Group #26021747 — subgroups

Subgroups — #26021747

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	C₂×#15667941	C₂	1	1	yes	maximal, characteristic
3	#19151497	C₂	1	2	yes	maximal
4	#19151360	C₂	1	2	yes	maximal
5	#340×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10623494	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₈₉	—	2	2	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
10	#15667946	—	2	2	no	—
11	#15667941	—	2	2	no	—
12	#15667919	—	2	2	no	—
13	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
15	C₂²×Dic₈₉	—	2	2	no	—
16	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
18	C₂×#10623494	—	2	1	no	—
19	C₂×#10623494	—	2	2	no	—
20	C₄⋊Dic₈₉	—	2	2	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₄⋊Dic₈₉	—	4	4	no	—
23	C₄×Dic₈₉	—	2	1	no	—
24	C₄×Dic₈₉	—	2	1	no	—
25	C₂²×C₄×C₈₉	—	2	2	no	—
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₄⋊C₄×C₈₉	Q₈	1	4	yes	—
31	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	—	2	4	no	—
33	C₂×C₄×C₈₉	—	4	2	no	—
34	C₂×C₄×C₈₉	—	4	2	no	—
35	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₈₉	#41	1	2	yes	—
38	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
39	C₂×Dic₈₉	—	2	1	no	—
40	C₂×Dic₈₉	—	2	1	no	—
41	C₂×Dic₈₉	—	2	1	no	—
42	C₂×Dic₈₉	—	2	2	no	—
43	C₂×Dic₈₉	—	4	2	no	—
44	C₂×Dic₈₉	—	4	2	no	—
45	C₂×Dic₈₉	—	4	2	no	—
46	C₂×Dic₈₉	Q₁₆	1	2	yes	—
47	C₈×C₈₉	—	2	2	no	—
48	#10623494	—	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	—	4	4	no	—
55	C₄×C₈₉	#123	1	1	yes	characteristic
56	C₄×C₈₉	#125	1	2	yes	—
57	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₈₉	—	2	2	no	—
59	Dic₈₉	—	4	1	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	—	89	1	no	—
70	#389	—	89	2	no	—
71	#340	—	89	2	no	—
72	#340	D₈₉	1	1	yes	characteristic
73	C₂²×C₈	—	89	1	no	—
74	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	89	2	no	—
76	C₂×C₄⋊C₄	—	178	2	no	—
77	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
78	C₂×C₄²	—	89	1	no	—
79	#107	—	178	2	no	—
80	C₄⋊C₈	—	178	2	no	—
81	#95	—	178	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	89	1	no	—
84	C₂²×C₄	—	89	1	no	—
85	C₂²×C₄	—	178	2	no	—
86	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	178	1	no	—
89	C₂×C₈	—	178	2	no	—
90	C₂×C₈	D₃₅₆	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	356	4	no	—
93	C₄⋊C₄	—	178	2	no	—
94	C₄⋊C₄	—	178	2	no	—
95	C₄⋊C₄	Dic₁₇₈	1	4	yes	—
96	C₄²	—	178	1	no	—
97	C₄²	—	178	1	no	—
98	C₂³	D₄×D₈₉	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	89	2	no	—
103	C₂×C₄	—	356	2	no	—
104	C₂×C₄	—	356	2	no	—
105	C₂×C₄	—	356	2	no	—
106	C₂×C₄	—	178	1	no	—
107	C₂×C₄	—	178	2	no	—
108	C₂×C₄	—	178	1	no	—
109	C₂×C₄	—	178	1	no	—
110	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
112	C₂×C₄	#13083573	1	2	yes	—
113	C₂×C₄	#13083575	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	356	1	no	—
116	C₂²	#15667987	1	1	yes	characteristic
117	C₂²	#15667989	1	2	yes	—
118	C₂²	Q₁₆×D₈₉	1	1	yes	characteristic
119	C₂²	#15668031	1	1	yes	characteristic
120	C₂²	#15668039	1	2	yes	—
121	C₄	—	4	4	no	—
122	C₄	—	356	1	no	—
123	C₄	—	178	2	no	—
124	C₄	—	712	2	no	—
125	C₄	#15667996	1	1	yes	characteristic
126	C₄	#15668003	1	2	yes	—
127	C₄	#15668012	1	1	yes	characteristic
128	C₂	#19151530	1	1	yes	characteristic
129	C₂	#19151840	1	1	yes	characteristic
130	C₂	#19151852	1	1	yes	characteristic
131	C₂	#19151891	1	2	yes	—
132	C₂	#19151896	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667946

C₂

yes

maximal, characteristic

C₂×#15667941

C₂

yes

maximal, characteristic

#19151497

C₂

yes

maximal

#19151360

C₂

yes

maximal

#340×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

—

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₄

yes

characteristic

C₂²×Dic₈₉

D₄

yes

characteristic

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

Q₈

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

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

—

C₂×Dic₈₉

—

C₂×Dic₈₉

Q₁₆

yes

—

C₈×C₈₉

—

#10623494

—

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

#123

yes

characteristic

C₄×C₈₉

#125

yes

—

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

Dic₈₉

—

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

—

#389

—

#340

—

#340

D₈₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

#107

—

178

—

C₄⋊C₈

—

178

—

#95

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

D₃₅₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

Dic₁₇₈

yes

—

C₄²

—

178

—

C₄²

—

178

—

C₂³

D₄×D₈₉

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

—

178

—

107

C₂×C₄

—

178

—

108

C₂×C₄

—

178

—

109

C₂×C₄

—

178

—

110

C₂×C₄

C₂×D₃₅₆

yes

characteristic

111

C₂×C₄

C₂×Dic₁₇₈

yes

—

112

C₂×C₄

#13083573

yes

—

113

C₂×C₄

#13083575

yes

characteristic

114

C₈

—

115

C₈

—

356

—

116

C₂²

#15667987

yes

characteristic

117

C₂²

#15667989

yes

—

118

C₂²

Q₁₆×D₈₉

yes

characteristic

119

C₂²

#15668031

yes

characteristic

120

C₂²

#15668039

yes

—

121

C₄

—

122

C₄

—

356

—

123

C₄

—

178

—

124

C₄

—

712

—

125

C₄

#15667996

yes

characteristic

126

C₄

#15668003

yes

—

127

C₄

#15668012

yes

characteristic

128

C₂

#19151530

yes

characteristic

129

C₂

#19151840

yes

characteristic

130

C₂

#19151852

yes

characteristic

131

C₂

#19151891

yes

—

132

C₂

#19151896

yes

—

	subgroup	G/H
center	C₂³	D₄×D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#340×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#340	D₈₉
hypocenter	C₈₉	#2087
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₈₉

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#340×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

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
C₂×#15667941	2
#19151497	2
#19151360	2
#340×C₈₉	2
#2087	89

subgroup

index

#340×C₈₉

Subgroups — #26021747

Conjugacy classes of subgroups

Complements

Maximal subgroups