Group #26019366 — subgroups

Subgroups — #26019366

Order 11392.

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

normal subgroups only

This group has 314 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	#19152233	C₂	1	1	yes	maximal, characteristic
2	#19151885	C₂	1	2	yes	maximal, complemented
3	#19151342	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151327	C₂	1	2	yes	maximal, complemented
5	#492×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₈₉	—	2	2	no	—
7	#15667983	—	2	2	no	—
8	#15667980	—	2	2	no	—
9	C₄×D₃₅₆	C₂²	1	2	yes	—
10	#15667943	—	2	2	no	—
11	C₄⋊#10623494	C₂²	1	2	yes	complemented
12	#15667932	—	2	2	no	—
13	C₂×SD₁₆×C₈₉	—	2	2	no	—
14	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
15	#127×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₈₉	—	2	2	no	—
18	C₂×D₃₅₆	D₄	1	2	yes	—
19	C₂×C₂²⋊D₈₉	—	2	4	no	—
20	D₄×D₈₉	—	4	4	no	—
21	D₄×D₈₉	—	4	4	no	—
22	C₂×C₄×D₈₉	—	2	2	no	—
23	C₂×#10623494	—	2	2	no	—
24	C₄⋊Dic₈₉	D₄	1	2	yes	—
25	C₂²⋊Dic₈₉	—	2	2	no	—
26	#13083551	—	2	2	no	—
27	C₂×Q₈×C₈₉	—	2	1	no	—
28	C₂×D₄×C₈₉	—	2	2	no	—
29	C₂×D₄×C₈₉	D₄	1	2	yes	—
30	SD₁₆×C₈₉	—	4	4	no	—
31	C₂×C₈×C₈₉	—	2	1	no	—
32	C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₂³×C₈₉	—	2	2	no	—
35	C₂²×D₈₉	—	2	2	no	—
36	C₂²×D₈₉	—	4	8	no	—
37	C₂²×D₈₉	—	8	2	no	—
38	Q₈×C₈₉	—	4	2	no	—
39	D₄×C₈₉	—	4	2	no	—
40	D₄×C₈₉	—	4	2	no	—
41	D₄×C₈₉	—	4	2	no	—
42	C₂×C₄×C₈₉	—	4	1	no	—
43	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂²⋊D₈₉	—	4	8	no	—
46	C₂×Dic₈₉	—	2	2	no	—
47	D₃₅₆	—	2	2	no	—
48	D₃₅₆	SD₁₆	1	4	yes	complemented
49	C₄×D₈₉	—	4	2	no	—
50	C₄×D₈₉	—	4	2	no	—
51	C₈×C₈₉	—	2	2	no	—
52	#10623494	—	4	2	no	—
53	C₂²×C₈₉	—	4	4	no	—
54	C₂²×C₈₉	—	4	2	no	—
55	C₂²×C₈₉	#120	1	1	yes	characteristic
56	C₂×D₈₉	—	2	4	no	—
57	C₂×D₈₉	—	4	2	no	—
58	C₂×D₈₉	—	4	2	no	—
59	C₂×D₈₉	—	8	8	no	—
60	C₄×C₈₉	—	2	1	no	—
61	C₄×C₈₉	—	8	1	no	—
62	C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
63	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
64	Dic₈₉	—	4	2	no	—
65	C₂×C₈₉	—	8	2	no	—
66	C₂×C₈₉	#450	1	2	yes	—
67	C₂×C₈₉	#453	1	1	yes	characteristic
68	D₈₉	—	4	4	no	—
69	#1629	—	89	1	no	maximal, Sylow
70	C₈₉	#1629	1	1	yes	characteristic, complemented, Sylow
71	D₄²	—	89	1	no	—
72	#492	D₈₉	1	1	yes	characteristic, complemented
73	#476	—	89	2	no	—
74	#331	—	89	1	no	—
75	D₄⋊C₈	—	89	2	no	—
76	C₂²×D₄	—	178	2	no	—
77	C₂×SD₁₆	—	2	2	no	—
78	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
79	#127	C₂×D₈₉	1	1	yes	characteristic
80	#121	—	178	2	no	—
81	#120	—	178	2	no	—
82	C₄×D₄	—	89	2	no	—
83	#106	—	178	2	no	—
84	C₄⋊C₈	—	89	2	no	—
85	#102	—	178	2	no	—
86	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
87	C₂⁴	—	178	2	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	—	89	2	no	—
91	C₂×D₄	—	356	4	no	—
92	C₂×D₄	—	356	4	no	—
93	C₂×D₄	—	178	4	no	—
94	C₂×D₄	C₂²⋊D₈₉	1	2	yes	—
95	C₂²×C₄	—	178	2	no	—
96	SD₁₆	—	4	4	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	178	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	89	2	no	—
101	C₂²⋊C₄	—	178	2	no	—
102	C₂²⋊C₄	—	178	2	no	—
103	C₄²	C₂²×D₈₉	1	1	yes	characteristic
104	C₂³	—	2	2	no	—
105	C₂³	—	356	8	no	—
106	C₂³	—	178	2	no	—
107	C₂³	—	712	2	no	—
108	Q₈	—	4	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	89	4	no	—
113	D₄	—	356	8	no	—
114	D₄	—	178	2	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	356	2	no	—
117	C₂×C₄	—	356	2	no	—
118	C₂×C₄	—	178	2	no	—
119	C₂×C₄	D₄×D₈₉	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	356	2	no	—
123	C₂²	—	4	4	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	356	2	no	—
126	C₂²	—	356	2	no	—
127	C₂²	—	178	4	no	—
128	C₂²	—	712	8	no	—
129	C₂²	#15667980	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	356	2	no	—
133	C₄	SD₁₆×D₈₉	1	2	yes	—
134	C₄	#15668039	1	2	yes	—
135	C₂	—	8	2	no	—
136	C₂	—	356	4	no	—
137	C₂	#19151747	1	2	yes	—
138	C₂	#19151765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152233

C₂

yes

maximal, characteristic

#19151885

C₂

yes

maximal, complemented

#19151342

C₂

yes

maximal, characteristic, complemented

#19151327

C₂

yes

maximal, complemented

#492×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₈₉

—

#15667983

—

#15667980

—

C₄×D₃₅₆

C₂²

yes

—

#15667943

—

C₄⋊#10623494

C₂²

yes

complemented

#15667932

—

C₂×SD₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic, complemented

#127×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic, complemented

C₂³×D₈₉

—

C₂×D₃₅₆

D₄

yes

—

C₂×C₂²⋊D₈₉

—

D₄×D₈₉

—

D₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₄

yes

—

C₂²⋊Dic₈₉

—

#13083551

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

D₄

yes

—

SD₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

SD₁₆

yes

complemented

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#450

yes

—

C₂×C₈₉

#453

yes

characteristic

D₈₉

—

#1629

—

maximal, Sylow

C₈₉

#1629

yes

characteristic, complemented, Sylow

D₄²

—

#492

D₈₉

yes

characteristic, complemented

#476

—

#331

—

D₄⋊C₈

—

C₂²×D₄

—

178

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic, complemented

#127

C₂×D₈₉

yes

characteristic

#121

—

178

—

#120

—

178

—

C₄×D₄

—

#106

—

178

—

C₄⋊C₈

—

#102

—

178

—

C₄×C₈

C₂×D₈₉

yes

characteristic, complemented

C₂⁴

—

178

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

178

—

C₂×D₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

—

178

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

178

—

102

C₂²⋊C₄

—

178

—

103

C₄²

C₂²×D₈₉

yes

characteristic

104

C₂³

—

105

C₂³

—

356

—

106

C₂³

—

178

—

107

C₂³

—

712

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

356

—

114

D₄

—

178

—

115

C₂×C₄

—

116

C₂×C₄

—

356

—

117

C₂×C₄

—

356

—

118

C₂×C₄

—

178

—

119

C₂×C₄

D₄×D₈₉

yes

—

120

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

121

C₈

—

122

C₈

—

356

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

356

—

126

C₂²

—

356

—

127

C₂²

—

178

—

128

C₂²

—

712

—

129

C₂²

#15667980

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

356

—

133

C₄

SD₁₆×D₈₉

yes

—

134

C₄

#15668039

yes

—

135

C₂

—

136

C₂

—

356

—

137

C₂

#19151747

yes

—

138

C₂

#19151765

yes

characteristic

	subgroup	G/H
center	C₂²	#15667980
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#492×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	#19152233	C₂
hypercenter	#492	D₈₉
hypocenter	C₈₉	#1629
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#15667980

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#492×C₈₉

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #19151885	C₂	semidirect
3. #19151342	C₂	semidirect
4. #19151327	C₂	semidirect
5. #492×C₈₉	C₂	semidirect
11. C₄⋊#10623494	C₂²	semidirect
14. C₄⋊Q₈×C₈₉	C₂²	semidirect
16. C₄×C₈×C₈₉	C₂²	semidirect
48. D₃₅₆	SD₁₆	semidirect
70. C₈₉	#1629	semidirect
72. #492	D₈₉	semidirect
78. C₄⋊Q₈	C₂×D₈₉	semidirect
86. C₄×C₈	C₂×D₈₉	semidirect

H ≅ G/N

type

2. #19151885

C₂

semidirect

3. #19151342

C₂

semidirect

4. #19151327

C₂

semidirect

5. #492×C₈₉

C₂

semidirect

11. C₄⋊#10623494

C₂²

semidirect

14. C₄⋊Q₈×C₈₉

C₂²

semidirect

16. C₄×C₈×C₈₉

C₂²

semidirect

48. D₃₅₆

SD₁₆

semidirect

70. C₈₉

#1629

semidirect

72. #492

D₈₉

semidirect

78. C₄⋊Q₈

C₂×D₈₉

semidirect

86. C₄×C₈

C₂×D₈₉

semidirect

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

subgroup	index
#19152233	2
#19151885	2
#19151342	2
#19151327	2
#492×C₈₉	2
#1629	89

subgroup

index

#492×C₈₉

Subgroups — #26019366

Conjugacy classes of subgroups

Complements

Maximal subgroups