Group #26021575 — subgroups

Subgroups — #26021575

Order 11392.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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	1	yes	maximal, characteristic
2	C₂×#15667937	C₂	1	2	yes	maximal
3	#19151504	C₂	1	2	yes	maximal
4	#19151361	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂×Dic₃₅₆	—	2	4	no	—
8	C₂×Dic₃₅₆	—	2	2	no	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	#15667937	—	2	4	no	—
11	#15667919	—	2	2	no	—
12	#15667919	—	2	2	no	—
13	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₈₉	—	2	2	no	—
17	C₂×Dic₁₇₈	—	2	2	no	—
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	D₄	1	4	yes	—
23	C₂²×Dic₈₉	—	2	2	no	—
24	C₂²×Dic₈₉	—	2	2	no	—
25	Dic₃₅₆	—	2	8	no	—
26	C₄⋊Dic₈₉	—	2	2	no	—
27	C₄⋊Dic₈₉	—	2	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	1	no	—
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₂×C₈×C₈₉	D₄	1	2	yes	—
35	C₄²×C₈₉	—	2	1	no	—
36	C₄²×C₈₉	—	2	1	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	2	no	—
42	C₂×C₄×C₈₉	Q₁₆	1	2	yes	complemented
43	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#41	1	2	yes	—
47	C₂×Dic₈₉	—	2	4	no	—
48	C₂×Dic₈₉	—	4	2	no	—
49	C₂×Dic₈₉	—	4	2	no	—
50	C₂×Dic₈₉	—	4	2	no	—
51	C₂×Dic₈₉	—	4	2	no	—
52	C₂×Dic₈₉	—	4	2	no	—
53	Dic₁₇₈	—	2	4	no	—
54	Dic₁₇₈	—	2	4	no	—
55	Dic₁₇₈	—	4	2	no	—
56	Dic₁₇₈	—	4	2	no	—
57	C₈×C₈₉	—	2	2	no	—
58	C₈×C₈₉	—	4	1	no	—
59	C₂²×C₈₉	#121	1	2	yes	—
60	C₂²×C₈₉	#121	1	1	yes	characteristic
61	C₂²×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₈₉	#135	1	1	yes	characteristic
63	C₂²×C₈₉	#137	1	2	yes	—
64	C₄×C₈₉	—	2	2	no	—
65	C₄×C₈₉	—	4	1	no	—
66	C₄×C₈₉	#120	1	1	yes	characteristic
67	C₄×C₈₉	#123	1	2	yes	—
68	C₄×C₈₉	#124	1	1	yes	characteristic
69	Dic₈₉	—	4	4	no	—
70	Dic₈₉	—	8	2	no	—
71	C₂×C₈₉	#394	1	1	yes	characteristic
72	C₂×C₈₉	#462	1	2	yes	—
73	C₂×C₈₉	#464	1	2	yes	—
74	C₂×C₈₉	#467	1	1	yes	characteristic
75	C₂×C₈₉	#470	1	1	yes	characteristic
76	#2056	—	89	1	no	maximal, Sylow
77	C₈₉	#2056	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₁₆	—	89	1	no	—
79	C₂×C₄⋊C₈	D₈₉	1	1	yes	characteristic
80	C₂×#103	—	89	2	no	—
81	#391	—	89	2	no	—
82	#340	—	89	1	no	—
83	C₂²×Q₈	—	89	2	no	—
84	C₂×Q₁₆	—	178	4	no	—
85	C₂×Q₁₆	—	178	2	no	—
86	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	89	2	no	—
89	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	178	4	no	—
92	#95	—	178	2	no	—
93	#95	—	178	2	no	—
94	C₂×Q₈	—	89	4	no	—
95	C₂×Q₈	—	178	2	no	—
96	C₂×Q₈	—	178	2	no	—
97	C₂×Q₈	—	178	2	no	—
98	C₂×Q₈	—	178	2	no	—
99	C₂×Q₈	—	178	2	no	—
100	C₂²×C₄	—	178	2	no	—
101	C₂²×C₄	—	178	2	no	—
102	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
103	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
105	Q₁₆	—	178	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
110	C₄⋊C₄	—	178	2	no	—
111	C₄⋊C₄	—	178	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
115	Q₈	—	356	2	no	—
116	Q₈	—	356	2	no	—
117	Q₈	—	178	4	no	—
118	Q₈	—	178	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	356	2	no	—
124	C₂×C₄	—	356	2	no	—
125	C₂×C₄	—	356	2	no	—
126	C₂×C₄	—	356	2	no	—
127	C₂×C₄	—	356	2	no	—
128	C₂×C₄	—	178	4	no	—
129	C₂×C₄	Dic₃₅₆	1	2	yes	complemented
130	C₂×C₄	D₄×D₈₉	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
132	C₂×C₄	#13083574	1	1	yes	characteristic
133	C₂×C₄	#13083575	1	2	yes	—
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#15667986	1	1	yes	characteristic
137	C₂²	#15667987	1	2	yes	—
138	C₂²	C₂×Dic₃₅₆	1	1	yes	characteristic
139	C₂²	#15668034	1	1	yes	characteristic
140	C₂²	#15668045	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	356	4	no	—
144	C₄	—	712	2	no	—
145	C₄	#15667980	1	1	yes	characteristic
146	C₄	#15667997	1	2	yes	—
147	C₄	#15668001	1	1	yes	characteristic
148	C₂	#19151515	1	1	yes	characteristic
149	C₂	#19151809	1	2	yes	—
150	C₂	#19151821	1	2	yes	—
151	C₂	#19151839	1	1	yes	characteristic
152	C₂	#19151856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₅₆

C₂

yes

maximal, characteristic

C₂×#15667937

C₂

yes

maximal

#19151504

C₂

yes

maximal

#19151361

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667937

—

#15667919

—

#15667919

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

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

—

C₂×C₈×C₈₉

D₄

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

Q₁₆

yes

complemented

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

#121

yes

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

C₂×Q₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#120

yes

characteristic

C₄×C₈₉

#123

yes

—

C₄×C₈₉

#124

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#394

yes

characteristic

C₂×C₈₉

#462

yes

—

C₂×C₈₉

#464

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊C₈

D₈₉

yes

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

#103

—

178

—

#95

—

178

—

#95

—

178

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

100

C₂²×C₄

—

178

—

101

C₂²×C₄

—

178

—

102

C₂²×C₄

D₃₅₆

yes

characteristic

103

C₂²×C₄

D₃₅₆

yes

characteristic

104

C₂²×C₄

C₂²×D₈₉

yes

characteristic

105

Q₁₆

—

178

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₈₉

yes

—

110

C₄⋊C₄

—

178

—

111

C₄⋊C₄

—

178

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₃₅₆

yes

characteristic

115

Q₈

—

356

—

116

Q₈

—

356

—

117

Q₈

—

178

—

118

Q₈

—

178

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

356

—

124

C₂×C₄

—

356

—

125

C₂×C₄

—

356

—

126

C₂×C₄

—

356

—

127

C₂×C₄

—

356

—

128

C₂×C₄

—

178

—

129

C₂×C₄

Dic₃₅₆

yes

complemented

130

C₂×C₄

D₄×D₈₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

132

C₂×C₄

#13083574

yes

characteristic

133

C₂×C₄

#13083575

yes

—

134

C₈

—

135

C₈

—

136

C₂²

#15667986

yes

characteristic

137

C₂²

#15667987

yes

—

138

C₂²

C₂×Dic₃₅₆

yes

characteristic

139

C₂²

#15668034

yes

characteristic

140

C₂²

#15668045

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

356

—

144

C₄

—

712

—

145

C₄

#15667980

yes

characteristic

146

C₄

#15667997

yes

—

147

C₄

#15668001

yes

characteristic

148

C₂

#19151515

yes

characteristic

149

C₂

#19151809

yes

—

150

C₂

#19151821

yes

—

151

C₂

#19151839

yes

characteristic

152

C₂

#19151856

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#2056
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×C₄⋊C₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₈₉

hypocenter

C₈₉

#2056

LCS

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
42. C₂×C₄×C₈₉	Q₁₆	semidirect
77. C₈₉	#2056	semidirect
129. C₂×C₄	Dic₃₅₆	semidirect

H ≅ G/N

type

42. C₂×C₄×C₈₉

Q₁₆

semidirect

77. C₈₉

#2056

semidirect

129. 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×#15667937	2
#19151504	2
#19151361	2
C₂×C₄⋊C₈×C₈₉	2
#2056	89

subgroup

index

C₂²×Dic₃₅₆

C₂×C₄⋊C₈×C₈₉

Subgroups — #26021575

Conjugacy classes of subgroups

Complements

Maximal subgroups