Group #26019312 — subgroups

Subgroups — #26019312

Order 11392.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19152334	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151974	C₂	1	1	yes	maximal, characteristic
3	#19151799	C₂	1	2	yes	maximal, complemented
4	#19151328	C₂	1	2	yes	maximal, complemented
5	#331×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083563	—	2	2	no	—
7	C₂×#13083559	—	2	2	no	—
8	#15668009	C₂²	1	1	yes	characteristic
9	#15667999	—	2	2	no	—
10	#15667993	—	2	2	no	—
11	C₄×Dic₁₇₈	C₂²	1	2	yes	—
12	D₄×Dic₈₉	—	2	2	no	—
13	#15667934	—	2	2	no	—
14	C₄×#10623494	C₂²	1	1	yes	characteristic
15	#127×C₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
17	C₂×Dic₁₇₈	D₄	1	2	yes	—
18	C₂×D₃₅₆	—	2	1	no	—
19	C₂×D₃₅₆	—	2	2	no	—
20	C₂²×Dic₈₉	—	2	2	no	—
21	#13083563	—	4	4	no	—
22	#13083559	—	4	4	no	—
23	C₂×#10623494	—	2	1	no	—
24	C₄⋊Dic₈₉	D₄	1	2	yes	—
25	#13083553	—	2	2	no	—
26	C₂²⋊Dic₈₉	—	2	4	no	—
27	C₂²⋊Dic₈₉	—	2	2	no	—
28	C₄×Dic₈₉	—	2	2	no	—
29	C₂×D₄×C₈₉	—	2	2	no	—
30	C₂×D₄×C₈₉	D₄	1	2	yes	—
31	C₂×C₈×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	D₄×C₈₉	—	4	2	no	—
36	D₄×C₈₉	—	4	2	no	—
37	D₄×C₈₉	—	4	2	no	—
38	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₈₉	—	2	2	no	—
41	C₂×Dic₈₉	—	2	2	no	—
42	C₂×Dic₈₉	—	8	2	no	—
43	D₃₅₆	—	4	2	no	—
44	D₃₅₆	—	4	2	no	—
45	D₃₅₆	—	4	2	no	—
46	Dic₁₇₈	—	2	2	no	—
47	Dic₁₇₈	—	2	2	no	—
48	C₈×C₈₉	—	4	2	no	—
49	#10623494	—	2	2	no	—
50	C₂²×C₈₉	—	4	4	no	—
51	C₂²×C₈₉	—	4	2	no	—
52	C₂²×C₈₉	#120	1	1	yes	characteristic
53	C₂×D₈₉	—	8	2	no	—
54	C₂×D₈₉	—	8	1	no	—
55	C₄×C₈₉	—	2	1	no	—
56	C₄×C₈₉	#135	1	2	yes	—
57	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
58	Dic₈₉	—	4	2	no	—
59	Dic₈₉	—	4	2	no	—
60	C₂×C₈₉	—	8	2	no	—
61	C₂×C₈₉	#449	1	2	yes	—
62	C₂×C₈₉	#453	1	1	yes	characteristic
63	D₈₉	—	16	1	no	—
64	#1620	—	89	1	no	maximal, Sylow
65	C₈₉	#1620	1	1	yes	characteristic, complemented, Sylow
66	#559	—	89	1	no	—
67	#493	—	89	1	no	—
68	#460	—	89	2	no	—
69	#331	D₈₉	1	1	yes	characteristic, complemented
70	Q₈⋊C₈	—	89	2	no	—
71	C₂×SD₁₆	—	178	2	no	—
72	C₂×D₈	—	178	2	no	—
73	#127	—	89	1	no	—
74	#127	C₂×D₈₉	1	1	yes	characteristic
75	#124	—	178	2	no	—
76	#123	—	178	2	no	—
77	C₄×Q₈	—	89	2	no	—
78	C₄×D₄	—	178	2	no	—
79	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
80	#102	—	178	2	no	—
81	C₄×C₈	—	89	1	no	—
82	C₂×Q₈	—	89	2	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	178	2	no	—
85	C₂×D₄	—	178	1	no	—
86	C₂×D₄	D₃₅₆	1	2	yes	—
87	C₂²×C₄	—	178	2	no	—
88	SD₁₆	—	356	4	no	—
89	D₈	—	356	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	178	1	no	—
92	C₄⋊C₄	—	89	2	no	—
93	C₄⋊C₄	—	178	2	no	—
94	C₂²⋊C₄	—	178	4	no	—
95	C₂²⋊C₄	—	178	2	no	—
96	C₄²	—	178	2	no	—
97	C₄²	C₂²×D₈₉	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	356	1	no	—
100	Q₈	—	178	2	no	—
101	Q₈	—	178	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	356	2	no	—
106	D₄	—	356	2	no	—
107	D₄	—	356	2	no	—
108	C₂×C₄	—	178	2	no	—
109	C₂×C₄	—	178	2	no	—
110	C₂×C₄	—	712	2	no	—
111	C₂×C₄	D₄×D₈₉	1	2	yes	—
112	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
113	C₈	—	4	2	no	—
114	C₈	—	178	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	712	2	no	—
118	C₂²	—	712	1	no	—
119	C₂²	#15667982	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	356	2	no	—
122	C₄	—	356	2	no	—
123	C₄	#15668033	1	2	yes	—
124	C₄	#15668036	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	1424	1	no	—
127	C₂	#19151744	1	2	yes	—
128	C₂	#19151768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152334

C₂

yes

maximal, characteristic, complemented

#19151974

C₂

yes

maximal, characteristic

#19151799

C₂

yes

maximal, complemented

#19151328

C₂

yes

maximal, complemented

#331×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083563

—

C₂×#13083559

—

#15668009

C₂²

yes

characteristic

#15667999

—

#15667993

—

C₄×Dic₁₇₈

C₂²

yes

—

D₄×Dic₈₉

—

#15667934

—

C₄×#10623494

C₂²

yes

characteristic

#127×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

D₄

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

—

#13083563

—

#13083559

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#449

yes

—

C₂×C₈₉

#453

yes

characteristic

D₈₉

—

#1620

—

maximal, Sylow

C₈₉

#1620

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic, complemented

Q₈⋊C₈

—

C₂×SD₁₆

—

178

—

C₂×D₈

—

178

—

#127

—

#127

C₂×D₈₉

yes

characteristic

#124

—

178

—

#123

—

178

—

C₄×Q₈

—

C₄×D₄

—

178

—

C₄⋊C₈

C₂×D₈₉

yes

—

#102

—

178

—

C₄×C₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

D₃₅₆

yes

—

C₂²×C₄

—

178

—

SD₁₆

—

356

—

D₈

—

356

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₄²

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

C₂³

—

356

—

100

Q₈

—

178

—

101

Q₈

—

178

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

356

—

106

D₄

—

356

—

107

D₄

—

356

—

108

C₂×C₄

—

178

—

109

C₂×C₄

—

178

—

110

C₂×C₄

—

712

—

111

C₂×C₄

D₄×D₈₉

yes

—

112

C₂×C₄

C₂×D₃₅₆

yes

characteristic

113

C₈

—

114

C₈

—

178

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

712

—

118

C₂²

—

712

—

119

C₂²

#15667982

yes

characteristic

120

C₄

—

121

C₄

—

356

—

122

C₄

—

356

—

123

C₄

#15668033

yes

—

124

C₄

#15668036

yes

—

125

C₂

—

126

C₂

—

1424

—

127

C₂

#19151744

yes

—

128

C₂

#19151768

yes

characteristic

	subgroup	G/H
center	C₂²	#15667982
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#331×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	#19151974	C₂
hypercenter	#331	D₈₉
hypocenter	C₈₉	#1620
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#15667982

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#331×C₈₉

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #19152334	C₂	semidirect
3. #19151799	C₂	semidirect
4. #19151328	C₂	semidirect
5. #331×C₈₉	C₂	semidirect
65. C₈₉	#1620	semidirect
69. #331	D₈₉	semidirect

H ≅ G/N

type

1. #19152334

C₂

semidirect

3. #19151799

C₂

semidirect

4. #19151328

C₂

semidirect

5. #331×C₈₉

C₂

semidirect

65. C₈₉

#1620

semidirect

69. #331

D₈₉

semidirect

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

subgroup	index
#19152334	2
#19151974	2
#19151799	2
#19151328	2
#331×C₈₉	2
#1620	89

subgroup

index

#331×C₈₉

Subgroups — #26019312

Conjugacy classes of subgroups

Complements

Maximal subgroups