Group #46819367 — subgroups

Subgroups — #46819367

Order 34432.

704 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 249 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×#22575500	C₂	1	1	yes	maximal, characteristic
2	#31771372	C₂	1	1	yes	maximal, characteristic, complemented
3	#31770587	C₂	1	6	yes	maximal, complemented
4	C₂×#22575371	C₂	1	6	yes	maximal, complemented
5	#568×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575500	—	2	4	no	—
7	#22575500	—	2	4	no	—
8	C₂×#16864402	C₂²	1	3	yes	—
9	C₂×#16864401	C₄	1	6	yes	—
10	C₂²×Dic₅₃₈	C₂²	1	3	yes	—
11	C₂×Q₈×D₂₆₉	C₄	1	2	yes	—
12	#22575460	—	2	4	no	—
13	C₂×#16864386	C₂²	1	6	yes	—
14	#22575371	C₂²	1	12	yes	—
15	#22575369	C₂²	1	4	yes	—
16	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic
17	#131×C₂₆₉	—	2	4	no	—
18	C₂×M₁₆×C₂₆₉	C₂²	1	6	yes	—
19	#16864402	—	2	6	no	—
20	#16864402	—	2	6	no	—
21	#16864401	—	2	24	no	—
22	C₂×Dic₅₃₈	—	2	6	no	—
23	C₂×Dic₅₃₈	—	2	6	no	—
24	C₂×Dic₅₃₈	C₂×C₄	1	6	yes	—
25	C₂×Dic₅₃₈	C₂³	1	3	yes	—
26	Q₈×D₂₆₉	—	2	8	no	—
27	C₂×D₁₀₇₆	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₂₆₉	C₂×C₄	1	6	yes	—
29	C₂²×Dic₂₆₉	C₂×C₄	1	6	yes	—
30	C₂×C₄×D₂₆₉	C₂×C₄	1	6	yes	—
31	#16864386	—	2	6	no	—
32	#16864386	C₂³	1	4	yes	—
33	C₂×#14524130	—	2	6	no	—
34	#41×C₂₆₉	D₄	1	4	yes	—
35	#41×C₂₆₉	D₄	1	4	yes	—
36	C₂×Q₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₂₆₉	C₂×C₄	1	3	yes	—
38	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
39	M₁₆×C₂₆₉	—	2	6	no	—
40	M₁₆×C₂₆₉	C₂³	1	4	yes	—
41	C₂×C₈×C₂₆₉	—	2	6	no	—
42	C₂²×D₂₆₉	C₂²×C₄	1	2	yes	—
43	Dic₅₃₈	—	2	12	no	—
44	Dic₅₃₈	—	2	3	no	—
45	Dic₅₃₈	—	2	3	no	—
46	D₁₀₇₆	—	2	1	no	—
47	D₁₀₇₆	—	2	1	no	—
48	C₂²⋊D₂₆₉	—	2	12	no	—
49	C₂×Dic₂₆₉	—	2	12	no	—
50	C₂×Dic₂₆₉	C₂²×C₄	1	6	yes	—
51	C₄×D₂₆₉	—	2	12	no	—
52	#14524130	—	2	4	no	—
53	C₂³×C₂₆₉	C₂²×C₄	1	3	yes	—
54	Q₈×C₂₆₉	C₂²⋊C₄	1	4	yes	—
55	D₄×C₂₆₉	C₂²⋊C₄	1	12	yes	—
56	C₂×C₄×C₂₆₉	C₂²×C₄	1	3	yes	—
57	C₂×C₄×C₂₆₉	C₂×D₄	1	6	yes	—
58	C₂×C₄×C₂₆₉	C₂×D₄	1	6	yes	—
59	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
60	C₈×C₂₆₉	—	2	4	no	—
61	C₂²×C₂₆₉	—	4	3	no	—
62	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	6	yes	—
63	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
64	C₂×D₂₆₉	—	2	2	no	—
65	C₂×D₂₆₉	—	4	2	no	—
66	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	6	yes	—
67	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
69	Dic₂₆₉	—	2	6	no	—
70	C₂×C₂₆₉	—	2	1	no	—
71	C₂×C₂₆₉	—	2	6	no	—
72	C₂×C₂₆₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
73	D₂₆₉	—	4	2	no	—
74	C₂₆₉	#2889	1	1	yes	characteristic, complemented, Sylow
75	#2889	—	269	1	no	maximal, Sylow
76	C₂×2₋¹⁺⁴	—	269	1	no	—
77	#568	—	269	1	no	—
78	#568	D₂₆₉	1	1	yes	characteristic, complemented
79	#413	—	269	6	no	—
80	C₂×#101	—	269	6	no	—
81	2₋¹⁺⁴	—	538	4	no	—
82	2₋¹⁺⁴	—	538	4	no	—
83	C₂×#41	—	269	6	no	—
84	C₂×#41	—	269	3	no	—
85	C₂×#41	C₂×D₂₆₉	1	1	yes	characteristic
86	C₂²×Q₈	—	269	2	no	—
87	C₂²×Q₈	—	269	3	no	—
88	#131	—	2	4	no	—
89	#131	—	538	4	no	—
90	C₂×M₁₆	—	269	6	no	—
91	C₂×M₁₆	C₂×D₂₆₉	1	6	yes	—
92	#101	—	269	12	no	—
93	#100	—	269	4	no	—
94	#41	—	538	24	no	—
95	#41	—	538	6	no	—
96	#41	—	538	6	no	—
97	#41	C₂²⋊D₂₆₉	1	4	yes	—
98	#41	D₁₀₇₆	1	4	yes	—
99	C₂×Q₈	—	269	6	no	—
100	C₂×Q₈	—	269	3	no	—
101	C₂×Q₈	—	538	8	no	—
102	C₂×Q₈	—	538	6	no	—
103	C₂×Q₈	—	538	6	no	—
104	C₂×Q₈	C₄×D₂₆₉	1	1	yes	characteristic
105	C₂×D₄	—	269	6	no	—
106	C₂×D₄	—	269	1	no	—
107	C₂×D₄	C₄×D₂₆₉	1	3	yes	—
108	C₂²×C₄	—	269	6	no	—
109	C₂²×C₄	—	269	6	no	—
110	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
111	M₁₆	—	2	6	no	—
112	M₁₆	—	269	4	no	—
113	M₁₆	—	538	6	no	—
114	M₁₆	C₂²×D₂₆₉	1	4	yes	—
115	C₂×C₈	—	2	6	no	—
116	C₂×C₈	—	538	6	no	—
117	C₂³	—	269	2	no	—
118	C₂³	C₂×C₄×D₂₆₉	1	3	yes	—
119	Q₈	—	538	12	no	—
120	Q₈	—	538	3	no	—
121	Q₈	—	538	3	no	—
122	Q₈	#16864379	1	4	yes	—
123	D₄	—	538	12	no	—
124	D₄	—	538	1	no	—
125	D₄	—	538	1	no	—
126	D₄	#16864379	1	12	yes	—
127	C₂×C₄	—	269	6	no	—
128	C₂×C₄	—	538	12	no	—
129	C₂×C₄	—	538	12	no	—
130	C₂×C₄	C₂×C₄×D₂₆₉	1	3	yes	—
131	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	6	yes	—
132	C₂×C₄	C₂×D₁₀₇₆	1	6	yes	—
133	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
134	C₈	—	2	4	no	—
135	C₈	—	538	4	no	—
136	C₂²	—	4	3	no	—
137	C₂²	—	538	2	no	—
138	C₂²	—	1076	2	no	—
139	C₂²	C₂×#16864379	1	6	yes	—
140	C₂²	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
141	C₄	—	538	6	no	—
142	C₄	C₂×#16864379	1	6	yes	—
143	C₄	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic
144	C₄	C₂²×D₁₀₇₆	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	6	no	—
147	C₂	—	1076	2	no	—
148	C₂	C₂²×#16864379	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575500

C₂

yes

maximal, characteristic

#31771372

C₂

yes

maximal, characteristic, complemented

#31770587

C₂

yes

maximal, complemented

C₂×#22575371

C₂

yes

maximal, complemented

#568×C₂₆₉

C₂

yes

maximal, characteristic, complemented

#22575500

—

#22575500

—

C₂×#16864402

C₂²

yes

—

C₂×#16864401

C₄

yes

—

C₂²×Dic₅₃₈

C₂²

yes

—

C₂×Q₈×D₂₆₉

C₄

yes

—

#22575460

—

C₂×#16864386

C₂²

yes

—

#22575371

C₂²

yes

—

#22575369

C₂²

yes

—

C₂×#41×C₂₆₉

C₂²

yes

characteristic

#131×C₂₆₉

—

C₂×M₁₆×C₂₆₉

C₂²

yes

—

#16864402

—

#16864402

—

#16864401

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

C₂×C₄

yes

—

C₂×Dic₅₃₈

C₂³

yes

—

Q₈×D₂₆₉

—

C₂×D₁₀₇₆

C₂³

yes

characteristic

C₂×C₂²⋊D₂₆₉

C₂×C₄

yes

—

C₂²×Dic₂₆₉

C₂×C₄

yes

—

C₂×C₄×D₂₆₉

C₂×C₄

yes

—

#16864386

—

#16864386

C₂³

yes

—

C₂×#14524130

—

#41×C₂₆₉

D₄

yes

—

#41×C₂₆₉

D₄

yes

—

C₂×Q₈×C₂₆₉

C₂×C₄

yes

characteristic

C₂×D₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

—

M₁₆×C₂₆₉

—

M₁₆×C₂₆₉

C₂³

yes

—

C₂×C₈×C₂₆₉

—

C₂²×D₂₆₉

C₂²×C₄

yes

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

D₁₀₇₆

—

D₁₀₇₆

—

C₂²⋊D₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₂²×C₄

yes

—

C₄×D₂₆₉

—

#14524130

—

C₂³×C₂₆₉

C₂²×C₄

yes

—

Q₈×C₂₆₉

C₂²⋊C₄

yes

—

D₄×C₂₆₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₆₉

C₂³×C₄

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₄×C₂₆₉

C₂×C₂²⋊C₄

yes

—

C₄×C₂₆₉

C₂²×D₄

yes

characteristic

C₄×C₂₆₉

C₂²×D₄

yes

characteristic

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂²×C₂²⋊C₄

yes

characteristic

D₂₆₉

—

C₂₆₉

#2889

yes

characteristic, complemented, Sylow

#2889

—

269

maximal, Sylow

C₂×2₋¹⁺⁴

—

269

—

#568

—

269

—

#568

D₂₆₉

yes

characteristic, complemented

#413

—

269

—

C₂×#101

—

269

—

2₋¹⁺⁴

—

538

—

2₋¹⁺⁴

—

538

—

C₂×#41

—

269

—

C₂×#41

—

269

—

C₂×#41

C₂×D₂₆₉

yes

characteristic

C₂²×Q₈

—

269

—

C₂²×Q₈

—

269

—

#131

—

#131

—

538

—

C₂×M₁₆

—

269

—

C₂×M₁₆

C₂×D₂₆₉

yes

—

#101

—

269

—

#100

—

269

—

#41

—

538

—

#41

—

538

—

#41

—

538

—

#41

C₂²⋊D₂₆₉

yes

—

#41

D₁₀₇₆

yes

—

C₂×Q₈

—

269

—

100

C₂×Q₈

—

269

—

101

C₂×Q₈

—

538

—

102

C₂×Q₈

—

538

—

103

C₂×Q₈

—

538

—

104

C₂×Q₈

C₄×D₂₆₉

yes

characteristic

105

C₂×D₄

—

269

—

106

C₂×D₄

—

269

—

107

C₂×D₄

C₄×D₂₆₉

yes

—

108

C₂²×C₄

—

269

—

109

C₂²×C₄

—

269

—

110

C₂²×C₄

C₂²×D₂₆₉

yes

—

111

M₁₆

—

112

M₁₆

—

269

—

113

M₁₆

—

538

—

114

M₁₆

C₂²×D₂₆₉

yes

—

115

C₂×C₈

—

116

C₂×C₈

—

538

—

117

C₂³

—

269

—

118

C₂³

C₂×C₄×D₂₆₉

yes

—

119

Q₈

—

538

—

120

Q₈

—

538

—

121

Q₈

—

538

—

122

Q₈

#16864379

yes

—

123

D₄

—

538

—

124

D₄

—

538

—

125

D₄

—

538

—

126

D₄

#16864379

yes

—

127

C₂×C₄

—

269

—

128

C₂×C₄

—

538

—

129

C₂×C₄

—

538

—

130

C₂×C₄

C₂×C₄×D₂₆₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

132

C₂×C₄

C₂×D₁₀₇₆

yes

—

133

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

134

C₈

—

135

C₈

—

538

—

136

C₂²

—

137

C₂²

—

538

—

138

C₂²

—

1076

—

139

C₂²

C₂×#16864379

yes

—

140

C₂²

C₂²×C₄×D₂₆₉

yes

characteristic

141

C₄

—

538

—

142

C₄

C₂×#16864379

yes

—

143

C₄

C₂²×C₂²⋊D₂₆₉

yes

characteristic

144

C₄

C₂²×D₁₀₇₆

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

1076

—

148

C₂

C₂²×#16864379

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#16864379
commutator	C₂²×C₂₆₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#568×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₂²×C₂²⋊C₄
prime radical	C₂×#22575500	C₂
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	#568	D₂₆₉
hypocenter	C₂₆₉	#2889
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₂

C₂²×#16864379

commutator

C₂²×C₂₆₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

#568×C₂₆₉

C₂

socle

C₂×C₂₆₉

C₂²×C₂²⋊C₄

prime radical

C₂×#22575500

C₂

cosocle

C₂×C₄×C₂₆₉

C₂⁴

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
2. #31771372	C₂	semidirect
3. #31770587	C₂	semidirect
4. C₂×#22575371	C₂	semidirect
5. #568×C₂₆₉	C₂	semidirect
74. C₂₆₉	#2889	semidirect
78. #568	D₂₆₉	semidirect

H ≅ G/N

type

2. #31771372

C₂

semidirect

3. #31770587

C₂

semidirect

4. C₂×#22575371

C₂

semidirect

5. #568×C₂₆₉

C₂

semidirect

74. C₂₆₉

#2889

semidirect

78. #568

D₂₆₉

semidirect

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

subgroup	index
C₂×#22575500	2
#31771372	2
#31770587	2
C₂×#22575371	2
#568×C₂₆₉	2
#2889	269

subgroup

index

#568×C₂₆₉

269

Subgroups — #46819367

Conjugacy classes of subgroups

Complements

Maximal subgroups