Group #44507112 — subgroups

Subgroups — #44507112

Order 30848.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#29508093	C₂	1	2	yes	maximal, complemented
2	#29508090	C₂	1	1	yes	maximal, characteristic, complemented
3	#29508063	C₂	1	2	yes	maximal, complemented
4	#29507438	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16838759	—	2	2	no	—
7	C₂×#16838755	—	2	4	no	—
8	#21440642	C₂²	1	1	yes	characteristic
9	#21440639	C₂²	1	1	yes	characteristic
10	#21440563	—	2	4	no	—
11	C₄×#14515349	C₂²	1	2	yes	—
12	C₂×D₈×C₂₄₁	—	2	2	no	—
13	#127×C₂₄₁	C₂²	1	2	yes	—
14	C₄×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₈₂	—	2	1	no	—
16	C₂×D₉₆₄	—	2	2	no	—
17	C₂×D₉₆₄	—	2	1	no	—
18	#16838759	—	4	4	no	—
19	#16838755	—	4	8	no	—
20	C₂×#14515349	D₄	1	4	yes	—
21	C₄⋊Dic₂₄₁	—	2	2	no	—
22	C₂×D₄×C₂₄₁	—	2	2	no	—
23	C₂×D₄×C₂₄₁	—	2	2	no	—
24	C₂×D₄×C₂₄₁	—	2	2	no	—
25	D₈×C₂₄₁	—	4	4	no	—
26	C₂×C₈×C₂₄₁	D₄	1	2	yes	—
27	C₄²×C₂₄₁	C₂³	1	1	yes	characteristic
28	C₂²×D₂₄₁	—	4	1	no	—
29	Dic₄₈₂	—	4	2	no	—
30	D₉₆₄	—	4	2	no	—
31	D₉₆₄	—	4	4	no	—
32	C₂×Dic₂₄₁	—	4	1	no	—
33	#14515349	—	2	4	no	—
34	C₂³×C₂₄₁	—	4	2	no	—
35	D₄×C₂₄₁	—	4	4	no	—
36	D₄×C₂₄₁	—	4	4	no	—
37	D₄×C₂₄₁	—	4	4	no	—
38	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₄₁	D₈	1	4	yes	complemented
41	C₂²×C₂₄₁	—	8	2	no	—
42	C₂²×C₂₄₁	—	8	2	no	—
43	C₂²×C₂₄₁	—	8	2	no	—
44	C₂²×C₂₄₁	#127	1	1	yes	characteristic
45	C₂×D₂₄₁	—	8	1	no	—
46	C₂×D₂₄₁	—	8	2	no	—
47	C₄×C₂₄₁	C₂×D₈	1	2	yes	—
48	C₄×C₂₄₁	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₂₄₁	—	8	1	no	—
50	C₂×C₂₄₁	—	16	2	no	—
51	C₂×C₂₄₁	#493	1	1	yes	characteristic
52	C₂×C₂₄₁	#496	1	2	yes	—
53	D₂₄₁	—	16	1	no	—
54	C₂₄₁	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	241	1	no	maximal, Sylow
56	#493	—	241	2	no	—
57	#493	D₂₄₁	1	1	yes	characteristic, complemented
58	#492	—	241	1	no	—
59	#486	—	241	2	no	—
60	#322	—	241	1	no	—
61	C₂×SD₁₆	—	482	2	no	—
62	C₂×D₈	—	2	2	no	—
63	C₂×D₈	—	482	4	no	—
64	C₄⋊Q₈	—	241	1	no	—
65	#127	—	241	1	no	—
66	#127	C₂×D₂₄₁	1	2	yes	—
67	#102	—	482	4	no	—
68	C₄×C₈	—	241	2	no	—
69	C₄×C₈	C₂×D₂₄₁	1	1	yes	characteristic
70	C₂×Q₈	—	482	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	482	2	no	—
75	C₂×D₄	—	482	1	no	—
76	SD₁₆	—	964	4	no	—
77	D₈	—	4	4	no	—
78	D₈	—	964	8	no	—
79	C₂×C₈	—	241	4	no	—
80	C₂×C₈	C₂²⋊D₂₄₁	1	2	yes	—
81	C₄⋊C₄	—	482	2	no	—
82	C₄²	C₂²×D₂₄₁	1	1	yes	characteristic
83	C₂³	—	4	2	no	—
84	C₂³	—	964	1	no	—
85	Q₈	—	964	2	no	—
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	4	no	—
89	D₄	—	964	2	no	—
90	D₄	—	964	4	no	—
91	C₂×C₄	—	964	1	no	—
92	C₂×C₄	D₄×D₂₄₁	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₄₁	1	1	yes	characteristic
94	C₈	—	482	4	no	—
95	C₈	#16838755	1	4	yes	complemented
96	C₂²	—	8	2	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	8	2	no	—
99	C₂²	—	1928	2	no	—
100	C₂²	—	1928	1	no	—
101	C₂²	#21440638	1	1	yes	characteristic
102	C₄	—	1928	1	no	—
103	C₄	C₂×#16838755	1	2	yes	—
104	C₄	#21440666	1	4	yes	—
105	C₂	—	16	2	no	—
106	C₂	—	3856	1	no	—
107	C₂	#29508093	1	1	yes	characteristic
108	C₂	#29508105	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#29508093

C₂

yes

maximal, complemented

#29508090

C₂

yes

maximal, characteristic, complemented

#29508063

C₂

yes

maximal, complemented

#29507438

C₂

yes

maximal, characteristic, complemented

#493×C₂₄₁

C₂

yes

maximal, characteristic, complemented

C₂×#16838759

—

C₂×#16838755

—

#21440642

C₂²

yes

characteristic

#21440639

C₂²

yes

characteristic

#21440563

—

C₄×#14515349

C₂²

yes

—

C₂×D₈×C₂₄₁

—

#127×C₂₄₁

C₂²

yes

—

C₄×C₈×C₂₄₁

C₂²

yes

characteristic

C₂×Dic₄₈₂

—

C₂×D₉₆₄

—

C₂×D₉₆₄

—

#16838759

—

#16838755

—

C₂×#14515349

D₄

yes

—

C₄⋊Dic₂₄₁

—

C₂×D₄×C₂₄₁

—

C₂×D₄×C₂₄₁

—

C₂×D₄×C₂₄₁

—

D₈×C₂₄₁

—

C₂×C₈×C₂₄₁

D₄

yes

—

C₄²×C₂₄₁

C₂³

yes

characteristic

C₂²×D₂₄₁

—

Dic₄₈₂

—

D₉₆₄

—

D₉₆₄

—

C₂×Dic₂₄₁

—

#14515349

—

C₂³×C₂₄₁

—

D₄×C₂₄₁

—

D₄×C₂₄₁

—

D₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂×D₄

yes

—

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₈×C₂₄₁

D₈

yes

complemented

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

#127

yes

characteristic

C₂×D₂₄₁

—

C₂×D₂₄₁

—

C₄×C₂₄₁

C₂×D₈

yes

—

C₄×C₂₄₁

AGL(1,ℤ/8)

yes

—

Dic₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

#493

yes

characteristic

C₂×C₂₄₁

#496

yes

—

D₂₄₁

—

C₂₄₁

#1721

yes

characteristic, complemented, Sylow

#1721

—

241

maximal, Sylow

#493

—

241

—

#493

D₂₄₁

yes

characteristic, complemented

#492

—

241

—

#486

—

241

—

#322

—

241

—

C₂×SD₁₆

—

482

—

C₂×D₈

—

C₂×D₈

—

482

—

C₄⋊Q₈

—

241

—

#127

—

241

—

#127

C₂×D₂₄₁

yes

—

#102

—

482

—

C₄×C₈

—

241

—

C₄×C₈

C₂×D₂₄₁

yes

characteristic

C₂×Q₈

—

482

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

482

—

C₂×D₄

—

482

—

SD₁₆

—

964

—

D₈

—

D₈

—

964

—

C₂×C₈

—

241

—

C₂×C₈

C₂²⋊D₂₄₁

yes

—

C₄⋊C₄

—

482

—

C₄²

C₂²×D₂₄₁

yes

characteristic

C₂³

—

C₂³

—

964

—

Q₈

—

964

—

D₄

—

D₄

—

D₄

—

D₄

—

964

—

D₄

—

964

—

C₂×C₄

—

964

—

C₂×C₄

D₄×D₂₄₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₄₁

yes

characteristic

C₈

—

482

—

C₈

#16838755

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

1928

—

100

C₂²

—

1928

—

101

C₂²

#21440638

yes

characteristic

102

C₄

—

1928

—

103

C₄

C₂×#16838755

yes

—

104

C₄

#21440666

yes

—

105

C₂

—

106

C₂

—

3856

—

107

C₂

#29508093

yes

characteristic

108

C₂

#29508105

yes

—

	subgroup	G/H
center	C₂²	#21440638
commutator, cosocle	C₄²×C₂₄₁	C₂³
Frattini	C₄²	C₂²×D₂₄₁
Fitting	#493×C₂₄₁	C₂
socle	C₂²×C₂₄₁	#127
hypercenter	#493	D₂₄₁
hypocenter	C₂₄₁	#1721
LCS	G ▷ C₄²×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#21440638

commutator, cosocle

C₄²×C₂₄₁

C₂³

Frattini

C₄²

C₂²×D₂₄₁

Fitting

#493×C₂₄₁

C₂

C₂²×C₂₄₁

D₂₄₁

C₂₄₁

G ▷ C₄²×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #29508093	C₂	semidirect
2. #29508090	C₂	semidirect
3. #29508063	C₂	semidirect
4. #29507438	C₂	semidirect
5. #493×C₂₄₁	C₂	semidirect
40. C₈×C₂₄₁	D₈	semidirect
54. C₂₄₁	#1721	semidirect
57. #493	D₂₄₁	semidirect
95. C₈	#16838755	semidirect

H ≅ G/N

type

1. #29508093

C₂

semidirect

2. #29508090

C₂

semidirect

3. #29508063

C₂

semidirect

4. #29507438

C₂

semidirect

5. #493×C₂₄₁

C₂

semidirect

40. C₈×C₂₄₁

D₈

semidirect

54. C₂₄₁

#1721

semidirect

57. #493

D₂₄₁

semidirect

95. C₈

#16838755

semidirect

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

subgroup	index
#29508093	2
#29508090	2
#29508063	2
#29507438	2
#493×C₂₄₁	2
#1721	241

subgroup

index

#493×C₂₄₁

241

Subgroups — #44507112

Conjugacy classes of subgroups

Complements

Maximal subgroups