Group #26021745 — subgroups

Subgroups — #26021745

Order 11392.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#15667946	C₂	1	1	yes	maximal, characteristic
2	#19151496	C₂	1	2	yes	maximal
3	#19151361	C₂	1	2	yes	maximal
4	#19151361	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
8	C₂×#13083553	—	2	2	no	—
9	#15667946	—	2	2	no	—
10	#15667919	—	2	2	no	—
11	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₉	—	2	2	no	—
15	C₂²×Dic₈₉	—	2	2	no	—
16	C₂²×Dic₈₉	—	2	2	no	—
17	C₄⋊Dic₈₉	—	2	2	no	—
18	C₄⋊Dic₈₉	—	2	2	no	—
19	C₄⋊Dic₈₉	—	2	2	no	—
20	C₄⋊Dic₈₉	Q₈	1	4	yes	—
21	#13083553	—	4	4	no	—
22	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₈₉	—	2	2	no	—
26	C₂×C₈×C₈₉	—	2	1	no	—
27	C₂×C₈×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	D₄	1	2	yes	—
29	C₄²×C₈₉	—	2	1	no	—
30	C₄²×C₈₉	—	2	1	no	—
31	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	—	2	1	no	—
33	C₂×C₄×C₈₉	—	2	1	no	—
34	C₂×C₄×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈₉	—	2	2	no	—
36	C₂×C₄×C₈₉	Q₁₆	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₈₉	#41	1	2	yes	—
40	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
41	C₂×Dic₈₉	—	2	4	no	—
42	C₂×Dic₈₉	—	4	2	no	—
43	C₂×Dic₈₉	—	4	2	no	—
44	C₂×Dic₈₉	—	4	2	no	—
45	C₂×Dic₈₉	—	4	2	no	—
46	C₂×Dic₈₉	—	4	2	no	—
47	C₈×C₈₉	—	2	2	no	—
48	C₈×C₈₉	—	4	1	no	—
49	C₂²×C₈₉	#121	1	1	yes	characteristic
50	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₂²×C₈₉	#135	1	1	yes	characteristic
53	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	—	4	1	no	—
56	C₄×C₈₉	#123	1	1	yes	characteristic
57	C₄×C₈₉	#125	1	2	yes	—
58	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₈₉	—	4	4	no	—
60	Dic₈₉	—	8	2	no	—
61	C₂×C₈₉	#398	1	1	yes	characteristic
62	C₂×C₈₉	#467	1	1	yes	characteristic
63	C₂×C₈₉	#469	1	1	yes	characteristic
64	C₂×C₈₉	#477	1	2	yes	—
65	C₂×C₈₉	#478	1	2	yes	—
66	#2087	—	89	1	no	maximal, Sylow
67	C₈₉	#2087	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	89	1	no	—
69	C₂×C₄⋊C₈	D₈₉	1	1	yes	characteristic
70	#389	—	89	2	no	—
71	#340	—	89	1	no	—
72	#340	—	89	2	no	—
73	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	89	2	no	—
76	C₂×C₄⋊C₄	—	89	2	no	—
77	C₂×C₄⋊C₄	—	178	2	no	—
78	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
79	#107	—	178	2	no	—
80	C₄⋊C₈	—	2	2	no	—
81	#95	—	178	2	no	—
82	C₂²×C₄	—	178	2	no	—
83	C₂²×C₄	—	178	2	no	—
84	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
85	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
91	C₄⋊C₄	—	89	4	no	—
92	C₄⋊C₄	—	356	4	no	—
93	C₄⋊C₄	—	178	2	no	—
94	C₄⋊C₄	—	178	2	no	—
95	C₄⋊C₄	—	178	2	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	356	2	no	—
104	C₂×C₄	—	356	2	no	—
105	C₂×C₄	—	356	2	no	—
106	C₂×C₄	—	356	2	no	—
107	C₂×C₄	—	356	2	no	—
108	C₂×C₄	—	178	4	no	—
109	C₂×C₄	Dic₃₅₆	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₈₉	1	2	yes	—
112	C₂×C₄	#13083573	1	2	yes	—
113	C₂×C₄	#13083574	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	#15667986	1	1	yes	characteristic
117	C₂²	#15667992	1	2	yes	—
118	C₂²	C₂×Dic₃₅₆	1	1	yes	characteristic
119	C₂²	#15668034	1	1	yes	characteristic
120	C₂²	#15668040	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	356	4	no	—
124	C₄	—	712	2	no	—
125	C₄	#15667993	1	1	yes	characteristic
126	C₄	#15668002	1	2	yes	—
127	C₄	#15668010	1	1	yes	characteristic
128	C₂	#19151528	1	1	yes	characteristic
129	C₂	#19151839	1	1	yes	characteristic
130	C₂	#19151851	1	1	yes	characteristic
131	C₂	#19151894	1	2	yes	—
132	C₂	#19151899	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667946

C₂

yes

maximal, characteristic

#19151496

C₂

yes

maximal

#19151361

C₂

yes

maximal

#19151361

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

—

#15667946

—

#15667919

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

Q₈

yes

—

#13083553

—

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×C₈×C₈₉

D₄

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

—

C₂²×C₈₉

C₂×Q₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#123

yes

characteristic

C₄×C₈₉

#125

yes

—

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#478

yes

—

#2087

—

maximal, Sylow

C₈₉

#2087

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×C₄⋊C₈

D₈₉

yes

—

—

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#107

—

178

—

C₄⋊C₈

—

#95

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

C₄²

—

C₂³

C₂×D₃₅₆

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

356

—

104

C₂×C₄

—

356

—

105

C₂×C₄

—

356

—

106

C₂×C₄

—

356

—

107

C₂×C₄

—

356

—

108

C₂×C₄

—

178

—

109

C₂×C₄

Dic₃₅₆

yes

—

110

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

111

C₂×C₄

Q₈×D₈₉

yes

—

112

C₂×C₄

#13083573

yes

—

113

C₂×C₄

#13083574

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#15667986

yes

characteristic

117

C₂²

#15667992

yes

—

118

C₂²

C₂×Dic₃₅₆

yes

characteristic

119

C₂²

#15668034

yes

characteristic

120

C₂²

#15668040

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

356

—

124

C₄

—

712

—

125

C₄

#15667993

yes

characteristic

126

C₄

#15668002

yes

—

127

C₄

#15668010

yes

characteristic

128

C₂

#19151528

yes

characteristic

129

C₂

#19151839

yes

characteristic

130

C₂

#19151851

yes

characteristic

131

C₂

#19151894

yes

—

132

C₂

#19151899

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#2087
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×C₄⋊C₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₈₉

hypocenter

C₈₉

#2087

LCS

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
67. C₈₉	#2087	semidirect

H ≅ G/N

type

67. C₈₉

#2087

semidirect

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

subgroup	index
C₂×#15667946	2
#19151496	2
#19151361	2
C₂×C₄⋊C₈×C₈₉	2
#2087	89

subgroup

index

C₂×C₄⋊C₈×C₈₉

Subgroups — #26021745

Conjugacy classes of subgroups

Complements

Maximal subgroups