Group #26019259 — subgroups

Subgroups — #26019259

Order 11392.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19152229	C₂	1	1	yes	maximal, characteristic
2	#19151928	C₂	1	2	yes	maximal
3	#19151334	C₂	1	2	yes	maximal
4	#19151324	C₂	1	1	yes	maximal, characteristic
5	#398×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668011	—	2	2	no	—
7	#15668003	—	2	2	no	—
8	#15667991	—	2	1	no	—
9	#15667991	C₂²	1	2	yes	—
10	#15667971	—	2	1	no	—
11	#15667943	—	2	2	no	—
12	#15667935	—	2	2	no	—
13	C₂²⋊#10623494	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
15	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
17	#95×C₈₉	—	2	1	no	—
18	#95×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₇₈	—	2	1	no	—
20	C₂×Dic₁₇₈	D₄	1	2	yes	—
21	C₂×#10623494	—	2	2	no	—
22	C₄⋊Dic₈₉	—	2	1	no	—
23	C₄⋊Dic₈₉	—	2	1	no	—
24	C₄⋊Dic₈₉	—	2	1	no	—
25	C₄⋊Dic₈₉	D₄	1	2	yes	—
26	#13083553	—	2	2	no	—
27	#13083553	—	2	2	no	—
28	C₂²⋊Dic₈₉	—	2	2	no	—
29	C₄×Dic₈₉	—	2	2	no	—
30	C₂²×C₄×C₈₉	—	2	1	no	—
31	C₂²×C₄×C₈₉	—	2	1	no	—
32	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₄⋊C₄×C₈₉	—	4	2	no	—
38	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈₉	—	2	1	no	—
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	—	2	1	no	—
43	C₂×C₄×C₈₉	—	4	1	no	—
44	C₂×C₄×C₈₉	—	4	1	no	—
45	C₂×C₄×C₈₉	—	4	1	no	—
46	C₂×C₄×C₈₉	—	4	1	no	—
47	C₂×C₄×C₈₉	—	4	1	no	—
48	C₂×C₄×C₈₉	—	4	1	no	—
49	C₂×C₄×C₈₉	—	4	1	no	—
50	C₂×C₄×C₈₉	SD₁₆	1	2	yes	—
51	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₂×Dic₈₉	—	2	2	no	—
53	C₂×Dic₈₉	—	2	2	no	—
54	Dic₁₇₈	—	4	2	no	—
55	Dic₁₇₈	—	8	1	no	—
56	#10623494	—	4	2	no	—
57	C₂²×C₈₉	—	2	1	no	—
58	C₂²×C₈₉	—	2	1	no	—
59	C₂²×C₈₉	#120	1	1	yes	characteristic
60	C₂²×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₈₉	#137	1	1	yes	characteristic
62	C₄×C₈₉	—	2	2	no	—
63	C₄×C₈₉	—	2	2	no	—
64	C₄×C₈₉	—	4	2	no	—
65	C₄×C₈₉	—	4	1	no	—
66	C₄×C₈₉	—	8	1	no	—
67	Dic₈₉	—	4	2	no	—
68	Dic₈₉	—	4	2	no	—
69	C₂×C₈₉	—	2	1	no	—
70	C₂×C₈₉	—	2	1	no	—
71	C₂×C₈₉	#448	1	1	yes	characteristic
72	C₂×C₈₉	#456	1	1	yes	characteristic
73	C₂×C₈₉	#458	1	1	yes	characteristic
74	#1611	—	89	1	no	maximal, Sylow
75	C₈₉	#1611	1	1	yes	characteristic, complemented, Sylow
76	#544	—	89	1	no	—
77	#483	—	89	2	no	—
78	#398	D₈₉	1	1	yes	characteristic
79	#328	—	89	2	no	—
80	#324	—	89	1	no	—
81	C₄⋊Q₈	—	178	2	no	—
82	#125	—	178	2	no	—
83	C₂²⋊Q₈	—	89	2	no	—
84	C₂²⋊Q₈	—	178	1	no	—
85	#117	—	178	1	no	—
86	C₂×C₄⋊C₄	—	2	1	no	—
87	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
89	#106	—	178	2	no	—
90	#103	—	178	2	no	—
91	C₂²⋊C₈	—	89	2	no	—
92	#95	—	2	1	no	—
93	#95	C₂×D₈₉	1	1	yes	characteristic
94	C₂×Q₈	—	89	2	no	—
95	C₂×Q₈	—	178	1	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
101	C₂×C₈	—	178	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	89	2	no	—
106	C₄⋊C₄	—	178	1	no	—
107	C₄⋊C₄	—	178	1	no	—
108	C₄⋊C₄	—	178	1	no	—
109	C₄⋊C₄	—	178	2	no	—
110	C₄⋊C₄	—	178	2	no	—
111	C₂²⋊C₄	—	178	2	no	—
112	C₄²	—	178	2	no	—
113	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
114	Q₈	—	356	2	no	—
115	Q₈	—	712	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	178	2	no	—
128	C₂×C₄	—	178	2	no	—
129	C₂×C₄	#13083562	1	2	yes	—
130	C₂×C₄	D₄×D₈₉	1	2	yes	—
131	C₈	—	356	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#15667980	1	1	yes	characteristic
135	C₂²	C₂×#13083562	1	1	yes	characteristic
136	C₂²	#15668042	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	4	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	356	2	no	—
143	C₄	—	356	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#19151736	1	1	yes	characteristic
147	C₂	#19151780	1	1	yes	characteristic
148	C₂	#19151787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152229

C₂

yes

maximal, characteristic

#19151928

C₂

yes

maximal

#19151334

C₂

yes

maximal

#19151324

C₂

yes

maximal, characteristic

#398×C₈₉

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

#95×C₈₉

—

#95×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂²×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

#137

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#544

—

#483

—

#398

D₈₉

yes

characteristic

#328

—

#324

—

C₄⋊Q₈

—

178

—

#125

—

178

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

178

—

#117

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

#106

—

178

—

#103

—

178

—

C₂²⋊C₈

—

#95

—

#95

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

100

C₂²×C₄

C₂²×D₈₉

yes

characteristic

101

C₂×C₈

—

178

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

—

178

—

108

C₄⋊C₄

—

178

—

109

C₄⋊C₄

—

178

—

110

C₄⋊C₄

—

178

—

111

C₂²⋊C₄

—

178

—

112

C₄²

—

178

—

113

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

114

Q₈

—

356

—

115

Q₈

—

712

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

178

—

128

C₂×C₄

—

178

—

129

C₂×C₄

#13083562

yes

—

130

C₂×C₄

D₄×D₈₉

yes

—

131

C₈

—

356

—

132

C₂²

—

133

C₂²

—

134

C₂²

#15667980

yes

characteristic

135

C₂²

C₂×#13083562

yes

characteristic

136

C₂²

#15668042

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

356

—

143

C₄

—

356

—

144

C₂

—

145

C₂

—

146

C₂

#19151736

yes

characteristic

147

C₂

#19151780

yes

characteristic

148

C₂

#19151787

yes

characteristic

	subgroup	G/H
center	C₂²	#15667980
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#398×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#398	D₈₉
hypocenter	C₈₉	#1611
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #398

subgroup

G/H

center

C₂²

#15667980

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#398×C₈₉

C₂

socle

C₂²×C₈₉

#120

prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #398

N	H ≅ G/N	type
75. C₈₉	#1611	semidirect

H ≅ G/N

type

75. C₈₉

#1611

semidirect

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

subgroup	index
#19152229	2
#19151928	2
#19151334	2
#19151324	2
#398×C₈₉	2
#1611	89

subgroup

index

#398×C₈₉

Subgroups — #26019259

Conjugacy classes of subgroups

Complements

Maximal subgroups