Group #26021509 — subgroups

Subgroups — #26021509

Order 11392.

296 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#19151624	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151604	C₂	1	2	yes	maximal, complemented
4	C₂×#15667928	C₂	1	2	yes	maximal, complemented
5	#341×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	C₂²	1	2	yes	—
7	C₂×D₇₁₂	—	2	4	no	—
8	C₂×D₇₁₂	—	2	2	no	—
9	C₂×#13083558	C₂²	1	1	yes	characteristic
10	#15667971	C₂²	1	1	yes	characteristic
11	#15667941	—	2	2	no	—
12	#15667934	—	2	4	no	—
13	#15667928	—	2	4	no	—
14	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂³×D₈₉	—	2	2	no	—
17	C₂×D₃₅₆	—	2	2	no	—
18	C₂×D₃₅₆	—	2	2	no	—
19	C₂×D₃₅₆	—	2	2	no	—
20	C₂×D₃₅₆	—	2	2	no	—
21	C₂×D₃₅₆	—	2	2	no	—
22	C₂×D₃₅₆	D₄	1	4	yes	—
23	D₇₁₂	—	2	8	no	—
24	#13083558	—	2	2	no	—
25	C₂×#10623494	—	2	1	no	—
26	C₄⋊Dic₈₉	D₄	1	2	yes	—
27	C₂²⋊Dic₈₉	—	2	1	no	—
28	C₄×Dic₈₉	—	2	1	no	—
29	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
30	M₁₆×C₈₉	—	2	2	no	—
31	M₁₆×C₈₉	—	2	2	no	—
32	C₂×C₈×C₈₉	—	2	2	no	—
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₂×C₈×C₈₉	D₄	1	2	yes	—
35	C₂³×C₈₉	#41	1	1	yes	characteristic
36	C₂²×D₈₉	—	2	4	no	—
37	C₂²×D₈₉	—	4	2	no	—
38	C₂²×D₈₉	—	4	2	no	—
39	C₂²×D₈₉	—	8	4	no	—
40	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	#41	1	2	yes	—
44	C₂×Dic₈₉	—	2	2	no	—
45	D₃₅₆	—	2	4	no	—
46	D₃₅₆	—	2	4	no	—
47	D₃₅₆	—	4	2	no	—
48	D₃₅₆	—	4	2	no	—
49	C₈×C₈₉	—	2	2	no	—
50	C₈×C₈₉	—	4	2	no	—
51	#10623494	—	4	1	no	—
52	C₂²×C₈₉	—	2	2	no	—
53	C₂²×C₈₉	#121	1	1	yes	characteristic
54	C₂²×C₈₉	#123	1	2	yes	—
55	C₂×D₈₉	—	4	4	no	—
56	C₂×D₈₉	—	8	4	no	—
57	C₂×D₈₉	—	8	4	no	—
58	C₂×D₈₉	—	8	4	no	—
59	C₄×C₈₉	#120	1	1	yes	characteristic
60	C₄×C₈₉	#121	1	2	yes	—
61	C₄×C₈₉	#124	1	1	yes	characteristic
62	Dic₈₉	—	4	2	no	—
63	C₂×C₈₉	—	2	2	no	—
64	C₂×C₈₉	#394	1	1	yes	characteristic
65	C₂×C₈₉	#472	1	2	yes	—
66	D₈₉	—	8	4	no	—
67	#2045	—	89	1	no	maximal, Sylow
68	C₈₉	#2045	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₈	—	89	1	no	—
70	#424	—	89	1	no	—
71	#418	—	89	2	no	—
72	C₂×#100	—	89	2	no	—
73	#341	D₈₉	1	1	yes	characteristic, complemented
74	C₂²×D₄	—	89	2	no	—
75	C₂×D₈	—	178	4	no	—
76	C₂×D₈	—	178	2	no	—
77	C₂×M₁₆	—	89	1	no	—
78	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
79	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
80	#117	—	89	1	no	—
81	C₄⋊C₈	—	178	2	no	—
82	#102	—	178	4	no	—
83	#100	—	178	4	no	—
84	C₂⁴	—	178	2	no	—
85	C₂×D₄	—	89	4	no	—
86	C₂×D₄	—	178	2	no	—
87	C₂×D₄	—	178	2	no	—
88	C₂×D₄	—	178	2	no	—
89	C₂×D₄	—	178	2	no	—
90	C₂×D₄	—	178	2	no	—
91	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
92	D₈	—	178	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	178	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	178	1	no	—
99	C₂×C₈	D₃₅₆	1	2	yes	—
100	C₄⋊C₄	—	89	2	no	—
101	C₂²⋊C₄	—	178	1	no	—
102	C₄²	—	178	1	no	—
103	C₂³	—	356	2	no	—
104	C₂³	—	356	2	no	—
105	C₂³	—	178	4	no	—
106	C₂³	—	712	4	no	—
107	C₂³	#13083575	1	1	yes	characteristic
108	D₄	—	356	2	no	—
109	D₄	—	356	2	no	—
110	D₄	—	178	4	no	—
111	D₄	—	178	4	no	—
112	C₂×C₄	—	178	2	no	—
113	C₂×C₄	D₄×D₈₉	1	2	yes	—
114	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
115	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
116	C₂×C₄	#13083574	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	356	1	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	356	4	no	—
122	C₂²	—	712	4	no	—
123	C₂²	—	712	4	no	—
124	C₂²	—	712	4	no	—
125	C₂²	#15667987	1	1	yes	characteristic
126	C₂²	#15667997	1	2	yes	—
127	C₄	—	356	2	no	—
128	C₄	#15667982	1	1	yes	characteristic
129	C₄	#15667985	1	2	yes	—
130	C₄	#15668000	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	712	4	no	—
133	C₂	#19151516	1	1	yes	characteristic
134	C₂	#19151869	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₇₁₂

C₂

yes

maximal, characteristic

#19151624

C₂

yes

maximal, characteristic, complemented

#19151604

C₂

yes

maximal, complemented

C₂×#15667928

C₂

yes

maximal, complemented

#341×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×#13083558

C₂²

yes

characteristic

#15667971

C₂²

yes

—

—

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

D₄

yes

—

D₇₁₂

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₄

yes

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂³×C₈₉

#41

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

#123

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#120

yes

characteristic

C₄×C₈₉

#121

yes

—

C₄×C₈₉

#124

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#394

yes

characteristic

C₂×C₈₉

#472

yes

—

D₈₉

—

#2045

—

maximal, Sylow

C₈₉

#2045

yes

characteristic, complemented, Sylow

C₂²×D₈

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂²×D₄

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

#117

—

C₄⋊C₈

—

178

—

#102

—

178

—

#100

—

178

—

C₂⁴

—

178

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

D₈

—

178

—

M₁₆

—

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

D₃₅₆

yes

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

178

—

102

C₄²

—

178

—

103

C₂³

—

356

—

104

C₂³

—

356

—

105

C₂³

—

178

—

106

C₂³

—

712

—

107

C₂³

#13083575

yes

characteristic

108

D₄

—

356

—

109

D₄

—

356

—

110

D₄

—

178

—

111

D₄

—

178

—

112

C₂×C₄

—

178

—

113

C₂×C₄

D₄×D₈₉

yes

—

114

C₂×C₄

D₄×D₈₉

yes

characteristic

115

C₂×C₄

C₂×D₃₅₆

yes

characteristic

116

C₂×C₄

#13083574

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

356

—

120

C₂²

—

121

C₂²

—

356

—

122

C₂²

—

712

—

123

C₂²

—

712

—

124

C₂²

—

712

—

125

C₂²

#15667987

yes

characteristic

126

C₂²

#15667997

yes

—

127

C₄

—

356

—

128

C₄

#15667982

yes

characteristic

129

C₄

#15667985

yes

—

130

C₄

#15668000

yes

characteristic

131

C₂

—

132

C₂

—

712

—

133

C₂

#19151516

yes

characteristic

134

C₂

#19151869

yes

—

	subgroup	G/H
center	C₂²	#15667987
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#341×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	C₂²×D₇₁₂	C₂
hypercenter	#341	D₈₉
hypocenter	C₈₉	#2045
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#15667987

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#341×C₈₉

C₂

socle

C₂²×C₈₉

#121

prime radical

C₂²×D₇₁₂

C₂

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. #19151624	C₂	semidirect
3. #19151604	C₂	semidirect
4. C₂×#15667928	C₂	semidirect
5. #341×C₈₉	C₂	semidirect
68. C₈₉	#2045	semidirect
73. #341	D₈₉	semidirect

H ≅ G/N

type

2. #19151624

C₂

semidirect

3. #19151604

C₂

semidirect

4. C₂×#15667928

C₂

semidirect

5. #341×C₈₉

C₂

semidirect

68. C₈₉

#2045

semidirect

73. #341

D₈₉

semidirect

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

subgroup	index
C₂²×D₇₁₂	2
#19151624	2
#19151604	2
C₂×#15667928	2
#341×C₈₉	2
#2045	89

subgroup

index

C₂²×D₇₁₂

#341×C₈₉

Subgroups — #26021509

Conjugacy classes of subgroups

Complements

Maximal subgroups