Group #26019634 — subgroups

Subgroups — #26019634

Order 11392.

150 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 61 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	#19151996	C₂	1	1	yes	maximal, characteristic
2	#19151893	C₂	1	1	yes	maximal, characteristic
3	#19151808	C₂	1	1	yes	maximal, characteristic
4	#19151349	C₂	1	1	yes	maximal, characteristic
5	#19151346	C₂	1	1	yes	maximal, characteristic
6	#19151343	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13083564	—	2	1	no	—
9	#15668012	C₂²	1	1	yes	characteristic
10	#15668012	C₂²	1	1	yes	characteristic
11	#15667945	—	2	1	no	—
12	#15667944	—	2	1	no	—
13	#15667944	—	2	1	no	—
14	C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	C₄⋊#10623494	C₂²	1	1	yes	characteristic
16	#15667936	—	2	1	no	—
17	#15667935	—	2	1	no	—
18	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₈₉	—	2	1	no	—
20	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₈₉	—	2	1	no	—
22	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₈	—	2	1	no	—
24	C₂×Dic₁₇₈	—	2	1	no	—
25	#13083564	—	4	2	no	—
26	C₂×#10623494	—	2	1	no	—
27	C₂×#10623494	—	2	1	no	—
28	C₄⋊Dic₈₉	—	2	1	no	—
29	C₄⋊Dic₈₉	—	2	1	no	—
30	C₄⋊Dic₈₉	—	2	1	no	—
31	C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₈₉	—	2	1	no	—
34	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₈₉	—	2	1	no	—
37	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
38	C₄²×C₈₉	—	2	1	no	—
39	C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	Q₈×C₈₉	—	2	1	no	—
41	Q₈×C₈₉	SD₁₆	1	2	yes	—
42	C₂×C₄×C₈₉	—	2	1	no	—
43	C₂×C₄×C₈₉	—	2	1	no	—
44	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
47	C₂×Dic₈₉	—	4	1	no	—
48	C₂×Dic₈₉	—	4	1	no	—
49	Dic₁₇₈	—	4	1	no	—
50	Dic₁₇₈	—	4	1	no	—
51	Dic₁₇₈	—	8	1	no	—
52	C₈×C₈₉	—	2	1	no	—
53	C₈×C₈₉	—	4	1	no	—
54	C₈×C₈₉	Q₁₆	1	2	yes	complemented
55	#10623494	—	4	1	no	—
56	#10623494	—	4	1	no	—
57	C₂²×C₈₉	#121	1	1	yes	characteristic
58	C₄×C₈₉	—	2	2	no	—
59	C₄×C₈₉	—	2	1	no	—
60	C₄×C₈₉	—	4	1	no	—
61	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₈₉	#135	1	1	yes	characteristic
64	C₄×C₈₉	#137	1	1	yes	characteristic
65	Dic₈₉	—	8	1	no	—
66	Dic₈₉	—	8	1	no	—
67	C₂×C₈₉	#462	1	1	yes	characteristic
68	C₂×C₈₉	#465	1	1	yes	characteristic
69	C₂×C₈₉	#471	1	1	yes	characteristic
70	#1672	—	89	1	no	maximal, Sylow
71	C₈₉	#1672	1	1	yes	characteristic, complemented, Sylow
72	#498	—	89	1	no	—
73	#477	—	89	1	no	—
74	#462	—	89	1	no	—
75	C₈×Q₈	D₈₉	1	1	yes	characteristic
76	#334	—	89	1	no	—
77	#333	—	89	1	no	—
78	#332	—	89	1	no	—
79	C₂×Q₁₆	—	178	1	no	—
80	C₄⋊Q₈	—	89	1	no	—
81	C₄⋊Q₈	—	89	1	no	—
82	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
83	#107	—	178	1	no	—
84	#106	—	178	1	no	—
85	#106	—	178	1	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	—	89	1	no	—
88	C₄⋊C₈	—	89	1	no	—
89	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
90	#103	—	178	1	no	—
91	#103	—	178	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
94	C₂×Q₈	—	178	1	no	—
95	C₂×Q₈	—	178	1	no	—
96	C₂×Q₈	D₃₅₆	1	1	yes	characteristic
97	Q₁₆	—	356	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	178	1	no	—
100	C₂×C₈	—	178	1	no	—
101	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	178	1	no	—
105	C₄⋊C₄	—	178	1	no	—
106	C₄⋊C₄	—	178	1	no	—
107	C₄⋊C₄	—	178	1	no	—
108	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₈₉	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	356	1	no	—
113	Q₈	—	356	1	no	—
114	Q₈	—	712	1	no	—
115	Q₈	#13083563	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	356	1	no	—
119	C₂×C₄	—	356	1	no	—
120	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
121	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
122	C₂×C₄	#13083574	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	356	1	no	—
126	C₈	—	356	1	no	—
127	C₈	#13083564	1	2	yes	complemented
128	C₂²	#15667986	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	712	1	no	—
133	C₄	—	712	1	no	—
134	C₄	C₂×#13083563	1	1	yes	characteristic
135	C₄	C₂×#13083564	1	1	yes	characteristic
136	C₄	#15668034	1	1	yes	characteristic
137	C₄	#15668042	1	1	yes	characteristic
138	C₂	#19151808	1	1	yes	characteristic
139	C₂	#19151827	1	1	yes	characteristic
140	C₂	#19151861	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151996

C₂

yes

maximal, characteristic

#19151893

C₂

yes

maximal, characteristic

#19151808

C₂

yes

maximal, characteristic

#19151349

C₂

yes

maximal, characteristic

#19151346

C₂

yes

maximal, characteristic

#19151343

C₂

yes

maximal, characteristic

C₈×Q₈×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083564

—

#15668012

C₂²

yes

characteristic

#15668012

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

characteristic

#15667936

—

#15667935

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

#13083564

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

Q₁₆

yes

complemented

#10623494

—

#10623494

—

C₂²×C₈₉

#121

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₄×C₈₉

C₂×Q₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#498

—

#477

—

#462

—

C₈×Q₈

D₈₉

yes

—

—

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₈₉

yes

characteristic

#107

—

178

—

#106

—

178

—

#106

—

178

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₉

yes

characteristic

#103

—

178

—

#103

—

178

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

D₃₅₆

yes

characteristic

Q₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

178

—

100

C₂×C₈

—

178

—

101

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

102

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

178

—

105

C₄⋊C₄

—

178

—

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

—

178

—

108

C₄⋊C₄

D₃₅₆

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₈₉

yes

characteristic

111

Q₈

—

112

Q₈

—

356

—

113

Q₈

—

356

—

114

Q₈

—

712

—

115

Q₈

#13083563

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

356

—

119

C₂×C₄

—

356

—

120

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

121

C₂×C₄

C₂×D₃₅₆

yes

characteristic

122

C₂×C₄

#13083574

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

356

—

126

C₈

—

356

—

127

C₈

#13083564

yes

complemented

128

C₂²

#15667986

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

712

—

133

C₄

—

712

—

134

C₄

C₂×#13083563

yes

characteristic

135

C₄

C₂×#13083564

yes

characteristic

136

C₄

#15668034

yes

characteristic

137

C₄

#15668042

yes

characteristic

138

C₂

#19151808

yes

characteristic

139

C₂

#19151827

yes

characteristic

140

C₂

#19151861

yes

characteristic

	subgroup	G/H
center	C₂²	#15667986
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#121
hypercenter	C₈×Q₈	D₈₉
hypocenter	C₈₉	#1672
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#15667986

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

C₈×Q₈×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

#121

hypercenter

C₈×Q₈

D₈₉

hypocenter

C₈₉

#1672

LCS

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
54. C₈×C₈₉	Q₁₆	semidirect
71. C₈₉	#1672	semidirect
127. C₈	#13083564	semidirect

H ≅ G/N

type

54. C₈×C₈₉

Q₁₆

semidirect

71. C₈₉

#1672

semidirect

127. C₈

#13083564

semidirect

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

subgroup	index
#19151996	2
#19151893	2
#19151808	2
#19151349	2
#19151346	2
#19151343	2
C₈×Q₈×C₈₉	2
#1672	89

subgroup

index

C₈×Q₈×C₈₉

Subgroups — #26019634

Conjugacy classes of subgroups

Complements

Maximal subgroups