Group #26021354 — subgroups

Subgroups — #26021354

Order 11392.

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

normal subgroups only

This group has 370 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	C₂²×Dic₃₅₆	C₂	1	2	yes	maximal
2	C₂×#15668012	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667937	C₂	1	2	yes	maximal
4	#19151504	C₂	1	1	yes	maximal, characteristic
5	#336×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
8	C₂×Dic₃₅₆	—	2	4	no	—
9	C₂×Dic₃₅₆	—	2	4	no	—
10	C₂×Dic₃₅₆	—	2	4	no	—
11	#15668012	—	2	4	no	—
12	C₂×C₄⋊Dic₈₉	—	2	2	no	—
13	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
14	#15667937	—	2	4	no	—
15	#15667919	—	2	2	no	—
16	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₇₈	—	2	2	no	—
19	C₂×Dic₁₇₈	—	2	2	no	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	—	2	2	no	—
22	C₂×Dic₁₇₈	—	2	2	no	—
23	C₂×Dic₁₇₈	—	2	2	no	—
24	C₂×Dic₁₇₈	—	2	2	no	—
25	C₂×Dic₁₇₈	—	2	1	no	—
26	C₂×Dic₁₇₈	D₄	1	4	yes	—
27	C₂×Dic₁₇₈	D₄	1	2	yes	—
28	C₂²×Dic₈₉	—	2	1	no	—
29	C₂²×Dic₈₉	—	2	2	no	—
30	C₂²×Dic₈₉	—	2	1	no	—
31	Dic₃₅₆	—	2	16	no	—
32	C₄⋊Dic₈₉	—	2	8	no	—
33	C₄⋊Dic₈₉	—	2	2	no	—
34	C₂²×C₄×C₈₉	D₄	1	2	yes	—
35	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₈₉	—	2	2	no	—
37	C₂×C₈×C₈₉	—	2	2	no	—
38	C₂×C₈×C₈₉	D₄	1	4	yes	—
39	C₄²×C₈₉	—	2	2	no	—
40	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	Q₁₆	1	4	yes	complemented
44	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₈₉	C₂×D₄	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	1	no	—
53	C₂×Dic₈₉	—	4	2	no	—
54	C₂×Dic₈₉	—	4	1	no	—
55	C₂×Dic₈₉	—	4	2	no	—
56	Dic₁₇₈	—	2	4	no	—
57	Dic₁₇₈	—	2	4	no	—
58	Dic₁₇₈	—	2	4	no	—
59	Dic₁₇₈	—	4	2	no	—
60	Dic₁₇₈	—	4	2	no	—
61	Dic₁₇₈	—	4	2	no	—
62	C₈×C₈₉	—	2	4	no	—
63	C₂²×C₈₉	#120	1	1	yes	characteristic
64	C₂²×C₈₉	#121	1	1	yes	characteristic
65	C₂²×C₈₉	#127	1	1	yes	characteristic
66	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₈₉	#137	1	2	yes	—
68	C₄×C₈₉	—	2	4	no	—
69	C₄×C₈₉	#120	1	2	yes	—
70	C₄×C₈₉	#121	1	2	yes	—
71	Dic₈₉	—	4	2	no	—
72	Dic₈₉	—	4	4	no	—
73	Dic₈₉	—	8	1	no	—
74	C₂×C₈₉	#392	1	1	yes	characteristic
75	C₂×C₈₉	#451	1	2	yes	—
76	C₂×C₈₉	#462	1	2	yes	—
77	C₂×C₈₉	#494	1	1	yes	characteristic
78	C₂×C₈₉	#497	1	1	yes	characteristic
79	#2016	—	89	1	no	maximal, Sylow
80	C₈₉	#2016	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₁₆	—	89	2	no	—
82	C₂×C₄⋊Q₈	—	89	1	no	—
83	C₂×#103	—	89	2	no	—
84	#391	—	89	1	no	—
85	#336	D₈₉	1	1	yes	characteristic
86	C₂²×Q₈	—	89	2	no	—
87	C₂²×Q₈	—	89	1	no	—
88	C₂×Q₁₆	—	178	4	no	—
89	C₂×Q₁₆	—	178	4	no	—
90	C₂×Q₁₆	—	178	4	no	—
91	C₂²×C₈	C₂×D₈₉	1	2	yes	—
92	C₄⋊Q₈	—	178	4	no	—
93	C₂×C₄⋊C₄	—	89	1	no	—
94	C₂×C₄⋊C₄	—	178	2	no	—
95	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
96	#103	—	178	4	no	—
97	#95	—	178	2	no	—
98	C₂×Q₈	—	89	4	no	—
99	C₂×Q₈	—	89	2	no	—
100	C₂×Q₈	—	178	2	no	—
101	C₂×Q₈	—	178	2	no	—
102	C₂×Q₈	—	178	2	no	—
103	C₂×Q₈	—	178	2	no	—
104	C₂×Q₈	—	178	2	no	—
105	C₂×Q₈	—	178	2	no	—
106	C₂×Q₈	—	178	2	no	—
107	C₂×Q₈	—	178	1	no	—
108	C₂²×C₄	—	178	1	no	—
109	C₂²×C₄	—	178	1	no	—
110	C₂²×C₄	—	178	2	no	—
111	C₂²×C₄	D₃₅₆	1	2	yes	—
112	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	Q₁₆	—	178	16	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	D₃₅₆	1	4	yes	—
117	C₄⋊C₄	—	178	8	no	—
118	C₄⋊C₄	—	178	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
121	Q₈	—	356	2	no	—
122	Q₈	—	356	2	no	—
123	Q₈	—	356	2	no	—
124	Q₈	—	178	4	no	—
125	Q₈	—	178	4	no	—
126	Q₈	—	178	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	356	1	no	—
130	C₂×C₄	—	356	2	no	—
131	C₂×C₄	—	356	1	no	—
132	C₂×C₄	—	356	2	no	—
133	C₂×C₄	—	356	1	no	—
134	C₂×C₄	—	356	2	no	—
135	C₂×C₄	—	178	4	no	—
136	C₂×C₄	—	178	2	no	—
137	C₂×C₄	Dic₃₅₆	1	4	yes	complemented
138	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
139	C₂×C₄	D₄×D₈₉	1	2	yes	—
140	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
141	C₂×C₄	#13083575	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₂²	#15667982	1	1	yes	characteristic
144	C₂²	#15667987	1	1	yes	characteristic
145	C₂²	#15668009	1	1	yes	characteristic
146	C₂²	C₂×Dic₃₅₆	1	2	yes	—
147	C₂²	#15668045	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	356	2	no	—
150	C₄	—	356	4	no	—
151	C₄	—	712	1	no	—
152	C₄	#15667982	1	2	yes	—
153	C₄	#15667987	1	2	yes	—
154	C₂	#19151508	1	1	yes	characteristic
155	C₂	#19151756	1	2	yes	—
156	C₂	#19151809	1	2	yes	—
157	C₂	#19151978	1	1	yes	characteristic
158	C₂	#19151992	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₅₆

C₂

yes

maximal

C₂×#15668012

C₂

yes

maximal, characteristic

C₂×#15667937

C₂

yes

maximal

#19151504

C₂

yes

maximal, characteristic

#336×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

#15667937

—

#15667919

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂×Dic₁₇₈

D₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

Q₁₆

yes

complemented

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

#137

yes

—

C₄×C₈₉

—

C₄×C₈₉

#120

yes

—

C₄×C₈₉

#121

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#392

yes

characteristic

C₂×C₈₉

#451

yes

—

C₂×C₈₉

#462

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

—

#336

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₂²×C₈

C₂×D₈₉

yes

—

C₄⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#103

—

178

—

#95

—

178

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

178

—

101

C₂×Q₈

—

178

—

102

C₂×Q₈

—

178

—

103

C₂×Q₈

—

178

—

104

C₂×Q₈

—

178

—

105

C₂×Q₈

—

178

—

106

C₂×Q₈

—

178

—

107

C₂×Q₈

—

178

—

108

C₂²×C₄

—

178

—

109

C₂²×C₄

—

178

—

110

C₂²×C₄

—

178

—

111

C₂²×C₄

D₃₅₆

yes

—

112

C₂²×C₄

C₂²×D₈₉

yes

characteristic

113

Q₁₆

—

178

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

D₃₅₆

yes

—

117

C₄⋊C₄

—

178

—

118

C₄⋊C₄

—

178

—

119

C₄²

—

120

C₂³

C₂×D₃₅₆

yes

characteristic

121

Q₈

—

356

—

122

Q₈

—

356

—

123

Q₈

—

356

—

124

Q₈

—

178

—

125

Q₈

—

178

—

126

Q₈

—

178

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

356

—

130

C₂×C₄

—

356

—

131

C₂×C₄

—

356

—

132

C₂×C₄

—

356

—

133

C₂×C₄

—

356

—

134

C₂×C₄

—

356

—

135

C₂×C₄

—

178

—

136

C₂×C₄

—

178

—

137

C₂×C₄

Dic₃₅₆

yes

complemented

138

C₂×C₄

D₄×D₈₉

yes

characteristic

139

C₂×C₄

D₄×D₈₉

yes

—

140

C₂×C₄

C₂×D₃₅₆

yes

—

141

C₂×C₄

#13083575

yes

characteristic

142

C₈

—

143

C₂²

#15667982

yes

characteristic

144

C₂²

#15667987

yes

characteristic

145

C₂²

#15668009

yes

characteristic

146

C₂²

C₂×Dic₃₅₆

yes

—

147

C₂²

#15668045

yes

—

148

C₄

—

149

C₄

—

356

—

150

C₄

—

356

—

151

C₄

—

712

—

152

C₄

#15667982

yes

—

153

C₄

#15667987

yes

—

154

C₂

#19151508

yes

characteristic

155

C₂

#19151756

yes

—

156

C₂

#19151809

yes

—

157

C₂

#19151978

yes

characteristic

158

C₂

#19151992

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	#2016
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
43. C₂×C₄×C₈₉	Q₁₆	semidirect
80. C₈₉	#2016	semidirect
137. C₂×C₄	Dic₃₅₆	semidirect

H ≅ G/N

type

43. C₂×C₄×C₈₉

Q₁₆

semidirect

80. C₈₉

#2016

semidirect

137. C₂×C₄

Dic₃₅₆

semidirect

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

subgroup	index
C₂²×Dic₃₅₆	2
C₂×#15668012	2
C₂×#15667937	2
#19151504	2
#336×C₈₉	2
#2016	89

subgroup

index

C₂²×Dic₃₅₆

#336×C₈₉

Subgroups — #26021354

Conjugacy classes of subgroups

Complements

Maximal subgroups