Group #44484712 — subgroups

Subgroups — #44484712

Order 30592.

248 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 141 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×#21439894	C₂	1	2	yes	maximal
2	C₂×#21439892	C₂	1	2	yes	maximal
3	#29504638	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₂₃₉	C₂²	1	4	yes	—
6	#21439894	—	2	4	no	—
7	#21439892	—	2	4	no	—
8	C₂₃₉⋊#95	—	2	4	no	—
9	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₂₃₉	C₄	1	4	yes	complemented
13	C₂²×Dic₂₃₉	—	2	4	no	—
14	C₄⋊Dic₂₃₉	—	2	4	no	—
15	C₄⋊Dic₂₃₉	—	2	4	no	—
16	C₂²×C₄×C₂₃₉	D₄	1	2	yes	—
17	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₂₃₉	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₂₃₉	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
21	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
22	C₂×C₈×C₂₃₉	Q₈	1	2	yes	—
23	C₂×C₈×C₂₃₉	Q₈	1	2	yes	—
24	C₄²×C₂₃₉	C₂×C₄	1	2	yes	—
25	C₄²×C₂₃₉	Q₈	1	2	yes	—
26	C₂×Dic₂₃₉	—	4	4	no	—
27	C₂×Dic₂₃₉	—	4	8	no	—
28	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₃₉	—	2	4	no	—
30	C₂×C₄×C₂₃₉	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
37	C₈×C₂₃₉	C₄⋊C₄	1	4	yes	—
38	C₈×C₂₃₉	C₄⋊C₄	1	4	yes	—
39	C₂²×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₃₉	#127	1	1	yes	characteristic
41	C₂²×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₂₃₉	#135	1	4	yes	—
43	C₄×C₂₃₉	—	2	4	no	—
44	C₄×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
48	Dic₂₃₉	—	8	4	no	—
49	C₂×C₂₃₉	#384	1	1	yes	characteristic
50	C₂×C₂₃₉	#427	1	2	yes	—
51	C₂×C₂₃₉	#495	1	2	yes	—
52	C₂×C₂₃₉	#499	1	2	yes	—
53	C₂₃₉	#1846	1	1	yes	characteristic, complemented, Sylow
54	#1846	—	239	1	no	maximal, Sylow
55	C₂×#107	—	239	2	no	—
56	C₂×#106	—	239	2	no	—
57	C₂×C₄×C₈	D₂₃₉	1	1	yes	characteristic
58	#382	—	239	2	no	—
59	C₂²×C₈	C₂×D₂₃₉	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₂₃₉	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	239	4	no	—
62	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
63	#107	—	478	4	no	—
64	#106	—	478	4	no	—
65	C₄×C₈	Dic₂₃₉	1	4	yes	complemented
66	#95	—	478	4	no	—
67	C₂²×C₄	—	478	4	no	—
68	C₂²×C₄	D₉₅₆	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
70	C₂×C₈	C₂×Dic₂₃₉	1	2	yes	—
71	C₂×C₈	C₂×Dic₂₃₉	1	2	yes	—
72	C₂×C₈	D₉₅₆	1	2	yes	—
73	C₂×C₈	D₉₅₆	1	2	yes	—
74	C₂×C₈	Dic₄₇₈	1	2	yes	—
75	C₂×C₈	Dic₄₇₈	1	2	yes	—
76	C₄⋊C₄	—	478	4	no	—
77	C₄⋊C₄	—	478	4	no	—
78	C₄²	C₂×Dic₂₃₉	1	2	yes	—
79	C₄²	Dic₄₇₈	1	2	yes	—
80	C₂³	C₂×D₉₅₆	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	956	4	no	—
83	C₂×C₄	—	956	8	no	—
84	C₂×C₄	C₄⋊Dic₂₃₉	1	4	yes	—
85	C₂×C₄	C₂²×Dic₂₃₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
87	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
89	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
90	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
91	C₈	C₄⋊Dic₂₃₉	1	4	yes	—
92	C₈	C₄⋊Dic₂₃₉	1	4	yes	—
93	C₂²	C₂×C₄⋊Dic₂₃₉	1	1	yes	characteristic
94	C₂²	#21439957	1	1	yes	characteristic
95	C₂²	#21439960	1	1	yes	characteristic
96	C₂²	#21439982	1	4	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	1912	4	no	—
99	C₄	C₂×C₄⋊Dic₂₃₉	1	1	yes	characteristic
100	C₄	C₂×C₄⋊Dic₂₃₉	1	1	yes	characteristic
101	C₄	#21439960	1	1	yes	characteristic
102	C₄	#21439960	1	1	yes	characteristic
103	C₂	#29504642	1	1	yes	characteristic
104	C₂	#29504810	1	2	yes	—
105	C₂	#29505156	1	2	yes	—
106	C₂	#29505174	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439894

C₂

yes

maximal

C₂×#21439892

C₂

yes

maximal

#29504638

C₂

yes

maximal

C₂×C₄×C₈×C₂₃₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

#21439894

—

#21439892

—

C₂₃₉⋊#95

—

C₂²×C₈×C₂₃₉

C₂²

yes

characteristic

C₂²×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₄×C₈×C₂₃₉

C₄

yes

complemented

C₂²×Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₂²×C₄×C₂₃₉

D₄

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₂×C₈×C₂₃₉

C₂×C₄

yes

—

C₂×C₈×C₂₃₉

C₂×C₄

yes

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₂×C₈×C₂₃₉

Q₈

yes

—

C₂×C₈×C₂₃₉

Q₈

yes

—

C₄²×C₂₃₉

C₂×C₄

yes

—

C₄²×C₂₃₉

Q₈

yes

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂³×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₈×C₂₃₉

C₄⋊C₄

yes

—

C₈×C₂₃₉

C₄⋊C₄

yes

—

C₂²×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃₉

#127

yes

characteristic

C₂²×C₂₃₉

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃₉

#135

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

Dic₂₃₉

—

C₂×C₂₃₉

#384

yes

characteristic

C₂×C₂₃₉

#427

yes

—

C₂×C₂₃₉

#495

yes

—

C₂×C₂₃₉

#499

yes

—

C₂₃₉

#1846

yes

characteristic, complemented, Sylow

#1846

—

239

maximal, Sylow

C₂×#107

—

239

—

C₂×#106

—

239

—

C₂×C₄×C₈

D₂₃₉

yes

characteristic

#382

—

239

—

C₂²×C₈

C₂×D₂₃₉

yes

characteristic

C₂²×C₈

C₂×D₂₃₉

yes

characteristic

C₂×C₄⋊C₄

—

239

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

#107

—

478

—

#106

—

478

—

C₄×C₈

Dic₂₃₉

yes

complemented

#95

—

478

—

C₂²×C₄

—

478

—

C₂²×C₄

D₉₅₆

yes

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

C₂×Dic₂₃₉

yes

—

C₂×C₈

C₂×Dic₂₃₉

yes

—

C₂×C₈

D₉₅₆

yes

—

C₂×C₈

D₉₅₆

yes

—

C₂×C₈

Dic₄₇₈

yes

—

C₂×C₈

Dic₄₇₈

yes

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄²

C₂×Dic₂₃₉

yes

—

C₄²

Dic₄₇₈

yes

—

C₂³

C₂×D₉₅₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

956

—

C₂×C₄

—

956

—

C₂×C₄

C₄⋊Dic₂₃₉

yes

—

C₂×C₄

C₂²×Dic₂₃₉

yes

characteristic

C₂×C₄

C₂×D₉₅₆

yes

characteristic

C₂×C₄

C₂×D₉₅₆

yes

characteristic

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

C₈

C₄⋊Dic₂₃₉

yes

—

C₈

C₄⋊Dic₂₃₉

yes

—

C₂²

C₂×C₄⋊Dic₂₃₉

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

1912

—

C₄

C₂×C₄⋊Dic₂₃₉

yes

characteristic

100

C₄

C₂×C₄⋊Dic₂₃₉

yes

characteristic

101

C₄

#21439960

yes

characteristic

102

C₄

#21439960

yes

characteristic

103

C₂

#29504642

yes

characteristic

104

C₂

#29504810

yes

—

105

C₂

#29505156

yes

—

106

C₂

#29505174

yes

—

	subgroup	G/H
center	C₂³	C₂×D₉₅₆
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₂×C₄×C₈×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	C₂×D₄
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂×C₄×C₈	D₂₃₉
hypocenter	C₂₃₉	#1846
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₉₅₆

commutator

C₂×C₄×C₂₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

C₂×C₄×C₈×C₂₃₉

C₂

socle, prime radical

C₂³×C₂₃₉

C₂×D₄

cosocle

C₂²×C₄×C₂₃₉

C₂³

hypercenter

C₂×C₄×C₈

D₂₃₉

hypocenter

C₂₃₉

#1846

LCS

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
12. C₄×C₈×C₂₃₉	C₄	semidirect
53. C₂₃₉	#1846	semidirect
65. C₄×C₈	Dic₂₃₉	semidirect

H ≅ G/N

type

12. C₄×C₈×C₂₃₉

C₄

semidirect

53. C₂₃₉

#1846

semidirect

65. C₄×C₈

Dic₂₃₉

semidirect

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

subgroup	index
C₂×#21439894	2
C₂×#21439892	2
#29504638	2
C₂×C₄×C₈×C₂₃₉	2
#1846	239

subgroup

index

C₂×C₄×C₈×C₂₃₉

239

Subgroups — #44484712

Conjugacy classes of subgroups

Complements

Maximal subgroups