Group #44507495 — subgroups

Subgroups — #44507495

Order 30848.

252 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 105 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×#21440578	C₂	1	2	yes	maximal, complemented
2	#29507701	C₂	1	2	yes	maximal
3	#29507475	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₂₄₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16838754	C₂²	1	4	yes	—
6	#21440578	—	2	4	no	—
7	#21440561	—	2	2	no	—
8	#21440559	—	2	2	no	—
9	C₂×#41×C₂₄₁	C₂²	1	1	yes	characteristic
10	#131×C₂₄₁	—	2	4	no	—
11	C₂×M₁₆×C₂₄₁	C₄	1	2	yes	—
12	C₂×M₁₆×C₂₄₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₄₁	C₄	1	2	yes	—
14	C₂²×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
15	#16838754	—	2	4	no	—
16	#16838754	—	2	4	no	—
17	C₂×#14515349	—	2	4	no	—
18	#41×C₂₄₁	—	2	2	no	—
19	#41×C₂₄₁	—	2	2	no	—
20	C₂×Q₈×C₂₄₁	D₄	1	1	yes	characteristic
21	C₂×D₄×C₂₄₁	D₄	1	1	yes	characteristic
22	C₂×D₄×C₂₄₁	Q₈	1	2	yes	—
23	C₂²×C₄×C₂₄₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
25	M₁₆×C₂₄₁	—	2	2	no	—
26	M₁₆×C₂₄₁	—	2	4	no	—
27	C₂×C₈×C₂₄₁	—	2	4	no	—
28	C₂×C₈×C₂₄₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₄₁	D₄	1	4	yes	—
32	#14515349	—	4	4	no	—
33	C₂³×C₂₄₁	C₄⋊C₄	1	2	yes	—
34	C₂³×C₂₄₁	C₂×Q₈	1	1	yes	characteristic
35	Q₈×C₂₄₁	—	2	2	no	—
36	D₄×C₂₄₁	—	2	4	no	—
37	D₄×C₂₄₁	—	2	2	no	—
38	C₂×C₄×C₂₄₁	—	2	2	no	—
39	C₂×C₄×C₂₄₁	—	2	2	no	—
40	C₂×C₄×C₂₄₁	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₄₁	#41	1	2	yes	—
45	C₈×C₂₄₁	—	2	2	no	—
46	C₈×C₂₄₁	C₂²⋊C₄	1	2	yes	—
47	C₈×C₂₄₁	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₂₄₁	—	2	2	no	—
49	C₂²×C₂₄₁	—	4	1	no	—
50	C₂²×C₂₄₁	—	4	2	no	—
51	C₂²×C₂₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₄₁	C₂²⋊Q₈	1	2	yes	—
53	C₄×C₂₄₁	—	2	2	no	—
54	C₄×C₂₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₂₄₁	#117	1	1	yes	characteristic
56	C₄×C₂₄₁	#121	1	2	yes	—
57	C₂×C₂₄₁	—	2	1	no	—
58	C₂×C₂₄₁	—	2	2	no	—
59	C₂×C₂₄₁	—	4	2	no	—
60	C₂×C₂₄₁	#380	1	1	yes	characteristic
61	C₂₄₁	#1813	1	1	yes	characteristic, complemented, Sylow
62	#1813	—	241	1	no	maximal, Sylow
63	C₂×#131	D₂₄₁	1	1	yes	characteristic
64	C₂×#108	—	241	2	no	—
65	#413	—	241	2	no	—
66	#338	—	241	2	no	—
67	C₂×#41	C₂×D₂₄₁	1	1	yes	characteristic
68	#131	—	2	4	no	—
69	C₂×M₁₆	—	241	4	no	—
70	C₂×M₁₆	Dic₂₄₁	1	2	yes	—
71	C₂×M₁₆	C₂×D₂₄₁	1	1	yes	characteristic
72	C₂²×C₈	Dic₂₄₁	1	2	yes	—
73	C₂²×C₈	C₂×D₂₄₁	1	1	yes	characteristic
74	#108	—	482	4	no	—
75	#101	—	482	2	no	—
76	#100	—	482	2	no	—
77	#41	—	2	2	no	—
78	#41	—	2	2	no	—
79	C₂×Q₈	D₉₆₄	1	1	yes	characteristic
80	C₂×D₄	D₉₆₄	1	1	yes	characteristic
81	C₂×D₄	Dic₄₈₂	1	2	yes	—
82	C₂²×C₄	C₂×Dic₂₄₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	4	no	—
86	M₁₆	—	482	4	no	—
87	M₁₆	—	482	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	482	4	no	—
90	C₂×C₈	C₂×Dic₂₄₁	1	2	yes	—
91	C₂×C₈	C₂×Dic₂₄₁	1	1	yes	characteristic
92	C₂×C₈	C₂×Dic₂₄₁	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₂₄₁	1	4	yes	—
94	C₂³	C₄⋊Dic₂₄₁	1	2	yes	—
95	C₂³	C₂×Dic₄₈₂	1	1	yes	characteristic
96	Q₈	—	2	2	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄⋊Dic₂₄₁	1	2	yes	—
102	C₂×C₄	C₂²×Dic₂₄₁	1	1	yes	characteristic
103	C₂×C₄	C₂×C₂²⋊D₂₄₁	1	2	yes	—
104	C₂×C₄	C₂×D₉₆₄	1	1	yes	characteristic
105	C₂×C₄	#16838770	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	964	4	no	—
108	C₈	C₂²⋊Dic₂₄₁	1	2	yes	—
109	C₈	C₂²⋊Dic₂₄₁	1	2	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	C₂×C₄⋊Dic₂₄₁	1	1	yes	characteristic
114	C₂²	#21440621	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	C₂×C₂²⋊Dic₂₄₁	1	1	yes	characteristic
117	C₄	#21440601	1	1	yes	characteristic
118	C₄	#21440616	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	2	no	—
121	C₂	—	4	2	no	—
122	C₂	#29507577	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21440578

C₂

yes

maximal, complemented

#29507701

C₂

yes

maximal

#29507475

C₂

yes

maximal, complemented

C₂×#131×C₂₄₁

C₂

yes

maximal, characteristic

C₂×#16838754

C₂²

yes

—

#21440578

—

#21440561

—

#21440559

—

C₂×#41×C₂₄₁

C₂²

yes

characteristic

#131×C₂₄₁

—

C₂×M₁₆×C₂₄₁

C₄

yes

—

C₂×M₁₆×C₂₄₁

C₂²

yes

characteristic

C₂²×C₈×C₂₄₁

C₄

yes

—

C₂²×C₈×C₂₄₁

C₂²

yes

—

—

—

#41×C₂₄₁

—

#41×C₂₄₁

—

C₂×Q₈×C₂₄₁

D₄

yes

characteristic

C₂×D₄×C₂₄₁

D₄

yes

characteristic

C₂×D₄×C₂₄₁

Q₈

yes

—

C₂²×C₄×C₂₄₁

C₂×C₄

yes

—

C₂²×C₄×C₂₄₁

C₂³

yes

characteristic

M₁₆×C₂₄₁

—

M₁₆×C₂₄₁

—

C₂×C₈×C₂₄₁

—

C₂×C₈×C₂₄₁

C₂×C₄

yes

—

C₂×C₈×C₂₄₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₄₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₄₁

D₄

yes

—

#14515349

—

C₂³×C₂₄₁

C₄⋊C₄

yes

—

C₂³×C₂₄₁

C₂×Q₈

yes

characteristic

Q₈×C₂₄₁

—

D₄×C₂₄₁

—

D₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

—

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

#41

yes

—

C₈×C₂₄₁

—

C₈×C₂₄₁

C₂²⋊C₄

yes

—

C₈×C₂₄₁

C₂²⋊C₄

yes

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₄₁

C₂²⋊Q₈

yes

—

C₄×C₂₄₁

—

C₄×C₂₄₁

C₂×C₂²⋊C₄

yes

C₄×C₂₄₁

yes

C₄×C₂₄₁

yes

—

C₂×C₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

#380

yes

characteristic

C₂₄₁

#1813

yes

characteristic, complemented, Sylow

#1813

—

241

maximal, Sylow

C₂×#131

D₂₄₁

yes

characteristic

C₂×#108

—

241

—

#413

—

241

—

#338

—

241

—

C₂×#41

C₂×D₂₄₁

yes

characteristic

#131

—

C₂×M₁₆

—

241

—

C₂×M₁₆

Dic₂₄₁

yes

—

C₂×M₁₆

C₂×D₂₄₁

yes

characteristic

C₂²×C₈

Dic₂₄₁

yes

—

C₂²×C₈

C₂×D₂₄₁

yes

characteristic

#108

—

482

—

#101

—

482

—

#100

—

482

—

#41

—

#41

—

C₂×Q₈

D₉₆₄

yes

characteristic

C₂×D₄

D₉₆₄

yes

characteristic

C₂×D₄

Dic₄₈₂

yes

—

C₂²×C₄

C₂×Dic₂₄₁

yes

—

C₂²×C₄

C₂²×D₂₄₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

482

—

M₁₆

—

482

—

C₂×C₈

—

C₂×C₈

—

482

—

C₂×C₈

C₂×Dic₂₄₁

yes

—

C₂×C₈

C₂×Dic₂₄₁

yes

characteristic

C₂×C₈

C₂×Dic₂₄₁

yes

characteristic

C₂×C₈

C₂²⋊D₂₄₁

yes

—

C₂³

C₄⋊Dic₂₄₁

yes

—

C₂³

C₂×Dic₄₈₂

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₂₄₁

yes

—

102

C₂×C₄

C₂²×Dic₂₄₁

yes

characteristic

103

C₂×C₄

C₂×C₂²⋊D₂₄₁

yes

—

104

C₂×C₄

C₂×D₉₆₄

yes

characteristic

105

C₂×C₄

#16838770

yes

—

106

C₈

—

107

C₈

—

964

—

108

C₈

C₂²⋊Dic₂₄₁

yes

—

109

C₈

C₂²⋊Dic₂₄₁

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₄⋊Dic₂₄₁

yes

characteristic

114

C₂²

#21440621

yes

—

115

C₄

—

116

C₄

C₂×C₂²⋊Dic₂₄₁

yes

characteristic

117

C₄

#21440601

yes

characteristic

118

C₄

#21440616

yes

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#29507577

yes

characteristic

	subgroup	G/H
center	C₄	#21440601
commutator	C₂×C₄×C₂₄₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₄₁
Fitting	C₂×#131×C₂₄₁	C₂
socle	C₂×C₂₄₁	#380
prime radical	C₂×#41×C₂₄₁	C₂²
cosocle	C₂²×C₄×C₂₄₁	C₂³
hypercenter	C₂×#131	D₂₄₁
hypocenter	C₂₄₁	#1813
LCS	G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#21440601

commutator

C₂×C₄×C₂₄₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₄₁

Fitting

C₂×#131×C₂₄₁

C₂

socle

C₂×C₂₄₁

#380

prime radical

C₂×#41×C₂₄₁

C₂²

cosocle

C₂²×C₄×C₂₄₁

C₂³

D₂₄₁

C₂₄₁

G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#21440578	C₂	semidirect
3. #29507475	C₂	semidirect
61. C₂₄₁	#1813	semidirect

H ≅ G/N

type

1. C₂×#21440578

C₂

semidirect

3. #29507475

C₂

semidirect

61. C₂₄₁

#1813

semidirect

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

subgroup	index
C₂×#21440578	2
#29507701	2
#29507475	2
C₂×#131×C₂₄₁	2
#1813	241

subgroup

index

C₂×#131×C₂₄₁

241

Subgroups — #44507495

Conjugacy classes of subgroups

Complements

Maximal subgroups