Group #44507571 — subgroups

Subgroups — #44507571

Order 30848.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₈×Dic₂₄₁	C₂	1	1	yes	maximal, characteristic
2	#29507586	C₂	1	2	yes	maximal
3	#29507480	C₂	1	1	yes	maximal, characteristic
4	#29507479	C₂	1	2	yes	maximal
5	#340×C₂₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14515349	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₄₁	C₂²	1	2	yes	—
8	C₂×#16838749	—	2	2	no	—
9	C₂×#16838749	—	2	2	no	—
10	C₂×C₄×Dic₂₄₁	C₂²	1	1	yes	characteristic
11	C₈×Dic₂₄₁	C₄	1	4	yes	complemented
12	C₂²×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
14	C₂²×Dic₂₄₁	—	2	2	no	—
15	C₂²×Dic₂₄₁	D₄	1	1	yes	characteristic
16	C₂²×Dic₂₄₁	D₄	1	1	yes	characteristic
17	C₂×#14515349	—	2	2	no	—
18	C₂×#14515349	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₂₄₁	—	2	2	no	—
20	C₄⋊Dic₂₄₁	—	2	2	no	—
21	#16838749	—	4	4	no	—
22	#16838749	—	4	4	no	—
23	C₄×Dic₂₄₁	C₂×C₄	1	2	yes	—
24	C₄×Dic₂₄₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₄₁	—	2	2	no	—
26	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₄₁	—	2	2	no	—
28	C₂×C₈×C₂₄₁	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₂₄₁	—	2	2	no	—
30	C₄⋊C₄×C₂₄₁	—	2	2	no	—
31	C₂×Dic₂₄₁	—	4	2	no	—
32	C₂×Dic₂₄₁	—	4	2	no	—
33	C₂×Dic₂₄₁	—	4	2	no	—
34	C₂×Dic₂₄₁	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₂₄₁	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₂₄₁	D₈	1	2	yes	—
37	C₂×Dic₂₄₁	SD₁₆	1	4	yes	—
38	C₂×Dic₂₄₁	Q₁₆	1	2	yes	—
39	#14515349	—	2	2	no	—
40	C₂³×C₂₄₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₄₁	—	4	2	no	—
42	C₂×C₄×C₂₄₁	—	4	2	no	—
43	C₂×C₄×C₂₄₁	—	4	2	no	—
44	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₄₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₄₁	#41	1	2	yes	—
47	C₂×C₄×C₂₄₁	#41	1	2	yes	—
48	C₈×C₂₄₁	—	2	2	no	—
49	C₂²×C₂₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₄₁	#124	1	2	yes	—
51	C₂²×C₂₄₁	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₂₄₁	C₂×SD₁₆	1	2	yes	—
53	C₂²×C₂₄₁	C₂×Q₁₆	1	1	yes	characteristic
54	C₄×C₂₄₁	—	8	2	no	—
55	C₄×C₂₄₁	#117	1	1	yes	characteristic
56	C₄×C₂₄₁	#117	1	1	yes	characteristic
57	C₄×C₂₄₁	#125	1	2	yes	—
58	Dic₂₄₁	—	8	2	no	—
59	Dic₂₄₁	#106	1	4	yes	—
60	Dic₂₄₁	#107	1	4	yes	—
61	C₂×C₂₄₁	#382	1	1	yes	characteristic
62	C₂×C₂₄₁	C₂×#106	1	1	yes	characteristic
63	C₂×C₂₄₁	C₂×#107	1	1	yes	characteristic
64	C₂×C₂₄₁	#486	1	2	yes	—
65	C₂×C₂₄₁	#487	1	2	yes	—
66	C₂₄₁	#1834	1	1	yes	characteristic, complemented, Sylow
67	#1834	—	241	1	no	maximal, Sylow
68	C₂×C₄×C₈	—	241	1	no	—
69	#384	—	241	2	no	—
70	#340	—	241	1	no	—
71	#340	—	241	2	no	—
72	#340	D₂₄₁	1	1	yes	characteristic
73	C₂²×C₈	—	241	1	no	—
74	C₂²×C₈	C₂×D₂₄₁	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	482	2	no	—
76	C₂×C₄⋊C₄	—	482	2	no	—
77	C₂×C₄⋊C₄	—	241	2	no	—
78	C₂×C₄⋊C₄	C₂×D₂₄₁	1	2	yes	—
79	C₂×C₄²	—	241	1	no	—
80	C₄×C₈	—	241	4	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	482	2	no	—
83	C₂²×C₄	—	241	1	no	—
84	C₂²×C₄	—	241	1	no	—
85	C₂²×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	482	2	no	—
88	C₂×C₈	—	241	2	no	—
89	C₂×C₈	C₄×D₂₄₁	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	964	4	no	—
93	C₄⋊C₄	—	964	4	no	—
94	C₄⋊C₄	—	482	2	no	—
95	C₄⋊C₄	—	482	2	no	—
96	C₄²	—	241	2	no	—
97	C₄²	—	241	2	no	—
98	C₂³	D₄×D₂₄₁	1	1	yes	characteristic
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	964	2	no	—
103	C₂×C₄	—	964	2	no	—
104	C₂×C₄	—	964	2	no	—
105	C₂×C₄	—	241	2	no	—
106	C₂×C₄	—	241	2	no	—
107	C₂×C₄	—	241	2	no	—
108	C₂×C₄	—	241	4	no	—
109	C₂×C₄	—	241	2	no	—
110	C₂×C₄	C₂×C₄×D₂₄₁	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₂₄₁	1	1	yes	characteristic
112	C₂×C₄	#16838769	1	2	yes	—
113	C₂×C₄	#16838770	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	482	2	no	—
116	C₂²	C₄⋊C₄×D₂₄₁	1	1	yes	characteristic
117	C₂²	#21440628	1	2	yes	—
118	C₂²	D₈×D₂₄₁	1	1	yes	characteristic
119	C₂²	SD₁₆×D₂₄₁	1	2	yes	—
120	C₂²	Q₁₆×D₂₄₁	1	1	yes	characteristic
121	C₄	—	8	2	no	—
122	C₄	—	1928	2	no	—
123	C₄	—	241	4	no	—
124	C₄	—	241	4	no	—
125	C₄	#21440598	1	1	yes	characteristic
126	C₄	#21440599	1	1	yes	characteristic
127	C₄	#21440632	1	2	yes	—
128	C₂	#29507581	1	1	yes	characteristic
129	C₂	#106×D₂₄₁	1	1	yes	characteristic
130	C₂	#107×D₂₄₁	1	1	yes	characteristic
131	C₂	#29508062	1	2	yes	—
132	C₂	#29508066	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₂₄₁

C₂

yes

maximal, characteristic

#29507586

C₂

yes

maximal

#29507480

C₂

yes

maximal, characteristic

#29507479

C₂

yes

maximal

#340×C₂₄₁

C₂

yes

maximal, characteristic

C₂²×#14515349

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₄₁

C₂²

yes

—

C₂×#16838749

—

C₂×#16838749

—

C₂×C₄×Dic₂₄₁

C₂²

yes

characteristic

C₈×Dic₂₄₁

C₄

yes

complemented

C₂²×C₈×C₂₄₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₄₁

C₂²

yes

—

C₂²×Dic₂₄₁

—

C₂²×Dic₂₄₁

D₄

yes

characteristic

C₂²×Dic₂₄₁

D₄

yes

characteristic

C₂×#14515349

—

C₂×#14515349

C₂×C₄

yes

—

C₄⋊Dic₂₄₁

—

C₄⋊Dic₂₄₁

—

#16838749

—

#16838749

—

C₄×Dic₂₄₁

C₂×C₄

yes

—

C₄×Dic₂₄₁

Q₈

yes

—

C₂²×C₄×C₂₄₁

—

C₂²×C₄×C₂₄₁

C₂³

yes

characteristic

C₂×C₈×C₂₄₁

—

C₂×C₈×C₂₄₁

C₂×C₄

yes

—

C₄⋊C₄×C₂₄₁

—

C₄⋊C₄×C₂₄₁

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

C₄⋊C₄

yes

—

C₂×Dic₂₄₁

C₄⋊C₄

yes

—

C₂×Dic₂₄₁

D₈

yes

—

C₂×Dic₂₄₁

SD₁₆

yes

—

C₂×Dic₂₄₁

Q₁₆

yes

—

#14515349

—

C₂³×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₄₁

#41

yes

—

C₂×C₄×C₂₄₁

#41

yes

—

C₈×C₂₄₁

—

C₂²×C₂₄₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₄₁

#124

yes

—

C₂²×C₂₄₁

C₂×D₈

yes

characteristic

C₂²×C₂₄₁

C₂×SD₁₆

yes

—

C₂²×C₂₄₁

C₂×Q₁₆

yes

characteristic

C₄×C₂₄₁

—

C₄×C₂₄₁

yes

C₄×C₂₄₁

yes

C₄×C₂₄₁

yes

—

Dic₂₄₁

—

Dic₂₄₁

#106

yes

—

Dic₂₄₁

#107

yes

—

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

—

C₂×C₂₄₁

#487

yes

—

C₂₄₁

#1834

yes

characteristic, complemented, Sylow

#1834

—

241

maximal, Sylow

C₂×C₄×C₈

—

241

—

#384

—

241

—

#340

—

241

—

#340

—

241

—

#340

D₂₄₁

yes

characteristic

C₂²×C₈

—

241

—

C₂²×C₈

C₂×D₂₄₁

yes

characteristic

C₂×C₄⋊C₄

—

482

—

C₂×C₄⋊C₄

—

482

—

C₂×C₄⋊C₄

—

241

—

C₂×C₄⋊C₄

C₂×D₂₄₁

yes

—

C₂×C₄²

—

241

—

C₄×C₈

—

241

—

C₂²×C₄

—

C₂²×C₄

—

482

—

C₂²×C₄

—

241

—

C₂²×C₄

—

241

—

C₂²×C₄

C₂²×D₂₄₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

482

—

C₂×C₈

—

241

—

C₂×C₈

C₄×D₂₄₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

964

—

C₄⋊C₄

—

964

—

C₄⋊C₄

—

482

—

C₄⋊C₄

—

482

—

C₄²

—

241

—

C₄²

—

241

—

C₂³

D₄×D₂₄₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

964

—

103

C₂×C₄

—

964

—

104

C₂×C₄

—

964

—

105

C₂×C₄

—

241

—

106

C₂×C₄

—

241

—

107

C₂×C₄

—

241

—

108

C₂×C₄

—

241

—

109

C₂×C₄

—

241

—

110

C₂×C₄

C₂×C₄×D₂₄₁

yes

characteristic

111

C₂×C₄

Q₈×D₂₄₁

yes

characteristic

112

C₂×C₄

#16838769

yes

—

113

C₂×C₄

#16838770

yes

—

114

C₈

—

115

C₈

—

482

—

116

C₂²

C₄⋊C₄×D₂₄₁

yes

characteristic

117

C₂²

#21440628

yes

—

118

C₂²

D₈×D₂₄₁

yes

characteristic

119

C₂²

SD₁₆×D₂₄₁

yes

—

120

C₂²

Q₁₆×D₂₄₁

yes

characteristic

121

C₄

—

122

C₄

—

1928

—

123

C₄

—

241

—

124

C₄

—

241

—

125

C₄

#21440598

yes

characteristic

126

C₄

#21440599

yes

characteristic

127

C₄

#21440632

yes

—

128

C₂

#29507581

yes

characteristic

129

C₂

#106×D₂₄₁

yes

characteristic

130

C₂

#107×D₂₄₁

yes

characteristic

131

C₂

#29508062

yes

—

132

C₂

#29508066

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₄₁
commutator	C₂×C₄×C₂₄₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₄₁
Fitting	#340×C₂₄₁	C₂
socle, prime radical	C₂³×C₂₄₁	C₂×D₄
cosocle	C₂²×C₄×C₂₄₁	C₂³
hypercenter	#340	D₂₄₁
hypocenter	C₂₄₁	#1834
LCS	G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₂₄₁

commutator

C₂×C₄×C₂₄₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₄₁

Fitting

#340×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 ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
11. C₈×Dic₂₄₁	C₄	semidirect
66. C₂₄₁	#1834	semidirect

H ≅ G/N

type

11. C₈×Dic₂₄₁

C₄

semidirect

66. C₂₄₁

#1834

semidirect

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

subgroup	index
C₂×C₈×Dic₂₄₁	2
#29507586	2
#29507480	2
#29507479	2
#340×C₂₄₁	2
#1834	241

subgroup

index

C₂×C₈×Dic₂₄₁

#340×C₂₄₁

241

Subgroups — #44507571

Conjugacy classes of subgroups

Complements

Maximal subgroups