Group #46810523 — subgroups

Subgroups — #46810523

Order 34432.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#31770977	C₂	1	1	yes	maximal, characteristic, complemented
2	#31770956	C₂	1	1	yes	maximal, characteristic, complemented
3	#31770951	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770951	C₂	1	1	yes	maximal, characteristic, complemented
5	#31770951	C₂	1	1	yes	maximal, characteristic, complemented
6	#31770327	C₂	1	1	yes	maximal, characteristic, complemented
7	#493×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	#22575378	—	2	2	no	—
12	#22575374	—	2	2	no	—
13	#22575374	—	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×D₈×C₂₆₉	—	2	2	no	—
18	#127×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×D₄×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	C₂×D₄×C₂₆₉	—	2	1	no	—
35	C₂×D₄×C₂₆₉	—	2	1	no	—
36	D₈×C₂₆₉	—	4	4	no	—
37	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
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	—	2	2	no	—
45	C₂³×C₂₆₉	—	4	1	no	—
46	C₂³×C₂₆₉	—	4	1	no	—
47	D₄×C₂₆₉	—	4	2	no	—
48	D₄×C₂₆₉	—	4	2	no	—
49	D₄×C₂₆₉	—	4	2	no	—
50	D₄×C₂₆₉	—	4	2	no	—
51	D₄×C₂₆₉	—	4	2	no	—
52	D₄×C₂₆₉	—	4	2	no	—
53	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
56	C₈×C₂₆₉	—	2	2	no	—
57	C₂²×C₂₆₉	—	8	1	no	—
58	C₂²×C₂₆₉	—	8	1	no	—
59	C₂²×C₂₆₉	—	8	1	no	—
60	C₂²×C₂₆₉	—	8	1	no	—
61	C₂²×C₂₆₉	—	8	1	no	—
62	C₂²×C₂₆₉	—	8	1	no	—
63	C₂²×C₂₆₉	#127	1	1	yes	characteristic
64	C₄×C₂₆₉	#135	1	2	yes	—
65	C₄×C₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
66	C₄×C₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
67	Dic₂₆₉	—	8	1	no	—
68	Dic₂₆₉	—	8	1	no	—
69	C₂×C₂₆₉	—	16	1	no	—
70	C₂×C₂₆₉	—	16	1	no	—
71	C₂×C₂₆₉	#495	1	1	yes	characteristic
72	C₂×C₂₆₉	#496	1	1	yes	characteristic
73	C₂×C₂₆₉	#496	1	1	yes	characteristic
74	C₂₆₉	#1717	1	1	yes	characteristic, complemented, Sylow
75	#1717	—	269	1	no	maximal, Sylow
76	#493	D₂₆₉	1	1	yes	characteristic
77	#492	—	269	1	no	—
78	#487	—	269	1	no	—
79	#486	—	269	1	no	—
80	#486	—	269	1	no	—
81	#486	—	269	1	no	—
82	#322	—	269	1	no	—
83	C₂×SD₁₆	—	538	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	269	1	no	—
86	C₄⋊Q₈	—	269	1	no	—
87	#127	C₂×D₂₆₉	1	1	yes	characteristic
88	#127	C₂×D₂₆₉	1	1	yes	characteristic
89	#103	—	538	2	no	—
90	#102	—	538	2	no	—
91	#102	—	538	2	no	—
92	#102	—	538	2	no	—
93	C₄×C₈	—	269	1	no	—
94	C₄×C₈	—	269	1	no	—
95	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
96	C₂×Q₈	—	538	1	no	—
97	C₂×Q₈	—	538	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	SD₁₆	—	1076	4	no	—
105	D₈	—	4	4	no	—
106	C₂×C₈	—	269	2	no	—
107	C₂×C₈	—	269	2	no	—
108	C₂×C₈	C₂²⋊D₂₆₉	1	2	yes	—
109	C₄⋊C₄	—	538	1	no	—
110	C₄⋊C₄	—	538	1	no	—
111	C₄⋊C₄	—	538	1	no	—
112	C₄⋊C₄	—	538	1	no	—
113	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
114	C₂³	—	4	1	no	—
115	C₂³	—	4	1	no	—
116	Q₈	—	1076	2	no	—
117	Q₈	—	1076	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	4	2	no	—
124	C₂×C₄	—	1076	1	no	—
125	C₂×C₄	—	1076	1	no	—
126	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
127	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	538	2	no	—
131	C₈	—	538	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	#22575449	1	1	yes	characteristic
139	C₄	—	2152	1	no	—
140	C₄	—	2152	1	no	—
141	C₄	#22575472	1	2	yes	—
142	C₄	#22575477	1	2	yes	—
143	C₄	#22575479	1	2	yes	—
144	C₂	—	16	1	no	—
145	C₂	—	16	1	no	—
146	C₂	#31770988	1	1	yes	characteristic
147	C₂	#31770992	1	1	yes	characteristic
148	C₂	#31770993	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31770977

C₂

yes

maximal, characteristic, complemented

#31770956

C₂

yes

maximal, characteristic, complemented

#31770951

C₂

yes

maximal, characteristic, complemented

#31770951

C₂

yes

maximal, characteristic, complemented

#31770951

C₂

yes

maximal, characteristic, complemented

#31770327

C₂

yes

maximal, characteristic, complemented

#493×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864390

—

#22575453

C₂²

yes

characteristic

#22575453

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#14524130

C₂²

yes

characteristic

C₂×D₈×C₂₆₉

—

#127×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×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

D₈×C₂₆₉

—

C₂×C₈×C₂₆₉

D₄

yes

—

C₄²×C₂₆₉

C₂³

yes

characteristic

Dic₅₃₈

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

#14524130

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

D₄×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

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

#127

yes

characteristic

C₄×C₂₆₉

#135

yes

—

C₄×C₂₆₉

AGL(1,ℤ/8)

yes

—

C₄×C₂₆₉

AGL(1,ℤ/8)

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#1717

—

269

maximal, Sylow

#493

D₂₆₉

yes

characteristic

#492

—

269

—

#487

—

269

—

#486

—

269

—

#486

—

269

—

#486

—

269

—

#322

—

269

—

C₂×SD₁₆

—

538

—

C₂×D₈

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

269

—

#127

C₂×D₂₆₉

yes

characteristic

#127

C₂×D₂₆₉

yes

characteristic

#103

—

538

—

#102

—

538

—

#102

—

538

—

#102

—

538

—

C₄×C₈

—

269

—

C₄×C₈

—

269

—

C₄×C₈

C₂×D₂₆₉

yes

characteristic

C₂×Q₈

—

538

—

C₂×Q₈

—

538

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

SD₁₆

—

1076

—

105

D₈

—

106

C₂×C₈

—

269

—

107

C₂×C₈

—

269

—

108

C₂×C₈

C₂²⋊D₂₆₉

yes

—

109

C₄⋊C₄

—

538

—

110

C₄⋊C₄

—

538

—

111

C₄⋊C₄

—

538

—

112

C₄⋊C₄

—

538

—

113

C₄²

C₂²×D₂₆₉

yes

characteristic

114

C₂³

—

115

C₂³

—

116

Q₈

—

1076

—

117

Q₈

—

1076

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

1076

—

125

C₂×C₄

—

1076

—

126

C₂×C₄

D₄×D₂₆₉

yes

characteristic

127

C₂×C₄

D₄×D₂₆₉

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

129

C₈

—

130

C₈

—

538

—

131

C₈

—

538

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#22575449

yes

characteristic

139

C₄

—

2152

—

140

C₄

—

2152

—

141

C₄

#22575472

yes

—

142

C₄

#22575477

yes

—

143

C₄

#22575479

yes

—

144

C₂

—

145

C₂

—

146

C₂

#31770988

yes

characteristic

147

C₂

#31770992

yes

characteristic

148

C₂

#31770993

yes

characteristic

	subgroup	G/H
center	C₂²	#22575449
commutator, cosocle	C₄²×C₂₆₉	C₂³
Frattini	C₄²	C₂²×D₂₆₉
Fitting, prime radical	#493×C₂₆₉	C₂
socle	C₂²×C₂₆₉	#127
hypercenter	#493	D₂₆₉
hypocenter	C₂₆₉	#1717
LCS	G ▷ C₄²×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#22575449

commutator, cosocle

C₄²×C₂₆₉

C₂³

Frattini

C₄²

C₂²×D₂₆₉

Fitting, prime radical

#493×C₂₆₉

C₂

C₂²×C₂₆₉

D₂₆₉

C₂₆₉

G ▷ C₄²×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #31770977	C₂	semidirect
2. #31770956	C₂	semidirect
3. #31770951	C₂	semidirect
4. #31770951	C₂	semidirect
5. #31770951	C₂	semidirect
6. #31770327	C₂	semidirect
74. C₂₆₉	#1717	semidirect

H ≅ G/N

type

1. #31770977

C₂

semidirect

2. #31770956

C₂

semidirect

3. #31770951

C₂

semidirect

4. #31770951

C₂

semidirect

5. #31770951

C₂

semidirect

6. #31770327

C₂

semidirect

74. C₂₆₉

#1717

semidirect

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

subgroup	index
#31770977	2
#31770956	2
#31770951	2
#31770327	2
#493×C₂₆₉	2
#1717	269

subgroup

index

#493×C₂₆₉

269

Subgroups — #46810523

Conjugacy classes of subgroups

Complements

Maximal subgroups