Group #26019409 — subgroups

Subgroups — #26019409

Order 11392.

188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19152327	C₂	1	1	yes	maximal, characteristic
2	#19151948	C₂	1	1	yes	maximal, characteristic
3	#19151890	C₂	1	1	yes	maximal, characteristic
4	#19151343	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#10623494	C₂	1	1	yes	maximal, characteristic
6	#19151328	C₂	1	1	yes	maximal, characteristic
7	#477×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#15668012	—	2	1	no	—
9	#15668012	C₂²	1	1	yes	characteristic
10	#15668003	—	2	2	no	—
11	#15668003	—	2	1	no	—
12	Q₈×Dic₈₉	—	2	1	no	—
13	C₄×Dic₁₇₈	—	2	1	no	—
14	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
15	#15667945	—	2	1	no	—
16	C₄⋊#10623494	C₂²	1	1	yes	characteristic
17	#15667936	—	2	2	no	—
18	#15667935	—	2	1	no	—
19	C₄×#10623494	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	#107×C₈₉	—	2	1	no	—
23	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#103×C₈₉	—	2	1	no	—
25	C₂×Dic₁₇₈	—	2	1	no	—
26	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
27	C₂×#10623494	—	2	1	no	—
28	C₂×#10623494	—	2	1	no	—
29	C₄⋊Dic₈₉	—	2	2	no	—
30	C₄⋊Dic₈₉	—	2	2	no	—
31	C₄⋊Dic₈₉	—	2	1	no	—
32	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
33	C₄⋊Dic₈₉	D₄	1	2	yes	—
34	#13083553	—	2	2	no	—
35	#13083553	—	2	1	no	—
36	C₄×Dic₈₉	—	2	2	no	—
37	C₄×Dic₈₉	—	2	1	no	—
38	C₂×Q₈×C₈₉	—	2	1	no	—
39	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	—	2	1	no	—
41	C₄⋊C₄×C₈₉	—	2	1	no	—
42	C₄⋊C₄×C₈₉	—	2	1	no	—
43	C₄⋊C₄×C₈₉	—	2	1	no	—
44	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
45	C₄²×C₈₉	—	2	1	no	—
46	C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	Q₈×C₈₉	—	2	1	no	—
48	Q₈×C₈₉	—	4	2	no	—
49	Q₈×C₈₉	Q₁₆	1	2	yes	—
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	4	1	no	—
53	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₈₉	—	2	1	no	—
57	C₂×Dic₈₉	—	2	1	no	—
58	C₂×Dic₈₉	—	2	2	no	—
59	Dic₁₇₈	—	2	1	no	—
60	Dic₁₇₈	—	2	1	no	—
61	Dic₁₇₈	—	4	1	no	—
62	Dic₁₇₈	—	4	1	no	—
63	C₈×C₈₉	—	4	1	no	—
64	#10623494	—	4	1	no	—
65	#10623494	—	4	1	no	—
66	C₂²×C₈₉	#120	1	1	yes	characteristic
67	C₄×C₈₉	—	2	2	no	—
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	—	4	1	no	—
70	C₄×C₈₉	—	8	1	no	—
71	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₄×C₈₉	#135	1	1	yes	characteristic
73	C₄×C₈₉	#137	1	1	yes	characteristic
74	C₄×C₈₉	#137	1	1	yes	characteristic
75	Dic₈₉	—	4	1	no	—
76	Dic₈₉	—	4	2	no	—
77	Dic₈₉	—	4	1	no	—
78	C₂×C₈₉	#451	1	1	yes	characteristic
79	C₂×C₈₉	#452	1	1	yes	characteristic
80	C₂×C₈₉	#454	1	1	yes	characteristic
81	#1636	—	89	1	no	maximal, Sylow
82	C₈₉	#1636	1	1	yes	characteristic, complemented, Sylow
83	#557	—	89	1	no	—
84	#487	—	89	1	no	—
85	#477	—	89	1	no	—
86	#477	D₈₉	1	1	yes	characteristic
87	#332	—	89	1	no	—
88	Q₈⋊C₈	—	89	1	no	—
89	Q₈⋊C₈	—	89	1	no	—
90	C₄⋊Q₈	—	89	1	no	—
91	C₄⋊Q₈	—	178	1	no	—
92	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
93	#125	—	178	2	no	—
94	#125	—	178	1	no	—
95	C₄×Q₈	—	89	1	no	—
96	C₄×Q₈	—	178	1	no	—
97	C₄×Q₈	—	178	1	no	—
98	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
99	#107	—	2	1	no	—
100	#107	—	178	1	no	—
101	C₄⋊C₈	—	89	1	no	—
102	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
103	#103	—	2	1	no	—
104	#103	—	178	1	no	—
105	#103	—	178	2	no	—
106	C₄×C₈	—	89	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	89	1	no	—
109	C₂×Q₈	—	178	1	no	—
110	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	178	1	no	—
113	C₂×C₈	—	178	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	89	2	no	—
118	C₄⋊C₄	—	89	1	no	—
119	C₄⋊C₄	—	178	2	no	—
120	C₄⋊C₄	—	178	2	no	—
121	C₄⋊C₄	—	178	2	no	—
122	C₄⋊C₄	—	178	1	no	—
123	C₄⋊C₄	—	178	1	no	—
124	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	—	178	2	no	—
127	C₄²	—	178	1	no	—
128	C₄²	C₂²×D₈₉	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	356	1	no	—
132	Q₈	—	356	1	no	—
133	Q₈	—	178	1	no	—
134	Q₈	—	178	1	no	—
135	Q₈	#13083564	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	178	1	no	—
140	C₂×C₄	—	178	2	no	—
141	C₂×C₄	—	178	1	no	—
142	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	356	1	no	—
147	C₈	—	356	1	no	—
148	C₂²	#15667980	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	8	1	no	—
153	C₄	—	356	2	no	—
154	C₄	—	356	1	no	—
155	C₄	—	356	1	no	—
156	C₄	C₂×#13083564	1	1	yes	characteristic
157	C₄	#15668035	1	1	yes	characteristic
158	C₄	#15668042	1	1	yes	characteristic
159	C₄	#15668044	1	1	yes	characteristic
160	C₂	#19151753	1	1	yes	characteristic
161	C₂	#19151759	1	1	yes	characteristic
162	C₂	#19151769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152327

C₂

yes

maximal, characteristic

#19151948

C₂

yes

maximal, characteristic

#19151890

C₂

yes

maximal, characteristic

#19151343

C₂

yes

maximal, characteristic

Q₈⋊#10623494

C₂

yes

maximal, characteristic

#19151328

C₂

yes

maximal, characteristic

#477×C₈₉

C₂

yes

maximal, characteristic

#15668012

—

#15668012

C₂²

yes

characteristic

#15668003

—

#15668003

—

Q₈×Dic₈₉

—

C₄×Dic₁₇₈

—

C₄×Dic₁₇₈

C₂²

yes

characteristic

#15667945

—

C₄⋊#10623494

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄×Q₈×C₈₉

C₂²

yes

characteristic

#107×C₈₉

—

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

#103×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

characteristic

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

#120

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic

#332

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

#125

—

178

—

#125

—

178

—

C₄×Q₈

—

C₄×Q₈

—

178

—

C₄×Q₈

—

178

—

C₄×Q₈

C₂×D₈₉

yes

characteristic

#107

—

100

#107

—

178

—

101

C₄⋊C₈

—

102

C₄⋊C₈

C₂×D₈₉

yes

characteristic

103

#103

—

104

#103

—

178

—

105

#103

—

178

—

106

C₄×C₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

178

—

110

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

178

—

113

C₂×C₈

—

178

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

178

—

120

C₄⋊C₄

—

178

—

121

C₄⋊C₄

—

178

—

122

C₄⋊C₄

—

178

—

123

C₄⋊C₄

—

178

—

124

C₄⋊C₄

C₂²⋊D₈₉

yes

characteristic

125

C₄²

—

126

C₄²

—

178

—

127

C₄²

—

178

—

128

C₄²

C₂²×D₈₉

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

356

—

132

Q₈

—

356

—

133

Q₈

—

178

—

134

Q₈

—

178

—

135

Q₈

#13083564

yes

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

178

—

140

C₂×C₄

—

178

—

141

C₂×C₄

—

178

—

142

C₂×C₄

D₄×D₈₉

yes

characteristic

143

C₂×C₄

D₄×D₈₉

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

145

C₈

—

146

C₈

—

356

—

147

C₈

—

356

—

148

C₂²

#15667980

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

356

—

154

C₄

—

356

—

155

C₄

—

356

—

156

C₄

C₂×#13083564

yes

characteristic

157

C₄

#15668035

yes

characteristic

158

C₄

#15668042

yes

characteristic

159

C₄

#15668044

yes

characteristic

160

C₂

#19151753

yes

characteristic

161

C₂

#19151759

yes

characteristic

162

C₂

#19151769

yes

characteristic

	subgroup	G/H
center	C₂²	#15667980
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#477×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#120
hypercenter	#477	D₈₉
hypocenter	C₈₉	#1636
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#15667980

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#477×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
82. C₈₉	#1636	semidirect

H ≅ G/N

type

82. C₈₉

#1636

semidirect

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

subgroup	index
#19152327	2
#19151948	2
#19151890	2
#19151343	2
Q₈⋊#10623494	2
#19151328	2
#477×C₈₉	2
#1636	89

subgroup

index

#477×C₈₉

Subgroups — #26019409

Conjugacy classes of subgroups

Complements

Maximal subgroups