Group #26019335 — subgroups

Subgroups — #26019335

Order 11392.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19151972	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151793	C₂	1	2	yes	maximal, complemented
3	#19151347	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊#10623494	C₂	1	2	yes	maximal, complemented
5	#555×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083561	—	2	2	no	—
7	C₂×#13083559	—	2	2	no	—
8	#15668009	C₂²	1	1	yes	characteristic
9	C₄⋊#10623494	C₂²	1	2	yes	—
10	#15667932	—	2	2	no	—
11	C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	#124×C₈₉	—	2	2	no	—
14	#121×C₈₉	—	2	2	no	—
15	C₄×D₄×C₈₉	C₂²	1	2	yes	—
16	#117×C₈₉	—	2	2	no	—
17	C₂×D₃₅₆	—	2	2	no	—
18	C₂×D₃₅₆	—	2	1	no	—
19	#13083561	—	4	4	no	—
20	#13083559	—	4	4	no	—
21	C₂×#10623494	—	2	1	no	—
22	C₂×#10623494	—	2	2	no	—
23	C₂×Q₈×C₈₉	D₄	1	2	yes	—
24	C₂×D₄×C₈₉	—	2	2	no	—
25	C₂×D₄×C₈₉	D₄	1	2	yes	—
26	C₂²×C₄×C₈₉	—	2	2	no	—
27	C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
29	C₂²⋊C₄×C₈₉	—	2	4	no	—
30	C₂²⋊C₄×C₈₉	—	2	2	no	—
31	C₄²×C₈₉	—	2	2	no	—
32	C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₂³×C₈₉	—	2	2	no	—
34	C₂²×D₈₉	—	4	1	no	—
35	Q₈×C₈₉	—	4	2	no	—
36	D₄×C₈₉	—	2	2	no	—
37	D₄×C₈₉	—	2	2	no	—
38	D₄×C₈₉	—	8	2	no	—
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	4	2	no	—
42	C₂×C₄×C₈₉	—	4	2	no	—
43	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
45	D₃₅₆	—	4	2	no	—
46	D₃₅₆	—	4	2	no	—
47	D₃₅₆	—	4	2	no	—
48	#10623494	—	2	2	no	—
49	#10623494	—	4	2	no	—
50	C₂²×C₈₉	—	4	2	no	—
51	C₂²×C₈₉	—	4	2	no	—
52	C₂²×C₈₉	—	4	2	no	—
53	C₂²×C₈₉	#120	1	1	yes	characteristic
54	C₂×D₈₉	—	8	2	no	—
55	C₂×D₈₉	—	8	1	no	—
56	C₄×C₈₉	—	2	1	no	—
57	C₄×C₈₉	—	4	2	no	—
58	C₄×C₈₉	—	4	2	no	—
59	C₄×C₈₉	#135	1	2	yes	—
60	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₈₉	—	8	2	no	—
62	C₂×C₈₉	#449	1	2	yes	—
63	C₂×C₈₉	#453	1	1	yes	characteristic
64	D₈₉	—	16	1	no	—
65	#1624	—	89	1	no	maximal, Sylow
66	C₈₉	#1624	1	1	yes	characteristic, complemented, Sylow
67	#555	D₈₉	1	1	yes	characteristic, complemented
68	#492	—	89	1	no	—
69	#459	—	89	2	no	—
70	#333	—	89	1	no	—
71	D₄⋊C₈	—	89	2	no	—
72	C₂×SD₁₆	—	178	2	no	—
73	C₂×D₈	—	178	2	no	—
74	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
75	#127	—	89	1	no	—
76	#124	—	2	2	no	—
77	#121	—	2	2	no	—
78	C₄×D₄	C₂×D₈₉	1	2	yes	—
79	#117	—	2	2	no	—
80	C₄⋊C₈	—	89	2	no	—
81	#102	—	178	2	no	—
82	C₄×C₈	—	89	1	no	—
83	C₂×Q₈	C₂²⋊D₈₉	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	178	2	no	—
86	C₂×D₄	—	178	1	no	—
87	C₂×D₄	C₂²⋊D₈₉	1	2	yes	—
88	C₂²×C₄	—	2	2	no	—
89	SD₁₆	—	356	4	no	—
90	D₈	—	356	4	no	—
91	C₂×C₈	—	178	2	no	—
92	C₂×C₈	—	178	1	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
95	C₂²⋊C₄	—	2	4	no	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	C₂²×D₈₉	1	1	yes	characteristic
99	C₂³	—	2	2	no	—
100	C₂³	—	356	1	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	2	2	no	—
103	D₄	—	2	2	no	—
104	D₄	—	8	2	no	—
105	D₄	—	356	2	no	—
106	D₄	—	356	2	no	—
107	D₄	—	356	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
114	C₈	—	356	2	no	—
115	C₈	—	178	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	2	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	712	2	no	—
120	C₂²	—	712	1	no	—
121	C₂²	#15667981	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	2	no	—
125	C₄	#15668032	1	2	yes	—
126	C₄	#15668037	1	2	yes	—
127	C₂	—	8	2	no	—
128	C₂	—	1424	1	no	—
129	C₂	#19151743	1	2	yes	—
130	C₂	#19151766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151972

C₂

yes

maximal, characteristic, complemented

#19151793

C₂

yes

maximal, complemented

#19151347

C₂

yes

maximal, characteristic, complemented

D₄⋊#10623494

C₂

yes

maximal, complemented

#555×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083561

—

C₂×#13083559

—

#15668009

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

#15667932

—

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#124×C₈₉

—

#121×C₈₉

—

C₄×D₄×C₈₉

C₂²

yes

—

#117×C₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

—

—

—

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₈₉

—

C₂×C₈₉

#449

yes

—

C₂×C₈₉

#453

yes

characteristic

D₈₉

—

#1624

—

maximal, Sylow

C₈₉

#1624

yes

characteristic, complemented, Sylow

#555

D₈₉

yes

characteristic, complemented

#492

—

#459

—

#333

—

D₄⋊C₈

—

C₂×SD₁₆

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

—

—

—

C₄×D₄

C₂×D₈₉

yes

—

#117

—

C₄⋊C₈

—

#102

—

178

—

C₄×C₈

—

C₂×Q₈

C₂²⋊D₈₉

yes

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

—

SD₁₆

—

356

—

D₈

—

356

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₈₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

100

C₂³

—

356

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

356

—

106

D₄

—

356

—

107

D₄

—

356

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

114

C₈

—

356

—

115

C₈

—

178

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

712

—

120

C₂²

—

712

—

121

C₂²

#15667981

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#15668032

yes

—

126

C₄

#15668037

yes

—

127

C₂

—

128

C₂

—

1424

—

129

C₂

#19151743

yes

—

130

C₂

#19151766

yes

characteristic

	subgroup	G/H
center	C₂²	#15667981
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#555×C₈₉	C₂
socle	C₂²×C₈₉	#120
hypercenter	#555	D₈₉
hypocenter	C₈₉	#1624
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #555

subgroup

G/H

center

C₂²

#15667981

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#555×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #555

N	H ≅ G/N	type
1. #19151972	C₂	semidirect
2. #19151793	C₂	semidirect
3. #19151347	C₂	semidirect
4. D₄⋊#10623494	C₂	semidirect
5. #555×C₈₉	C₂	semidirect
66. C₈₉	#1624	semidirect
67. #555	D₈₉	semidirect

H ≅ G/N

type

1. #19151972

C₂

semidirect

2. #19151793

C₂

semidirect

3. #19151347

C₂

semidirect

4. D₄⋊#10623494

C₂

semidirect

5. #555×C₈₉

C₂

semidirect

66. C₈₉

#1624

semidirect

67. #555

D₈₉

semidirect

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

subgroup	index
#19151972	2
#19151793	2
#19151347	2
D₄⋊#10623494	2
#555×C₈₉	2
#1624	89

subgroup

index

#555×C₈₉

Subgroups — #26019335

Conjugacy classes of subgroups

Complements

Maximal subgroups