Group #44485513 — subgroups

Subgroups — #44485513

Order 30592.

252 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 109 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×#21439952	C₂	1	1	yes	maximal, characteristic
2	#29504638	C₂	1	1	yes	maximal, characteristic
3	#29504535	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	#21439952	—	2	2	no	—
6	#21439952	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
9	C₂×Dic₂₃₉⋊C₄	—	2	2	no	—
10	C₂₃₉⋊#95	—	2	2	no	—
11	C₂²×C₈×C₂₃₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₂₃₉	—	2	2	no	—
14	C₂²×Dic₂₃₉	—	2	2	no	—
15	C₂²×Dic₂₃₉	—	2	2	no	—
16	C₄⋊Dic₂₃₉	—	2	2	no	—
17	C₄⋊Dic₂₃₉	—	2	4	no	—
18	C₄⋊Dic₂₃₉	C₂×C₄	1	4	yes	—
19	Dic₂₃₉⋊C₄	—	2	8	no	—
20	C₂²×C₄×C₂₃₉	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₂₃₉	—	2	2	no	—
23	C₂×C₈×C₂₃₉	—	2	2	no	—
24	C₂×C₈×C₂₃₉	Q₈	1	4	yes	—
25	C₄²×C₂₃₉	C₂×C₄	1	2	yes	—
26	C₄²×C₂₃₉	D₄	1	2	yes	—
27	C₂×Dic₂₃₉	—	2	4	no	—
28	C₂×Dic₂₃₉	—	4	4	no	—
29	C₂×Dic₂₃₉	—	4	2	no	—
30	C₂×Dic₂₃₉	—	4	4	no	—
31	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃₉	—	2	4	no	—
33	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₂₃₉	#41	1	2	yes	—
38	C₈×C₂₃₉	—	2	4	no	—
39	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₃₉	#124	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	C₄×Q₈	1	2	yes	—
45	C₄×C₂₃₉	C₂²⋊Q₈	1	2	yes	—
46	Dic₂₃₉	—	4	4	no	—
47	Dic₂₃₉	—	8	2	no	—
48	C₂×C₂₃₉	#391	1	1	yes	characteristic
49	C₂×C₂₃₉	#416	1	2	yes	—
50	C₂×C₂₃₉	#488	1	2	yes	—
51	C₂×C₂₃₉	#499	1	2	yes	—
52	C₂₃₉	#1991	1	1	yes	characteristic, complemented, Sylow
53	#1991	—	239	1	no	maximal, Sylow
54	C₂×#125	—	239	1	no	—
55	C₂×C₄×C₈	D₂₃₉	1	1	yes	characteristic
56	#382	—	239	1	no	—
57	#340	—	239	4	no	—
58	C₂²×C₈	C₂×D₂₃₉	1	2	yes	—
59	#125	—	478	2	no	—
60	#125	—	239	4	no	—
61	C₂×C₄⋊C₄	—	478	2	no	—
62	C₂×C₄⋊C₄	—	239	2	no	—
63	C₂×C₄⋊C₄	—	239	2	no	—
64	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
65	C₄×C₈	—	2	2	no	—
66	#95	—	478	2	no	—
67	C₂²×C₄	—	478	2	no	—
68	C₂²×C₄	—	478	2	no	—
69	C₂²×C₄	D₉₅₆	1	2	yes	—
70	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	Dic₄₇₈	1	4	yes	—
74	C₄⋊C₄	—	478	4	no	—
75	C₄⋊C₄	—	478	8	no	—
76	C₄⋊C₄	—	478	2	no	—
77	C₄⋊C₄	—	239	4	no	—
78	C₄²	C₄×D₂₃₉	1	2	yes	—
79	C₄²	C₂²⋊D₂₃₉	1	2	yes	—
80	C₂³	C₂×D₉₅₆	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	956	2	no	—
83	C₂×C₄	—	956	4	no	—
84	C₂×C₄	—	956	4	no	—
85	C₂×C₄	—	478	4	no	—
86	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
89	C₂×C₄	C₂×Dic₄₇₈	1	2	yes	—
90	C₂×C₄	#16838548	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
93	C₂²	#21439948	1	1	yes	characteristic
94	C₂²	#21439960	1	1	yes	characteristic
95	C₂²	#21439982	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	1912	2	no	—
98	C₄	—	956	4	no	—
99	C₄	C₄×Dic₄₇₈	1	2	yes	—
100	C₄	#21439939	1	2	yes	—
101	C₂	#29504678	1	1	yes	characteristic
102	C₂	#29504769	1	2	yes	—
103	C₂	#29505127	1	2	yes	—
104	C₂	#29505174	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439952

C₂

yes

maximal, characteristic

#29504638

C₂

yes

maximal, characteristic

#29504535

C₂

yes

maximal

C₂×C₄×C₈×C₂₃₉

C₂

yes

maximal, characteristic

#21439952

—

#21439952

C₄

yes

complemented

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

C₂×Dic₂₃₉⋊C₄

—

C₂₃₉⋊#95

—

C₂²×C₈×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₄×C₈×C₂₃₉

—

C₂²×Dic₂₃₉

—

C₂²×Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

C₂×C₄

yes

—

Dic₂₃₉⋊C₄

—

C₂²×C₄×C₂₃₉

D₄

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

Q₈

yes

—

C₄²×C₂₃₉

C₂×C₄

yes

—

C₄²×C₂₃₉

D₄

yes

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂³×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

—

C₂×C₄×C₂₃₉

#41

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃₉

#124

yes

characteristic

C₂²×C₂₃₉

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃₉

#135

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₄×Q₈

yes

—

C₄×C₂₃₉

C₂²⋊Q₈

yes

—

Dic₂₃₉

—

Dic₂₃₉

—

C₂×C₂₃₉

#391

yes

characteristic

C₂×C₂₃₉

#416

yes

—

C₂×C₂₃₉

#488

yes

—

C₂×C₂₃₉

#499

yes

—

C₂₃₉

#1991

yes

characteristic, complemented, Sylow

#1991

—

239

maximal, Sylow

C₂×#125

—

239

—

C₂×C₄×C₈

D₂₃₉

yes

characteristic

#382

—

239

—

#340

—

239

—

C₂²×C₈

C₂×D₂₃₉

yes

—

#125

—

478

—

#125

—

239

—

C₂×C₄⋊C₄

—

478

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

C₄×C₈

—

#95

—

478

—

C₂²×C₄

—

478

—

C₂²×C₄

—

478

—

C₂²×C₄

D₉₅₆

yes

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₇₈

yes

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

239

—

C₄²

C₄×D₂₃₉

yes

—

C₄²

C₂²⋊D₂₃₉

yes

—

C₂³

C₂×D₉₅₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

956

—

C₂×C₄

—

956

—

C₂×C₄

—

956

—

C₂×C₄

—

478

—

C₂×C₄

(C₂×D₂₃₉)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₃₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

C₂×C₄

C₂×Dic₄₇₈

yes

—

C₂×C₄

#16838548

yes

—

C₈

—

C₂²

C₂×(C₂×D₂₃₉)⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

1912

—

C₄

—

956

—

C₄

C₄×Dic₄₇₈

yes

—

100

C₄

#21439939

yes

—

101

C₂

#29504678

yes

characteristic

102

C₂

#29504769

yes

—

103

C₂

#29505127

yes

—

104

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	#1991
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

#1991

LCS

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
6. #21439952	C₄	semidirect
52. C₂₃₉	#1991	semidirect

H ≅ G/N

type

6. #21439952

C₄

semidirect

52. C₂₃₉

#1991

semidirect

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

subgroup	index
C₂×#21439952	2
#29504638	2
#29504535	2
C₂×C₄×C₈×C₂₃₉	2
#1991	239

subgroup

index

C₂×C₄×C₈×C₂₃₉

239

Subgroups — #44485513

Conjugacy classes of subgroups

Complements

Maximal subgroups