Group #26018088 — subgroups

Subgroups — #26018088

Order 11392.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄×#13083558	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
3	#431×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13083558	C₄	1	2	yes	—
5	C₂²×#10623494	—	2	1	no	—
6	#15667923	—	2	1	no	—
7	C₄×#10623494	—	2	1	no	—
8	C₄×#10623494	—	2	1	no	—
9	C₄×#10623494	C₄	1	2	yes	—
10	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₈₉	C₄	1	2	yes	—
12	C₂²⋊C₈×C₈₉	—	2	1	no	—
13	C₄×C₈×C₈₉	—	2	1	no	—
14	#13083558	—	2	2	no	—
15	#13083558	C₈	1	4	yes	complemented
16	C₂×#10623494	—	2	4	no	—
17	C₂×#10623494	—	2	2	no	—
18	C₂×#10623494	—	2	2	no	—
19	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₈₉	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₈₉	—	2	2	no	—
23	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₈₉	D₄	1	1	yes	characteristic
26	C₄²×C₈₉	D₄	1	1	yes	characteristic
27	C₂³×C₈₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₈₉	—	2	2	no	—
29	C₂×C₄×C₈₉	—	2	2	no	—
30	C₂×C₄×C₈₉	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₈₉	M₁₆	1	2	yes	—
38	C₈×C₈₉	—	4	2	no	—
39	#10623494	—	2	4	no	—
40	#10623494	—	2	4	no	—
41	C₂²×C₈₉	—	2	1	no	—
42	C₂²×C₈₉	—	2	1	no	—
43	C₂²×C₈₉	#95	1	1	yes	characteristic
44	C₂²×C₈₉	C₄⋊C₈	1	2	yes	—
45	C₄×C₈₉	—	2	1	no	—
46	C₄×C₈₉	—	2	1	no	—
47	C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
48	C₄×C₈₉	#97	1	1	yes	characteristic
49	C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
50	C₄×C₈₉	#104	1	2	yes	—
51	C₄×C₈₉	#108	1	2	yes	—
52	C₂×C₈₉	—	2	2	no	—
53	C₂×C₈₉	#336	1	1	yes	characteristic
54	C₂×C₈₉	#339	1	1	yes	characteristic
55	C₂×C₈₉	#341	1	1	yes	characteristic
56	#1278	—	89	1	no	maximal, Sylow
57	C₈₉	#1278	1	1	yes	characteristic, complemented, Sylow
58	#431	D₈₉	1	1	yes	characteristic
59	C₄×M₁₆	—	89	1	no	—
60	C₂×C₄×C₈	—	89	1	no	—
61	C₂×M₁₆	—	89	2	no	—
62	C₂²×C₈	—	178	1	no	—
63	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
64	C₄⋊C₈	Dic₈₉	1	2	yes	—
65	C₂²⋊C₈	—	2	1	no	—
66	#97	—	178	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	89	2	no	—
69	C₄×C₈	—	178	1	no	—
70	C₄×C₈	—	178	1	no	—
71	C₂²×C₄	Dic₁₇₈	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
74	M₁₆	—	89	4	no	—
75	M₁₆	—	178	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	178	4	no	—
78	C₂×C₈	—	178	2	no	—
79	C₂×C₈	—	178	2	no	—
80	C₄²	C₄×D₈₉	1	1	yes	characteristic
81	C₄²	D₃₅₆	1	1	yes	characteristic
82	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
83	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
84	C₂³	#13083553	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄×Dic₈₉	1	1	yes	characteristic
88	C₂×C₄	#13083551	1	1	yes	characteristic
89	C₂×C₄	#13083551	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
91	C₂×C₄	#13083553	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
93	C₂×C₄	C₈×D₈₉	1	2	yes	—
94	C₂×C₄	#13083557	1	2	yes	—
95	C₈	—	4	2	no	—
96	C₈	—	178	4	no	—
97	C₈	—	178	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#15667919	1	1	yes	characteristic
101	C₂²	#15667941	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₈₉	1	1	yes	characteristic
105	C₄	#15667922	1	1	yes	characteristic
106	C₄	#15667924	1	2	yes	—
107	C₄	#15667939	1	2	yes	—
108	C₄	#15667947	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#19151352	1	1	yes	characteristic
111	C₂	#19151357	1	1	yes	characteristic
112	C₂	#19151362	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13083558

C₂

yes

maximal, characteristic

C₂×C₄×#10623494

C₂

yes

maximal, characteristic

#431×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₄

yes

—

—

—

—

—

C₄

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₄

yes

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

#13083558

—

#13083558

C₈

yes

—

—

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

Q₈

yes

characteristic

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₂³×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄²

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×C₈

yes

—

C₂×C₄×C₈₉

M₁₆

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#95

yes

characteristic

C₂²×C₈₉

C₄⋊C₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×C₈

yes

characteristic

C₄×C₈₉

#97

yes

characteristic

C₄×C₈₉

C₂²⋊C₈

yes

—

C₄×C₈₉

#104

yes

—

C₄×C₈₉

#108

yes

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#431

D₈₉

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

C₂×M₁₆

—

C₂²×C₈

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

Dic₈₉

yes

—

C₂²⋊C₈

—

#97

—

178

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

178

—

C₄×C₈

—

178

—

C₂²×C₄

Dic₁₇₈

yes

characteristic

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄²

C₄×D₈₉

yes

characteristic

C₄²

D₃₅₆

yes

characteristic

C₄²

C₂×Dic₈₉

yes

characteristic

C₄²

C₂²⋊D₈₉

yes

characteristic

C₂³

#13083553

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₈₉

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₈₉

yes

characteristic

C₂×C₄

#13083553

yes

characteristic

C₂×C₄

C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

C₈×D₈₉

yes

—

C₂×C₄

#13083557

yes

—

C₈

—

C₈

—

178

—

C₈

—

178

—

C₂²

—

C₂²

—

100

C₂²

#15667919

yes

characteristic

101

C₂²

#15667941

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₈₉

yes

characteristic

105

C₄

#15667922

yes

characteristic

106

C₄

#15667924

yes

—

107

C₄

#15667939

yes

—

108

C₄

#15667947

yes

—

109

C₂

—

110

C₂

#19151352

yes

characteristic

111

C₂

#19151357

yes

characteristic

112

C₂

#19151362

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₈₉
commutator	C₄×C₈₉	C₄×C₈
Frattini	C₂×C₄²	C₂×D₈₉
Fitting	#431×C₈₉	C₂
socle	C₂²×C₈₉	#95
prime radical	C₂³×C₈₉	C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂²
hypercenter	#431	D₈₉
hypocenter	C₈₉	#1278
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₈₉

commutator

C₄×C₈₉

C₄×C₈

Frattini

C₂×C₄²

C₂×D₈₉

Fitting

#431×C₈₉

C₂

socle

C₂²×C₈₉

#95

prime radical

C₂³×C₈₉

C₄⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
15. #13083558	C₈	semidirect
57. C₈₉	#1278	semidirect

H ≅ G/N

type

15. #13083558

C₈

semidirect

57. C₈₉

#1278

semidirect

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

subgroup	index
C₄×#13083558	2
C₂×C₄×#10623494	2
#431×C₈₉	2
#1278	89

subgroup

index

C₄×#13083558

C₂×C₄×#10623494

#431×C₈₉

#1278

Subgroups — #26018088

Conjugacy classes of subgroups

Complements

Maximal subgroups