Group #26018149 — subgroups

Subgroups — #26018149

Order 11392.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19151651	C₂	1	1	yes	maximal, characteristic
2	#19151622	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13083558	—	2	1	no	—
5	C₄⋊#10623494	—	2	1	no	—
6	C₄⋊#10623494	—	2	1	no	—
7	C₄⋊#10623494	C₄	1	2	yes	—
8	C₂²⋊#10623494	C₄	1	2	yes	—
9	C₄×#10623494	—	2	1	no	—
10	C₂²×C₈×C₈₉	—	2	1	no	—
11	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₈₉	—	2	1	no	—
13	C₄⋊C₈×C₈₉	C₄	1	2	yes	—
14	#13083558	—	4	1	no	—
15	#13083558	—	4	1	no	—
16	C₂×#10623494	—	2	2	no	—
17	C₂×#10623494	—	2	2	no	—
18	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₈₉	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₈₉	—	2	2	no	—
22	C₂×C₈×C₈₉	—	4	1	no	—
23	C₂×C₈×C₈₉	—	4	1	no	—
24	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₈₉	D₄	1	1	yes	characteristic
27	C₄²×C₈₉	D₄	1	1	yes	characteristic
28	C₂³×C₈₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₈₉	—	2	2	no	—
30	C₂×C₄×C₈₉	—	4	1	no	—
31	C₂×C₄×C₈₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₈₉	D₈	1	1	yes	characteristic
38	C₂×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
41	C₈×C₈₉	—	4	2	no	—
42	#10623494	—	4	2	no	—
43	#10623494	—	4	2	no	—
44	C₂²×C₈₉	—	2	1	no	—
45	C₂²×C₈₉	—	2	1	no	—
46	C₂²×C₈₉	#95	1	1	yes	characteristic
47	C₂²×C₈₉	#106	1	1	yes	characteristic
48	C₂²×C₈₉	#107	1	1	yes	characteristic
49	C₄×C₈₉	—	2	1	no	—
50	C₄×C₈₉	—	2	1	no	—
51	C₄×C₈₉	—	2	1	no	—
52	C₄×C₈₉	#102	1	1	yes	characteristic
53	C₄×C₈₉	#102	1	1	yes	characteristic
54	C₄×C₈₉	#103	1	1	yes	characteristic
55	C₄×C₈₉	#103	1	1	yes	characteristic
56	C₄×C₈₉	#108	1	2	yes	—
57	C₂×C₈₉	—	2	1	no	—
58	C₂×C₈₉	—	2	1	no	—
59	C₂×C₈₉	#340	1	1	yes	characteristic
60	C₂×C₈₉	#341	1	1	yes	characteristic
61	C₂×C₈₉	#344	1	1	yes	characteristic
62	#1301	—	89	1	no	maximal, Sylow
63	C₈₉	#1301	1	1	yes	characteristic, complemented, Sylow
64	#431	—	89	1	no	—
65	#423	—	89	1	no	—
66	C₂×C₄⋊C₈	D₈₉	1	1	yes	characteristic
67	C₂×M₁₆	—	178	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	89	2	no	—
72	C₄⋊C₈	—	178	1	no	—
73	C₄⋊C₈	—	178	1	no	—
74	C₄⋊C₈	Dic₈₉	1	2	yes	—
75	C₂²⋊C₈	—	89	2	no	—
76	C₄×C₈	—	178	1	no	—
77	C₂²×C₄	Dic₁₇₈	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
80	M₁₆	—	356	1	no	—
81	M₁₆	—	356	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	178	2	no	—
86	C₂×C₈	—	178	2	no	—
87	C₄²	C₄×D₈₉	1	1	yes	characteristic
88	C₄²	D₃₅₆	1	1	yes	characteristic
89	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
90	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
91	C₂³	#13083553	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₈₉	1	1	yes	characteristic
95	C₂×C₄	#13083551	1	1	yes	characteristic
96	C₂×C₄	#13083551	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
98	C₂×C₄	#13083553	1	1	yes	characteristic
99	C₂×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
100	C₂×C₄	#13083559	1	1	yes	characteristic
101	C₂×C₄	#13083561	1	1	yes	characteristic
102	C₂×C₄	#13083562	1	1	yes	characteristic
103	C₂×C₄	#13083564	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	356	2	no	—
106	C₈	—	356	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#15667919	1	1	yes	characteristic
110	C₂²	#15667943	1	1	yes	characteristic
111	C₂²	#15667945	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#15667932	1	1	yes	characteristic
116	C₄	#15667933	1	1	yes	characteristic
117	C₄	#15667935	1	1	yes	characteristic
118	C₄	#15667936	1	1	yes	characteristic
119	C₄	#15667948	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#19151360	1	1	yes	characteristic
123	C₂	#19151363	1	1	yes	characteristic
124	C₂	#19151370	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151651

C₂

yes

maximal, characteristic

#19151622

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₈₉

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

C₂²⋊#10623494

C₄

yes

—

C₄×#10623494

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

C₄

yes

—

—

—

—

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

Q₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₂³×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

D₈

yes

characteristic

C₂×C₄×C₈₉

SD₁₆

yes

characteristic

C₂×C₄×C₈₉

SD₁₆

yes

characteristic

C₂×C₄×C₈₉

Q₁₆

yes

characteristic

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#431

—

#423

—

C₂×C₄⋊C₈

D₈₉

yes

characteristic

C₂×M₁₆

—

178

—

C₂²×C₈

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

—

178

—

C₄⋊C₈

Dic₈₉

yes

—

C₂²⋊C₈

—

C₄×C₈

—

178

—

C₂²×C₄

Dic₁₇₈

yes

characteristic

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

M₁₆

—

356

—

M₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄²

C₄×D₈₉

yes

characteristic

C₄²

D₃₅₆

yes

characteristic

C₄²

C₂×Dic₈₉

yes

characteristic

C₄²

C₂²⋊D₈₉

yes

characteristic

C₂³

#13083553

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₈₉

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₈₉

yes

characteristic

C₂×C₄

#13083553

yes

characteristic

C₂×C₄

C₄⋊Dic₈₉

yes

characteristic

100

C₂×C₄

#13083559

yes

characteristic

101

C₂×C₄

#13083561

yes

characteristic

102

C₂×C₄

#13083562

yes

characteristic

103

C₂×C₄

#13083564

yes

characteristic

104

C₈

—

105

C₈

—

356

—

106

C₈

—

356

—

107

C₂²

—

108

C₂²

—

109

C₂²

#15667919

yes

characteristic

110

C₂²

#15667943

yes

characteristic

111

C₂²

#15667945

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#15667932

yes

characteristic

116

C₄

#15667933

yes

characteristic

117

C₄

#15667935

yes

characteristic

118

C₄

#15667936

yes

characteristic

119

C₄

#15667948

yes

—

120

C₂

—

121

C₂

—

122

C₂

#19151360

yes

characteristic

123

C₂

#19151363

yes

characteristic

124

C₂

#19151370

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13083551
commutator	C₂×C₄×C₈₉	C₄²
Frattini	C₂×C₄²	C₂×D₈₉
Fitting	C₂×C₄⋊C₈×C₈₉	C₂
socle	C₂²×C₈₉	#95
prime radical	C₂³×C₈₉	C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂²
hypercenter	C₂×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#1301
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

#13083551

commutator

C₂×C₄×C₈₉

C₄²

Frattini

C₂×C₄²

C₂×D₈₉

Fitting

C₂×C₄⋊C₈×C₈₉

C₂

socle

C₂²×C₈₉

#95

prime radical

C₂³×C₈₉

C₄⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂²

hypercenter

C₂×C₄⋊C₈

D₈₉

hypocenter

C₈₉

#1301

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
63. C₈₉	#1301	semidirect

H ≅ G/N

type

63. C₈₉

#1301

semidirect

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

subgroup	index
#19151651	2
#19151622	2
C₂×C₄⋊C₈×C₈₉	2
#1301	89

subgroup

index

#19151651

#19151622

C₂×C₄⋊C₈×C₈₉

#1301

Subgroups — #26018149

Conjugacy classes of subgroups

Complements

Maximal subgroups