Group #26021374 — subgroups

Subgroups — #26021374

Order 11392.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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×#15668045	C₂	1	2	yes	maximal
2	C₂²×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15667930	C₂	1	2	yes	maximal, complemented
4	#19151564	C₂	1	1	yes	maximal, characteristic
5	#341×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
8	#15668045	—	2	4	no	—
9	#15668045	—	2	4	no	—
10	C₂×Dic₃₅₆	—	2	2	no	—
11	C₂×Dic₃₅₆	—	2	4	no	—
12	C₂×Dic₃₅₆	—	2	2	no	—
13	C₂×#13083563	—	2	2	no	—
14	C₂×#13083558	C₂²	1	1	yes	characteristic
15	#15667930	—	2	4	no	—
16	#15667924	—	2	2	no	—
17	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#13083574	—	2	2	no	—
20	#13083574	—	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	2	no	—
26	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
27	C₂×Dic₁₇₈	D₄	1	4	yes	—
28	C₂×D₃₅₆	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊D₈₉	—	2	1	no	—
30	C₂²×Dic₈₉	—	2	2	no	—
31	C₂×C₄×D₈₉	—	2	1	no	—
32	Dic₃₅₆	—	2	8	no	—
33	Dic₃₅₆	—	2	8	no	—
34	#13083563	—	2	8	no	—
35	#13083558	—	2	2	no	—
36	C₂×#10623494	—	2	1	no	—
37	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	M₁₆×C₈₉	—	2	2	no	—
39	M₁₆×C₈₉	D₄	1	4	yes	—
40	C₂×C₈×C₈₉	—	2	2	no	—
41	C₂×C₈×C₈₉	—	2	2	no	—
42	C₂×C₈×C₈₉	D₄	1	2	yes	—
43	C₂³×C₈₉	#41	1	1	yes	characteristic
44	C₂²×D₈₉	—	2	1	no	—
45	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
49	C₂²⋊D₈₉	—	4	2	no	—
50	C₂×Dic₈₉	—	2	1	no	—
51	C₂×Dic₈₉	—	2	4	no	—
52	C₂×Dic₈₉	—	4	2	no	—
53	C₂×Dic₈₉	—	4	2	no	—
54	D₃₅₆	—	2	2	no	—
55	C₄×D₈₉	—	4	1	no	—
56	C₄×D₈₉	—	4	1	no	—
57	Dic₁₇₈	—	2	4	no	—
58	Dic₁₇₈	—	2	4	no	—
59	Dic₁₇₈	—	2	2	no	—
60	Dic₁₇₈	—	4	2	no	—
61	Dic₁₇₈	—	4	2	no	—
62	C₈×C₈₉	—	2	2	no	—
63	C₈×C₈₉	—	2	4	no	—
64	#10623494	—	4	1	no	—
65	C₂²×C₈₉	—	2	2	no	—
66	C₂²×C₈₉	#121	1	2	yes	—
67	C₂²×C₈₉	#121	1	1	yes	characteristic
68	C₂×D₈₉	—	4	1	no	—
69	C₂×D₈₉	—	4	2	no	—
70	C₄×C₈₉	#120	1	1	yes	characteristic
71	C₄×C₈₉	#120	1	2	yes	—
72	C₄×C₈₉	#127	1	1	yes	characteristic
73	Dic₈₉	—	4	4	no	—
74	Dic₈₉	—	4	1	no	—
75	C₂×C₈₉	—	2	2	no	—
76	C₂×C₈₉	#392	1	1	yes	characteristic
77	C₂×C₈₉	#473	1	2	yes	—
78	D₈₉	—	8	1	no	—
79	#2020	—	89	1	no	maximal, Sylow
80	C₈₉	#2020	1	1	yes	characteristic, complemented, Sylow
81	C₂×#137	—	89	2	no	—
82	C₂²×Q₁₆	—	89	1	no	—
83	C₂×#101	—	89	2	no	—
84	#408	—	89	1	no	—
85	#341	D₈₉	1	1	yes	characteristic, complemented
86	C₂×#41	—	89	1	no	—
87	C₂²×Q₈	—	89	2	no	—
88	#137	—	178	4	no	—
89	#137	—	178	4	no	—
90	C₂×Q₁₆	—	178	4	no	—
91	C₂×Q₁₆	—	178	2	no	—
92	C₂×Q₁₆	—	178	2	no	—
93	C₂×SD₁₆	—	178	2	no	—
94	C₂×M₁₆	—	89	1	no	—
95	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
96	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
97	#101	—	178	4	no	—
98	C₂²⋊C₈	—	178	2	no	—
99	#41	—	178	2	no	—
100	#41	—	178	2	no	—
101	C₂×Q₈	—	89	1	no	—
102	C₂×Q₈	—	89	4	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	2	no	—
108	C₂×D₄	—	89	1	no	—
109	C₂×D₄	—	178	1	no	—
110	C₂²×C₄	—	178	2	no	—
111	C₂²×C₄	—	178	1	no	—
112	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	Q₁₆	—	178	8	no	—
114	Q₁₆	—	178	8	no	—
115	SD₁₆	—	178	8	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	—	178	2	no	—
118	M₁₆	D₃₅₆	1	4	yes	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	178	1	no	—
122	C₂×C₈	D₃₅₆	1	2	yes	—
123	C₂³	—	178	1	no	—
124	C₂³	#13083575	1	1	yes	characteristic
125	Q₈	—	356	2	no	—
126	Q₈	—	356	2	no	—
127	Q₈	—	178	4	no	—
128	Q₈	—	178	4	no	—
129	Q₈	—	178	2	no	—
130	D₄	—	356	2	no	—
131	D₄	—	178	2	no	—
132	C₂×C₄	—	356	2	no	—
133	C₂×C₄	—	356	2	no	—
134	C₂×C₄	—	356	1	no	—
135	C₂×C₄	—	356	1	no	—
136	C₂×C₄	—	178	4	no	—
137	C₂×C₄	—	178	1	no	—
138	C₂×C₄	D₄×D₈₉	1	2	yes	—
139	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
140	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
141	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	4	no	—
144	C₈	—	356	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	356	2	no	—
147	C₂²	—	356	1	no	—
148	C₂²	#15667987	1	2	yes	—
149	C₂²	#15667987	1	1	yes	characteristic
150	C₄	—	356	1	no	—
151	C₄	—	356	4	no	—
152	C₄	#15667982	1	1	yes	characteristic
153	C₄	#15667982	1	2	yes	—
154	C₄	#15668009	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	712	1	no	—
157	C₂	#19151508	1	1	yes	characteristic
158	C₂	#19151874	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668045

C₂

yes

maximal

C₂²×Dic₃₅₆

C₂

yes

maximal, characteristic, complemented

C₂×#15667930

C₂

yes

maximal, complemented

#19151564

C₂

yes

maximal, characteristic

#341×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668045

—

#15668045

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×#13083563

—

C₂×#13083558

C₂²

yes

characteristic

#15667930

—

#15667924

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#13083574

—

#13083574

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×Dic₁₇₈

D₄

yes

—

C₂×D₃₅₆

D₄

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

—

C₂×C₄×D₈₉

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083563

—

#13083558

—

C₂×#10623494

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂³×C₈₉

#41

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

—

C₂²×C₈₉

#121

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#120

yes

characteristic

C₄×C₈₉

#120

yes

—

C₄×C₈₉

#127

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#392

yes

characteristic

C₂×C₈₉

#473

yes

—

D₈₉

—

#2020

—

maximal, Sylow

C₈₉

#2020

yes

characteristic, complemented, Sylow

C₂×#137

—

C₂²×Q₁₆

—

C₂×#101

—

#408

—

#341

D₈₉

yes

characteristic, complemented

C₂×#41

—

C₂²×Q₈

—

#137

—

178

—

#137

—

178

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₂×SD₁₆

—

178

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

#101

—

178

—

C₂²⋊C₈

—

178

—

#41

—

178

—

100

#41

—

178

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

178

—

104

C₂×Q₈

—

178

—

105

C₂×Q₈

—

178

—

106

C₂×Q₈

—

178

—

107

C₂×Q₈

—

178

—

108

C₂×D₄

—

109

C₂×D₄

—

178

—

110

C₂²×C₄

—

178

—

111

C₂²×C₄

—

178

—

112

C₂²×C₄

C₂²×D₈₉

yes

characteristic

113

Q₁₆

—

178

—

114

Q₁₆

—

178

—

115

SD₁₆

—

178

—

116

M₁₆

—

117

M₁₆

—

178

—

118

M₁₆

D₃₅₆

yes

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

178

—

122

C₂×C₈

D₃₅₆

yes

—

123

C₂³

—

178

—

124

C₂³

#13083575

yes

characteristic

125

Q₈

—

356

—

126

Q₈

—

356

—

127

Q₈

—

178

—

128

Q₈

—

178

—

129

Q₈

—

178

—

130

D₄

—

356

—

131

D₄

—

178

—

132

C₂×C₄

—

356

—

133

C₂×C₄

—

356

—

134

C₂×C₄

—

356

—

135

C₂×C₄

—

356

—

136

C₂×C₄

—

178

—

137

C₂×C₄

—

178

—

138

C₂×C₄

D₄×D₈₉

yes

—

139

C₂×C₄

D₄×D₈₉

yes

characteristic

140

C₂×C₄

C₂×D₃₅₆

yes

—

141

C₂×C₄

C₂×D₃₅₆

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

356

—

145

C₂²

—

146

C₂²

—

356

—

147

C₂²

—

356

—

148

C₂²

#15667987

yes

—

149

C₂²

#15667987

yes

characteristic

150

C₄

—

356

—

151

C₄

—

356

—

152

C₄

#15667982

yes

characteristic

153

C₄

#15667982

yes

—

154

C₄

#15668009

yes

characteristic

155

C₂

—

156

C₂

—

712

—

157

C₂

#19151508

yes

characteristic

158

C₂

#19151874

yes

—

	subgroup	G/H
center	C₂²	#15667987
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#341×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	C₂×#13083574	C₂²
hypercenter	#341	D₈₉
hypocenter	C₈₉	#2020
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#15667987

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#341×C₈₉

C₂

C₂²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂²×Dic₃₅₆	C₂	semidirect
3. C₂×#15667930	C₂	semidirect
5. #341×C₈₉	C₂	semidirect
80. C₈₉	#2020	semidirect
85. #341	D₈₉	semidirect

H ≅ G/N

type

2. C₂²×Dic₃₅₆

C₂

semidirect

3. C₂×#15667930

C₂

semidirect

5. #341×C₈₉

C₂

semidirect

80. C₈₉

#2020

semidirect

85. #341

D₈₉

semidirect

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

subgroup	index
C₂×#15668045	2
C₂²×Dic₃₅₆	2
C₂×#15667930	2
#19151564	2
#341×C₈₉	2
#2020	89

subgroup

index

C₂²×Dic₃₅₆

#341×C₈₉

Subgroups — #26021374

Conjugacy classes of subgroups

Complements

Maximal subgroups