Group #44507162 — subgroups

Subgroups — #44507162

Order 30848.

424 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 205 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×#21440601	C₂	1	2	yes	maximal
2	#29507480	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₂₄₁	C₂	1	1	yes	maximal, characteristic
4	#21440601	—	2	4	no	—
5	#21440601	C₄	1	8	yes	complemented
6	C₂×C₄⋊Dic₂₄₁	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₂₄₁	—	2	2	no	—
8	C₂×C₄×Dic₂₄₁	—	2	2	no	—
9	C₂³×C₄×C₂₄₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₄₁	—	2	4	no	—
11	C₂²×C₈×C₂₄₁	C₄	1	4	yes	complemented
12	C₂²×C₈×C₂₄₁	C₂²	1	2	yes	—
13	C₂²×Dic₂₄₁	—	2	4	no	—
14	C₄⋊Dic₂₄₁	—	2	4	no	—
15	C₄⋊Dic₂₄₁	C₂×C₄	1	8	yes	—
16	C₂²⋊Dic₂₄₁	—	2	8	no	—
17	C₄×Dic₂₄₁	—	2	8	no	—
18	C₂⁴×C₂₄₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₄₁	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₂₄₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₄₁	D₄	1	4	yes	—
22	C₂²×C₄×C₂₄₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₄₁	Q₈	1	2	yes	—
24	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₂₄₁	—	2	8	no	—
26	C₂×C₈×C₂₄₁	—	2	4	no	—
27	C₂×C₈×C₂₄₁	C₂×C₄	1	4	yes	—
28	C₂×Dic₂₄₁	—	2	8	no	—
29	C₂×Dic₂₄₁	—	4	8	no	—
30	C₂³×C₂₄₁	—	2	4	no	—
31	C₂³×C₂₄₁	C₂²⋊C₄	1	4	yes	—
32	C₂³×C₂₄₁	C₄⋊C₄	1	2	yes	—
33	C₂³×C₂₄₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₄₁	C₄²	1	4	yes	—
35	C₂×C₄×C₂₄₁	C₂²⋊C₄	1	4	yes	—
36	C₂×C₄×C₂₄₁	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₂₄₁	C₄⋊C₄	1	8	yes	—
38	C₂×C₄×C₂₄₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₄₁	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₂₄₁	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₂₄₁	—	2	4	no	—
44	C₂²×C₂₄₁	—	2	8	no	—
45	C₂²×C₂₄₁	—	2	4	no	—
46	C₂²×C₂₄₁	#95	1	4	yes	—
47	C₂²×C₂₄₁	C₂×C₂²⋊C₄	1	2	yes	—
48	C₂²×C₂₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₄₁	#135	1	4	yes	—
50	C₄×C₂₄₁	#95	1	4	yes	—
51	C₄×C₂₄₁	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₂₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₄₁	C₂×C₄⋊C₄	1	2	yes	—
54	Dic₂₄₁	—	4	8	no	—
55	C₂×C₂₄₁	—	2	4	no	—
56	C₂×C₂₄₁	C₂×#95	1	1	yes	characteristic
57	C₂×C₂₄₁	#416	1	4	yes	—
58	C₂×C₂₄₁	#427	1	2	yes	—
59	C₂₄₁	#1735	1	1	yes	characteristic, complemented, Sylow
60	#1735	—	241	1	no	maximal, Sylow
61	C₂³×C₈	D₂₄₁	1	1	yes	characteristic
62	C₂×#117	—	241	2	no	—
63	#340	—	241	4	no	—
64	C₂³×C₄	C₂×D₂₄₁	1	1	yes	characteristic
65	C₂²×C₈	—	2	4	no	—
66	C₂²×C₈	Dic₂₄₁	1	4	yes	complemented
67	C₂²×C₈	C₂×D₂₄₁	1	2	yes	—
68	#117	—	482	4	no	—
69	#117	—	241	8	no	—
70	C₂×C₄⋊C₄	—	241	4	no	—
71	C₂×C₂²⋊C₄	—	482	2	no	—
72	C₂×C₄²	—	482	2	no	—
73	C₂⁴	D₉₆₄	1	1	yes	characteristic
74	C₂²×C₄	—	482	4	no	—
75	C₂²×C₄	C₄×D₂₄₁	1	4	yes	—
76	C₂²×C₄	C₂×Dic₂₄₁	1	2	yes	—
77	C₂²×C₄	C₂²⋊D₂₄₁	1	4	yes	—
78	C₂²×C₄	D₉₆₄	1	1	yes	characteristic
79	C₂²×C₄	Dic₄₈₂	1	2	yes	—
80	C₂²×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₂×Dic₂₄₁	1	4	yes	—
84	C₄⋊C₄	—	482	4	no	—
85	C₄⋊C₄	—	241	8	no	—
86	C₂²⋊C₄	—	482	8	no	—
87	C₄²	—	482	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	#16838747	1	4	yes	—
90	C₂³	C₄⋊Dic₂₄₁	1	2	yes	—
91	C₂³	C₂×D₉₆₄	1	1	yes	characteristic
92	C₂×C₄	—	964	8	no	—
93	C₂×C₄	—	482	8	no	—
94	C₂×C₄	C₄×Dic₂₄₁	1	4	yes	—
95	C₂×C₄	#16838747	1	4	yes	—
96	C₂×C₄	C₂²⋊Dic₂₄₁	1	4	yes	—
97	C₂×C₄	#16838749	1	8	yes	—
98	C₂×C₄	C₄⋊Dic₂₄₁	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₂₄₁	1	2	yes	—
100	C₂×C₄	C₂²×Dic₂₄₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₂₄₁	1	2	yes	—
102	C₂×C₄	C₂×Dic₄₈₂	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	4	no	—
106	C₂²	#21440549	1	4	yes	—
107	C₂²	C₂×#16838747	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₂₄₁	1	1	yes	characteristic
109	C₂²	#21440664	1	4	yes	—
110	C₄	—	964	8	no	—
111	C₄	#21440549	1	4	yes	—
112	C₄	C₂×C₄×Dic₂₄₁	1	1	yes	characteristic
113	C₄	C₂×C₂²⋊Dic₂₄₁	1	1	yes	characteristic
114	C₄	C₂×#16838749	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	C₂×#21440549	1	1	yes	characteristic
117	C₂	#29507714	1	4	yes	—
118	C₂	#29507755	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21440601

C₂

yes

maximal

#29507480

C₂

yes

maximal, complemented

C₂³×C₈×C₂₄₁

C₂

yes

maximal, characteristic

#21440601

—

#21440601

C₄

yes

complemented

C₂×C₄⋊Dic₂₄₁

C₂²

yes

—

C₂×C₂²⋊Dic₂₄₁

—

C₂×C₄×Dic₂₄₁

—

C₂³×C₄×C₂₄₁

C₂²

yes

characteristic

C₂²×C₈×C₂₄₁

—

C₂²×C₈×C₂₄₁

C₄

yes

complemented

C₂²×C₈×C₂₄₁

C₂²

yes

—

C₂²×Dic₂₄₁

—

C₄⋊Dic₂₄₁

—

C₄⋊Dic₂₄₁

C₂×C₄

yes

—

C₂²⋊Dic₂₄₁

—

C₄×Dic₂₄₁

—

C₂⁴×C₂₄₁

D₄

yes

characteristic

C₂²×C₄×C₂₄₁

C₂×C₄

yes

—

C₂²×C₄×C₂₄₁

C₂×C₄

yes

—

C₂²×C₄×C₂₄₁

D₄

yes

—

C₂²×C₄×C₂₄₁

D₄

yes

characteristic

C₂²×C₄×C₂₄₁

Q₈

yes

—

C₂²×C₄×C₂₄₁

C₂³

yes

characteristic

C₂×C₈×C₂₄₁

—

C₂×C₈×C₂₄₁

—

C₂×C₈×C₂₄₁

C₂×C₄

yes

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

—

C₂³×C₂₄₁

—

C₂³×C₂₄₁

C₂²⋊C₄

yes

—

C₂³×C₂₄₁

C₄⋊C₄

yes

—

C₂³×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

C₄²

yes

—

C₂×C₄×C₂₄₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₄₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₄₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₄₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

—

C₂×C₄×C₂₄₁

C₂×Q₈

yes

characteristic

C₈×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

#95

yes

—

C₂²×C₂₄₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₄₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₄₁

#135

yes

—

C₄×C₂₄₁

#95

yes

—

C₄×C₂₄₁

C₂×C₄²

yes

characteristic

C₄×C₂₄₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₄₁

C₂×C₄⋊C₄

yes

—

Dic₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

C₂×#95

yes

characteristic

C₂×C₂₄₁

#416

yes

—

C₂×C₂₄₁

#427

yes

—

C₂₄₁

#1735

yes

characteristic, complemented, Sylow

#1735

—

241

maximal, Sylow

C₂³×C₈

D₂₄₁

yes

characteristic

C₂×#117

—

241

—

#340

—

241

—

C₂³×C₄

C₂×D₂₄₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₂₄₁

yes

complemented

C₂²×C₈

C₂×D₂₄₁

yes

—

#117

—

482

—

#117

—

241

—

C₂×C₄⋊C₄

—

241

—

C₂×C₂²⋊C₄

—

482

—

C₂×C₄²

—

482

—

C₂⁴

D₉₆₄

yes

characteristic

C₂²×C₄

—

482

—

C₂²×C₄

C₄×D₂₄₁

yes

—

C₂²×C₄

C₂×Dic₂₄₁

yes

—

C₂²×C₄

C₂²⋊D₂₄₁

yes

—

C₂²×C₄

D₉₆₄

yes

characteristic

C₂²×C₄

Dic₄₈₂

yes

—

C₂²×C₄

C₂²×D₂₄₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₄₁

yes

—

C₄⋊C₄

—

482

—

C₄⋊C₄

—

241

—

C₂²⋊C₄

—

482

—

C₄²

—

482

—

C₂³

—

C₂³

#16838747

yes

—

C₂³

C₄⋊Dic₂₄₁

yes

—

C₂³

C₂×D₉₆₄

yes

characteristic

C₂×C₄

—

964

—

C₂×C₄

—

482

—

C₂×C₄

C₄×Dic₂₄₁

yes

—

C₂×C₄

#16838747

yes

—

C₂×C₄

C₂²⋊Dic₂₄₁

yes

—

C₂×C₄

#16838749

yes

—

C₂×C₄

C₄⋊Dic₂₄₁

yes

—

C₂×C₄

C₂×C₄×D₂₄₁

yes

—

100

C₂×C₄

C₂²×Dic₂₄₁

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₂₄₁

yes

—

102

C₂×C₄

C₂×Dic₄₈₂

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

#21440549

yes

—

107

C₂²

C₂×#16838747

yes

—

108

C₂²

C₂×C₄⋊Dic₂₄₁

yes

characteristic

109

C₂²

#21440664

yes

—

110

C₄

—

964

—

111

C₄

#21440549

yes

—

112

C₄

C₂×C₄×Dic₂₄₁

yes

characteristic

113

C₄

C₂×C₂²⋊Dic₂₄₁

yes

characteristic

114

C₄

C₂×#16838749

yes

—

115

C₂

—

116

C₂

C₂×#21440549

yes

characteristic

117

C₂

#29507714

yes

—

118

C₂

#29507755

yes

—

	subgroup	G/H
center	C₂²×C₄	D₉₆₄
commutator	C₄×C₂₄₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₄₁
Fitting	C₂³×C₈×C₂₄₁	C₂
socle	C₂³×C₂₄₁	C₂×D₄
prime radical	C₂⁴×C₂₄₁	D₄
cosocle	C₂²×C₄×C₂₄₁	C₂³
hypercenter	C₂³×C₈	D₂₄₁
hypocenter	C₂₄₁	#1735
LCS	G ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

D₉₆₄

commutator

C₄×C₂₄₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₄₁

Fitting

C₂³×C₈×C₂₄₁

C₂

socle

C₂³×C₂₄₁

C₂×D₄

prime radical

C₂⁴×C₂₄₁

D₄

cosocle

C₂²×C₄×C₂₄₁

C₂³

hypercenter

C₂³×C₈

D₂₄₁

hypocenter

C₂₄₁

#1735

LCS

G ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. #29507480	C₂	semidirect
5. #21440601	C₄	semidirect
11. C₂²×C₈×C₂₄₁	C₄	semidirect
59. C₂₄₁	#1735	semidirect
66. C₂²×C₈	Dic₂₄₁	semidirect

H ≅ G/N

type

2. #29507480

C₂

semidirect

5. #21440601

C₄

semidirect

11. C₂²×C₈×C₂₄₁

C₄

semidirect

59. C₂₄₁

#1735

semidirect

66. C₂²×C₈

Dic₂₄₁

semidirect

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

subgroup	index
C₂×#21440601	2
#29507480	2
C₂³×C₈×C₂₄₁	2
#1735	241

subgroup

index

C₂×#21440601

#29507480

C₂³×C₈×C₂₄₁

#1735

241

Subgroups — #44507162

Conjugacy classes of subgroups

Complements

Maximal subgroups