Group #26020083 — subgroups

Subgroups — #26020083

Order 11392.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667937	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151361	C₂	1	2	yes	maximal
5	C₂×#97×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₇₈	—	2	2	no	—
8	C₄×Dic₁₇₈	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	C₂×#13083553	—	2	1	no	—
12	C₂×C₄×Dic₈₉	—	2	1	no	—
13	C₂×C₄×Dic₈₉	—	2	1	no	—
14	#15667937	C₄	1	8	yes	complemented
15	#15667919	—	2	1	no	—
16	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	#97×C₈₉	—	2	2	no	—
19	C₂×Dic₁₇₈	—	2	1	no	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
22	C₂×Dic₁₇₈	C₂×C₄	1	4	yes	—
23	C₂²×Dic₈₉	—	2	1	no	—
24	C₂²×Dic₈₉	—	2	2	no	—
25	C₂²×Dic₈₉	—	2	1	no	—
26	C₄⋊Dic₈₉	—	2	1	no	—
27	C₄⋊Dic₈₉	—	2	2	no	—
28	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
29	C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
30	#13083553	—	2	4	no	—
31	C₄×Dic₈₉	—	2	4	no	—
32	C₄×Dic₈₉	—	4	2	no	—
33	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
39	C₄²×C₈₉	C₂×C₄	1	2	yes	—
40	C₄²×C₈₉	D₄	1	2	yes	—
41	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	—	2	1	no	—
43	C₂×C₄×C₈₉	—	2	1	no	—
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#41	1	2	yes	—
51	C₂×Dic₈₉	—	2	2	no	—
52	C₂×Dic₈₉	—	2	4	no	—
53	C₂×Dic₈₉	—	2	2	no	—
54	C₂×Dic₈₉	—	2	4	no	—
55	C₂×Dic₈₉	—	4	2	no	—
56	C₂×Dic₈₉	—	4	1	no	—
57	C₂×Dic₈₉	—	4	1	no	—
58	C₂×Dic₈₉	—	4	2	no	—
59	Dic₁₇₈	—	2	2	no	—
60	Dic₁₇₈	—	2	4	no	—
61	Dic₁₇₈	C₄²	1	4	yes	complemented
62	C₈×C₈₉	—	2	4	no	—
63	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₈₉	C₄×D₄	1	2	yes	—
65	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₈₉	#137	1	1	yes	characteristic
67	C₂²×C₈₉	#137	1	2	yes	—
68	C₄×C₈₉	—	2	2	no	—
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
71	C₄×C₈₉	#117	1	1	yes	characteristic
72	C₄×C₈₉	C₄×D₄	1	2	yes	—
73	Dic₈₉	—	2	4	no	—
74	Dic₈₉	—	4	2	no	—
75	Dic₈₉	—	4	4	no	—
76	C₂×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈₉	#417	1	1	yes	characteristic
78	C₂×C₈₉	#419	1	1	yes	characteristic
79	C₂×C₈₉	#440	1	2	yes	—
80	C₂×C₈₉	#441	1	2	yes	—
81	#1763	—	89	1	no	maximal, Sylow
82	C₈₉	#1763	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄×Q₈	—	89	1	no	—
84	C₂×#103	—	89	2	no	—
85	C₂×#97	D₈₉	1	1	yes	characteristic
86	C₄×C₄⋊C₄	—	89	1	no	—
87	#340	—	89	2	no	—
88	C₂²×Q₈	—	89	1	no	—
89	C₂²×C₈	C₂×D₈₉	1	2	yes	—
90	C₄×Q₈	—	89	4	no	—
91	C₄×Q₈	—	178	2	no	—
92	C₂×C₄⋊C₄	—	89	1	no	—
93	C₂×C₄⋊C₄	—	89	2	no	—
94	C₂×C₄⋊C₄	—	178	1	no	—
95	C₂×C₄²	—	178	1	no	—
96	C₂×C₄²	—	178	1	no	—
97	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
98	#103	—	89	8	no	—
99	#97	—	2	2	no	—
100	#95	—	178	1	no	—
101	C₂×Q₈	—	89	4	no	—
102	C₂×Q₈	—	89	2	no	—
103	C₂×Q₈	—	178	1	no	—
104	C₂×Q₈	—	178	2	no	—
105	C₂²×C₄	—	178	2	no	—
106	C₂²×C₄	—	178	1	no	—
107	C₂²×C₄	—	178	1	no	—
108	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
109	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	C₄×D₈₉	1	4	yes	—
114	C₄⋊C₄	—	89	4	no	—
115	C₄⋊C₄	—	89	2	no	—
116	C₄⋊C₄	—	178	4	no	—
117	C₄⋊C₄	—	178	1	no	—
118	C₄⋊C₄	—	178	2	no	—
119	C₄²	—	356	2	no	—
120	C₄²	—	178	4	no	—
121	C₄²	C₄×D₈₉	1	2	yes	—
122	C₄²	C₂²⋊D₈₉	1	2	yes	—
123	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
124	Q₈	—	89	4	no	—
125	Q₈	—	178	2	no	—
126	Q₈	—	178	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	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	—	178	4	no	—
135	C₂×C₄	—	178	2	no	—
136	C₂×C₄	—	178	4	no	—
137	C₂×C₄	—	178	2	no	—
138	C₂×C₄	#13083551	1	2	yes	—
139	C₂×C₄	#13083551	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
143	C₂×C₄	#13083574	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₂²	C₂×#13083551	1	1	yes	characteristic
146	C₂²	C₄×D₃₅₆	1	2	yes	—
147	C₂²	#15668040	1	1	yes	characteristic
148	C₂²	#15668045	1	2	yes	—
149	C₂²	#15668045	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	356	4	no	—
153	C₄	—	356	2	no	—
154	C₄	—	178	4	no	—
155	C₄	C₄²×D₈₉	1	1	yes	characteristic
156	C₄	#15667969	1	1	yes	characteristic
157	C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
158	C₂	C₄×#13083551	1	1	yes	characteristic
159	C₂	#19151600	1	1	yes	characteristic
160	C₂	#19151608	1	1	yes	characteristic
161	C₂	#19151700	1	2	yes	—
162	C₂	#19151706	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₇₈

C₂

yes

maximal, characteristic

C₂×#15667937

C₂

yes

maximal

C₄×C₄⋊Dic₈₉

C₂

yes

maximal, characteristic

#19151361

C₂

yes

maximal

C₂×#97×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₄×Dic₁₇₈

—

C₄×Dic₁₇₈

C₄

yes

complemented

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

—

C₂×C₄×Dic₈₉

—

C₂×C₄×Dic₈₉

—

#15667937

C₄

yes

complemented

#15667919

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#97×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083553

—

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×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

C₄²

yes

complemented

C₈×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

C₄×D₄

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄²

yes

characteristic

C₄×C₈₉

#117

yes

characteristic

C₄×C₈₉

C₄×D₄

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₄×C₂²⋊C₄

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#441

yes

—

#1763

—

maximal, Sylow

C₈₉

#1763

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₂×#103

—

C₂×#97

D₈₉

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₈₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

—

178

—

C₂×C₄²

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#103

—

#97

—

100

#95

—

178

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

178

—

104

C₂×Q₈

—

178

—

105

C₂²×C₄

—

178

—

106

C₂²×C₄

—

178

—

107

C₂²×C₄

—

178

—

108

C₂²×C₄

D₃₅₆

yes

characteristic

109

C₂²×C₄

D₃₅₆

yes

characteristic

110

C₂²×C₄

C₂²×D₈₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄×D₈₉

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

178

—

117

C₄⋊C₄

—

178

—

118

C₄⋊C₄

—

178

—

119

C₄²

—

356

—

120

C₄²

—

178

—

121

C₄²

C₄×D₈₉

yes

—

122

C₄²

C₂²⋊D₈₉

yes

—

123

C₂³

C₂×D₃₅₆

yes

characteristic

124

Q₈

—

125

Q₈

—

178

—

126

Q₈

—

178

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

356

—

131

C₂×C₄

—

356

—

132

C₂×C₄

—

356

—

133

C₂×C₄

—

356

—

134

C₂×C₄

—

178

—

135

C₂×C₄

—

178

—

136

C₂×C₄

—

178

—

137

C₂×C₄

—

178

—

138

C₂×C₄

#13083551

yes

—

139

C₂×C₄

#13083551

yes

—

140

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

141

C₂×C₄

C₂×C₄×D₈₉

yes

—

142

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

143

C₂×C₄

#13083574

yes

—

144

C₈

—

145

C₂²

C₂×#13083551

yes

characteristic

146

C₂²

C₄×D₃₅₆

yes

—

147

C₂²

#15668040

yes

characteristic

148

C₂²

#15668045

yes

—

149

C₂²

#15668045

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

356

—

153

C₄

—

356

—

154

C₄

—

178

—

155

C₄

C₄²×D₈₉

yes

characteristic

156

C₄

#15667969

yes

characteristic

157

C₄

C₄×C₂²⋊D₈₉

yes

—

158

C₂

C₄×#13083551

yes

characteristic

159

C₂

#19151600

yes

characteristic

160

C₂

#19151608

yes

characteristic

161

C₂

#19151700

yes

—

162

C₂

#19151706

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator	C₄×C₈₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×#97×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×#97	D₈₉
hypocenter	C₈₉	#1763
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator

C₄×C₈₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×#97×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂×#97

D₈₉

hypocenter

C₈₉

#1763

LCS

G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
8. C₄×Dic₁₇₈	C₄	semidirect
14. #15667937	C₄	semidirect
61. Dic₁₇₈	C₄²	semidirect
82. C₈₉	#1763	semidirect

H ≅ G/N

type

8. C₄×Dic₁₇₈

C₄

semidirect

14. #15667937

C₄

semidirect

61. Dic₁₇₈

C₄²

semidirect

82. C₈₉

#1763

semidirect

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

subgroup	index
C₂×C₄×Dic₁₇₈	2
C₂×#15667937	2
C₄×C₄⋊Dic₈₉	2
#19151361	2
C₂×#97×C₈₉	2
#1763	89

subgroup

index

C₂×C₄×Dic₁₇₈

C₂×#15667937

C₄×C₄⋊Dic₈₉

#19151361

C₂×#97×C₈₉

#1763

Subgroups — #26020083

Conjugacy classes of subgroups

Complements

Maximal subgroups