Group #44508199 — subgroups

Subgroups — #44508199

Order 30848.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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	1	yes	maximal, characteristic
2	#29507744	C₂	1	2	yes	maximal
3	C₂×#21440552	C₂	1	1	yes	maximal, characteristic
4	#29507476	C₂	1	2	yes	maximal
5	#427×C₂₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16838754	C₂²	1	2	yes	—
7	C₂²×#14515349	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₄₁	C₂²	1	1	yes	characteristic
9	#21440578	—	2	2	no	—
10	C₄⋊#14515349	—	2	4	no	—
11	#21440552	—	2	2	no	—
12	C₂²×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
13	#117×C₂₄₁	C₂²	1	2	yes	—
14	#107×C₂₄₁	C₄	1	2	yes	complemented
15	#106×C₂₄₁	C₄	1	2	yes	complemented
16	C₂²×Dic₂₄₁	—	2	1	no	—
17	#16838754	—	2	2	no	—
18	#16838754	—	2	2	no	—
19	C₂×#14515349	—	2	1	no	—
20	C₂×#14515349	—	2	2	no	—
21	C₂×#14515349	—	2	1	no	—
22	C₂×#14515349	D₄	1	2	yes	—
23	C₄×Dic₂₄₁	—	2	1	no	—
24	C₄×Dic₂₄₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₂₄₁	D₄	1	2	yes	—
29	C₂×C₈×C₂₄₁	Q₈	1	2	yes	—
30	C₄⋊C₄×C₂₄₁	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₂₄₁	—	2	2	no	—
32	C₄²×C₂₄₁	—	2	2	no	—
33	C₂×Dic₂₄₁	—	2	2	no	—
34	C₂×Dic₂₄₁	—	4	2	no	—
35	#14515349	—	2	2	no	—
36	#14515349	—	4	2	no	—
37	C₂³×C₂₄₁	#41	1	1	yes	characteristic
38	C₂×C₄×C₂₄₁	—	2	4	no	—
39	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₄₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₂₄₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₄₁	#41	1	1	yes	characteristic
45	C₈×C₂₄₁	C₄⋊C₄	1	2	yes	—
46	C₈×C₂₄₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₂₄₁	—	2	2	no	—
48	C₂²×C₂₄₁	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₂₄₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₄₁	#125	1	1	yes	characteristic
51	C₄×C₂₄₁	—	4	4	no	—
52	C₄×C₂₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₂₄₁	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₂₄₁	#121	1	1	yes	characteristic
55	C₄×C₂₄₁	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₂₄₁	—	4	2	no	—
57	C₂×C₂₄₁	—	2	1	no	—
58	C₂×C₂₄₁	—	2	1	no	—
59	C₂×C₂₄₁	#389	1	1	yes	characteristic
60	C₂×C₂₄₁	#444	1	2	yes	—
61	C₂₄₁	#1946	1	1	yes	characteristic, complemented, Sylow
62	#1946	—	241	1	no	maximal, Sylow
63	C₂×#108	—	241	1	no	—
64	#427	D₂₄₁	1	1	yes	characteristic
65	#424	—	241	2	no	—
66	C₂×#97	—	241	1	no	—
67	#339	—	241	2	no	—
68	C₂×M₁₆	—	241	2	no	—
69	C₂²×C₈	—	241	1	no	—
70	C₂²×C₈	C₂×D₂₄₁	1	1	yes	characteristic
71	#117	C₂×D₂₄₁	1	2	yes	—
72	C₂×C₄²	—	241	1	no	—
73	#108	—	482	2	no	—
74	#107	Dic₂₄₁	1	2	yes	complemented
75	#106	Dic₂₄₁	1	2	yes	complemented
76	C₄⋊C₈	—	482	4	no	—
77	#97	—	482	2	no	—
78	C₂²×C₄	—	482	1	no	—
79	C₂²×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
80	M₁₆	—	482	2	no	—
81	M₁₆	—	482	2	no	—
82	C₂×C₈	—	482	2	no	—
83	C₂×C₈	—	482	1	no	—
84	C₂×C₈	—	482	1	no	—
85	C₂×C₈	—	241	2	no	—
86	C₂×C₈	C₂×Dic₂₄₁	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₂₄₁	1	1	yes	characteristic
88	C₂×C₈	D₉₆₄	1	2	yes	—
89	C₂×C₈	Dic₄₈₂	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₂₄₁	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	482	1	no	—
94	C₄²	—	241	2	no	—
95	C₂³	#16838769	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	964	2	no	—
98	C₂×C₄	—	482	2	no	—
99	C₂×C₄	C₂²×Dic₂₄₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₄₁	1	1	yes	characteristic
101	C₂×C₄	C₂×D₉₆₄	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₂₄₁	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₄₈₂	1	1	yes	characteristic
104	C₂×C₄	#16838771	1	1	yes	characteristic
105	C₈	—	964	2	no	—
106	C₈	—	482	2	no	—
107	C₈	C₄⋊Dic₂₄₁	1	2	yes	—
108	C₈	C₄⋊Dic₂₄₁	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₂₄₁	1	1	yes	characteristic
111	C₂²	#21440622	1	1	yes	characteristic
112	C₂²	#21440633	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	964	2	no	—
115	C₄	C₂×C₄⋊Dic₂₄₁	1	1	yes	characteristic
116	C₄	Q₈×Dic₂₄₁	1	1	yes	characteristic
117	C₄	#21440617	1	1	yes	characteristic
118	C₄	#21440641	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#29507616	1	1	yes	characteristic
122	C₂	#29507839	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21440578

C₂

yes

maximal, characteristic

#29507744

C₂

yes

maximal

C₂×#21440552

C₂

yes

maximal, characteristic

#29507476

C₂

yes

maximal

#427×C₂₄₁

C₂

yes

maximal, characteristic

C₂×#16838754

C₂²

yes

—

C₂²×#14515349

C₂²

yes

characteristic

C₂×C₄×Dic₂₄₁

C₂²

yes

—

—

—

C₂²×C₈×C₂₄₁

C₂²

yes

characteristic

#117×C₂₄₁

C₂²

yes

—

#107×C₂₄₁

C₄

yes

complemented

#106×C₂₄₁

C₄

yes

complemented

C₂²×Dic₂₄₁

—

—

—

—

—

—

D₄

yes

—

C₄×Dic₂₄₁

—

C₄×Dic₂₄₁

Q₈

yes

—

C₂²×C₄×C₂₄₁

C₂³

yes

characteristic

C₂×C₈×C₂₄₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₄₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₄₁

D₄

yes

—

C₂×C₈×C₂₄₁

Q₈

yes

—

C₄⋊C₄×C₂₄₁

C₂×C₄

yes

—

C₂²⋊C₄×C₂₄₁

—

C₄²×C₂₄₁

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

—

#14515349

—

#14515349

—

C₂³×C₂₄₁

#41

yes

characteristic

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₄₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₄₁

#41

yes

characteristic

C₈×C₂₄₁

C₄⋊C₄

yes

—

C₈×C₂₄₁

C₄⋊C₄

yes

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

C₄×D₄

yes

characteristic

C₂²×C₂₄₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₄₁

#125

yes

characteristic

C₄×C₂₄₁

—

C₄×C₂₄₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₄₁

C₄×Q₈

yes

characteristic

C₄×C₂₄₁

#121

yes

characteristic

C₄×C₂₄₁

C₄⋊Q₈

yes

characteristic

Dic₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

#389

yes

characteristic

C₂×C₂₄₁

#444

yes

—

C₂₄₁

#1946

yes

characteristic, complemented, Sylow

#1946

—

241

maximal, Sylow

C₂×#108

—

241

—

#427

D₂₄₁

yes

characteristic

#424

—

241

—

C₂×#97

—

241

—

#339

—

241

—

C₂×M₁₆

—

241

—

C₂²×C₈

—

241

—

C₂²×C₈

C₂×D₂₄₁

yes

characteristic

#117

C₂×D₂₄₁

yes

—

C₂×C₄²

—

241

—

#108

—

482

—

#107

Dic₂₄₁

yes

complemented

#106

Dic₂₄₁

yes

complemented

C₄⋊C₈

—

482

—

#97

—

482

—

C₂²×C₄

—

482

—

C₂²×C₄

C₂²×D₂₄₁

yes

characteristic

M₁₆

—

482

—

M₁₆

—

482

—

C₂×C₈

—

482

—

C₂×C₈

—

482

—

C₂×C₈

—

482

—

C₂×C₈

—

241

—

C₂×C₈

C₂×Dic₂₄₁

yes

characteristic

C₂×C₈

C₂×Dic₂₄₁

yes

characteristic

C₂×C₈

D₉₆₄

yes

—

C₂×C₈

Dic₄₈₂

yes

—

C₄⋊C₄

C₂×Dic₂₄₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

482

—

C₄²

—

241

—

C₂³

#16838769

yes

characteristic

C₂×C₄

—

C₂×C₄

—

964

—

C₂×C₄

—

482

—

C₂×C₄

C₂²×Dic₂₄₁

yes

characteristic

100

C₂×C₄

D₄×D₂₄₁

yes

characteristic

101

C₂×C₄

C₂×D₉₆₄

yes

characteristic

102

C₂×C₄

Q₈×D₂₄₁

yes

characteristic

103

C₂×C₄

C₂×Dic₄₈₂

yes

characteristic

104

C₂×C₄

#16838771

yes

characteristic

105

C₈

—

964

—

106

C₈

—

482

—

107

C₈

C₄⋊Dic₂₄₁

yes

—

108

C₈

C₄⋊Dic₂₄₁

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₂₄₁

yes

characteristic

111

C₂²

#21440622

yes

characteristic

112

C₂²

#21440633

yes

characteristic

113

C₄

—

114

C₄

—

964

—

115

C₄

C₂×C₄⋊Dic₂₄₁

yes

characteristic

116

C₄

Q₈×Dic₂₄₁

yes

characteristic

117

C₄

#21440617

yes

characteristic

118

C₄

#21440641

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#29507616

yes

characteristic

122

C₂

#29507839

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₄₁
commutator	C₂×C₄×C₂₄₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₄₁
Fitting	#427×C₂₄₁	C₂
socle	C₂²×C₂₄₁	C₄×D₄
prime radical	C₂³×C₂₄₁	#41
cosocle	C₂²×C₄×C₂₄₁	C₂³
hypercenter	#427	D₂₄₁
hypocenter	C₂₄₁	#1946
LCS	G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₄₁

commutator

C₂×C₄×C₂₄₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₄₁

Fitting

#427×C₂₄₁

C₂

socle

C₂²×C₂₄₁

C₄×D₄

prime radical

C₂³×C₂₄₁

#41

cosocle

C₂²×C₄×C₂₄₁

C₂³

D₂₄₁

C₂₄₁

G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₂₄₁	C₄	semidirect
15. #106×C₂₄₁	C₄	semidirect
61. C₂₄₁	#1946	semidirect
74. #107	Dic₂₄₁	semidirect
75. #106	Dic₂₄₁	semidirect

H ≅ G/N

type

14. #107×C₂₄₁

C₄

semidirect

15. #106×C₂₄₁

C₄

semidirect

61. C₂₄₁

#1946

semidirect

74. #107

Dic₂₄₁

semidirect

75. #106

Dic₂₄₁

semidirect

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

subgroup	index
C₂×#21440578	2
#29507744	2
C₂×#21440552	2
#29507476	2
#427×C₂₄₁	2
#1946	241

subgroup

index

#427×C₂₄₁

241

Subgroups — #44508199

Conjugacy classes of subgroups

Complements

Maximal subgroups