Group #44486058 — subgroups

Subgroups — #44486058

Order 30592.

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

normal subgroups only

This group has 176 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	#29504815	C₂	1	2	yes	maximal
2	C₂×#21439892	C₂	1	1	yes	maximal, characteristic
3	#29504540	C₂	1	2	yes	maximal
4	#29504537	C₂	1	1	yes	maximal, characteristic
5	#424×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16838532	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
8	#21439893	—	2	2	no	—
9	#21439892	—	2	2	no	—
10	#21439891	—	2	2	no	—
11	C₂×M₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
13	#117×C₂₃₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₃₉	—	2	2	no	—
15	C₂²×Dic₂₃₉	—	2	2	no	—
16	#16838532	—	2	2	no	—
17	#16838532	Q₈	1	4	yes	—
18	C₂×C₂₃₉⋊C₈	—	2	2	no	—
19	C₄⋊Dic₂₃₉	—	2	2	no	—
20	C₄⋊Dic₂₃₉	—	2	2	no	—
21	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₂₃₉	—	2	2	no	—
23	C₂×C₈×C₂₃₉	—	2	2	no	—
24	C₂×C₈×C₂₃₉	—	2	1	no	—
25	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₂₃₉	D₄	1	2	yes	—
27	C₂²⋊C₄×C₂₃₉	—	2	1	no	—
28	C₄²×C₂₃₉	—	2	1	no	—
29	C₂×Dic₂₃₉	—	4	2	no	—
30	C₂×Dic₂₃₉	—	8	2	no	—
31	C₂₃₉⋊C₈	—	2	4	no	—
32	C₂³×C₂₃₉	#41	1	1	yes	characteristic
33	C₂×C₄×C₂₃₉	—	2	2	no	—
34	C₂×C₄×C₂₃₉	C₂×D₄	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	2	no	—
39	C₈×C₂₃₉	—	4	1	no	—
40	C₂²×C₂₃₉	—	2	2	no	—
41	C₂²×C₂₃₉	#121	1	1	yes	characteristic
42	C₂²×C₂₃₉	#125	1	2	yes	—
43	C₄×C₂₃₉	—	4	2	no	—
44	C₄×C₂₃₉	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₂₃₉	#123	1	1	yes	characteristic
46	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₂₃₉	—	8	2	no	—
48	C₂×C₂₃₉	—	2	2	no	—
49	C₂×C₂₃₉	#398	1	1	yes	characteristic
50	C₂×C₂₃₉	#471	1	2	yes	—
51	C₂₃₉	#2090	1	1	yes	characteristic, complemented, Sylow
52	#2090	—	239	1	no	maximal, Sylow
53	#428	—	239	2	no	—
54	C₂×#106	—	239	1	no	—
55	#424	D₂₃₉	1	1	yes	characteristic
56	#343	—	239	2	no	—
57	#341	—	239	1	no	—
58	C₂×M₁₆	—	239	2	no	—
59	C₂×M₁₆	C₂×D₂₃₉	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₂₃₉	1	1	yes	characteristic
61	#117	C₂×D₂₃₉	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	239	2	no	—
63	#107	—	478	2	no	—
64	#106	—	478	2	no	—
65	#106	—	478	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	478	2	no	—
68	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	478	2	no	—
71	M₁₆	—	239	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	478	2	no	—
75	C₂×C₈	C₂²⋊D₂₃₉	1	2	yes	—
76	C₄⋊C₄	—	478	2	no	—
77	C₄⋊C₄	—	478	2	no	—
78	C₄⋊C₄	D₉₅₆	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#16838548	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	1912	2	no	—
84	C₂×C₄	—	956	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₂₃₉	1	2	yes	—
88	C₂×C₄	#16838547	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	478	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#21439934	1	1	yes	characteristic
94	C₂²	#21439950	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	1912	2	no	—
97	C₄	#21439940	1	2	yes	—
98	C₄	#21439941	1	1	yes	characteristic
99	C₄	#21439958	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#29504702	1	1	yes	characteristic
102	C₂	#29505035	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#29504815

C₂

yes

maximal

C₂×#21439892

C₂

yes

maximal, characteristic

#29504540

C₂

yes

maximal

#29504537

C₂

yes

maximal, characteristic

#424×C₂₃₉

C₂

yes

maximal, characteristic

C₂×#16838532

C₂²

yes

—

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

#21439893

—

#21439892

—

#21439891

—

C₂×M₁₆×C₂₃₉

C₂²

yes

characteristic

C₂²×C₈×C₂₃₉

C₂²

yes

characteristic

#117×C₂₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₃₉

—

C₂²×Dic₂₃₉

—

#16838532

—

#16838532

Q₈

yes

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

M₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₄⋊C₄×C₂₃₉

D₄

yes

—

C₂²⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

#41

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

—

C₂×C₄×C₂₃₉

#41

yes

—

C₈×C₂₃₉

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

#121

yes

characteristic

C₂²×C₂₃₉

#125

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂²⋊Q₈

yes

—

C₄×C₂₃₉

#123

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#398

yes

characteristic

C₂×C₂₃₉

#471

yes

—

C₂₃₉

#2090

yes

characteristic, complemented, Sylow

#2090

—

239

maximal, Sylow

#428

—

239

—

C₂×#106

—

239

—

#424

D₂₃₉

yes

characteristic

#343

—

239

—

#341

—

239

—

C₂×M₁₆

—

239

—

C₂×M₁₆

C₂×D₂₃₉

yes

characteristic

C₂²×C₈

C₂×D₂₃₉

yes

characteristic

#117

C₂×D₂₃₉

yes

characteristic

C₂×C₄⋊C₄

—

239

—

#107

—

478

—

#106

—

478

—

#106

—

478

—

C₄⋊C₈

—

C₂²×C₄

—

478

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

M₁₆

—

M₁₆

—

478

—

M₁₆

—

239

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

478

—

C₂×C₈

C₂²⋊D₂₃₉

yes

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄⋊C₄

D₉₅₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#16838548

yes

characteristic

C₂×C₄

—

C₂×C₄

—

1912

—

C₂×C₄

—

956

—

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

C₂×C₄

C₂×D₉₅₆

yes

characteristic

C₂×C₄

Q₈×D₂₃₉

yes

—

C₂×C₄

#16838547

yes

—

C₈

—

C₈

—

C₈

—

478

—

C₂²

—

C₂²

#21439934

yes

characteristic

C₂²

#21439950

yes

—

C₄

—

C₄

—

1912

—

C₄

#21439940

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#29504702

yes

characteristic

102

C₂

#29505035

yes

—

	subgroup	G/H
center	C₂²	#21439934
commutator, cosocle	C₂²×C₄×C₂₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	#424×C₂₃₉	C₂
socle	C₂²×C₂₃₉	#121
prime radical	C₂³×C₂₃₉	#41
hypercenter	#424	D₂₃₉
hypocenter	C₂₃₉	#2090
LCS	G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#21439934

commutator, cosocle

C₂²×C₄×C₂₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

#424×C₂₃₉

C₂

C₂²×C₂₃₉

C₂³×C₂₃₉

D₂₃₉

C₂₃₉

G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
51. C₂₃₉	#2090	semidirect

H ≅ G/N

type

51. C₂₃₉

#2090

semidirect

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

subgroup	index
#29504815	2
C₂×#21439892	2
#29504540	2
#29504537	2
#424×C₂₃₉	2
#2090	239

subgroup

index

#424×C₂₃₉

239

Subgroups — #44486058

Conjugacy classes of subgroups

Complements

Maximal subgroups