Group #26020376 — subgroups

Subgroups — #26020376

Order 11392.

172 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂×#15667948	C₂	1	1	yes	maximal, characteristic
2	#19151636	C₂	1	1	yes	maximal, characteristic
3	#19151369	C₂	1	2	yes	maximal
4	#19151354	C₂	1	2	yes	maximal
5	#405×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	#15667971	C₂²	1	2	yes	—
8	#15667948	—	2	1	no	—
9	#15667948	—	2	1	no	—
10	#15667946	—	2	1	no	—
11	#15667944	—	2	1	no	—
12	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#117×C₈₉	C₂²	1	1	yes	characteristic
15	#97×C₈₉	C₄	1	2	yes	complemented
16	C₄×C₈×C₈₉	C₄	1	2	yes	complemented
17	#13083558	—	2	2	no	—
18	#13083558	—	2	2	no	—
19	C₂×#10623494	—	2	2	no	—
20	C₄⋊Dic₈₉	—	2	2	no	—
21	C₂²⋊Dic₈₉	—	2	2	no	—
22	C₄×Dic₈₉	—	2	2	no	—
23	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
24	M₁₆×C₈₉	—	2	1	no	—
25	M₁₆×C₈₉	—	2	1	no	—
26	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₈₉	Q₈	1	1	yes	characteristic
32	C₂×C₈×C₈₉	Q₈	1	1	yes	characteristic
33	C₄⋊C₄×C₈₉	—	2	1	no	—
34	C₂²⋊C₄×C₈₉	—	2	1	no	—
35	C₄²×C₈₉	C₂×C₄	1	2	yes	—
36	C₂³×C₈₉	#41	1	1	yes	characteristic
37	C₂×C₄×C₈₉	—	2	2	no	—
38	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
44	C₂×Dic₈₉	—	4	2	no	—
45	C₈×C₈₉	—	2	2	no	—
46	C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
50	#10623494	—	4	2	no	—
51	C₂²×C₈₉	—	4	1	no	—
52	C₂²×C₈₉	#117	1	1	yes	characteristic
53	C₂²×C₈₉	#124	1	1	yes	characteristic
54	C₂²×C₈₉	#125	1	1	yes	characteristic
55	C₄×C₈₉	—	4	2	no	—
56	C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₈₉	#117	1	1	yes	characteristic
58	C₄×C₈₉	#124	1	1	yes	characteristic
59	C₄×C₈₉	#125	1	1	yes	characteristic
60	Dic₈₉	—	8	2	no	—
61	C₂×C₈₉	—	2	1	no	—
62	C₂×C₈₉	—	2	1	no	—
63	C₂×C₈₉	—	2	1	no	—
64	C₂×C₈₉	#382	1	1	yes	characteristic
65	#1833	—	89	1	no	maximal, Sylow
66	C₈₉	#1833	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	—	89	1	no	—
68	#427	—	89	1	no	—
69	#405	D₈₉	1	1	yes	characteristic
70	#344	—	89	2	no	—
71	H(3,ℤ/4)	—	89	2	no	—
72	C₂×M₁₆	—	89	2	no	—
73	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
75	#117	—	89	2	no	—
76	#117	C₂×D₈₉	1	1	yes	characteristic
77	#108	—	178	1	no	—
78	#108	—	178	1	no	—
79	#107	—	178	1	no	—
80	#106	—	178	1	no	—
81	#97	Dic₈₉	1	2	yes	complemented
82	C₄×C₈	Dic₈₉	1	2	yes	complemented
83	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
84	M₁₆	—	2	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	178	2	no	—
87	M₁₆	—	178	2	no	—
88	C₂×C₈	—	178	2	no	—
89	C₂×C₈	Dic₁₇₈	1	1	yes	characteristic
90	C₂×C₈	Dic₁₇₈	1	1	yes	characteristic
91	C₂×C₈	D₃₅₆	1	1	yes	characteristic
92	C₂×C₈	D₃₅₆	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₈₉	1	2	yes	—
94	C₂×C₈	C₂×Dic₈₉	1	1	yes	characteristic
95	C₂×C₈	C₂×Dic₈₉	1	1	yes	characteristic
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	178	2	no	—
98	C₂²⋊C₄	—	2	1	no	—
99	C₂²⋊C₄	—	178	2	no	—
100	C₄²	—	178	2	no	—
101	C₄²	C₂×Dic₈₉	1	2	yes	—
102	C₂³	#13083574	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	356	2	no	—
105	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
106	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
108	C₂×C₄	#13083574	1	1	yes	characteristic
109	C₂×C₄	#13083574	1	1	yes	characteristic
110	C₂×C₄	#13083574	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	356	2	no	—
113	C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
114	C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
115	C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
116	C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#15667971	1	1	yes	characteristic
119	C₂²	#15668000	1	1	yes	characteristic
120	C₂²	#15668004	1	1	yes	characteristic
121	C₄	—	4	2	no	—
122	C₄	—	712	2	no	—
123	C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
124	C₄	#15667971	1	1	yes	characteristic
125	C₄	#15668000	1	1	yes	characteristic
126	C₄	#15668004	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#19151464	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667948

C₂

yes

maximal, characteristic

#19151636

C₂

yes

maximal, characteristic

#19151369

C₂

yes

maximal

#19151354

C₂

yes

maximal

#405×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

#15667971

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

#97×C₈₉

C₄

yes

complemented

C₄×C₈×C₈₉

C₄

yes

—

—

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

Q₈

yes

characteristic

C₂×C₈×C₈₉

Q₈

yes

characteristic

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

—

C₂³×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₄⋊C₄

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄⋊C₄

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×#108

—

#427

—

#405

D₈₉

yes

characteristic

#344

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

#117

—

#117

C₂×D₈₉

yes

characteristic

#108

—

178

—

#108

—

178

—

#107

—

178

—

#106

—

178

—

#97

Dic₈₉

yes

complemented

C₄×C₈

Dic₈₉

yes

complemented

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

178

—

C₂×C₈

Dic₁₇₈

yes

characteristic

C₂×C₈

Dic₁₇₈

yes

characteristic

C₂×C₈

D₃₅₆

yes

characteristic

C₂×C₈

D₃₅₆

yes

characteristic

C₂×C₈

C₂×Dic₈₉

yes

—

C₂×C₈

C₂×Dic₈₉

yes

characteristic

C₂×C₈

C₂×Dic₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

C₂²⋊C₄

—

178

—

100

C₄²

—

178

—

101

C₄²

C₂×Dic₈₉

yes

—

102

C₂³

#13083574

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

356

—

105

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

106

C₂×C₄

C₂×D₃₅₆

yes

characteristic

107

C₂×C₄

C₂×Dic₁₇₈

yes

characteristic

108

C₂×C₄

#13083574

yes

characteristic

109

C₂×C₄

#13083574

yes

characteristic

110

C₂×C₄

#13083574

yes

characteristic

111

C₈

—

112

C₈

—

356

—

113

C₈

C₄⋊Dic₈₉

yes

characteristic

114

C₈

C₄⋊Dic₈₉

yes

characteristic

115

C₈

C₄⋊Dic₈₉

yes

characteristic

116

C₈

C₄⋊Dic₈₉

yes

characteristic

117

C₂²

—

118

C₂²

#15667971

yes

characteristic

119

C₂²

#15668000

yes

characteristic

120

C₂²

#15668004

yes

characteristic

121

C₄

—

122

C₄

—

712

—

123

C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

124

C₄

#15667971

yes

characteristic

125

C₄

#15668000

yes

characteristic

126

C₄

#15668004

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#19151464

yes

characteristic

	subgroup	G/H
center	C₄	#15667971
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#405×C₈₉	C₂
socle	C₂×C₈₉	#382
prime radical	C₂³×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#405	D₈₉
hypocenter	C₈₉	#1833
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405

subgroup

G/H

center

C₄

#15667971

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#405×C₈₉

C₂

C₂×C₈₉

C₂³×C₈₉

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ #117 ◁ #405

N	H ≅ G/N	type
15. #97×C₈₉	C₄	semidirect
16. C₄×C₈×C₈₉	C₄	semidirect
66. C₈₉	#1833	semidirect
81. #97	Dic₈₉	semidirect
82. C₄×C₈	Dic₈₉	semidirect

H ≅ G/N

type

15. #97×C₈₉

C₄

semidirect

16. C₄×C₈×C₈₉

C₄

semidirect

66. C₈₉

#1833

semidirect

81. #97

Dic₈₉

semidirect

82. C₄×C₈

Dic₈₉

semidirect

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

subgroup	index
C₂×#15667948	2
#19151636	2
#19151369	2
#19151354	2
#405×C₈₉	2
#1833	89

subgroup

index

#405×C₈₉

Subgroups — #26020376

Conjugacy classes of subgroups

Complements

Maximal subgroups