Group #44507326 — subgroups

Subgroups — #44507326

Order 30848.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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	#29507673	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₄₁	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	C₄	1	4	yes	complemented
10	#21440553	—	2	2	no	—
11	C₄×#14515349	—	2	2	no	—
12	C₈×Dic₂₄₁	C₄	1	4	yes	complemented
13	C₂²×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
14	#117×C₂₄₁	C₂²	1	2	yes	—
15	#107×C₂₄₁	C₄	1	2	yes	complemented
16	#106×C₂₄₁	C₄	1	2	yes	complemented
17	C₂²×Dic₂₄₁	—	2	1	no	—
18	#16838754	—	2	2	no	—
19	#16838754	C₂×C₄	1	4	yes	—
20	C₂×#14515349	—	2	2	no	—
21	C₂×#14515349	C₂×C₄	1	2	yes	—
22	C₂×#14515349	D₄	1	1	yes	characteristic
23	C₂×#14515349	D₄	1	1	yes	characteristic
24	C₂×#14515349	Q₈	1	2	yes	—
25	C₄×Dic₂₄₁	—	2	1	no	—
26	C₄×Dic₂₄₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₂₄₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₄₁	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₄×C₂₄₁	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₂₄₁	—	2	2	no	—
34	C₄²×C₂₄₁	—	2	2	no	—
35	C₂×Dic₂₄₁	—	2	4	no	—
36	C₂×Dic₂₄₁	—	2	2	no	—
37	#14515349	—	2	4	no	—
38	#14515349	C₄⋊C₄	1	2	yes	—
39	#14515349	C₄⋊C₄	1	2	yes	—
40	C₂³×C₂₄₁	#41	1	1	yes	characteristic
41	C₂×C₄×C₂₄₁	—	2	4	no	—
42	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₄₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₄₁	#41	1	1	yes	characteristic
48	C₈×C₂₄₁	C₄²	1	2	yes	—
49	C₈×C₂₄₁	C₄²	1	2	yes	—
50	C₂²×C₂₄₁	—	2	2	no	—
51	C₂²×C₂₄₁	#117	1	1	yes	characteristic
52	C₂²×C₂₄₁	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₄₁	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₂₄₁	—	4	4	no	—
55	C₄×C₂₄₁	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₂₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₂₄₁	C₄×D₄	1	1	yes	characteristic
58	C₄×C₂₄₁	C₄×Q₈	1	1	yes	characteristic
59	Dic₂₄₁	—	2	4	no	—
60	C₂×C₂₄₁	—	2	1	no	—
61	C₂×C₂₄₁	—	2	1	no	—
62	C₂×C₂₄₁	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₂₄₁	#443	1	2	yes	—
64	C₂₄₁	#1772	1	1	yes	characteristic, complemented, Sylow
65	#1772	—	241	1	no	maximal, Sylow
66	C₂×#108	—	241	1	no	—
67	#427	D₂₄₁	1	1	yes	characteristic
68	#405	—	241	2	no	—
69	C₂×C₄×C₈	—	241	1	no	—
70	#339	—	241	2	no	—
71	C₂×M₁₆	—	241	2	no	—
72	C₂²×C₈	—	241	1	no	—
73	C₂²×C₈	C₂×D₂₄₁	1	1	yes	characteristic
74	#117	C₂×D₂₄₁	1	2	yes	—
75	C₂×C₄²	—	241	1	no	—
76	#108	—	241	4	no	—
77	#107	Dic₂₄₁	1	2	yes	complemented
78	#106	Dic₂₄₁	1	2	yes	complemented
79	#97	—	482	2	no	—
80	C₄×C₈	—	482	2	no	—
81	C₄×C₈	—	241	4	no	—
82	C₂²×C₄	—	482	1	no	—
83	C₂²×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
84	M₁₆	—	482	2	no	—
85	M₁₆	—	241	4	no	—
86	C₂×C₈	—	482	2	no	—
87	C₂×C₈	—	241	2	no	—
88	C₂×C₈	—	241	1	no	—
89	C₂×C₈	—	241	1	no	—
90	C₂×C₈	—	241	2	no	—
91	C₂×C₈	C₄×D₂₄₁	1	2	yes	—
92	C₂×C₈	C₄×D₂₄₁	1	2	yes	—
93	C₂×C₈	C₂×Dic₂₄₁	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₂₄₁	1	1	yes	characteristic
95	C₄⋊C₄	C₂×Dic₂₄₁	1	4	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	482	1	no	—
99	C₄²	—	241	2	no	—
100	C₂³	#16838769	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	482	4	no	—
103	C₂×C₄	—	482	2	no	—
104	C₂×C₄	C₂×C₄×D₂₄₁	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₂₄₁	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₂₄₁	1	1	yes	characteristic
107	C₂×C₄	D₄×D₂₄₁	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₂₄₁	1	1	yes	characteristic
109	C₂×C₄	#16838771	1	1	yes	characteristic
110	C₈	—	482	4	no	—
111	C₈	—	241	2	no	—
112	C₈	—	241	2	no	—
113	C₈	C₄×Dic₂₄₁	1	2	yes	—
114	C₈	C₄×Dic₂₄₁	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#21440600	1	1	yes	characteristic
117	C₂²	D₄×Dic₂₄₁	1	1	yes	characteristic
118	C₂²	#21440607	1	1	yes	characteristic
119	C₄	—	4	4	no	—
120	C₄	—	482	4	no	—
121	C₄	C₂×C₄×Dic₂₄₁	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₂₄₁	1	1	yes	characteristic
123	C₄	#21440604	1	1	yes	characteristic
124	C₄	Q₈×Dic₂₄₁	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₂₄₁	1	1	yes	characteristic
128	C₂	#29507834	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21440578

C₂

yes

maximal, characteristic

#29507673

C₂

yes

maximal

C₂×C₈×Dic₂₄₁

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

characteristic

#21440578

C₄

yes

complemented

#21440553

—

C₄×#14515349

—

C₈×Dic₂₄₁

C₄

yes

complemented

C₂²×C₈×C₂₄₁

C₂²

yes

characteristic

#117×C₂₄₁

C₂²

yes

—

#107×C₂₄₁

C₄

yes

complemented

#106×C₂₄₁

C₄

yes

complemented

C₂²×Dic₂₄₁

—

#16838754

—

#16838754

C₂×C₄

yes

—

C₂×#14515349

—

C₂×#14515349

C₂×C₄

yes

—

C₂×#14515349

D₄

yes

characteristic

C₂×#14515349

D₄

yes

characteristic

C₂×#14515349

Q₈

yes

—

C₄×Dic₂₄₁

—

C₄×Dic₂₄₁

C₂×C₄

yes

—

C₂²×C₄×C₂₄₁

C₂³

yes

characteristic

C₂×C₈×C₂₄₁

C₂×C₄

yes

—

C₂×C₈×C₂₄₁

C₂×C₄

yes

—

C₂×C₈×C₂₄₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₄₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₄₁

C₂×C₄

yes

—

C₂²⋊C₄×C₂₄₁

—

C₄²×C₂₄₁

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

—

#14515349

—

#14515349

C₄⋊C₄

yes

—

#14515349

C₄⋊C₄

yes

—

C₂³×C₂₄₁

#41

yes

characteristic

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₄₁

#41

yes

characteristic

C₈×C₂₄₁

C₄²

yes

—

C₈×C₂₄₁

C₄²

yes

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

#117

yes

characteristic

C₂²×C₂₄₁

C₄×D₄

yes

characteristic

C₂²×C₂₄₁

C₄×Q₈

yes

characteristic

C₄×C₂₄₁

—

C₄×C₂₄₁

C₂×C₄²

yes

characteristic

C₄×C₂₄₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₄₁

C₄×D₄

yes

characteristic

C₄×C₂₄₁

C₄×Q₈

yes

characteristic

Dic₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₄₁

#443

yes

—

C₂₄₁

#1772

yes

characteristic, complemented, Sylow

#1772

—

241

maximal, Sylow

C₂×#108

—

241

—

#427

D₂₄₁

yes

characteristic

#405

—

241

—

C₂×C₄×C₈

—

241

—

#339

—

241

—

C₂×M₁₆

—

241

—

C₂²×C₈

—

241

—

C₂²×C₈

C₂×D₂₄₁

yes

characteristic

#117

C₂×D₂₄₁

yes

—

C₂×C₄²

—

241

—

#108

—

241

—

#107

Dic₂₄₁

yes

complemented

#106

Dic₂₄₁

yes

complemented

#97

—

482

—

C₄×C₈

—

482

—

C₄×C₈

—

241

—

C₂²×C₄

—

482

—

C₂²×C₄

C₂²×D₂₄₁

yes

characteristic

M₁₆

—

482

—

M₁₆

—

241

—

C₂×C₈

—

482

—

C₂×C₈

—

241

—

C₂×C₈

—

241

—

C₂×C₈

—

241

—

C₂×C₈

—

241

—

C₂×C₈

C₄×D₂₄₁

yes

—

C₂×C₈

C₄×D₂₄₁

yes

—

C₂×C₈

C₂×Dic₂₄₁

yes

characteristic

C₂×C₈

C₂×Dic₂₄₁

yes

characteristic

C₄⋊C₄

C₂×Dic₂₄₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

482

—

C₄²

—

241

—

100

C₂³

#16838769

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

482

—

103

C₂×C₄

—

482

—

104

C₂×C₄

C₂×C₄×D₂₄₁

yes

characteristic

105

C₂×C₄

C₂×C₄×D₂₄₁

yes

characteristic

106

C₂×C₄

C₂²×Dic₂₄₁

yes

characteristic

107

C₂×C₄

D₄×D₂₄₁

yes

characteristic

108

C₂×C₄

Q₈×D₂₄₁

yes

characteristic

109

C₂×C₄

#16838771

yes

characteristic

110

C₈

—

482

—

111

C₈

—

241

—

112

C₈

—

241

—

113

C₈

C₄×Dic₂₄₁

yes

—

114

C₈

C₄×Dic₂₄₁

yes

—

115

C₂²

—

116

C₂²

#21440600

yes

characteristic

117

C₂²

D₄×Dic₂₄₁

yes

characteristic

118

C₂²

#21440607

yes

characteristic

119

C₄

—

120

C₄

—

482

—

121

C₄

C₂×C₄×Dic₂₄₁

yes

characteristic

122

C₄

C₄⋊C₄×D₂₄₁

yes

characteristic

123

C₄

#21440604

yes

characteristic

124

C₄

Q₈×Dic₂₄₁

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₂₄₁

yes

characteristic

128

C₂

#29507834

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₄₁
commutator	C₄×C₂₄₁	C₂×C₄²
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	#1772
LCS	G ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₄₁

commutator

C₄×C₂₄₁

C₂×C₄²

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 ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
9. #21440578	C₄	semidirect
12. C₈×Dic₂₄₁	C₄	semidirect
15. #107×C₂₄₁	C₄	semidirect
16. #106×C₂₄₁	C₄	semidirect
64. C₂₄₁	#1772	semidirect
77. #107	Dic₂₄₁	semidirect
78. #106	Dic₂₄₁	semidirect

H ≅ G/N

type

9. #21440578

C₄

semidirect

12. C₈×Dic₂₄₁

C₄

semidirect

15. #107×C₂₄₁

C₄

semidirect

16. #106×C₂₄₁

C₄

semidirect

64. C₂₄₁

#1772

semidirect

77. #107

Dic₂₄₁

semidirect

78. #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
#29507673	2
C₂×C₈×Dic₂₄₁	2
#29507476	2
#427×C₂₄₁	2
#1772	241

subgroup

index

C₂×C₈×Dic₂₄₁

#427×C₂₄₁

241

Subgroups — #44507326

Conjugacy classes of subgroups

Complements

Maximal subgroups