Group #26019871 — subgroups

Subgroups — #26019871

Order 11392.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19152003	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151994	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151970	C₂	1	1	yes	maximal, characteristic
4	#19151950	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151944	C₂	1	1	yes	maximal, characteristic
6	#19151319	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13083562	—	2	2	no	—
9	#15668012	C₂²	1	1	yes	characteristic
10	#15668012	C₂²	1	1	yes	characteristic
11	#15667946	—	2	2	no	—
12	#15667943	—	2	2	no	—
13	#15667935	—	2	2	no	—
14	#15667933	—	2	2	no	—
15	C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₈₉	—	2	2	no	—
18	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	#127×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₇₈	—	2	1	no	—
22	C₂×Dic₁₇₈	—	2	1	no	—
23	#13083562	—	4	4	no	—
24	C₂×#10623494	D₄	1	2	yes	—
25	C₂×#10623494	D₄	1	2	yes	—
26	C₄⋊Dic₈₉	—	2	1	no	—
27	C₄⋊Dic₈₉	—	2	1	no	—
28	C₄⋊Dic₈₉	—	2	1	no	—
29	C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×Q₈×C₈₉	—	2	1	no	—
31	C₂×D₄×C₈₉	—	2	1	no	—
32	C₂×D₄×C₈₉	—	2	1	no	—
33	C₂×D₄×C₈₉	—	2	1	no	—
34	SD₁₆×C₈₉	—	4	4	no	—
35	C₂×C₈×C₈₉	D₄	1	2	yes	—
36	C₄⋊C₄×C₈₉	—	2	1	no	—
37	C₄⋊C₄×C₈₉	—	2	1	no	—
38	C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	C₂³×C₈₉	—	4	1	no	—
40	Q₈×C₈₉	—	4	2	no	—
41	D₄×C₈₉	—	4	2	no	—
42	D₄×C₈₉	—	4	2	no	—
43	D₄×C₈₉	—	4	2	no	—
44	C₂×C₄×C₈₉	—	4	1	no	—
45	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₈₉	—	4	1	no	—
49	C₂×Dic₈₉	—	4	1	no	—
50	Dic₁₇₈	—	4	2	no	—
51	Dic₁₇₈	—	4	2	no	—
52	C₈×C₈₉	SD₁₆	1	4	yes	complemented
53	#10623494	—	2	2	no	—
54	#10623494	—	2	2	no	—
55	C₂²×C₈₉	—	8	1	no	—
56	C₂²×C₈₉	—	8	1	no	—
57	C₂²×C₈₉	—	8	1	no	—
58	C₂²×C₈₉	#127	1	1	yes	characteristic
59	C₄×C₈₉	—	8	1	no	—
60	C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
61	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₈₉	#137	1	2	yes	—
63	Dic₈₉	—	8	1	no	—
64	Dic₈₉	—	8	1	no	—
65	C₂×C₈₉	—	16	1	no	—
66	C₂×C₈₉	#492	1	1	yes	characteristic
67	C₂×C₈₉	#496	1	1	yes	characteristic
68	C₂×C₈₉	#497	1	1	yes	characteristic
69	#1714	—	89	1	no	maximal, Sylow
70	C₈₉	#1714	1	1	yes	characteristic, complemented, Sylow
71	#500	—	89	1	no	—
72	#498	—	89	1	no	—
73	#492	—	89	1	no	—
74	#492	D₈₉	1	1	yes	characteristic
75	#487	—	89	1	no	—
76	#486	—	89	1	no	—
77	#322	—	89	1	no	—
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×SD₁₆	—	178	2	no	—
80	C₄⋊Q₈	—	89	1	no	—
81	C₄⋊Q₈	—	89	1	no	—
82	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
83	#127	C₂×D₈₉	1	1	yes	characteristic
84	#107	—	178	2	no	—
85	#106	—	178	2	no	—
86	#103	—	178	2	no	—
87	#102	—	178	2	no	—
88	C₄×C₈	—	89	1	no	—
89	C₄×C₈	—	89	1	no	—
90	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	178	1	no	—
93	C₂×Q₈	—	178	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	4	4	no	—
98	SD₁₆	—	356	4	no	—
99	C₂×C₈	—	89	2	no	—
100	C₂×C₈	—	89	2	no	—
101	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	178	1	no	—
105	C₄⋊C₄	—	178	1	no	—
106	C₄⋊C₄	—	178	1	no	—
107	C₄⋊C₄	—	178	1	no	—
108	C₄²	C₂²×D₈₉	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	356	2	no	—
112	Q₈	—	356	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	356	1	no	—
118	C₂×C₄	—	356	1	no	—
119	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
120	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
122	C₈	—	178	2	no	—
123	C₈	—	178	2	no	—
124	C₈	#13083562	1	4	yes	complemented
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#15668008	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	712	1	no	—
131	C₄	—	712	1	no	—
132	C₄	C₂×#13083562	1	2	yes	—
133	C₄	#15668039	1	2	yes	—
134	C₄	#15668041	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#19151970	1	1	yes	characteristic
137	C₂	#19151984	1	1	yes	characteristic
138	C₂	#19151989	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152003

C₂

yes

maximal, characteristic, complemented

#19151994

C₂

yes

maximal, characteristic, complemented

#19151970

C₂

yes

maximal, characteristic

#19151950

C₂

yes

maximal, characteristic, complemented

#19151944

C₂

yes

maximal, characteristic

#19151319

C₂

yes

maximal, characteristic, complemented

#492×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083562

—

#15668012

C₂²

yes

characteristic

#15668012

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₂×SD₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#127×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

#13083562

—

C₂×#10623494

D₄

yes

—

C₂×#10623494

D₄

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

SD₁₆×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

SD₁₆

yes

complemented

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#127

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₄×C₈₉

#137

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

—

—

—

C₂×SD₁₆

—

C₂×SD₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

#127

C₂×D₈₉

yes

characteristic

#107

—

178

—

#106

—

178

—

#103

—

178

—

#102

—

178

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

356

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₈₉

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

178

—

105

C₄⋊C₄

—

178

—

106

C₄⋊C₄

—

178

—

107

C₄⋊C₄

—

178

—

108

C₄²

C₂²×D₈₉

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

356

—

112

Q₈

—

356

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

356

—

118

C₂×C₄

—

356

—

119

C₂×C₄

D₄×D₈₉

yes

characteristic

120

C₂×C₄

D₄×D₈₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

122

C₈

—

178

—

123

C₈

—

178

—

124

C₈

#13083562

yes

complemented

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#15668008

yes

characteristic

129

C₄

—

130

C₄

—

712

—

131

C₄

—

712

—

132

C₄

C₂×#13083562

yes

—

133

C₄

#15668039

yes

—

134

C₄

#15668041

yes

—

135

C₂

—

136

C₂

#19151970

yes

characteristic

137

C₂

#19151984

yes

characteristic

138

C₂

#19151989

yes

characteristic

	subgroup	G/H
center	C₂²	#15668008
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#492×C₈₉	C₂
socle	C₂²×C₈₉	#127
prime radical	#127×C₈₉	C₂²
hypercenter	#492	D₈₉
hypocenter	C₈₉	#1714
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#15668008

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#492×C₈₉

C₂

socle

C₂²×C₈₉

#127

prime radical

#127×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #19152003	C₂	semidirect
2. #19151994	C₂	semidirect
4. #19151950	C₂	semidirect
6. #19151319	C₂	semidirect
52. C₈×C₈₉	SD₁₆	semidirect
70. C₈₉	#1714	semidirect
124. C₈	#13083562	semidirect

H ≅ G/N

type

1. #19152003

C₂

semidirect

2. #19151994

C₂

semidirect

4. #19151950

C₂

semidirect

6. #19151319

C₂

semidirect

52. C₈×C₈₉

SD₁₆

semidirect

70. C₈₉

#1714

semidirect

124. C₈

#13083562

semidirect

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

subgroup	index
#19152003	2
#19151994	2
#19151970	2
#19151950	2
#19151944	2
#19151319	2
#492×C₈₉	2
#1714	89

subgroup

index

#492×C₈₉

Subgroups — #26019871

Conjugacy classes of subgroups

Complements

Maximal subgroups