Group #26021113 — subgroups

Subgroups — #26021113

Order 11392.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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×#15667946	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667937	C₂	1	2	yes	maximal
4	#19151504	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂×Dic₃₅₆	—	2	2	no	—
8	C₂×Dic₃₅₆	—	2	2	no	—
9	C₂×Dic₃₅₆	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	#15667946	—	2	2	no	—
12	#15667937	—	2	4	no	—
13	#15667919	—	2	4	no	—
14	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₈₉	—	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	—	2	2	no	—
23	C₂×Dic₁₇₈	C₂×C₄	1	4	yes	—
24	C₂²×Dic₈₉	—	2	2	no	—
25	C₂²×Dic₈₉	—	2	2	no	—
26	Dic₃₅₆	—	2	8	no	—
27	C₄⋊Dic₈₉	—	2	2	no	—
28	C₄⋊Dic₈₉	—	2	2	no	—
29	C₂²×C₄×C₈₉	D₄	1	2	yes	—
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₈₉	—	2	1	no	—
32	C₂×C₈×C₈₉	—	2	1	no	—
33	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₈₉	D₄	1	2	yes	—
35	C₂×C₈×C₈₉	D₄	1	2	yes	—
36	C₂×C₈×C₈₉	D₄	1	2	yes	—
37	C₄²×C₈₉	—	2	1	no	—
38	C₄²×C₈₉	—	2	1	no	—
39	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
48	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
49	C₂×Dic₈₉	—	2	4	no	—
50	C₂×Dic₈₉	—	4	2	no	—
51	C₂×Dic₈₉	—	4	2	no	—
52	C₂×Dic₈₉	—	4	2	no	—
53	C₂×Dic₈₉	—	4	4	no	—
54	Dic₁₇₈	—	2	4	no	—
55	Dic₁₇₈	—	2	4	no	—
56	Dic₁₇₈	—	4	2	no	—
57	Dic₁₇₈	—	4	2	no	—
58	C₈×C₈₉	—	2	2	no	—
59	C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₈₉	#121	1	1	yes	characteristic
62	C₂²×C₈₉	#127	1	1	yes	characteristic
63	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₈₉	#135	1	2	yes	—
65	C₄×C₈₉	—	2	4	no	—
66	C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
68	C₄×C₈₉	#121	1	1	yes	characteristic
69	C₄×C₈₉	#124	1	1	yes	characteristic
70	Dic₈₉	—	4	4	no	—
71	Dic₈₉	—	8	2	no	—
72	C₂×C₈₉	#390	1	1	yes	characteristic
73	C₂×C₈₉	C₄×Q₁₆	1	2	yes	—
74	C₂×C₈₉	#467	1	2	yes	—
75	C₂×C₈₉	#494	1	1	yes	characteristic
76	C₂×C₈₉	#495	1	1	yes	characteristic
77	#1969	—	89	1	no	maximal, Sylow
78	C₈₉	#1969	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	—	89	1	no	—
80	C₂×#107	—	89	1	no	—
81	C₂×#103	—	89	2	no	—
82	C₂×C₄×C₈	D₈₉	1	1	yes	characteristic
83	#391	—	89	2	no	—
84	C₂²×Q₈	—	89	2	no	—
85	C₂×Q₁₆	—	89	4	no	—
86	C₂×Q₁₆	—	178	2	no	—
87	C₂×Q₁₆	—	178	2	no	—
88	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	89	2	no	—
91	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
92	#107	—	178	2	no	—
93	#103	—	178	4	no	—
94	C₄×C₈	—	2	2	no	—
95	#95	—	178	4	no	—
96	C₂×Q₈	—	89	4	no	—
97	C₂×Q₈	—	178	2	no	—
98	C₂×Q₈	—	178	2	no	—
99	C₂×Q₈	—	178	2	no	—
100	C₂×Q₈	—	178	2	no	—
101	C₂×Q₈	—	178	2	no	—
102	C₂²×C₄	—	178	2	no	—
103	C₂²×C₄	—	178	2	no	—
104	C₂²×C₄	D₃₅₆	1	2	yes	—
105	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
106	Q₁₆	—	178	8	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₄×D₈₉	1	2	yes	—
110	C₂×C₈	D₃₅₆	1	2	yes	—
111	C₂×C₈	D₃₅₆	1	2	yes	—
112	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
113	C₄⋊C₄	—	178	2	no	—
114	C₄⋊C₄	—	178	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
118	Q₈	—	356	2	no	—
119	Q₈	—	356	2	no	—
120	Q₈	—	178	4	no	—
121	Q₈	—	178	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	356	2	no	—
125	C₂×C₄	—	356	2	no	—
126	C₂×C₄	—	356	2	no	—
127	C₂×C₄	—	356	4	no	—
128	C₂×C₄	—	178	4	no	—
129	C₂×C₄	Dic₃₅₆	1	4	yes	complemented
130	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
132	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
134	C₂×C₄	#13083574	1	1	yes	characteristic
135	C₂×C₄	#13083574	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	#13083551	1	4	yes	—
138	C₂²	C₄×D₃₅₆	1	1	yes	characteristic
139	C₂²	#15667986	1	1	yes	characteristic
140	C₂²	#15668009	1	1	yes	characteristic
141	C₂²	C₂×Dic₃₅₆	1	2	yes	—
142	C₂²	#15668034	1	2	yes	—
143	C₄	—	2	4	no	—
144	C₄	—	356	4	no	—
145	C₄	—	712	2	no	—
146	C₄	C₂×#13083551	1	1	yes	characteristic
147	C₄	C₄×D₃₅₆	1	1	yes	characteristic
148	C₄	#15667986	1	1	yes	characteristic
149	C₄	#15668000	1	1	yes	characteristic
150	C₂	#19151500	1	1	yes	characteristic
151	C₂	C₄×Dic₃₅₆	1	2	yes	—
152	C₂	#19151839	1	2	yes	—
153	C₂	#19151978	1	1	yes	characteristic
154	C₂	#19151982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₅₆

C₂

yes

maximal, characteristic

C₂×#15667946

C₂

yes

maximal, characteristic

C₂×#15667937

C₂

yes

maximal

#19151504

C₂

yes

maximal

C₂×C₄×C₈×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

C₄

yes

complemented

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667946

—

#15667937

—

#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

C₂×C₄

yes

—

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

C₂×C₄

yes

—

C₂×C₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

D₄

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

Q₁₆

yes

complemented

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₈₉

C₄×D₄

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

#135

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₉

C₄×D₄

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#390

yes

characteristic

C₂×C₈₉

C₄×Q₁₆

yes

—

C₂×C₈₉

#467

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₈₉

yes

characteristic

#391

—

C₂²×Q₈

—

C₂×Q₁₆

—

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

#107

—

178

—

#103

—

178

—

C₄×C₈

—

#95

—

178

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

100

C₂×Q₈

—

178

—

101

C₂×Q₈

—

178

—

102

C₂²×C₄

—

178

—

103

C₂²×C₄

—

178

—

104

C₂²×C₄

D₃₅₆

yes

—

105

C₂²×C₄

C₂²×D₈₉

yes

characteristic

106

Q₁₆

—

178

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₈₉

yes

—

110

C₂×C₈

D₃₅₆

yes

—

111

C₂×C₈

D₃₅₆

yes

—

112

C₂×C₈

C₂²⋊D₈₉

yes

—

113

C₄⋊C₄

—

178

—

114

C₄⋊C₄

—

178

—

115

C₄²

—

116

C₄²

—

117

C₂³

C₂×D₃₅₆

yes

characteristic

118

Q₈

—

356

—

119

Q₈

—

356

—

120

Q₈

—

178

—

121

Q₈

—

178

—

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

—

178

—

129

C₂×C₄

Dic₃₅₆

yes

complemented

130

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

132

C₂×C₄

C₂×D₃₅₆

yes

characteristic

133

C₂×C₄

C₂×D₃₅₆

yes

characteristic

134

C₂×C₄

#13083574

yes

characteristic

135

C₂×C₄

#13083574

yes

characteristic

136

C₈

—

137

C₈

#13083551

yes

—

138

C₂²

C₄×D₃₅₆

yes

characteristic

139

C₂²

#15667986

yes

characteristic

140

C₂²

#15668009

yes

characteristic

141

C₂²

C₂×Dic₃₅₆

yes

—

142

C₂²

#15668034

yes

—

143

C₄

—

144

C₄

—

356

—

145

C₄

—

712

—

146

C₄

C₂×#13083551

yes

characteristic

147

C₄

C₄×D₃₅₆

yes

characteristic

148

C₄

#15667986

yes

characteristic

149

C₄

#15668000

yes

characteristic

150

C₂

#19151500

yes

characteristic

151

C₂

C₄×Dic₃₅₆

yes

—

152

C₂

#19151839

yes

—

153

C₂

#19151978

yes

characteristic

154

C₂

#19151982

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×C₄×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×C₄×C₈	D₈₉
hypocenter	C₈₉	#1969
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×C₄×C₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂×C₄×C₈

D₈₉

hypocenter

C₈₉

#1969

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₂×Dic₃₅₆	C₄	semidirect
42. C₂×C₄×C₈₉	Q₁₆	semidirect
78. C₈₉	#1969	semidirect
129. C₂×C₄	Dic₃₅₆	semidirect

H ≅ G/N

type

9. C₂×Dic₃₅₆

C₄

semidirect

42. C₂×C₄×C₈₉

Q₁₆

semidirect

78. C₈₉

#1969

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×#15667946	2
C₂×#15667937	2
#19151504	2
C₂×C₄×C₈×C₈₉	2
#1969	89

subgroup

index

C₂²×Dic₃₅₆

C₂×C₄×C₈×C₈₉

Subgroups — #26021113

Conjugacy classes of subgroups

Complements

Maximal subgroups