Group #26033257 — subgroups

Subgroups — #26033257

Order 11392.

348 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19152326	C₂	1	2	yes	maximal
2	#19151978	C₂	1	1	yes	maximal, characteristic
3	#19151976	C₂	1	2	yes	maximal
4	#19151808	C₂	1	2	yes	maximal
5	#19151807	C₂	1	4	yes	maximal
6	C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
7	#19151692	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₅₆	C₂²	1	2	yes	—
10	C₂×Dic₃₅₆	C₂²	1	2	yes	—
11	C₂×#13083564	C₂²	1	4	yes	—
12	#15668012	C₂²	1	2	yes	—
13	#15668010	C₂²	1	4	yes	—
14	#15668004	—	2	2	no	—
15	#15668002	—	2	4	no	—
16	C₄×Dic₁₇₈	—	2	2	no	—
17	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
18	#15667977	C₂²	1	4	yes	—
19	C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	#15667941	C₂²	1	2	yes	—
21	#15667935	C₂²	1	4	yes	—
22	C₈×Dic₈₉	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₈₉	C₂²	1	2	yes	—
25	#107×C₈₉	C₂²	1	1	yes	characteristic
26	#103×C₈₉	C₂²	1	2	yes	—
27	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
29	C₂×Dic₁₇₈	C₂³	1	4	yes	—
30	Dic₃₅₆	—	2	4	no	—
31	Dic₃₅₆	—	4	2	no	—
32	#13083564	—	4	4	no	—
33	C₂×#10623494	C₂³	1	2	yes	—
34	C₄⋊Dic₈₉	—	2	2	no	—
35	C₄⋊Dic₈₉	—	2	2	no	—
36	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
37	#13083553	—	2	4	no	—
38	#13083553	—	2	4	no	—
39	#13083553	—	2	4	no	—
40	#13083553	D₄	1	2	yes	—
41	C₄×Dic₈₉	—	2	4	no	—
42	C₄×Dic₈₉	C₂³	1	2	yes	—
43	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
44	Q₁₆×C₈₉	—	2	2	no	—
45	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₈₉	—	2	2	no	—
48	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
49	C₄²×C₈₉	—	2	2	no	—
50	C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	Q₈×C₈₉	—	2	2	no	—
52	Q₈×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	—	2	2	no	—
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₈₉	—	2	4	no	—
59	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
60	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
61	Dic₁₇₈	—	2	4	no	—
62	Dic₁₇₈	—	2	4	no	—
63	Dic₁₇₈	Q₁₆	1	4	yes	complemented
64	C₈×C₈₉	—	2	1	no	—
65	C₈×C₈₉	#41	1	2	yes	—
66	#10623494	—	2	2	no	—
67	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	—	4	2	no	—
70	C₄×C₈₉	—	4	2	no	—
71	C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
72	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
73	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
74	Dic₈₉	—	2	2	no	—
75	Dic₈₉	—	4	4	no	—
76	Dic₈₉	C₂×Q₁₆	1	4	yes	—
77	C₂×C₈₉	#550	1	1	yes	characteristic
78	C₂×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
79	C₂×C₈₉	#578	1	1	yes	characteristic
80	#3395	—	89	1	no	maximal, Sylow
81	C₈₉	#3395	1	1	yes	characteristic, complemented, Sylow
82	#557	—	89	2	no	—
83	#494	—	89	2	no	—
84	#494	—	89	1	no	—
85	#462	—	89	2	no	—
86	#462	—	89	4	no	—
87	C₈×Q₈	—	89	1	no	—
88	C₄×Q₁₆	—	89	2	no	—
89	C₄×Q₁₆	D₈₉	1	1	yes	characteristic
90	C₂×Q₁₆	—	89	4	no	—
91	C₂×Q₁₆	—	89	2	no	—
92	C₂×Q₁₆	—	89	2	no	—
93	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
94	C₄⋊Q₈	—	89	4	no	—
95	C₄⋊Q₈	—	89	2	no	—
96	#125	—	178	2	no	—
97	#125	—	178	4	no	—
98	C₄×Q₈	—	89	1	no	—
99	C₄×Q₈	—	89	4	no	—
100	C₄×Q₈	—	178	2	no	—
101	C₄×Q₈	C₂×D₈₉	1	2	yes	—
102	#107	C₂×D₈₉	1	1	yes	characteristic
103	C₄⋊C₈	—	89	2	no	—
104	C₄⋊C₈	—	89	1	no	—
105	#103	—	89	4	no	—
106	#103	C₂×D₈₉	1	2	yes	—
107	C₄×C₈	—	89	2	no	—
108	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
109	C₂×Q₈	—	89	1	no	—
110	C₂×Q₈	—	89	4	no	—
111	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	356	2	no	—
114	Q₁₆	—	356	4	no	—
115	Q₁₆	—	178	4	no	—
116	C₂×C₈	—	89	2	no	—
117	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	89	2	no	—
121	C₄⋊C₄	—	89	1	no	—
122	C₄⋊C₄	—	178	4	no	—
123	C₄⋊C₄	—	178	4	no	—
124	C₄⋊C₄	—	178	4	no	—
125	C₄⋊C₄	—	178	2	no	—
126	C₄⋊C₄	—	178	2	no	—
127	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
128	C₄²	—	2	2	no	—
129	C₄²	—	89	2	no	—
130	C₄²	—	178	4	no	—
131	C₄²	C₂²×D₈₉	1	1	yes	characteristic
132	Q₈	—	2	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	89	4	no	—
135	Q₈	—	178	4	no	—
136	Q₈	—	178	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	89	2	no	—
140	C₂×C₄	—	89	2	no	—
141	C₂×C₄	—	178	4	no	—
142	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
144	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	178	2	no	—
147	C₈	#13083574	1	2	yes	—
148	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	89	4	no	—
153	C₄	—	356	4	no	—
154	C₄	—	178	2	no	—
155	C₄	Q₁₆×D₈₉	1	2	yes	—
156	C₄	C₂×#13083574	1	1	yes	characteristic
157	C₄	#15668058	1	1	yes	characteristic
158	C₂	#19152269	1	1	yes	characteristic
159	C₂	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
160	C₂	#19152412	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152326

C₂

yes

maximal

#19151978

C₂

yes

maximal, characteristic

#19151976

C₂

yes

maximal

#19151808

C₂

yes

maximal

#19151807

C₂

yes

maximal

C₈×Dic₁₇₈

C₂

yes

maximal, characteristic

#19151692

C₂

yes

maximal

C₄×Q₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

—

C₂×Dic₃₅₆

C₂²

yes

—

C₂×#13083564

C₂²

yes

—

#15668012

C₂²

yes

—

#15668010

C₂²

yes

—

#15668004

—

#15668002

—

C₄×Dic₁₇₈

—

C₄×Dic₁₇₈

C₂²

yes

characteristic

#15667977

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

characteristic

#15667941

C₂²

yes

—

#15667935

C₂²

yes

—

C₈×Dic₈₉

C₂²

yes

—

C₂×Q₁₆×C₈₉

C₂²

yes

characteristic

C₄×Q₈×C₈₉

C₂²

yes

—

#107×C₈₉

C₂²

yes

characteristic

#103×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×Dic₁₇₈

C₂³

yes

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083564

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

—

—

D₄

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂³

yes

—

C₂×Q₈×C₈₉

C₂³

yes

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

Q₁₆

yes

complemented

C₈×C₈₉

—

C₈×C₈₉

#41

yes

—

#10623494

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₁₆

yes

—

C₄×C₈₉

C₂×#41

yes

characteristic

C₄×C₈₉

H(5,𝔽₂)

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₂×Q₁₆

yes

—

C₂×C₈₉

#550

yes

characteristic

C₂×C₈₉

C₂²×Q₁₆

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

178

—

#125

—

178

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

—

178

—

101

C₄×Q₈

C₂×D₈₉

yes

—

102

#107

C₂×D₈₉

yes

characteristic

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

106

#103

C₂×D₈₉

yes

—

107

C₄×C₈

—

108

C₄×C₈

C₂×D₈₉

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

C₂²×D₈₉

yes

—

112

Q₁₆

—

113

Q₁₆

—

356

—

114

Q₁₆

—

356

—

115

Q₁₆

—

178

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₈₉

yes

characteristic

118

C₂×C₈

C₂²×D₈₉

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

178

—

123

C₄⋊C₄

—

178

—

124

C₄⋊C₄

—

178

—

125

C₄⋊C₄

—

178

—

126

C₄⋊C₄

—

178

—

127

C₄⋊C₄

C₂²×D₈₉

yes

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

178

—

131

C₄²

C₂²×D₈₉

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

178

—

136

Q₈

—

178

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

178

—

142

C₂×C₄

D₄×D₈₉

yes

characteristic

143

C₂×C₄

D₄×D₈₉

yes

characteristic

144

C₂×C₄

C₂³×D₈₉

yes

characteristic

145

C₈

—

146

C₈

—

178

—

147

C₈

#13083574

yes

—

148

C₂²

C₂×D₄×D₈₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

356

—

154

C₄

—

178

—

155

C₄

Q₁₆×D₈₉

yes

—

156

C₄

C₂×#13083574

yes

characteristic

157

C₄

#15668058

yes

characteristic

158

C₂

#19152269

yes

characteristic

159

C₂

C₂×Q₁₆×D₈₉

yes

characteristic

160

C₂

#19152412

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₄×Q₁₆×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂²×D₄
hypercenter	C₄×Q₁₆	D₈₉
hypocenter	C₈₉	#3395
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₄×Q₁₆×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₂²×D₄

hypercenter

C₄×Q₁₆

D₈₉

hypocenter

C₈₉

#3395

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
63. Dic₁₇₈	Q₁₆	semidirect
81. C₈₉	#3395	semidirect

H ≅ G/N

type

63. Dic₁₇₈

Q₁₆

semidirect

81. C₈₉

#3395

semidirect

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

subgroup	index
#19152326	2
#19151978	2
#19151976	2
#19151808	2
#19151807	2
C₈×Dic₁₇₈	2
#19151692	2
C₄×Q₁₆×C₈₉	2
#3395	89

subgroup

index

C₈×Dic₁₇₈

C₄×Q₁₆×C₈₉

Subgroups — #26033257

Conjugacy classes of subgroups

Complements

Maximal subgroups