Group #26018363 — subgroups

Subgroups — #26018363

Order 11392.

124 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 69 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×#15667952	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667948	C₂	1	1	yes	maximal, characteristic
3	C₂×#108×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13083558	—	2	1	no	—
5	#15667952	—	2	1	no	—
6	#15667952	C₄	1	2	yes	—
7	C₂×#13083549	—	2	1	no	—
8	#15667948	—	2	1	no	—
9	#15667948	C₄	1	2	yes	—
10	C₂×M₁₆×C₈₉	—	2	1	no	—
11	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	#108×C₈₉	—	2	1	no	—
13	#108×C₈₉	C₄	1	2	yes	—
14	#13083558	—	2	2	no	—
15	#13083558	—	4	1	no	—
16	C₂×#10623494	—	4	1	no	—
17	#13083549	—	2	2	no	—
18	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
19	M₁₆×C₈₉	—	2	2	no	—
20	M₁₆×C₈₉	—	4	1	no	—
21	C₂×C₈×C₈₉	—	4	1	no	—
22	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₈₉	Q₈	1	1	yes	characteristic
28	C₂³×C₈₉	Q₁₆	1	1	yes	characteristic
29	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	D₈	1	1	yes	characteristic
33	C₂×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
35	C₈×C₈₉	—	4	2	no	—
36	C₈×C₈₉	C₄²	1	1	yes	characteristic
37	C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
40	#10623494	—	4	2	no	—
41	C₂²×C₈₉	—	4	1	no	—
42	C₂²×C₈₉	#103	1	1	yes	characteristic
43	C₂²×C₈₉	#103	1	1	yes	characteristic
44	C₂²×C₈₉	#107	1	1	yes	characteristic
45	C₄×C₈₉	#95	1	1	yes	characteristic
46	C₄×C₈₉	#102	1	1	yes	characteristic
47	C₄×C₈₉	#102	1	1	yes	characteristic
48	C₄×C₈₉	#106	1	1	yes	characteristic
49	C₂×C₈₉	—	2	1	no	—
50	C₂×C₈₉	—	2	1	no	—
51	C₂×C₈₉	—	2	1	no	—
52	C₂×C₈₉	#340	1	1	yes	characteristic
53	#1388	—	89	1	no	maximal, Sylow
54	C₈₉	#1388	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	89	1	no	—
56	C₂×#108	—	89	1	no	—
57	C₂×#108	D₈₉	1	1	yes	characteristic
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	—	178	1	no	—
60	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
61	M₃₂	—	89	2	no	—
62	M₃₂	—	178	1	no	—
63	C₂×C₁₆	—	178	1	no	—
64	#108	—	2	1	no	—
65	#108	—	89	2	no	—
66	#108	—	178	1	no	—
67	#108	Dic₈₉	1	2	yes	—
68	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	4	1	no	—
71	M₁₆	—	356	1	no	—
72	M₁₆	—	178	2	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	356	1	no	—
75	C₂×C₈	Dic₁₇₈	1	1	yes	characteristic
76	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
77	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
78	C₂×C₈	D₃₅₆	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₈₉	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
81	C₁₆	—	178	2	no	—
82	C₂³	#13083564	1	1	yes	characteristic
83	C₂×C₄	#13083551	1	1	yes	characteristic
84	C₂×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
85	C₂×C₄	#13083553	1	1	yes	characteristic
86	C₂×C₄	#13083559	1	1	yes	characteristic
87	C₂×C₄	#13083561	1	1	yes	characteristic
88	C₂×C₄	#13083562	1	1	yes	characteristic
89	C₈	—	4	2	no	—
90	C₈	—	356	2	no	—
91	C₈	C₄×Dic₈₉	1	1	yes	characteristic
92	C₈	#13083551	1	1	yes	characteristic
93	C₈	#13083553	1	1	yes	characteristic
94	C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
95	C₂²	—	4	1	no	—
96	C₂²	#15667935	1	1	yes	characteristic
97	C₂²	#15667936	1	1	yes	characteristic
98	C₂²	#15667945	1	1	yes	characteristic
99	C₄	#15667919	1	1	yes	characteristic
100	C₄	#15667932	1	1	yes	characteristic
101	C₄	#15667933	1	1	yes	characteristic
102	C₄	#15667943	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	—	2	1	no	—
106	C₂	#19151360	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667952

C₂

yes

maximal, characteristic

C₂×#15667948

C₂

yes

maximal, characteristic

C₂×#108×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

—

#15667952

—

#15667952

C₄

yes

—

C₂×#13083549

—

#15667948

—

#15667948

C₄

yes

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#108×C₈₉

—

#108×C₈₉

C₄

yes

—

—

—

—

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

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

C₂³×C₈₉

Q₁₆

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

D₈

yes

characteristic

C₂×C₄×C₈₉

SD₁₆

yes

characteristic

C₂×C₄×C₈₉

SD₁₆

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

C₄²

yes

characteristic

C₈×C₈₉

C₂²⋊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

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#108

—

C₂×#108

D₈₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

178

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

M₃₂

—

M₃₂

—

178

—

C₂×C₁₆

—

178

—

#108

—

#108

—

#108

—

178

—

#108

Dic₈₉

yes

—

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

356

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

Dic₁₇₈

yes

characteristic

C₂×C₈

C₄×D₈₉

yes

characteristic

C₂×C₈

C₄×D₈₉

yes

characteristic

C₂×C₈

D₃₅₆

yes

characteristic

C₂×C₈

C₂×Dic₈₉

yes

characteristic

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

C₁₆

—

178

—

C₂³

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₈₉

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

356

—

C₈

C₄×Dic₈₉

yes

C₈

yes

C₈

yes

C₈

C₄⋊Dic₈₉

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

yes

100

C₄

#15667932

yes

characteristic

101

C₄

#15667933

yes

characteristic

102

C₄

#15667943

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

106

C₂

#19151360

yes

characteristic

	subgroup	G/H
center	C₄	#15667933
commutator	C₈×C₈₉	C₄²
Frattini	C₂²×C₈	C₂×D₈₉
Fitting	C₂×#108×C₈₉	C₂
socle	C₂×C₈₉	#340
prime radical	C₂³×C₈₉	Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂²
hypercenter	C₂×#108	D₈₉
hypocenter	C₈₉	#1388
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

subgroup

G/H

center

C₄

#15667933

commutator

C₈×C₈₉

C₄²

Frattini

C₂²×C₈

C₂×D₈₉

Fitting

C₂×#108×C₈₉

C₂

socle

C₂×C₈₉

#340

prime radical

C₂³×C₈₉

Q₁₆

cosocle

C₂²×C₈×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

N	H ≅ G/N	type
54. C₈₉	#1388	semidirect

H ≅ G/N

type

54. C₈₉

#1388

semidirect

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

subgroup	index
C₂×#15667952	2
C₂×#15667948	2
C₂×#108×C₈₉	2
#1388	89

subgroup

index

C₂×#15667952

C₂×#15667948

C₂×#108×C₈₉

#1388

Subgroups — #26018363

Conjugacy classes of subgroups

Complements

Maximal subgroups