Group #46811836 — subgroups

Subgroups — #46811836

Order 34432.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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×#22575453	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
3	#31770368	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₃₈	—	2	1	no	—
6	#22575453	—	2	2	no	—
7	#22575453	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₂₆₉	—	2	1	no	—
9	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₆₉	—	2	1	no	—
12	#22575360	—	2	1	no	—
13	C₂²×C₈×C₂₆₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₂₆₉	—	2	2	no	—
16	C₂×Dic₅₃₈	—	2	4	no	—
17	C₂×Dic₅₃₈	—	4	2	no	—
18	C₂×Dic₅₃₈	—	4	2	no	—
19	C₂²×Dic₂₆₉	—	2	2	no	—
20	C₂²×Dic₂₆₉	—	2	2	no	—
21	C₄⋊Dic₂₆₉	—	2	4	no	—
22	C₄⋊Dic₂₆₉	—	2	2	no	—
23	C₄⋊Dic₂₆₉	—	2	2	no	—
24	C₄⋊Dic₂₆₉	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₂₆₉	Q₈	1	4	yes	complemented
26	C₄×Dic₂₆₉	—	4	2	no	—
27	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₂₆₉	—	2	2	no	—
31	C₂×C₈×C₂₆₉	—	2	4	no	—
32	C₄²×C₂₆₉	C₂×C₄	1	2	yes	—
33	C₄²×C₂₆₉	D₄	1	2	yes	—
34	Dic₅₃₈	—	4	8	no	—
35	C₂×Dic₂₆₉	—	2	4	no	—
36	C₂×Dic₂₆₉	—	2	4	no	—
37	C₂×Dic₂₆₉	—	4	2	no	—
38	C₂×Dic₂₆₉	—	4	2	no	—
39	C₂×Dic₂₆₉	—	4	2	no	—
40	C₂×Dic₂₆₉	—	4	2	no	—
41	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₆₉	—	2	2	no	—
43	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₆₉	D₈	1	2	yes	—
46	C₂×C₄×C₂₆₉	SD₁₆	1	2	yes	—
47	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₂₆₉	#41	1	2	yes	—
51	C₈×C₂₆₉	—	4	2	no	—
52	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₂₆₉	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₂₆₉	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₂₆₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₂₆₉	#137	1	2	yes	—
57	C₄×C₂₆₉	—	2	2	no	—
58	C₄×C₂₆₉	#102	1	4	yes	—
59	C₄×C₂₆₉	C₄×Q₈	1	2	yes	—
60	C₄×C₂₆₉	#124	1	1	yes	characteristic
61	C₄×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₂₆₉	—	4	4	no	—
63	Dic₂₆₉	—	4	4	no	—
64	C₂×C₂₆₉	#391	1	1	yes	characteristic
65	C₂×C₂₆₉	C₂×#102	1	1	yes	characteristic
66	C₂×C₂₆₉	#419	1	1	yes	characteristic
67	C₂×C₂₆₉	#474	1	2	yes	—
68	C₂×C₂₆₉	#475	1	2	yes	—
69	C₂₆₉	#1988	1	1	yes	characteristic, complemented, Sylow
70	#1988	—	269	1	no	maximal, Sylow
71	C₂×C₄⋊Q₈	—	269	1	no	—
72	C₂×C₄⋊C₈	D₂₆₉	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	269	1	no	—
74	#340	—	269	4	no	—
75	C₂²×Q₈	—	538	1	no	—
76	C₂²×C₈	C₂×D₂₆₉	1	2	yes	—
77	C₄⋊Q₈	—	269	4	no	—
78	C₄⋊Q₈	—	538	2	no	—
79	C₂×C₄⋊C₄	—	269	2	no	—
80	C₂×C₄⋊C₄	—	269	2	no	—
81	C₂×C₄⋊C₄	—	538	1	no	—
82	C₂×C₄²	—	538	1	no	—
83	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
84	C₄⋊C₈	—	2	2	no	—
85	#95	—	538	1	no	—
86	C₂×Q₈	—	538	4	no	—
87	C₂×Q₈	—	1076	2	no	—
88	C₂×Q₈	—	1076	2	no	—
89	C₂²×C₄	—	538	2	no	—
90	C₂²×C₄	—	538	2	no	—
91	C₂²×C₄	D₁₀₇₆	1	1	yes	characteristic
92	C₂²×C₄	D₁₀₇₆	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₄⋊C₄	—	269	4	no	—
97	C₄⋊C₄	—	269	4	no	—
98	C₄⋊C₄	—	538	4	no	—
99	C₄⋊C₄	—	538	2	no	—
100	C₄⋊C₄	—	538	2	no	—
101	C₄²	—	1076	2	no	—
102	C₄²	C₄×D₂₆₉	1	2	yes	—
103	C₄²	C₂²⋊D₂₆₉	1	2	yes	—
104	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
105	Q₈	—	1076	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	538	4	no	—
108	C₂×C₄	—	538	4	no	—
109	C₂×C₄	—	1076	2	no	—
110	C₂×C₄	—	1076	2	no	—
111	C₂×C₄	—	1076	2	no	—
112	C₂×C₄	—	1076	2	no	—
113	C₂×C₄	#16864379	1	2	yes	—
114	C₂×C₄	#16864379	1	2	yes	—
115	C₂×C₄	D₂₁₅₂	1	2	yes	—
116	C₂×C₄	#16864391	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₂₆₉	1	2	yes	—
120	C₂×C₄	#16864401	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₂²	C₂×#16864379	1	1	yes	characteristic
123	C₂²	#22575433	1	2	yes	—
124	C₂²	C₂×D₂₁₅₂	1	1	yes	characteristic
125	C₂²	C₂×#16864391	1	1	yes	characteristic
126	C₂²	#22575486	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	1076	4	no	—
129	C₄	—	1076	4	no	—
130	C₄	#22575375	1	4	yes	—
131	C₄	#22575418	1	2	yes	—
132	C₄	#22575440	1	1	yes	characteristic
133	C₄	#22575451	1	1	yes	characteristic
134	C₂	#31770510	1	1	yes	characteristic
135	C₂	C₂×#22575375	1	1	yes	characteristic
136	C₂	#31770615	1	1	yes	characteristic
137	C₂	#31770886	1	2	yes	—
138	C₂	#31770891	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575453

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₂₆₉

C₂

yes

maximal, characteristic

#31770368

C₂

yes

maximal

C₂×C₄⋊C₈×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₃₈

—

#22575453

—

#22575453

C₄

yes

complemented

C₂×C₄⋊Dic₂₆₉

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×C₄×Dic₂₆₉

—

#22575360

—

C₂²×C₈×C₂₆₉

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₉

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂²×Dic₂₆₉

—

C₂²×Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂×C₄

yes

—

C₄⋊Dic₂₆₉

Q₈

yes

complemented

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₄²×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

D₄

yes

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₆₉

D₈

yes

—

C₂×C₄×C₂₆₉

SD₁₆

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

—

C₂×C₄×C₂₆₉

#41

yes

—

C₈×C₂₆₉

—

C₂²×C₂₆₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₆₉

C₂²⋊Q₈

yes

—

C₂²×C₂₆₉

C₂×D₈

yes

characteristic

C₂²×C₂₆₉

C₂×SD₁₆

yes

characteristic

C₂²×C₂₆₉

#137

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#102

yes

—

C₄×C₂₆₉

C₄×Q₈

yes

—

C₄×C₂₆₉

#124

yes

characteristic

C₄×C₂₆₉

C₄⋊Q₈

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

—

C₂×C₂₆₉

#475

yes

—

C₂₆₉

#1988

yes

characteristic, complemented, Sylow

#1988

—

269

maximal, Sylow

C₂×C₄⋊Q₈

—

269

—

C₂×C₄⋊C₈

D₂₆₉

yes

characteristic

C₄×C₄⋊C₄

—

269

—

#340

—

269

—

C₂²×Q₈

—

538

—

C₂²×C₈

C₂×D₂₆₉

yes

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

—

538

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄²

—

538

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

C₄⋊C₈

—

#95

—

538

—

C₂×Q₈

—

538

—

C₂×Q₈

—

1076

—

C₂×Q₈

—

1076

—

C₂²×C₄

—

538

—

C₂²×C₄

—

538

—

C₂²×C₄

D₁₀₇₆

yes

characteristic

C₂²×C₄

D₁₀₇₆

yes

characteristic

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

538

—

100

C₄⋊C₄

—

538

—

101

C₄²

—

1076

—

102

C₄²

C₄×D₂₆₉

yes

—

103

C₄²

C₂²⋊D₂₆₉

yes

—

104

C₂³

C₂×D₁₀₇₆

yes

characteristic

105

Q₈

—

1076

—

106

C₂×C₄

—

107

C₂×C₄

—

538

—

108

C₂×C₄

—

538

—

109

C₂×C₄

—

1076

—

110

C₂×C₄

—

1076

—

111

C₂×C₄

—

1076

—

112

C₂×C₄

—

1076

—

113

C₂×C₄

#16864379

yes

—

114

C₂×C₄

#16864379

yes

—

115

C₂×C₄

D₂₁₅₂

yes

—

116

C₂×C₄

#16864391

yes

—

117

C₂×C₄

C₂×C₄×D₂₆₉

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

119

C₂×C₄

Q₈×D₂₆₉

yes

—

120

C₂×C₄

#16864401

yes

—

121

C₈

—

122

C₂²

C₂×#16864379

yes

characteristic

123

C₂²

#22575433

yes

—

124

C₂²

C₂×D₂₁₅₂

yes

characteristic

125

C₂²

C₂×#16864391

yes

characteristic

126

C₂²

#22575486

yes

—

127

C₄

—

128

C₄

—

1076

—

129

C₄

—

1076

—

130

C₄

#22575375

yes

—

131

C₄

#22575418

yes

—

132

C₄

#22575440

yes

characteristic

133

C₄

#22575451

yes

characteristic

134

C₂

#31770510

yes

characteristic

135

C₂

C₂×#22575375

yes

characteristic

136

C₂

#31770615

yes

characteristic

137

C₂

#31770886

yes

—

138

C₂

#31770891

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₀₇₆
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	C₂×C₄⋊C₈×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂×D₄
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₆₉
hypocenter	C₂₆₉	#1988
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₀₇₆

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

C₂×C₄⋊C₈×C₂₆₉

C₂

socle, prime radical

C₂³×C₂₆₉

C₂×D₄

cosocle

C₂²×C₄×C₂₆₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₆₉

hypocenter

C₂₆₉

#1988

LCS

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #22575453	C₄	semidirect
25. C₄⋊Dic₂₆₉	Q₈	semidirect
69. C₂₆₉	#1988	semidirect

H ≅ G/N

type

7. #22575453

C₄

semidirect

25. C₄⋊Dic₂₆₉

Q₈

semidirect

69. C₂₆₉

#1988

semidirect

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

subgroup	index
C₂×#22575453	2
C₄×C₄⋊Dic₂₆₉	2
#31770368	2
C₂×C₄⋊C₈×C₂₆₉	2
#1988	269

subgroup

index

C₂×#22575453

C₄×C₄⋊Dic₂₆₉

#31770368

C₂×C₄⋊C₈×C₂₆₉

#1988

269

Subgroups — #46811836

Conjugacy classes of subgroups

Complements

Maximal subgroups