Group #49094980 — subgroups

Subgroups — #49094980

Order 35968.

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	#31823868	C₂	1	1	yes	maximal, characteristic
2	#31823868	C₂	1	1	yes	maximal, characteristic
3	#31823861	C₂	1	1	yes	maximal, characteristic
4	#31823843	C₂	1	1	yes	maximal, characteristic
5	#31823814	C₂	1	1	yes	maximal, characteristic
6	#31823814	C₂	1	1	yes	maximal, characteristic
7	#322×C₂₈₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₁₂₄	—	2	2	no	—
9	#22582879	C₂²	1	1	yes	characteristic
10	#22582879	C₂²	1	1	yes	characteristic
11	#22582879	C₂²	1	1	yes	characteristic
12	#22582879	C₂²	1	1	yes	characteristic
13	#22582813	—	2	2	no	—
14	#22582813	—	2	2	no	—
15	#22582811	—	2	2	no	—
16	#22582804	—	2	2	no	—
17	#22582804	—	2	2	no	—
18	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₆₂	—	2	1	no	—
22	C₂×Dic₅₆₂	—	2	1	no	—
23	C₂×Dic₅₆₂	—	2	1	no	—
24	C₂×Dic₅₆₂	—	2	1	no	—
25	Dic₁₁₂₄	—	4	4	no	—
26	C₄⋊Dic₂₈₁	—	2	1	no	—
27	C₄⋊Dic₂₈₁	—	2	1	no	—
28	C₄⋊Dic₂₈₁	—	2	1	no	—
29	C₄⋊Dic₂₈₁	—	2	1	no	—
30	C₄⋊Dic₂₈₁	—	2	1	no	—
31	C₄⋊Dic₂₈₁	—	2	1	no	—
32	C₄⋊Dic₂₈₁	—	2	1	no	—
33	C₄⋊Dic₂₈₁	—	2	1	no	—
34	C₂×C₈×C₂₈₁	D₄	1	2	yes	—
35	C₂×C₈×C₂₈₁	D₄	1	2	yes	—
36	C₂×C₈×C₂₈₁	D₄	1	2	yes	—
37	C₄²×C₂₈₁	C₂³	1	1	yes	characteristic
38	Dic₅₆₂	—	4	2	no	—
39	Dic₅₆₂	—	4	2	no	—
40	Dic₅₆₂	—	4	2	no	—
41	Dic₅₆₂	—	4	2	no	—
42	C₂×Dic₂₈₁	—	4	1	no	—
43	C₂×Dic₂₈₁	—	4	1	no	—
44	C₂×Dic₂₈₁	—	4	1	no	—
45	C₂×Dic₂₈₁	—	4	1	no	—
46	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₈₁	—	2	2	no	—
50	C₈×C₂₈₁	—	2	2	no	—
51	C₈×C₂₈₁	Q₁₆	1	4	yes	complemented
52	C₂²×C₂₈₁	#127	1	1	yes	characteristic
53	C₄×C₂₈₁	C₂×Q₁₆	1	2	yes	—
54	C₄×C₂₈₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₂₈₁	#137	1	2	yes	—
56	Dic₂₈₁	—	8	1	no	—
57	Dic₂₈₁	—	8	1	no	—
58	Dic₂₈₁	—	8	1	no	—
59	Dic₂₈₁	—	8	1	no	—
60	C₂×C₂₈₁	#494	1	1	yes	characteristic
61	C₂×C₂₈₁	#496	1	1	yes	characteristic
62	C₂×C₂₈₁	#497	1	1	yes	characteristic
63	C₂₈₁	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	281	1	no	maximal, Sylow
65	#500	—	281	1	no	—
66	#500	—	281	1	no	—
67	#498	—	281	1	no	—
68	#494	—	281	1	no	—
69	#487	—	281	1	no	—
70	#487	—	281	1	no	—
71	#322	D₂₈₁	1	1	yes	characteristic
72	C₂×Q₁₆	—	562	2	no	—
73	C₄⋊Q₈	—	281	1	no	—
74	C₄⋊Q₈	—	281	1	no	—
75	C₄⋊Q₈	—	281	1	no	—
76	C₄⋊Q₈	—	281	1	no	—
77	#107	—	562	2	no	—
78	#107	—	562	2	no	—
79	#106	—	562	2	no	—
80	#103	—	562	2	no	—
81	#103	—	562	2	no	—
82	C₄×C₈	C₂×D₂₈₁	1	1	yes	characteristic
83	C₄×C₈	C₂×D₂₈₁	1	1	yes	characteristic
84	C₄×C₈	C₂×D₂₈₁	1	1	yes	characteristic
85	C₂×Q₈	—	562	1	no	—
86	C₂×Q₈	—	562	1	no	—
87	C₂×Q₈	—	562	1	no	—
88	C₂×Q₈	—	562	1	no	—
89	Q₁₆	—	1124	4	no	—
90	C₂×C₈	D₁₁₂₄	1	2	yes	—
91	C₂×C₈	D₁₁₂₄	1	2	yes	—
92	C₂×C₈	D₁₁₂₄	1	2	yes	—
93	C₄⋊C₄	—	562	1	no	—
94	C₄⋊C₄	—	562	1	no	—
95	C₄⋊C₄	—	562	1	no	—
96	C₄⋊C₄	—	562	1	no	—
97	C₄⋊C₄	—	562	1	no	—
98	C₄⋊C₄	—	562	1	no	—
99	C₄⋊C₄	—	562	1	no	—
100	C₄⋊C₄	—	562	1	no	—
101	C₄²	C₂²×D₂₈₁	1	1	yes	characteristic
102	Q₈	—	1124	2	no	—
103	Q₈	—	1124	2	no	—
104	Q₈	—	1124	2	no	—
105	Q₈	—	1124	2	no	—
106	C₂×C₄	—	1124	1	no	—
107	C₂×C₄	—	1124	1	no	—
108	C₂×C₄	—	1124	1	no	—
109	C₂×C₄	—	1124	1	no	—
110	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
111	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
112	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₁₁₂₄	1	4	yes	complemented
116	C₂²	#22582876	1	1	yes	characteristic
117	C₄	—	2248	1	no	—
118	C₄	—	2248	1	no	—
119	C₄	—	2248	1	no	—
120	C₄	—	2248	1	no	—
121	C₄	C₂×Dic₁₁₂₄	1	2	yes	—
122	C₄	#22582907	1	2	yes	—
123	C₄	#22582912	1	2	yes	—
124	C₂	#31823843	1	1	yes	characteristic
125	C₂	#31823852	1	1	yes	characteristic
126	C₂	#31823857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823868

C₂

yes

maximal, characteristic

#31823868

C₂

yes

maximal, characteristic

#31823861

C₂

yes

maximal, characteristic

#31823843

C₂

yes

maximal, characteristic

#31823814

C₂

yes

maximal, characteristic

#31823814

C₂

yes

maximal, characteristic

#322×C₂₈₁

C₂

yes

maximal, characteristic

C₂×Dic₁₁₂₄

—

#22582879

C₂²

yes

characteristic

#22582879

C₂²

yes

characteristic

#22582879

C₂²

yes

characteristic

#22582879

C₂²

yes

—

—

—

—

—

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

C₂×Dic₅₆₂

—

C₂×Dic₅₆₂

—

C₂×Dic₅₆₂

—

C₂×Dic₅₆₂

—

Dic₁₁₂₄

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₂×C₈×C₂₈₁

D₄

yes

—

C₂×C₈×C₂₈₁

D₄

yes

—

C₂×C₈×C₂₈₁

D₄

yes

—

C₄²×C₂₈₁

C₂³

yes

characteristic

Dic₅₆₂

—

Dic₅₆₂

—

Dic₅₆₂

—

Dic₅₆₂

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₈×C₂₈₁

—

C₈×C₂₈₁

—

C₈×C₂₈₁

Q₁₆

yes

complemented

C₂²×C₂₈₁

#127

yes

characteristic

C₄×C₂₈₁

C₂×Q₁₆

yes

—

C₄×C₂₈₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₈₁

#137

yes

—

Dic₂₈₁

—

Dic₂₈₁

—

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂₈₁

yes

characteristic, complemented, Sylow

#1723

—

281

maximal, Sylow

#500

—

281

—

#500

—

281

—

#498

—

281

—

#494

—

281

—

#487

—

281

—

#487

—

281

—

#322

D₂₈₁

yes

characteristic

C₂×Q₁₆

—

562

—

C₄⋊Q₈

—

281

—

C₄⋊Q₈

—

281

—

C₄⋊Q₈

—

281

—

C₄⋊Q₈

—

281

—

#107

—

562

—

#107

—

562

—

#106

—

562

—

#103

—

562

—

#103

—

562

—

C₄×C₈

C₂×D₂₈₁

yes

characteristic

C₄×C₈

C₂×D₂₈₁

yes

characteristic

C₄×C₈

C₂×D₂₈₁

yes

characteristic

C₂×Q₈

—

562

—

C₂×Q₈

—

562

—

C₂×Q₈

—

562

—

C₂×Q₈

—

562

—

Q₁₆

—

1124

—

C₂×C₈

D₁₁₂₄

yes

—

C₂×C₈

D₁₁₂₄

yes

—

C₂×C₈

D₁₁₂₄

yes

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

100

C₄⋊C₄

—

562

—

101

C₄²

C₂²×D₂₈₁

yes

characteristic

102

Q₈

—

1124

—

103

Q₈

—

1124

—

104

Q₈

—

1124

—

105

Q₈

—

1124

—

106

C₂×C₄

—

1124

—

107

C₂×C₄

—

1124

—

108

C₂×C₄

—

1124

—

109

C₂×C₄

—

1124

—

110

C₂×C₄

C₂×D₁₁₂₄

yes

characteristic

111

C₂×C₄

C₂×D₁₁₂₄

yes

characteristic

112

C₂×C₄

C₂×D₁₁₂₄

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₁₁₂₄

yes

complemented

116

C₂²

#22582876

yes

characteristic

117

C₄

—

2248

—

118

C₄

—

2248

—

119

C₄

—

2248

—

120

C₄

—

2248

—

121

C₄

C₂×Dic₁₁₂₄

yes

—

122

C₄

#22582907

yes

—

123

C₄

#22582912

yes

—

124

C₂

#31823843

yes

characteristic

125

C₂

#31823852

yes

characteristic

126

C₂

#31823857

yes

characteristic

	subgroup	G/H
center	C₂²	#22582876
commutator, cosocle	C₄²×C₂₈₁	C₂³
Frattini	C₄²	C₂²×D₂₈₁
Fitting	#322×C₂₈₁	C₂
socle, prime radical	C₂²×C₂₈₁	#127
hypercenter	#322	D₂₈₁
hypocenter	C₂₈₁	#1723
LCS	G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#22582876

commutator, cosocle

C₄²×C₂₈₁

C₂³

Frattini

C₄²

C₂²×D₂₈₁

Fitting

#322×C₂₈₁

C₂

C₂²×C₂₈₁

D₂₈₁

C₂₈₁

G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
51. C₈×C₂₈₁	Q₁₆	semidirect
63. C₂₈₁	#1723	semidirect
115. C₈	Dic₁₁₂₄	semidirect

H ≅ G/N

type

51. C₈×C₂₈₁

Q₁₆

semidirect

63. C₂₈₁

#1723

semidirect

115. C₈

Dic₁₁₂₄

semidirect

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

subgroup	index
#31823868	2
#31823861	2
#31823843	2
#31823814	2
#322×C₂₈₁	2
#1723	281

subgroup

index

#322×C₂₈₁

281

Subgroups — #49094980

Conjugacy classes of subgroups

Complements

Maximal subgroups