Group #49122815 — subgroups

Subgroups — #49122815

Order 36224.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#31826514	C₂	1	1	yes	maximal, characteristic, complemented
2	#31826492	C₂	1	2	yes	maximal, complemented
3	#31826483	C₂	1	2	yes	maximal
4	#31826433	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₂₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17985392	C₂²	1	1	yes	characteristic
7	C₂×#17985376	C₂²	1	2	yes	—
8	C₂²×C₂₈₃⋊C₈	C₂²	1	1	yes	characteristic
9	#22583546	—	2	2	no	—
10	C₄⋊(C₂₈₃⋊C₈)	—	2	4	no	—
11	#22583538	—	2	4	no	—
12	(C₂×D₂₈₃)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₂₈₃	C₂²	1	1	yes	characteristic
14	#117×C₂₈₃	C₂²	1	2	yes	—
15	#17985392	—	2	2	no	—
16	#17985392	—	2	2	no	—
17	C₂×Dic₅₆₆	Q₈	1	1	yes	characteristic
18	C₂×D₁₁₃₂	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₈₃	—	2	1	no	—
20	C₂×C₄×D₂₈₃	—	2	1	no	—
21	#17985376	—	2	2	no	—
22	#17985376	—	2	2	no	—
23	C₂×C₂₈₃⋊C₈	—	2	1	no	—
24	C₂×C₂₈₃⋊C₈	—	2	2	no	—
25	C₂×C₂₈₃⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₂₈₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₈₃	—	2	2	no	—
28	C₂×C₈×C₂₈₃	D₄	1	2	yes	—
29	C₄⋊C₄×C₂₈₃	—	2	2	no	—
30	C₂²⋊C₄×C₂₈₃	—	2	2	no	—
31	C₄²×C₂₈₃	D₄	1	4	yes	—
32	C₂²×D₂₈₃	—	2	1	no	—
33	Dic₅₆₆	—	2	2	no	—
34	D₁₁₃₂	—	2	2	no	—
35	C₂²⋊D₂₈₃	—	4	2	no	—
36	C₂×Dic₂₈₃	—	2	1	no	—
37	C₄×D₂₈₃	—	4	1	no	—
38	C₄×D₂₈₃	—	4	1	no	—
39	C₂₈₃⋊C₈	—	4	1	no	—
40	C₂₈₃⋊C₈	—	4	2	no	—
41	C₂³×C₂₈₃	#41	1	1	yes	characteristic
42	C₂×C₄×C₂₈₃	—	2	4	no	—
43	C₂×C₄×C₂₈₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₈₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₈₃	#41	1	2	yes	—
47	C₈×C₂₈₃	—	2	2	no	—
48	C₂²×C₂₈₃	—	4	1	no	—
49	C₂²×C₂₈₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₈₃	#123	1	2	yes	—
51	C₂×D₂₈₃	—	4	2	no	—
52	C₂×D₂₈₃	—	4	1	no	—
53	C₄×C₂₈₃	—	4	4	no	—
54	C₄×C₂₈₃	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₂₈₃	#123	1	1	yes	characteristic
56	C₄×C₂₈₃	#127	1	1	yes	characteristic
57	Dic₂₈₃	—	4	1	no	—
58	C₂×C₂₈₃	—	2	2	no	—
59	C₂×C₂₈₃	—	2	1	no	—
60	C₂×C₂₈₃	#399	1	1	yes	characteristic
61	D₂₈₃	—	8	1	no	—
62	C₂₈₃	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	283	1	no	maximal, Sylow
64	#430	—	283	1	no	—
65	#424	—	283	2	no	—
66	#421	—	283	2	no	—
67	#408	—	283	1	no	—
68	H(3,ℤ/4)	D₂₈₃	1	1	yes	characteristic, complemented
69	C₂×#41	—	283	1	no	—
70	C₂×M₁₆	—	283	2	no	—
71	C₂×M₁₆	C₂×D₂₈₃	1	1	yes	characteristic
72	C₂²×C₈	—	283	1	no	—
73	#117	C₂×D₂₈₃	1	2	yes	—
74	#108	—	566	2	no	—
75	C₄⋊C₈	—	566	4	no	—
76	#104	—	566	4	no	—
77	C₂²⋊C₈	—	566	2	no	—
78	#41	—	566	2	no	—
79	#41	—	566	2	no	—
80	C₂×Q₈	—	283	1	no	—
81	C₂×D₄	—	283	1	no	—
82	C₂×D₄	—	566	1	no	—
83	C₂²×C₄	—	566	1	no	—
84	C₂²×C₄	C₂²×D₂₈₃	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	566	2	no	—
87	M₁₆	—	566	2	no	—
88	C₂×C₈	—	566	1	no	—
89	C₂×C₈	—	566	2	no	—
90	C₂×C₈	—	566	2	no	—
91	C₂×C₈	D₁₁₃₂	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₁₁₃₂	1	4	yes	—
95	C₂³	—	566	1	no	—
96	C₂³	#17985391	1	1	yes	characteristic
97	Q₈	—	566	2	no	—
98	D₄	—	566	2	no	—
99	D₄	—	1132	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	566	1	no	—
102	C₂×C₄	—	1132	1	no	—
103	C₂×C₄	—	1132	1	no	—
104	C₂×C₄	C₂×D₁₁₃₂	1	2	yes	—
105	C₂×C₄	C₂×D₁₁₃₂	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₈₃	1	1	yes	characteristic
107	C₂×C₄	#17985391	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	1132	1	no	—
110	C₈	—	1132	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	1132	2	no	—
113	C₂²	—	1132	1	no	—
114	C₂²	#22583590	1	1	yes	characteristic
115	C₂²	#22583594	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	1132	1	no	—
118	C₄	#22583590	1	2	yes	—
119	C₄	#22583594	1	1	yes	characteristic
120	C₄	#22583607	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	2264	1	no	—
124	C₂	#31826402	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31826514

C₂

yes

maximal, characteristic, complemented

#31826492

C₂

yes

maximal, complemented

#31826483

C₂

yes

maximal

#31826433

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₂₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#17985392

C₂²

yes

characteristic

C₂×#17985376

C₂²

yes

—

C₂²×C₂₈₃⋊C₈

C₂²

yes

characteristic

#22583546

—

C₄⋊(C₂₈₃⋊C₈)

—

#22583538

—

(C₂×D₂₈₃)⋊C₈

—

C₂×M₁₆×C₂₈₃

C₂²

yes

characteristic

#117×C₂₈₃

C₂²

yes

—

#17985392

—

#17985392

—

C₂×Dic₅₆₆

Q₈

yes

characteristic

C₂×D₁₁₃₂

Q₈

yes

characteristic

C₂×C₂²⋊D₂₈₃

—

C₂×C₄×D₂₈₃

—

#17985376

—

#17985376

—

C₂×C₂₈₃⋊C₈

—

C₂×C₂₈₃⋊C₈

—

C₂×C₂₈₃⋊C₈

—

C₂²×C₄×C₂₈₃

C₂³

yes

characteristic

M₁₆×C₂₈₃

—

C₂×C₈×C₂₈₃

D₄

yes

—

C₄⋊C₄×C₂₈₃

—

C₂²⋊C₄×C₂₈₃

—

C₄²×C₂₈₃

D₄

yes

—

C₂²×D₂₈₃

—

Dic₅₆₆

—

D₁₁₃₂

—

C₂²⋊D₂₈₃

—

C₂×Dic₂₈₃

—

C₄×D₂₈₃

—

C₄×D₂₈₃

—

C₂₈₃⋊C₈

—

C₂₈₃⋊C₈

—

C₂³×C₂₈₃

#41

yes

characteristic

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₈₃

#41

yes

—

C₈×C₂₈₃

—

C₂²×C₂₈₃

—

C₂²×C₂₈₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₈₃

#123

yes

—

C₂×D₂₈₃

—

C₂×D₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

C₂²⋊Q₈

yes

—

C₄×C₂₈₃

yes

C₄×C₂₈₃

yes

Dic₂₈₃

—

C₂×C₂₈₃

—

C₂×C₂₈₃

—

C₂×C₂₈₃

#399

yes

characteristic

D₂₈₃

—

C₂₈₃

#2098

yes

characteristic, complemented, Sylow

#2098

—

283

maximal, Sylow

#430

—

283

—

#424

—

283

—

#421

—

283

—

#408

—

283

—

H(3,ℤ/4)

D₂₈₃

yes

characteristic, complemented

C₂×#41

—

283

—

C₂×M₁₆

—

283

—

C₂×M₁₆

C₂×D₂₈₃

yes

characteristic

C₂²×C₈

—

283

—

#117

C₂×D₂₈₃

yes

—

#108

—

566

—

C₄⋊C₈

—

566

—

#104

—

566

—

C₂²⋊C₈

—

566

—

#41

—

566

—

#41

—

566

—

C₂×Q₈

—

283

—

C₂×D₄

—

283

—

C₂×D₄

—

566

—

C₂²×C₄

—

566

—

C₂²×C₄

C₂²×D₂₈₃

yes

characteristic

M₁₆

—

M₁₆

—

566

—

M₁₆

—

566

—

C₂×C₈

—

566

—

C₂×C₈

—

566

—

C₂×C₈

—

566

—

C₂×C₈

D₁₁₃₂

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₁₁₃₂

yes

—

C₂³

—

566

—

C₂³

#17985391

yes

characteristic

Q₈

—

566

—

D₄

—

566

—

D₄

—

1132

—

100

C₂×C₄

—

101

C₂×C₄

—

566

—

102

C₂×C₄

—

1132

—

103

C₂×C₄

—

1132

—

104

C₂×C₄

C₂×D₁₁₃₂

yes

—

105

C₂×C₄

C₂×D₁₁₃₂

yes

characteristic

106

C₂×C₄

Q₈×D₂₈₃

yes

characteristic

107

C₂×C₄

#17985391

yes

—

108

C₈

—

109

C₈

—

1132

—

110

C₈

—

1132

—

111

C₂²

—

112

C₂²

—

1132

—

113

C₂²

—

1132

—

114

C₂²

#22583590

yes

characteristic

115

C₂²

#22583594

yes

—

116

C₄

—

117

C₄

—

1132

—

118

C₄

#22583590

yes

—

119

C₄

#22583594

yes

characteristic

120

C₄

#22583607

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

2264

—

124

C₂

#31826402

yes

characteristic

	subgroup	G/H
center	C₄	#22583607
commutator, cosocle	C₂²×C₄×C₂₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₈₃
Fitting	H(3,ℤ/4)×C₂₈₃	C₂
socle	C₂×C₂₈₃	#399
prime radical	C₂×#17985392	C₂²
hypercenter	H(3,ℤ/4)	D₂₈₃
hypocenter	C₂₈₃	#2098
LCS	G ▷ C₂²×C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#22583607

commutator, cosocle

C₂²×C₄×C₂₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₈₃

Fitting

H(3,ℤ/4)×C₂₈₃

C₂

C₂×C₂₈₃

C₂²

D₂₈₃

C₂₈₃

G ▷ C₂²×C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #31826514	C₂	semidirect
2. #31826492	C₂	semidirect
5. H(3,ℤ/4)×C₂₈₃	C₂	semidirect
62. C₂₈₃	#2098	semidirect
68. H(3,ℤ/4)	D₂₈₃	semidirect

H ≅ G/N

type

1. #31826514

C₂

semidirect

2. #31826492

C₂

semidirect

5. H(3,ℤ/4)×C₂₈₃

C₂

semidirect

62. C₂₈₃

#2098

semidirect

68. H(3,ℤ/4)

D₂₈₃

semidirect

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

subgroup	index
#31826514	2
#31826492	2
#31826483	2
#31826433	2
H(3,ℤ/4)×C₂₈₃	2
#2098	283

subgroup

index

H(3,ℤ/4)×C₂₈₃

283

Subgroups — #49122815

Conjugacy classes of subgroups

Complements

Maximal subgroups