Group #46810505 — subgroups

Subgroups — #46810505

Order 34432.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#31771008	C₂	1	1	yes	maximal, characteristic, complemented
2	#31771001	C₂	1	1	yes	maximal, characteristic, complemented
3	#31770977	C₂	1	1	yes	maximal, characteristic
4	#31770956	C₂	1	1	yes	maximal, characteristic, complemented
5	#31770951	C₂	1	1	yes	maximal, characteristic
6	#31770327	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#16864390	—	2	2	no	—
9	#22575453	C₂²	1	1	yes	characteristic
10	#22575453	C₂²	1	1	yes	characteristic
11	#22575386	—	2	2	no	—
12	#22575384	—	2	2	no	—
13	#22575378	—	2	2	no	—
14	#22575374	—	2	2	no	—
15	C₄×#14524130	C₂²	1	1	yes	characteristic
16	C₄×#14524130	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₂₆₉	—	2	2	no	—
18	C₄⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
19	#127×C₂₆₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₃₈	—	2	1	no	—
22	C₂×Dic₅₃₈	—	2	1	no	—
23	#16864390	—	4	4	no	—
24	C₂×#14524130	D₄	1	2	yes	—
25	C₂×#14524130	D₄	1	2	yes	—
26	C₄⋊Dic₂₆₉	—	2	1	no	—
27	C₄⋊Dic₂₆₉	—	2	1	no	—
28	C₄⋊Dic₂₆₉	—	2	1	no	—
29	C₄⋊Dic₂₆₉	—	2	1	no	—
30	C₂×Q₈×C₂₆₉	—	2	1	no	—
31	C₂×D₄×C₂₆₉	—	2	1	no	—
32	C₂×D₄×C₂₆₉	—	2	1	no	—
33	C₂×D₄×C₂₆₉	—	2	1	no	—
34	SD₁₆×C₂₆₉	—	4	4	no	—
35	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
36	C₄⋊C₄×C₂₆₉	—	2	1	no	—
37	C₄⋊C₄×C₂₆₉	—	2	1	no	—
38	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
39	Dic₅₃₈	—	4	2	no	—
40	Dic₅₃₈	—	4	2	no	—
41	C₂×Dic₂₆₉	—	4	1	no	—
42	C₂×Dic₂₆₉	—	4	1	no	—
43	#14524130	—	2	2	no	—
44	#14524130	SD₁₆	1	4	yes	complemented
45	C₂³×C₂₆₉	—	4	1	no	—
46	Q₈×C₂₆₉	—	4	2	no	—
47	D₄×C₂₆₉	—	4	2	no	—
48	D₄×C₂₆₉	—	4	2	no	—
49	D₄×C₂₆₉	—	4	2	no	—
50	C₂×C₄×C₂₆₉	—	4	1	no	—
51	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
54	C₈×C₂₆₉	—	2	2	no	—
55	C₂²×C₂₆₉	—	8	1	no	—
56	C₂²×C₂₆₉	—	8	1	no	—
57	C₂²×C₂₆₉	—	8	1	no	—
58	C₂²×C₂₆₉	#127	1	1	yes	characteristic
59	C₄×C₂₆₉	—	8	1	no	—
60	C₄×C₂₆₉	C₂×SD₁₆	1	2	yes	—
61	C₄×C₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₂₆₉	#137	1	2	yes	—
63	Dic₂₆₉	—	8	1	no	—
64	Dic₂₆₉	—	8	1	no	—
65	C₂×C₂₆₉	—	16	1	no	—
66	C₂×C₂₆₉	#492	1	1	yes	characteristic
67	C₂×C₂₆₉	#496	1	1	yes	characteristic
68	C₂×C₂₆₉	#497	1	1	yes	characteristic
69	C₂₆₉	#1714	1	1	yes	characteristic, complemented, Sylow
70	#1714	—	269	1	no	maximal, Sylow
71	#500	—	269	1	no	—
72	#498	—	269	1	no	—
73	#492	—	269	1	no	—
74	#492	D₂₆₉	1	1	yes	characteristic
75	#487	—	269	1	no	—
76	#486	—	269	1	no	—
77	#322	—	269	1	no	—
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×SD₁₆	—	538	2	no	—
80	C₄⋊Q₈	—	269	1	no	—
81	C₄⋊Q₈	—	269	1	no	—
82	C₄⋊Q₈	C₂×D₂₆₉	1	1	yes	characteristic
83	#127	C₂×D₂₆₉	1	1	yes	characteristic
84	#107	—	538	2	no	—
85	#106	—	538	2	no	—
86	#103	—	538	2	no	—
87	#102	—	538	2	no	—
88	C₄×C₈	—	269	1	no	—
89	C₄×C₈	—	269	1	no	—
90	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	538	1	no	—
93	C₂×Q₈	—	538	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	4	4	no	—
98	SD₁₆	—	1076	4	no	—
99	C₂×C₈	—	269	2	no	—
100	C₂×C₈	—	269	2	no	—
101	C₂×C₈	C₂²⋊D₂₆₉	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	538	1	no	—
105	C₄⋊C₄	—	538	1	no	—
106	C₄⋊C₄	—	538	1	no	—
107	C₄⋊C₄	—	538	1	no	—
108	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	1076	2	no	—
112	Q₈	—	1076	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	1076	1	no	—
118	C₂×C₄	—	1076	1	no	—
119	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
120	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	269	4	no	—
124	C₈	—	538	2	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#22575449	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	2152	1	no	—
131	C₄	—	2152	1	no	—
132	C₄	SD₁₆×D₂₆₉	1	2	yes	—
133	C₄	#22575479	1	2	yes	—
134	C₄	#22575482	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#31770976	1	1	yes	characteristic
137	C₂	#31770992	1	1	yes	characteristic
138	C₂	#31770996	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31771008

C₂

yes

maximal, characteristic, complemented

#31771001

C₂

yes

maximal, characteristic, complemented

#31770977

C₂

yes

maximal, characteristic

#31770956

C₂

yes

maximal, characteristic, complemented

#31770951

C₂

yes

maximal, characteristic

#31770327

C₂

yes

maximal, characteristic, complemented

#492×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864390

—

#22575453

C₂²

yes

characteristic

#22575453

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#14524130

C₂²

yes

characteristic

C₂×SD₁₆×C₂₆₉

—

C₄⋊Q₈×C₂₆₉

C₂²

yes

characteristic

#127×C₂₆₉

C₂²

yes

characteristic

C₄×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

#16864390

—

C₂×#14524130

D₄

yes

—

C₂×#14524130

D₄

yes

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₂×Q₈×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

SD₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

D₄

yes

—

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

C₂³

yes

characteristic

Dic₅₃₈

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

#14524130

SD₁₆

yes

complemented

C₂³×C₂₆₉

—

Q₈×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

#127

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×SD₁₆

yes

—

C₄×C₂₆₉

AGL(1,ℤ/8)

yes

—

C₄×C₂₆₉

#137

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#1714

—

269

maximal, Sylow

#500

—

269

—

#498

—

269

—

#492

—

269

—

#492

D₂₆₉

yes

characteristic

#487

—

269

—

#486

—

269

—

#322

—

269

—

C₂×SD₁₆

—

C₂×SD₁₆

—

538

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

C₂×D₂₆₉

yes

characteristic

#127

C₂×D₂₆₉

yes

characteristic

#107

—

538

—

#106

—

538

—

#103

—

538

—

#102

—

538

—

C₄×C₈

—

269

—

C₄×C₈

—

269

—

C₄×C₈

C₂×D₂₆₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

538

—

C₂×Q₈

—

538

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

1076

—

C₂×C₈

—

269

—

100

C₂×C₈

—

269

—

101

C₂×C₈

C₂²⋊D₂₆₉

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

538

—

105

C₄⋊C₄

—

538

—

106

C₄⋊C₄

—

538

—

107

C₄⋊C₄

—

538

—

108

C₄²

C₂²×D₂₆₉

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

1076

—

112

Q₈

—

1076

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

1076

—

118

C₂×C₄

—

1076

—

119

C₂×C₄

D₄×D₂₆₉

yes

characteristic

120

C₂×C₄

D₄×D₂₆₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

122

C₈

—

123

C₈

—

269

—

124

C₈

—

538

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#22575449

yes

characteristic

129

C₄

—

130

C₄

—

2152

—

131

C₄

—

2152

—

132

C₄

SD₁₆×D₂₆₉

yes

—

133

C₄

#22575479

yes

—

134

C₄

#22575482

yes

—

135

C₂

—

136

C₂

#31770976

yes

characteristic

137

C₂

#31770992

yes

characteristic

138

C₂

#31770996

yes

characteristic

	subgroup	G/H
center	C₂²	#22575449
commutator, cosocle	C₄²×C₂₆₉	C₂³
Frattini	C₄²	C₂²×D₂₆₉
Fitting	#492×C₂₆₉	C₂
socle	C₂²×C₂₆₉	#127
prime radical	#127×C₂₆₉	C₂²
hypercenter	#492	D₂₆₉
hypocenter	C₂₆₉	#1714
LCS	G ▷ C₄²×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#22575449

commutator, cosocle

C₄²×C₂₆₉

C₂³

Frattini

C₄²

C₂²×D₂₆₉

Fitting

#492×C₂₆₉

C₂

socle

C₂²×C₂₆₉

#127

prime radical

#127×C₂₆₉

C₂²

D₂₆₉

C₂₆₉

G ▷ C₄²×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #31771008	C₂	semidirect
2. #31771001	C₂	semidirect
4. #31770956	C₂	semidirect
6. #31770327	C₂	semidirect
44. #14524130	SD₁₆	semidirect
69. C₂₆₉	#1714	semidirect

H ≅ G/N

type

1. #31771008

C₂

semidirect

2. #31771001

C₂

semidirect

4. #31770956

C₂

semidirect

6. #31770327

C₂

semidirect

44. #14524130

SD₁₆

semidirect

69. C₂₆₉

#1714

semidirect

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

subgroup	index
#31771008	2
#31771001	2
#31770977	2
#31770956	2
#31770951	2
#31770327	2
#492×C₂₆₉	2
#1714	269

subgroup

index

#492×C₂₆₉

269

Subgroups — #46810505

Conjugacy classes of subgroups

Complements

Maximal subgroups