Group #26019728 — subgroups

Subgroups — #26019728

Order 11392.

198 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 55 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	#19151790	C₂	1	2	yes	maximal, complemented
2	#19151677	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151349	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151342	C₂	1	2	yes	maximal, complemented
5	#493×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083559	—	2	2	no	—
7	C₂×#13083557	—	2	1	no	—
8	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
9	C₄⋊#10623494	C₂²	1	1	yes	characteristic
10	C₄⋊#10623494	C₂²	1	2	yes	—
11	#15667933	—	2	2	no	—
12	#15667924	—	2	1	no	—
13	C₂×D₈×C₈₉	—	2	1	no	—
14	C₂×D₈×C₈₉	—	2	1	no	—
15	#127×C₈₉	C₂²	1	2	yes	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×D₃₅₆	D₄	1	1	yes	characteristic
18	C₂×C₄×D₈₉	—	2	1	no	—
19	#13083559	—	4	4	no	—
20	#13083557	—	4	2	no	—
21	C₂×#10623494	—	2	1	no	—
22	C₂×#10623494	—	2	2	no	—
23	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
24	#13083551	—	2	1	no	—
25	C₂×D₄×C₈₉	—	2	2	no	—
26	C₂×D₄×C₈₉	—	2	2	no	—
27	C₂×D₄×C₈₉	—	2	2	no	—
28	D₈×C₈₉	—	4	2	no	—
29	D₈×C₈₉	—	4	2	no	—
30	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₂³×C₈₉	—	4	2	no	—
34	C₂²×D₈₉	—	2	1	no	—
35	D₄×C₈₉	—	4	2	no	—
36	D₄×C₈₉	—	4	2	no	—
37	D₄×C₈₉	—	4	2	no	—
38	D₄×C₈₉	—	4	2	no	—
39	D₄×C₈₉	—	8	2	no	—
40	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
43	C₂×Dic₈₉	—	2	1	no	—
44	D₃₅₆	—	2	2	no	—
45	C₄×D₈₉	—	4	1	no	—
46	C₄×D₈₉	—	4	1	no	—
47	C₈×C₈₉	—	2	1	no	—
48	C₈×C₈₉	D₈	1	2	yes	complemented
49	#10623494	—	4	1	no	—
50	#10623494	—	4	2	no	—
51	C₂²×C₈₉	—	8	2	no	—
52	C₂²×C₈₉	—	8	2	no	—
53	C₂²×C₈₉	—	8	2	no	—
54	C₂²×C₈₉	#121	1	1	yes	characteristic
55	C₂×D₈₉	—	4	1	no	—
56	C₂×D₈₉	—	4	1	no	—
57	C₂×D₈₉	—	4	1	no	—
58	C₄×C₈₉	—	2	1	no	—
59	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
60	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
62	Dic₈₉	—	4	1	no	—
63	C₂×C₈₉	—	16	2	no	—
64	C₂×C₈₉	#459	1	1	yes	characteristic
65	C₂×C₈₉	#469	1	1	yes	characteristic
66	C₂×C₈₉	#472	1	1	yes	characteristic
67	D₈₉	—	8	1	no	—
68	#1687	—	89	1	no	maximal, Sylow
69	C₈₉	#1687	1	1	yes	characteristic, complemented, Sylow
70	#493	D₈₉	1	1	yes	characteristic, complemented
71	#459	—	89	2	no	—
72	#436	—	89	1	no	—
73	#334	—	89	1	no	—
74	#331	—	89	2	no	—
75	C₂×D₈	—	2	1	no	—
76	C₂×D₈	—	2	1	no	—
77	C₂×D₈	—	178	2	no	—
78	C₂×M₁₆	—	178	1	no	—
79	#127	C₂×D₈₉	1	2	yes	—
80	C₄×D₄	—	89	1	no	—
81	C₄⋊C₈	—	89	1	no	—
82	C₄⋊C₈	—	89	2	no	—
83	#102	—	178	2	no	—
84	C₂²⋊C₈	—	178	1	no	—
85	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	89	1	no	—
90	C₂²×C₄	—	178	1	no	—
91	D₈	—	4	2	no	—
92	D₈	—	4	2	no	—
93	D₈	—	356	4	no	—
94	M₁₆	—	356	2	no	—
95	C₂×C₈	—	178	2	no	—
96	C₂×C₈	—	178	1	no	—
97	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
99	C₄⋊C₄	—	89	1	no	—
100	C₂²⋊C₄	—	178	1	no	—
101	C₄²	C₂²×D₈₉	1	1	yes	characteristic
102	C₂³	—	4	2	no	—
103	C₂³	—	178	1	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	D₄	—	8	2	no	—
109	D₄	—	178	2	no	—
110	C₂×C₄	—	356	1	no	—
111	C₂×C₄	—	356	1	no	—
112	C₂×C₄	—	178	1	no	—
113	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
115	C₂×C₄	#13083573	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	356	1	no	—
118	C₈	—	356	2	no	—
119	C₈	#13083559	1	2	yes	complemented
120	C₂²	—	8	2	no	—
121	C₂²	—	8	2	no	—
122	C₂²	—	8	2	no	—
123	C₂²	—	356	1	no	—
124	C₂²	—	356	1	no	—
125	C₂²	—	356	1	no	—
126	C₂²	#15667983	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	356	1	no	—
129	C₄	C₂×#13083559	1	1	yes	characteristic
130	C₄	#15668036	1	2	yes	—
131	C₄	#15668038	1	1	yes	characteristic
132	C₂	—	16	2	no	—
133	C₂	—	712	1	no	—
134	C₂	#19151790	1	1	yes	characteristic
135	C₂	#19151848	1	1	yes	characteristic
136	C₂	#19151865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151790

C₂

yes

maximal, complemented

#19151677

C₂

yes

maximal, characteristic, complemented

#19151349

C₂

yes

maximal, characteristic, complemented

#19151342

C₂

yes

maximal, complemented

#493×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083559

—

C₂×#13083557

—

C₄×D₃₅₆

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

#15667933

—

#15667924

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

#127×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×D₃₅₆

D₄

yes

characteristic

C₂×C₄×D₈₉

—

—

—

—

—

C₄⋊Dic₈₉

D₄

yes

characteristic

#13083551

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

D₈

yes

complemented

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×D₈

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#1687

—

maximal, Sylow

C₈₉

#1687

yes

characteristic, complemented, Sylow

#493

D₈₉

yes

characteristic, complemented

—

—

—

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

178

—

C₂×M₁₆

—

178

—

#127

C₂×D₈₉

yes

—

C₄×D₄

—

C₄⋊C₈

—

C₄⋊C₈

—

#102

—

178

—

C₂²⋊C₈

—

178

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

178

—

D₈

—

D₈

—

D₈

—

356

—

M₁₆

—

356

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

C₄⋊C₄

—

100

C₂²⋊C₄

—

178

—

101

C₄²

C₂²×D₈₉

yes

characteristic

102

C₂³

—

103

C₂³

—

178

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

178

—

110

C₂×C₄

—

356

—

111

C₂×C₄

—

356

—

112

C₂×C₄

—

178

—

113

C₂×C₄

D₄×D₈₉

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

115

C₂×C₄

#13083573

yes

characteristic

116

C₈

—

117

C₈

—

356

—

118

C₈

—

356

—

119

C₈

#13083559

yes

complemented

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

356

—

124

C₂²

—

356

—

125

C₂²

—

356

—

126

C₂²

#15667983

yes

characteristic

127

C₄

—

128

C₄

—

356

—

129

C₄

C₂×#13083559

yes

characteristic

130

C₄

#15668036

yes

—

131

C₄

#15668038

yes

characteristic

132

C₂

—

133

C₂

—

712

—

134

C₂

#19151790

yes

characteristic

135

C₂

#19151848

yes

characteristic

136

C₂

#19151865

yes

characteristic

	subgroup	G/H
center	C₂²	#15667983
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#493×C₈₉	C₂
socle	C₂²×C₈₉	#121
hypercenter	#493	D₈₉
hypocenter	C₈₉	#1687
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#15667983

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#493×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #19151790	C₂	semidirect
2. #19151677	C₂	semidirect
3. #19151349	C₂	semidirect
4. #19151342	C₂	semidirect
5. #493×C₈₉	C₂	semidirect
48. C₈×C₈₉	D₈	semidirect
69. C₈₉	#1687	semidirect
70. #493	D₈₉	semidirect
119. C₈	#13083559	semidirect

H ≅ G/N

type

1. #19151790

C₂

semidirect

2. #19151677

C₂

semidirect

3. #19151349

C₂

semidirect

4. #19151342

C₂

semidirect

5. #493×C₈₉

C₂

semidirect

48. C₈×C₈₉

D₈

semidirect

69. C₈₉

#1687

semidirect

70. #493

D₈₉

semidirect

119. C₈

#13083559

semidirect

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

subgroup	index
#19151790	2
#19151677	2
#19151349	2
#19151342	2
#493×C₈₉	2
#1687	89

subgroup

index

#493×C₈₉

Subgroups — #26019728

Conjugacy classes of subgroups

Complements

Maximal subgroups