes

—

135

C₂×C₄

#13083574

yes

—

136

C₂×C₄

#13083575

yes

characteristic

137

C₈

—

138

C₂²

#15667987

yes

characteristic

139

C₂²

#15667996

yes

characteristic

140

C₂²

#15668000

yes

characteristic

141

C₂²

C₂×#13083563

yes

characteristic

142

C₂²

C₂×Dic₃₅₆

yes

characteristic

143

C₂²

#15668040

yes

characteristic

144

C₂²

#15668045

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

356

—

148

C₄

—

712

—

149

C₄

—

712

—

150

C₄

#15667994

yes

—

151

C₄

#15668007

yes

—

152

C₂

#19151525

yes

characteristic

153

C₂

#19151799

yes

characteristic

154

C₂

#19151809

yes

characteristic

155

C₂

#19151929

yes

characteristic

156

C₂

#19151934

yes

characteristic

157

C₂

#19151949

yes

characteristic

158

C₂

#19151958

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	#2072
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
80. C₈₉	#2072	semidirect

H ≅ G/N

type

80. C₈₉

#2072

semidirect

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

subgroup	index
C₂×#15667937	2
#19151504	2
#19151468	2
#19151361	2
#336×C₈₉	2
#2072	89

subgroup

index

#336×C₈₉

Subgroups — #26021662

Conjugacy classes of subgroups

Complements

Maximal subgroups