Group #49122105 — subgroups

Subgroups — #49122105

Order 36224.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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×#22583610	C₂	1	1	yes	maximal, characteristic
2	#31826489	C₂	1	1	yes	maximal, characteristic
3	C₂×#22583528	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₂₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₆₆	C₂²	1	2	yes	—
6	C₂×#17985376	C₂²	1	2	yes	—
7	#22583610	—	2	1	no	—
8	#22583610	—	2	1	no	—
9	#22583610	—	2	2	no	—
10	C₂×C₄⋊Dic₂₈₃	C₄	1	4	yes	—
11	C₄⋊(C₂₈₃⋊C₈)	—	2	2	no	—
12	#22583528	—	2	8	no	—
13	C₂×M₁₆×C₂₈₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₈₃	C₂²	1	1	yes	characteristic
15	#97×C₂₈₃	—	2	1	no	—
16	C₄×C₈×C₂₈₃	—	2	1	no	—
17	C₂×Dic₅₆₆	—	2	2	no	—
18	C₂×Dic₅₆₆	—	2	2	no	—
19	C₂×Dic₅₆₆	—	2	4	no	—
20	C₂×Dic₅₆₆	—	2	4	no	—
21	C₂²×Dic₂₈₃	C₂×C₄	1	4	yes	—
22	#17985376	—	2	4	no	—
23	C₂×C₂₈₃⋊C₈	—	2	2	no	—
24	C₄⋊Dic₂₈₃	—	2	4	no	—
25	C₄⋊Dic₂₈₃	—	2	4	no	—
26	C₂²×C₄×C₂₈₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₈₃	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₈₃	—	2	2	no	—
29	M₁₆×C₂₈₃	D₄	1	4	yes	—
30	C₂×C₈×C₂₈₃	—	2	2	no	—
31	C₄²×C₂₈₃	D₄	1	1	yes	characteristic
32	C₄²×C₂₈₃	D₄	1	1	yes	characteristic
33	C₄²×C₂₈₃	D₄	1	2	yes	—
34	Dic₅₆₆	—	4	4	no	—
35	Dic₅₆₆	—	4	2	no	—
36	Dic₅₆₆	—	4	2	no	—
37	C₂×Dic₂₈₃	—	2	4	no	—
38	C₂×Dic₂₈₃	—	2	8	no	—
39	C₂₈₃⋊C₈	—	4	2	no	—
40	C₂³×C₂₈₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₈₃	—	2	4	no	—
42	C₂×C₄×C₂₈₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₈₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₈₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₈₃	#41	1	2	yes	—
48	C₈×C₂₈₃	—	2	4	no	—
49	C₂²×C₂₈₃	—	2	2	no	—
50	C₂²×C₂₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₂₈₃	C₄×D₄	1	2	yes	—
52	C₄×C₂₈₃	—	2	2	no	—
53	C₄×C₂₈₃	#101	1	4	yes	complemented
54	C₄×C₂₈₃	#121	1	2	yes	—
55	C₄×C₂₈₃	#124	1	1	yes	characteristic
56	C₄×C₂₈₃	#127	1	1	yes	characteristic
57	Dic₂₈₃	—	4	4	no	—
58	C₂×C₂₈₃	—	2	2	no	—
59	C₂×C₂₈₃	#390	1	1	yes	characteristic
60	C₂×C₂₈₃	C₂×#101	1	2	yes	—
61	C₂₈₃	#1966	1	1	yes	characteristic, complemented, Sylow
62	#1966	—	283	1	no	maximal, Sylow
63	C₂×C₄⋊Q₈	—	283	1	no	—
64	#423	—	283	1	no	—
65	C₂×#101	—	283	4	no	—
66	C₄×M₁₆	D₂₈₃	1	1	yes	characteristic
67	C₂²×Q₈	—	283	2	no	—
68	C₂×M₁₆	—	283	2	no	—
69	C₂×M₁₆	C₂×D₂₈₃	1	2	yes	—
70	C₄⋊Q₈	—	566	1	no	—
71	C₄⋊Q₈	—	566	1	no	—
72	C₄⋊Q₈	—	566	2	no	—
73	C₂×C₄⋊C₄	—	283	4	no	—
74	C₂×C₄²	C₂×D₂₈₃	1	1	yes	characteristic
75	C₄⋊C₈	—	566	2	no	—
76	#101	—	566	8	no	—
77	#97	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₂×Q₈	—	566	2	no	—
80	C₂×Q₈	—	566	4	no	—
81	C₂×Q₈	—	566	2	no	—
82	C₂×Q₈	—	566	4	no	—
83	C₂²×C₄	—	283	4	no	—
84	C₂²×C₄	C₄×D₂₈₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₈₃	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	566	4	no	—
88	M₁₆	D₁₁₃₂	1	4	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	566	2	no	—
91	C₄⋊C₄	—	566	4	no	—
92	C₄⋊C₄	—	566	4	no	—
93	C₄²	C₂²⋊D₂₈₃	1	1	yes	characteristic
94	C₄²	C₂²⋊D₂₈₃	1	1	yes	characteristic
95	C₄²	D₁₁₃₂	1	2	yes	—
96	C₂³	C₂×C₄×D₂₈₃	1	1	yes	characteristic
97	Q₈	—	1132	4	no	—
98	Q₈	—	1132	2	no	—
99	Q₈	—	1132	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	566	4	no	—
102	C₂×C₄	—	566	8	no	—
103	C₂×C₄	(C₂×D₂₈₃)⋊C₄	1	2	yes	—
104	C₂×C₄	(C₂×D₂₈₃)⋊C₄	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₂₈₃	1	1	yes	characteristic
106	C₂×C₄	C₂×D₁₁₃₂	1	2	yes	—
107	C₂×C₄	C₂×D₁₁₃₂	1	1	yes	characteristic
108	C₂×C₄	#17985392	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	1132	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×(C₂×D₂₈₃)⋊C₄	1	1	yes	characteristic
113	C₂²	C₄×D₁₁₃₂	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	1132	4	no	—
116	C₄	#22583528	1	4	yes	complemented
117	C₄	#22583584	1	2	yes	—
118	C₄	#22583598	1	1	yes	characteristic
119	C₄	#22583607	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#31826367	1	1	yes	characteristic
122	C₂	C₂×#22583528	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22583610

C₂

yes

maximal, characteristic

#31826489

C₂

yes

maximal, characteristic

C₂×#22583528

C₂

yes

maximal

C₄×M₁₆×C₂₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₆₆

C₂²

yes

—

C₂×#17985376

C₂²

yes

—

#22583610

—

#22583610

—

#22583610

—

C₂×C₄⋊Dic₂₈₃

C₄

yes

—

C₄⋊(C₂₈₃⋊C₈)

—

#22583528

—

C₂×M₁₆×C₂₈₃

C₂²

yes

—

C₂×C₄²×C₂₈₃

C₂²

yes

characteristic

#97×C₂₈₃

—

C₄×C₈×C₂₈₃

—

C₂×Dic₅₆₆

—

C₂×Dic₅₆₆

—

C₂×Dic₅₆₆

—

C₂×Dic₅₆₆

—

C₂²×Dic₂₈₃

C₂×C₄

yes

—

#17985376

—

C₂×C₂₈₃⋊C₈

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₂²×C₄×C₂₈₃

C₂×C₄

yes

—

C₂²×C₄×C₂₈₃

C₂³

yes

characteristic

M₁₆×C₂₈₃

—

M₁₆×C₂₈₃

D₄

yes

—

C₂×C₈×C₂₈₃

—

C₄²×C₂₈₃

D₄

yes

characteristic

C₄²×C₂₈₃

D₄

yes

characteristic

C₄²×C₂₈₃

D₄

yes

—

Dic₅₆₆

—

Dic₅₆₆

—

Dic₅₆₆

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₂₈₃⋊C₈

—

C₂³×C₂₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂×D₄

yes

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

#41

yes

—

C₈×C₂₈₃

—

C₂²×C₂₈₃

—

C₂²×C₂₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₈₃

C₄×D₄

yes

—

C₄×C₂₈₃

—

C₄×C₂₈₃

#101

yes

complemented

C₄×C₂₈₃

#121

yes

—

C₄×C₂₈₃

yes

C₄×C₂₈₃

yes

Dic₂₈₃

—

C₂×C₂₈₃

—

C₂×C₂₈₃

#390

yes

characteristic

C₂×C₂₈₃

C₂×#101

yes

—

C₂₈₃

#1966

yes

characteristic, complemented, Sylow

#1966

—

283

maximal, Sylow

C₂×C₄⋊Q₈

—

283

—

#423

—

283

—

C₂×#101

—

283

—

C₄×M₁₆

D₂₈₃

yes

characteristic

C₂²×Q₈

—

283

—

C₂×M₁₆

—

283

—

C₂×M₁₆

C₂×D₂₈₃

yes

—

C₄⋊Q₈

—

566

—

C₄⋊Q₈

—

566

—

C₄⋊Q₈

—

566

—

C₂×C₄⋊C₄

—

283

—

C₂×C₄²

C₂×D₂₈₃

yes

characteristic

C₄⋊C₈

—

566

—

#101

—

566

—

#97

—

C₄×C₈

—

C₂×Q₈

—

566

—

C₂×Q₈

—

566

—

C₂×Q₈

—

566

—

C₂×Q₈

—

566

—

C₂²×C₄

—

283

—

C₂²×C₄

C₄×D₂₈₃

yes

—

C₂²×C₄

C₂²×D₂₈₃

yes

characteristic

M₁₆

—

M₁₆

—

566

—

M₁₆

D₁₁₃₂

yes

—

C₂×C₈

—

C₂×C₈

—

566

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄²

C₂²⋊D₂₈₃

yes

characteristic

C₄²

C₂²⋊D₂₈₃

yes

characteristic

C₄²

D₁₁₃₂

yes

—

C₂³

C₂×C₄×D₂₈₃

yes

characteristic

Q₈

—

1132

—

Q₈

—

1132

—

Q₈

—

1132

—

100

C₂×C₄

—

101

C₂×C₄

—

566

—

102

C₂×C₄

—

566

—

103

C₂×C₄

(C₂×D₂₈₃)⋊C₄

yes

—

104

C₂×C₄

(C₂×D₂₈₃)⋊C₄

yes

—

105

C₂×C₄

C₂×C₂²⋊D₂₈₃

yes

characteristic

106

C₂×C₄

C₂×D₁₁₃₂

yes

—

107

C₂×C₄

C₂×D₁₁₃₂

yes

characteristic

108

C₂×C₄

#17985392

yes

—

109

C₈

—

110

C₈

—

1132

—

111

C₂²

—

112

C₂²

C₂×(C₂×D₂₈₃)⋊C₄

yes

characteristic

113

C₂²

C₄×D₁₁₃₂

yes

—

114

C₄

—

115

C₄

—

1132

—

116

C₄

#22583528

yes

complemented

117

C₄

#22583584

yes

—

118

C₄

#22583598

yes

characteristic

119

C₄

#22583607

yes

characteristic

120

C₂

—

121

C₂

#31826367

yes

characteristic

122

C₂

C₂×#22583528

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₂₈₃)⋊C₄
commutator, prime radical	C₂³×C₂₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₃
Fitting	C₄×M₁₆×C₂₈₃	C₂
socle	C₂²×C₂₈₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₈₃	C₂³
hypercenter	C₄×M₁₆	D₂₈₃
hypocenter	C₂₈₃	#1966
LCS	G ▷ C₂³×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×(C₂×D₂₈₃)⋊C₄

commutator, prime radical

C₂³×C₂₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₃

Fitting

C₄×M₁₆×C₂₈₃

C₂

socle

C₂²×C₂₈₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₈₃

C₂³

hypercenter

C₄×M₁₆

D₂₈₃

hypocenter

C₂₈₃

#1966

LCS

G ▷ C₂³×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
53. C₄×C₂₈₃	#101	semidirect
61. C₂₈₃	#1966	semidirect
116. C₄	#22583528	semidirect

H ≅ G/N

type

53. C₄×C₂₈₃

#101

semidirect

61. C₂₈₃

#1966

semidirect

116. C₄

#22583528

semidirect

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

subgroup	index
C₂×#22583610	2
#31826489	2
C₂×#22583528	2
C₄×M₁₆×C₂₈₃	2
#1966	283

subgroup

index

C₄×M₁₆×C₂₈₃

283

Subgroups — #49122105

Conjugacy classes of subgroups

Complements

Maximal subgroups