Group #26021006 — subgroups

Subgroups — #26021006

Order 11392.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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	#19151625	C₂	1	2	yes	maximal
3	C₂×#15667922	C₂	1	1	yes	maximal, characteristic
4	#19151357	C₂	1	2	yes	maximal
5	#427×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	C₂²×#10623494	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
9	#15667948	—	2	2	no	—
10	C₄⋊#10623494	—	2	4	no	—
11	#15667922	—	2	2	no	—
12	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	#117×C₈₉	C₂²	1	2	yes	—
14	#107×C₈₉	C₄	1	2	yes	complemented
15	#106×C₈₉	C₄	1	2	yes	complemented
16	C₂²×Dic₈₉	—	2	1	no	—
17	#13083558	—	2	2	no	—
18	#13083558	—	2	2	no	—
19	C₂×#10623494	—	2	1	no	—
20	C₂×#10623494	—	2	1	no	—
21	C₂×#10623494	—	2	2	no	—
22	C₂×#10623494	D₄	1	2	yes	—
23	C₄×Dic₈₉	—	2	1	no	—
24	C₄×Dic₈₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₈₉	D₄	1	2	yes	—
29	C₂×C₈×C₈₉	Q₈	1	2	yes	—
30	C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₈₉	—	2	2	no	—
32	C₄²×C₈₉	—	2	2	no	—
33	C₂³×C₈₉	#41	1	1	yes	characteristic
34	C₂×C₄×C₈₉	—	2	4	no	—
35	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
41	C₂×Dic₈₉	—	2	2	no	—
42	C₂×Dic₈₉	—	4	2	no	—
43	C₈×C₈₉	C₄⋊C₄	1	2	yes	—
44	C₈×C₈₉	C₄⋊C₄	1	2	yes	—
45	#10623494	—	2	2	no	—
46	#10623494	—	4	2	no	—
47	C₂²×C₈₉	—	2	2	no	—
48	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₈₉	#125	1	1	yes	characteristic
51	C₄×C₈₉	—	4	4	no	—
52	C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₈₉	#121	1	1	yes	characteristic
55	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₈₉	—	4	2	no	—
57	C₂×C₈₉	—	2	1	no	—
58	C₂×C₈₉	—	2	1	no	—
59	C₂×C₈₉	#389	1	1	yes	characteristic
60	C₂×C₈₉	#444	1	2	yes	—
61	#1946	—	89	1	no	maximal, Sylow
62	C₈₉	#1946	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	—	89	1	no	—
64	#427	D₈₉	1	1	yes	characteristic
65	#424	—	89	2	no	—
66	C₂×#97	—	89	1	no	—
67	#339	—	89	2	no	—
68	C₂×M₁₆	—	89	2	no	—
69	C₂²×C₈	—	89	1	no	—
70	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
71	#117	C₂×D₈₉	1	2	yes	—
72	C₂×C₄²	—	89	1	no	—
73	#108	—	178	2	no	—
74	#107	Dic₈₉	1	2	yes	complemented
75	#106	Dic₈₉	1	2	yes	complemented
76	C₄⋊C₈	—	178	4	no	—
77	#97	—	178	2	no	—
78	C₂²×C₄	—	178	1	no	—
79	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
80	M₁₆	—	178	2	no	—
81	M₁₆	—	178	2	no	—
82	C₂×C₈	—	89	2	no	—
83	C₂×C₈	—	178	2	no	—
84	C₂×C₈	—	178	1	no	—
85	C₂×C₈	—	178	1	no	—
86	C₂×C₈	Dic₁₇₈	1	2	yes	—
87	C₂×C₈	D₃₅₆	1	2	yes	—
88	C₂×C₈	C₂×Dic₈₉	1	1	yes	characteristic
89	C₂×C₈	C₂×Dic₈₉	1	1	yes	characteristic
90	C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	89	2	no	—
94	C₄²	—	178	1	no	—
95	C₂³	#13083573	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	356	2	no	—
98	C₂×C₄	—	178	2	no	—
99	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
104	C₂×C₄	#13083575	1	1	yes	characteristic
105	C₈	—	356	2	no	—
106	C₈	—	178	2	no	—
107	C₈	C₄⋊Dic₈₉	1	2	yes	—
108	C₈	C₄⋊Dic₈₉	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₈₉	1	1	yes	characteristic
111	C₂²	#15667992	1	1	yes	characteristic
112	C₂²	#15668003	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	356	2	no	—
115	C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
116	C₄	Q₈×Dic₈₉	1	1	yes	characteristic
117	C₄	#15667987	1	1	yes	characteristic
118	C₄	#15668011	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#19151497	1	1	yes	characteristic
122	C₂	#19151720	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667948

C₂

yes

maximal, characteristic

#19151625

C₂

yes

maximal

C₂×#15667922

C₂

yes

maximal, characteristic

#19151357

C₂

yes

maximal

#427×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

—

—

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

—

#107×C₈₉

C₄

yes

complemented

#106×C₈₉

C₄

yes

complemented

C₂²×Dic₈₉

—

—

—

—

—

—

D₄

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

Q₈

yes

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

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×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

C₄⋊C₄

yes

—

C₈×C₈₉

C₄⋊C₄

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×D₄

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₉

#125

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₉

C₄×Q₈

yes

characteristic

C₄×C₈₉

#121

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#389

yes

characteristic

C₂×C₈₉

#444

yes

—

#1946

—

maximal, Sylow

C₈₉

#1946

yes

characteristic, complemented, Sylow

C₂×#108

—

#427

D₈₉

yes

characteristic

#424

—

C₂×#97

—

#339

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

#117

C₂×D₈₉

yes

—

C₂×C₄²

—

#108

—

178

—

#107

Dic₈₉

yes

complemented

#106

Dic₈₉

yes

complemented

C₄⋊C₈

—

178

—

#97

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

Dic₁₇₈

yes

—

C₂×C₈

D₃₅₆

yes

—

C₂×C₈

C₂×Dic₈₉

yes

characteristic

C₂×C₈

C₂×Dic₈₉

yes

characteristic

C₄⋊C₄

C₂×Dic₈₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

178

—

C₂³

#13083573

yes

characteristic

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

100

C₂×C₄

D₄×D₈₉

yes

characteristic

101

C₂×C₄

C₂×D₃₅₆

yes

characteristic

102

C₂×C₄

Q₈×D₈₉

yes

characteristic

103

C₂×C₄

C₂×Dic₁₇₈

yes

characteristic

104

C₂×C₄

#13083575

yes

characteristic

105

C₈

—

356

—

106

C₈

—

178

—

107

C₈

C₄⋊Dic₈₉

yes

—

108

C₈

C₄⋊Dic₈₉

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₈₉

yes

characteristic

111

C₂²

#15667992

yes

characteristic

112

C₂²

#15668003

yes

characteristic

113

C₄

—

114

C₄

—

356

—

115

C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

116

C₄

Q₈×Dic₈₉

yes

characteristic

117

C₄

#15667987

yes

characteristic

118

C₄

#15668011

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#19151497

yes

characteristic

122

C₂

#19151720

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#427×C₈₉	C₂
socle	C₂²×C₈₉	C₄×D₄
prime radical	C₂³×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#427	D₈₉
hypocenter	C₈₉	#1946
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#427×C₈₉

C₂

socle

C₂²×C₈₉

C₄×D₄

prime radical

C₂³×C₈₉

#41

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₈₉	C₄	semidirect
15. #106×C₈₉	C₄	semidirect
62. C₈₉	#1946	semidirect
74. #107	Dic₈₉	semidirect
75. #106	Dic₈₉	semidirect

H ≅ G/N

type

14. #107×C₈₉

C₄

semidirect

15. #106×C₈₉

C₄

semidirect

62. C₈₉

#1946

semidirect

74. #107

Dic₈₉

semidirect

75. #106

Dic₈₉

semidirect

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

subgroup	index
C₂×#15667948	2
#19151625	2
C₂×#15667922	2
#19151357	2
#427×C₈₉	2
#1946	89

subgroup

index

#427×C₈₉

Subgroups — #26021006

Conjugacy classes of subgroups

Complements

Maximal subgroups