Group #49120938 — subgroups

Subgroups — #49120938

Order 36224.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#31826870	C₂	1	1	yes	maximal, characteristic
2	#31826870	C₂	1	1	yes	maximal, characteristic
3	#31826863	C₂	1	1	yes	maximal, characteristic
4	#31826845	C₂	1	1	yes	maximal, characteristic
5	#31826816	C₂	1	1	yes	maximal, characteristic
6	#31826816	C₂	1	1	yes	maximal, characteristic
7	#322×C₂₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₁₃₂	—	2	2	no	—
9	#22583610	C₂²	1	1	yes	characteristic
10	#22583610	C₂²	1	1	yes	characteristic
11	#22583610	C₂²	1	1	yes	characteristic
12	#22583610	C₂²	1	1	yes	characteristic
13	#22583544	—	2	2	no	—
14	#22583544	—	2	2	no	—
15	#22583542	—	2	2	no	—
16	#22583535	—	2	2	no	—
17	#22583535	—	2	2	no	—
18	C₄×C₈×C₂₈₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₂₈₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₂₈₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₆₆	—	2	1	no	—
22	C₂×Dic₅₆₆	—	2	1	no	—
23	C₂×Dic₅₆₆	—	2	1	no	—
24	C₂×Dic₅₆₆	—	2	1	no	—
25	Dic₁₁₃₂	—	4	4	no	—
26	C₄⋊Dic₂₈₃	—	2	1	no	—
27	C₄⋊Dic₂₈₃	—	2	1	no	—
28	C₄⋊Dic₂₈₃	—	2	1	no	—
29	C₄⋊Dic₂₈₃	—	2	1	no	—
30	C₄⋊Dic₂₈₃	—	2	1	no	—
31	C₄⋊Dic₂₈₃	—	2	1	no	—
32	C₄⋊Dic₂₈₃	—	2	1	no	—
33	C₄⋊Dic₂₈₃	—	2	1	no	—
34	C₂×C₈×C₂₈₃	D₄	1	2	yes	—
35	C₂×C₈×C₂₈₃	D₄	1	2	yes	—
36	C₂×C₈×C₂₈₃	D₄	1	2	yes	—
37	C₄²×C₂₈₃	C₂³	1	1	yes	characteristic
38	Dic₅₆₆	—	4	2	no	—
39	Dic₅₆₆	—	4	2	no	—
40	Dic₅₆₆	—	4	2	no	—
41	Dic₅₆₆	—	4	2	no	—
42	C₂×Dic₂₈₃	—	4	1	no	—
43	C₂×Dic₂₈₃	—	4	1	no	—
44	C₂×Dic₂₈₃	—	4	1	no	—
45	C₂×Dic₂₈₃	—	4	1	no	—
46	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₈₃	—	2	2	no	—
50	C₈×C₂₈₃	—	2	2	no	—
51	C₈×C₂₈₃	Q₁₆	1	4	yes	complemented
52	C₂²×C₂₈₃	#127	1	1	yes	characteristic
53	C₄×C₂₈₃	C₂×Q₁₆	1	2	yes	—
54	C₄×C₂₈₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₂₈₃	#137	1	2	yes	—
56	Dic₂₈₃	—	8	1	no	—
57	Dic₂₈₃	—	8	1	no	—
58	Dic₂₈₃	—	8	1	no	—
59	Dic₂₈₃	—	8	1	no	—
60	C₂×C₂₈₃	#494	1	1	yes	characteristic
61	C₂×C₂₈₃	#496	1	1	yes	characteristic
62	C₂×C₂₈₃	#497	1	1	yes	characteristic
63	C₂₈₃	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	283	1	no	maximal, Sylow
65	#500	—	283	1	no	—
66	#500	—	283	1	no	—
67	#498	—	283	1	no	—
68	#494	—	283	1	no	—
69	#487	—	283	1	no	—
70	#487	—	283	1	no	—
71	#322	D₂₈₃	1	1	yes	characteristic
72	C₂×Q₁₆	—	566	2	no	—
73	C₄⋊Q₈	—	283	1	no	—
74	C₄⋊Q₈	—	283	1	no	—
75	C₄⋊Q₈	—	283	1	no	—
76	C₄⋊Q₈	—	283	1	no	—
77	#107	—	566	2	no	—
78	#107	—	566	2	no	—
79	#106	—	566	2	no	—
80	#103	—	566	2	no	—
81	#103	—	566	2	no	—
82	C₄×C₈	C₂×D₂₈₃	1	1	yes	characteristic
83	C₄×C₈	C₂×D₂₈₃	1	1	yes	characteristic
84	C₄×C₈	C₂×D₂₈₃	1	1	yes	characteristic
85	C₂×Q₈	—	566	1	no	—
86	C₂×Q₈	—	566	1	no	—
87	C₂×Q₈	—	566	1	no	—
88	C₂×Q₈	—	566	1	no	—
89	Q₁₆	—	1132	4	no	—
90	C₂×C₈	D₁₁₃₂	1	2	yes	—
91	C₂×C₈	D₁₁₃₂	1	2	yes	—
92	C₂×C₈	D₁₁₃₂	1	2	yes	—
93	C₄⋊C₄	—	566	1	no	—
94	C₄⋊C₄	—	566	1	no	—
95	C₄⋊C₄	—	566	1	no	—
96	C₄⋊C₄	—	566	1	no	—
97	C₄⋊C₄	—	566	1	no	—
98	C₄⋊C₄	—	566	1	no	—
99	C₄⋊C₄	—	566	1	no	—
100	C₄⋊C₄	—	566	1	no	—
101	C₄²	C₂²×D₂₈₃	1	1	yes	characteristic
102	Q₈	—	1132	2	no	—
103	Q₈	—	1132	2	no	—
104	Q₈	—	1132	2	no	—
105	Q₈	—	1132	2	no	—
106	C₂×C₄	—	1132	1	no	—
107	C₂×C₄	—	1132	1	no	—
108	C₂×C₄	—	1132	1	no	—
109	C₂×C₄	—	1132	1	no	—
110	C₂×C₄	C₂×D₁₁₃₂	1	1	yes	characteristic
111	C₂×C₄	C₂×D₁₁₃₂	1	1	yes	characteristic
112	C₂×C₄	C₂×D₁₁₃₂	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₁₁₃₂	1	4	yes	complemented
116	C₂²	#22583607	1	1	yes	characteristic
117	C₄	—	2264	1	no	—
118	C₄	—	2264	1	no	—
119	C₄	—	2264	1	no	—
120	C₄	—	2264	1	no	—
121	C₄	C₂×Dic₁₁₃₂	1	2	yes	—
122	C₄	#22583638	1	2	yes	—
123	C₄	#22583643	1	2	yes	—
124	C₂	#31826845	1	1	yes	characteristic
125	C₂	#31826854	1	1	yes	characteristic
126	C₂	#31826859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31826870

C₂

yes

maximal, characteristic

#31826870

C₂

yes

maximal, characteristic

#31826863

C₂

yes

maximal, characteristic

#31826845

C₂

yes

maximal, characteristic

#31826816

C₂

yes

maximal, characteristic

#31826816

C₂

yes

maximal, characteristic

#322×C₂₈₃

C₂

yes

maximal, characteristic

C₂×Dic₁₁₃₂

—

#22583610

C₂²

yes

characteristic

#22583610

C₂²

yes

characteristic

#22583610

C₂²

yes

characteristic

#22583610

C₂²

yes

—

—

—

—

—

C₄×C₈×C₂₈₃

C₂²

yes

characteristic

C₄×C₈×C₂₈₃

C₂²

yes

characteristic

C₄×C₈×C₂₈₃

C₂²

yes

characteristic

C₂×Dic₅₆₆

—

C₂×Dic₅₆₆

—

C₂×Dic₅₆₆

—

C₂×Dic₅₆₆

—

Dic₁₁₃₂

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₂×C₈×C₂₈₃

D₄

yes

—

C₂×C₈×C₂₈₃

D₄

yes

—

C₂×C₈×C₂₈₃

D₄

yes

—

C₄²×C₂₈₃

C₂³

yes

characteristic

Dic₅₆₆

—

Dic₅₆₆

—

Dic₅₆₆

—

Dic₅₆₆

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₈×C₂₈₃

—

C₈×C₂₈₃

—

C₈×C₂₈₃

Q₁₆

yes

complemented

C₂²×C₂₈₃

#127

yes

characteristic

C₄×C₂₈₃

C₂×Q₁₆

yes

—

C₄×C₂₈₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₈₃

#137

yes

—

Dic₂₈₃

—

Dic₂₈₃

—

Dic₂₈₃

—

Dic₂₈₃

—

C₂×C₂₈₃

yes

C₂×C₂₈₃

yes

C₂×C₂₈₃

yes

C₂₈₃

yes

characteristic, complemented, Sylow

#1723

—

283

maximal, Sylow

#500

—

283

—

#500

—

283

—

#498

—

283

—

#494

—

283

—

#487

—

283

—

#487

—

283

—

#322

D₂₈₃

yes

characteristic

C₂×Q₁₆

—

566

—

C₄⋊Q₈

—

283

—

C₄⋊Q₈

—

283

—

C₄⋊Q₈

—

283

—

C₄⋊Q₈

—

283

—

#107

—

566

—

#107

—

566

—

#106

—

566

—

#103

—

566

—

#103

—

566

—

C₄×C₈

C₂×D₂₈₃

yes

characteristic

C₄×C₈

C₂×D₂₈₃

yes

characteristic

C₄×C₈

C₂×D₂₈₃

yes

characteristic

C₂×Q₈

—

566

—

C₂×Q₈

—

566

—

C₂×Q₈

—

566

—

C₂×Q₈

—

566

—

Q₁₆

—

1132

—

C₂×C₈

D₁₁₃₂

yes

—

C₂×C₈

D₁₁₃₂

yes

—

C₂×C₈

D₁₁₃₂

yes

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

100

C₄⋊C₄

—

566

—

101

C₄²

C₂²×D₂₈₃

yes

characteristic

102

Q₈

—

1132

—

103

Q₈

—

1132

—

104

Q₈

—

1132

—

105

Q₈

—

1132

—

106

C₂×C₄

—

1132

—

107

C₂×C₄

—

1132

—

108

C₂×C₄

—

1132

—

109

C₂×C₄

—

1132

—

110

C₂×C₄

C₂×D₁₁₃₂

yes

characteristic

111

C₂×C₄

C₂×D₁₁₃₂

yes

characteristic

112

C₂×C₄

C₂×D₁₁₃₂

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₁₁₃₂

yes

complemented

116

C₂²

#22583607

yes

characteristic

117

C₄

—

2264

—

118

C₄

—

2264

—

119

C₄

—

2264

—

120

C₄

—

2264

—

121

C₄

C₂×Dic₁₁₃₂

yes

—

122

C₄

#22583638

yes

—

123

C₄

#22583643

yes

—

124

C₂

#31826845

yes

characteristic

125

C₂

#31826854

yes

characteristic

126

C₂

#31826859

yes

characteristic

	subgroup	G/H
center	C₂²	#22583607
commutator, cosocle	C₄²×C₂₈₃	C₂³
Frattini	C₄²	C₂²×D₂₈₃
Fitting	#322×C₂₈₃	C₂
socle, prime radical	C₂²×C₂₈₃	#127
hypercenter	#322	D₂₈₃
hypocenter	C₂₈₃	#1723
LCS	G ▷ C₄²×C₂₈₃ ▷ C₂²×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#22583607

commutator, cosocle

C₄²×C₂₈₃

C₂³

Frattini

C₄²

C₂²×D₂₈₃

Fitting

#322×C₂₈₃

C₂

C₂²×C₂₈₃

D₂₈₃

C₂₈₃

G ▷ C₄²×C₂₈₃ ▷ C₂²×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
51. C₈×C₂₈₃	Q₁₆	semidirect
63. C₂₈₃	#1723	semidirect
115. C₈	Dic₁₁₃₂	semidirect

H ≅ G/N

type

51. C₈×C₂₈₃

Q₁₆

semidirect

63. C₂₈₃

#1723

semidirect

115. C₈

Dic₁₁₃₂

semidirect

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

subgroup	index
#31826870	2
#31826863	2
#31826845	2
#31826816	2
#322×C₂₈₃	2
#1723	283

subgroup

index

#322×C₂₈₃

283

Subgroups — #49120938

Conjugacy classes of subgroups

Complements

Maximal subgroups