Group #44507601 — subgroups

Subgroups — #44507601

Order 30848.

280 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 149 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×#21440573	C₂	1	4	yes	maximal
2	C₂×C₄×#14515349	C₂	1	1	yes	maximal, characteristic
3	#29507587	C₂	1	1	yes	maximal, characteristic
4	#384×C₂₄₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14515349	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₄₁	—	2	2	no	—
7	C₂×C₄⋊Dic₂₄₁	C₂²	1	2	yes	—
8	#21440573	—	2	8	no	—
9	C₄×#14515349	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₂₄₁	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₄₁	—	2	2	no	—
14	C₂×#14515349	C₂×C₄	1	4	yes	—
15	C₂×#14515349	D₄	1	4	yes	—
16	C₂×#14515349	Q₈	1	4	yes	—
17	C₄⋊Dic₂₄₁	—	2	4	no	—
18	C₄⋊Dic₂₄₁	—	4	4	no	—
19	C₂²×C₄×C₂₄₁	—	2	2	no	—
20	C₂²×C₄×C₂₄₁	D₄	1	2	yes	—
21	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₂₄₁	—	2	4	no	—
23	C₄⋊C₄×C₂₄₁	—	4	4	no	—
24	C₄²×C₂₄₁	C₂×C₄	1	2	yes	—
25	C₄²×C₂₄₁	Q₈	1	2	yes	—
26	C₂×Dic₂₄₁	—	4	2	no	—
27	C₂×Dic₂₄₁	—	4	4	no	—
28	#14515349	C₄⋊C₄	1	8	yes	complemented
29	C₂³×C₂₄₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₂₄₁	—	4	2	no	—
31	C₂×C₄×C₂₄₁	—	4	4	no	—
32	C₂×C₄×C₂₄₁	C₄⋊C₄	1	4	yes	—
33	C₂×C₄×C₂₄₁	SD₁₆	1	4	yes	—
34	C₂×C₄×C₂₄₁	SD₁₆	1	4	yes	—
35	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₂₄₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₂₄₁	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₂₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₄₁	#127	1	1	yes	characteristic
41	C₂²×C₂₄₁	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₂₄₁	C₂×SD₁₆	1	2	yes	—
43	C₂²×C₂₄₁	C₂×SD₁₆	1	2	yes	—
44	C₄×C₂₄₁	—	8	2	no	—
45	C₄×C₂₄₁	#106	1	8	yes	complemented
46	C₄×C₂₄₁	C₂×C₄⋊C₄	1	2	yes	—
47	C₄×C₂₄₁	C₄⋊Q₈	1	2	yes	—
48	Dic₂₄₁	—	8	2	no	—
49	C₂×C₂₄₁	#384	1	1	yes	characteristic
50	C₂×C₂₄₁	C₂×#106	1	2	yes	—
51	C₂×C₂₄₁	#492	1	1	yes	characteristic
52	C₂×C₂₄₁	#492	1	1	yes	characteristic
53	C₂×C₂₄₁	#498	1	2	yes	—
54	C₂₄₁	#1844	1	1	yes	characteristic, complemented, Sylow
55	#1844	—	241	1	no	maximal, Sylow
56	C₂×#106	—	241	4	no	—
57	C₂×C₄×C₈	—	241	1	no	—
58	#384	—	241	1	no	—
59	#384	D₂₄₁	1	1	yes	characteristic
60	C₂²×C₈	—	241	2	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	482	2	no	—
63	C₂×C₄⋊C₄	—	241	2	no	—
64	C₂×C₄⋊C₄	C₂×D₂₄₁	1	2	yes	—
65	C₂×C₄²	C₂×D₂₄₁	1	1	yes	characteristic
66	#106	—	482	8	no	—
67	C₄×C₈	—	241	4	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	482	2	no	—
70	C₂²×C₄	C₂²⋊D₂₄₁	1	2	yes	—
71	C₂²×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
72	C₂×C₈	—	241	4	no	—
73	C₂×C₈	—	241	4	no	—
74	C₂×C₈	—	241	4	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	964	4	no	—
78	C₄⋊C₄	—	482	4	no	—
79	C₄²	C₄×D₂₄₁	1	2	yes	—
80	C₄²	Dic₄₈₂	1	2	yes	—
81	C₂³	C₂×C₂²⋊D₂₄₁	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	964	2	no	—
85	C₂×C₄	—	964	4	no	—
86	C₂×C₄	#16838749	1	4	yes	—
87	C₂×C₄	#16838757	1	4	yes	—
88	C₂×C₄	#16838758	1	4	yes	—
89	C₂×C₄	C₂×C₄×D₂₄₁	1	1	yes	characteristic
90	C₂×C₄	D₄×D₂₄₁	1	2	yes	—
91	C₂×C₄	Q₈×D₂₄₁	1	2	yes	—
92	C₂×C₄	C₂×Dic₄₈₂	1	1	yes	characteristic
93	C₈	—	241	8	no	—
94	C₂²	C₂×#16838749	1	1	yes	characteristic
95	C₂²	#21440638	1	1	yes	characteristic
96	C₂²	#21440640	1	1	yes	characteristic
97	C₂²	C₂×#16838757	1	2	yes	—
98	C₂²	C₂×#16838758	1	2	yes	—
99	C₄	—	8	2	no	—
100	C₄	—	1928	2	no	—
101	C₄	#21440573	1	8	yes	complemented
102	C₄	C₄⋊C₄×D₂₄₁	1	2	yes	—
103	C₄	#21440641	1	2	yes	—
104	C₂	#29507586	1	1	yes	characteristic
105	C₂	C₂×#21440573	1	2	yes	—
106	C₂	#29508089	1	1	yes	characteristic
107	C₂	#29508091	1	1	yes	characteristic
108	C₂	#29508113	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21440573

C₂

yes

maximal

C₂×C₄×#14515349

C₂

yes

maximal, characteristic

#29507587

C₂

yes

maximal, characteristic

#384×C₂₄₁

C₂

yes

maximal, characteristic

C₂²×#14515349

C₂²

yes

—

C₂×C₄⋊Dic₂₄₁

—

C₂×C₄⋊Dic₂₄₁

C₂²

yes

—

#21440573

—

C₄×#14515349

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₄₁

—

C₂×C₄⋊C₄×C₂₄₁

C₂²

yes

—

C₂×C₄²×C₂₄₁

C₂²

yes

characteristic

C₂²×Dic₂₄₁

—

C₂×#14515349

C₂×C₄

yes

—

C₂×#14515349

D₄

yes

—

C₂×#14515349

Q₈

yes

—

C₄⋊Dic₂₄₁

—

C₄⋊Dic₂₄₁

—

C₂²×C₄×C₂₄₁

—

C₂²×C₄×C₂₄₁

D₄

yes

—

C₂²×C₄×C₂₄₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₄₁

—

C₄⋊C₄×C₂₄₁

—

C₄²×C₂₄₁

C₂×C₄

yes

—

C₄²×C₂₄₁

Q₈

yes

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

—

#14515349

C₄⋊C₄

yes

complemented

C₂³×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₄₁

SD₁₆

yes

—

C₂×C₄×C₂₄₁

SD₁₆

yes

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

—

C₂×C₄×C₂₄₁

C₂×Q₈

yes

—

C₂×C₄×C₂₄₁

C₂×Q₈

yes

characteristic

C₂²×C₂₄₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₄₁

#127

yes

characteristic

C₂²×C₂₄₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₄₁

C₂×SD₁₆

yes

—

C₂²×C₂₄₁

C₂×SD₁₆

yes

—

C₄×C₂₄₁

—

C₄×C₂₄₁

#106

yes

complemented

C₄×C₂₄₁

C₂×C₄⋊C₄

yes

—

C₄×C₂₄₁

C₄⋊Q₈

yes

—

Dic₂₄₁

—

C₂×C₂₄₁

#384

yes

characteristic

C₂×C₂₄₁

C₂×#106

yes

—

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

—

C₂₄₁

#1844

yes

characteristic, complemented, Sylow

#1844

—

241

maximal, Sylow

C₂×#106

—

241

—

C₂×C₄×C₈

—

241

—

#384

—

241

—

#384

D₂₄₁

yes

characteristic

C₂²×C₈

—

241

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

482

—

C₂×C₄⋊C₄

—

241

—

C₂×C₄⋊C₄

C₂×D₂₄₁

yes

—

C₂×C₄²

C₂×D₂₄₁

yes

characteristic

#106

—

482

—

C₄×C₈

—

241

—

C₂²×C₄

—

C₂²×C₄

—

482

—

C₂²×C₄

C₂²⋊D₂₄₁

yes

—

C₂²×C₄

C₂²×D₂₄₁

yes

characteristic

C₂×C₈

—

241

—

C₂×C₈

—

241

—

C₂×C₈

—

241

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

964

—

C₄⋊C₄

—

482

—

C₄²

C₄×D₂₄₁

yes

—

C₄²

Dic₄₈₂

yes

—

C₂³

C₂×C₂²⋊D₂₄₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

964

—

C₂×C₄

—

964

—

C₂×C₄

#16838749

yes

—

C₂×C₄

#16838757

yes

—

C₂×C₄

#16838758

yes

—

C₂×C₄

C₂×C₄×D₂₄₁

yes

characteristic

C₂×C₄

D₄×D₂₄₁

yes

—

C₂×C₄

Q₈×D₂₄₁

yes

—

C₂×C₄

C₂×Dic₄₈₂

yes

characteristic

C₈

—

241

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×#16838758

yes

—

C₄

—

100

C₄

—

1928

—

101

C₄

#21440573

yes

complemented

102

C₄

C₄⋊C₄×D₂₄₁

yes

—

103

C₄

#21440641

yes

—

104

C₂

#29507586

yes

characteristic

105

C₂

C₂×#21440573

yes

—

106

C₂

#29508089

yes

characteristic

107

C₂

#29508091

yes

characteristic

108

C₂

#29508113

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₄₁
commutator	C₂×C₄×C₂₄₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₄₁
Fitting	#384×C₂₄₁	C₂
socle, prime radical	C₂³×C₂₄₁	C₂×D₄
cosocle	C₂²×C₄×C₂₄₁	C₂³
hypercenter	#384	D₂₄₁
hypocenter	C₂₄₁	#1844
LCS	G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₄₁

commutator

C₂×C₄×C₂₄₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₄₁

Fitting

#384×C₂₄₁

C₂

socle, prime radical

C₂³×C₂₄₁

C₂×D₄

cosocle

C₂²×C₄×C₂₄₁

C₂³

D₂₄₁

C₂₄₁

G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#14515349	C₄	semidirect
28. #14515349	C₄⋊C₄	semidirect
45. C₄×C₂₄₁	#106	semidirect
54. C₂₄₁	#1844	semidirect
101. C₄	#21440573	semidirect

H ≅ G/N

type

9. C₄×#14515349

C₄

semidirect

28. #14515349

C₄⋊C₄

semidirect

45. C₄×C₂₄₁

#106

semidirect

54. C₂₄₁

#1844

semidirect

101. C₄

#21440573

semidirect

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

subgroup	index
C₂×#21440573	2
C₂×C₄×#14515349	2
#29507587	2
#384×C₂₄₁	2
#1844	241

subgroup

index

#384×C₂₄₁

241

Subgroups — #44507601

Conjugacy classes of subgroups

Complements

Maximal subgroups