Group #46814065 — subgroups

Subgroups — #46814065

Order 34432.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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	#31770509	C₂	1	4	yes	maximal
3	#31770490	C₂	1	4	yes	maximal
4	#31770488	C₂	1	4	yes	maximal
5	#31770473	C₂	1	1	yes	maximal, characteristic
6	#383×C₂₆₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₃₈	C₂²	1	2	yes	—
8	#22575453	C₄	1	8	yes	complemented
9	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
11	C₂×#16864381	C₂²	1	4	yes	—
12	C₂×C₄×Dic₂₆₉	C₂²	1	4	yes	—
13	#22575360	C₂²	1	2	yes	—
14	#22575360	C₂²	1	2	yes	—
15	#22575360	C₂²	1	2	yes	—
16	#22575360	C₂²	1	8	yes	—
17	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
18	#95×C₂₆₉	C₂²	1	2	yes	—
19	#95×C₂₆₉	C₂²	1	2	yes	—
20	#95×C₂₆₉	C₂²	1	2	yes	—
21	C₂×Dic₅₃₈	—	4	2	no	—
22	C₂×Dic₅₃₈	—	4	2	no	—
23	C₂×Dic₅₃₈	C₂×C₄	1	8	yes	—
24	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
25	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
26	C₄⋊Dic₂₆₉	C₂×C₄	1	8	yes	—
27	C₄⋊Dic₂₆₉	C₂×C₄	1	8	yes	—
28	#16864381	—	4	4	no	—
29	C₄×Dic₂₆₉	—	4	4	no	—
30	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
34	C₄²×C₂₆₉	C₂×C₄	1	4	yes	—
35	Dic₅₃₈	—	4	8	no	—
36	C₂×Dic₂₆₉	—	2	4	no	—
37	C₂×Dic₂₆₉	—	2	4	no	—
38	C₂×Dic₂₆₉	—	2	4	no	—
39	C₂×Dic₂₆₉	—	2	4	no	—
40	C₂×Dic₂₆₉	—	2	4	no	—
41	C₂×Dic₂₆₉	C₂²×C₄	1	8	yes	—
42	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
43	C₂×C₄×C₂₆₉	—	2	2	no	—
44	C₂×C₄×C₂₆₉	—	2	2	no	—
45	C₂×C₄×C₂₆₉	—	2	2	no	—
46	C₂×C₄×C₂₆₉	—	2	4	no	—
47	C₂×C₄×C₂₆₉	—	2	4	no	—
48	C₂×C₄×C₂₆₉	—	2	4	no	—
49	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
52	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
54	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
55	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
56	C₄×C₂₆₉	—	2	2	no	—
57	C₄×C₂₆₉	—	2	2	no	—
58	C₄×C₂₆₉	—	2	2	no	—
59	C₄×C₂₆₉	—	4	4	no	—
60	Dic₂₆₉	—	2	8	no	—
61	Dic₂₆₉	—	4	4	no	—
62	C₂×C₂₆₉	#518	1	1	yes	characteristic
63	C₂×C₂₆₉	#519	1	1	yes	characteristic
64	C₂×C₂₆₉	#519	1	1	yes	characteristic
65	C₂×C₂₆₉	#563	1	4	yes	—
66	C₂₆₉	#2381	1	1	yes	characteristic, complemented, Sylow
67	#2381	—	269	1	no	maximal, Sylow
68	C₂×C₄⋊Q₈	—	269	1	no	—
69	#391	—	269	4	no	—
70	#387	—	269	4	no	—
71	#387	—	269	4	no	—
72	#383	—	269	1	no	—
73	#383	D₂₆₉	1	1	yes	characteristic
74	C₂²×Q₈	—	269	2	no	—
75	C₄⋊Q₈	—	269	8	no	—
76	C₂×C₄⋊C₄	—	269	2	no	—
77	C₂×C₄⋊C₄	—	269	2	no	—
78	C₂×C₄⋊C₄	—	269	4	no	—
79	C₂×C₄²	—	269	4	no	—
80	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
81	#95	—	269	2	no	—
82	#95	—	269	2	no	—
83	#95	—	269	2	no	—
84	#95	—	269	8	no	—
85	#95	C₂×D₂₆₉	1	2	yes	—
86	#95	C₂×D₂₆₉	1	2	yes	—
87	#95	C₂×D₂₆₉	1	2	yes	—
88	C₂×Q₈	—	269	8	no	—
89	C₂×Q₈	—	1076	2	no	—
90	C₂×Q₈	—	1076	2	no	—
91	C₂²×C₄	—	269	4	no	—
92	C₂²×C₄	—	269	4	no	—
93	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
94	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
97	C₄⋊C₄	—	269	8	no	—
98	C₄⋊C₄	—	269	8	no	—
99	C₄⋊C₄	—	1076	4	no	—
100	C₄²	—	1076	4	no	—
101	C₄²	C₄×D₂₆₉	1	4	yes	—
102	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
103	Q₈	—	1076	8	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	269	8	no	—
111	C₂×C₄	—	538	4	no	—
112	C₂×C₄	—	538	4	no	—
113	C₂×C₄	—	538	4	no	—
114	C₂×C₄	—	538	4	no	—
115	C₂×C₄	—	538	4	no	—
116	C₂×C₄	C₂×C₄×D₂₆₉	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₂₆₉	1	2	yes	—
118	C₂×C₄	C₂×C₄×D₂₆₉	1	2	yes	—
119	C₂²	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
120	C₂²	#22575499	1	2	yes	—
121	C₂²	#22575500	1	2	yes	—
122	C₂²	#22575500	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	538	8	no	—
128	C₄	—	1076	4	no	—
129	C₂	#31771079	1	1	yes	characteristic
130	C₂	#31771082	1	1	yes	characteristic
131	C₂	#31771082	1	1	yes	characteristic
132	C₂	#31771357	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575453

C₂

yes

maximal, characteristic

#31770509

C₂

yes

maximal

#31770490

C₂

yes

maximal

#31770488

C₂

yes

maximal

#31770473

C₂

yes

maximal, characteristic

#383×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₃₈

C₂²

yes

—

#22575453

C₄

yes

complemented

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×#16864381

C₂²

yes

—

C₂×C₄×Dic₂₆₉

C₂²

yes

—

#22575360

C₂²

yes

—

#22575360

C₂²

yes

—

#22575360

C₂²

yes

—

#22575360

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

#95×C₂₆₉

C₂²

yes

—

#95×C₂₆₉

C₂²

yes

—

#95×C₂₆₉

C₂²

yes

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

C₂×C₄

yes

—

C₂²×Dic₂₆₉

C₂³

yes

—

C₂²×Dic₂₆₉

C₂³

yes

—

C₄⋊Dic₂₆₉

C₂×C₄

yes

—

C₄⋊Dic₂₆₉

C₂×C₄

yes

—

#16864381

—

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

C₂³

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₄²×C₂₆₉

C₂×C₄

yes

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₂²×C₄

yes

—

C₂³×C₂₆₉

C₂⁴

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂²×C₂₆₉

C₂³×C₄

yes

characteristic

C₂²×C₂₆₉

H(5,𝔽₂)

yes

—

C₂²×C₂₆₉

2₋¹⁺⁴

yes

—

C₂²×C₂₆₉

2₋¹⁺⁴

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

—

C₂₆₉

#2381

yes

characteristic, complemented, Sylow

#2381

—

269

maximal, Sylow

C₂×C₄⋊Q₈

—

269

—

#391

—

269

—

#387

—

269

—

#387

—

269

—

#383

—

269

—

#383

D₂₆₉

yes

characteristic

C₂²×Q₈

—

269

—

C₄⋊Q₈

—

269

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄²

—

269

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#95

—

269

—

#95

—

269

—

#95

—

269

—

#95

—

269

—

#95

C₂×D₂₆₉

yes

—

#95

C₂×D₂₆₉

yes

—

#95

C₂×D₂₆₉

yes

—

C₂×Q₈

—

269

—

C₂×Q₈

—

1076

—

C₂×Q₈

—

1076

—

C₂²×C₄

—

269

—

C₂²×C₄

—

269

—

C₂²×C₄

C₂²×D₂₆₉

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

1076

—

100

C₄²

—

1076

—

101

C₄²

C₄×D₂₆₉

yes

—

102

C₂³

C₂³×D₂₆₉

yes

characteristic

103

Q₈

—

1076

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

269

—

111

C₂×C₄

—

538

—

112

C₂×C₄

—

538

—

113

C₂×C₄

—

538

—

114

C₂×C₄

—

538

—

115

C₂×C₄

—

538

—

116

C₂×C₄

C₂×C₄×D₂₆₉

yes

—

117

C₂×C₄

C₂×C₄×D₂₆₉

yes

—

118

C₂×C₄

C₂×C₄×D₂₆₉

yes

—

119

C₂²

C₂²×C₄×D₂₆₉

yes

characteristic

120

C₂²

#22575499

yes

—

121

C₂²

#22575500

yes

—

122

C₂²

#22575500

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

538

—

128

C₄

—

1076

—

129

C₂

#31771079

yes

characteristic

130

C₂

#31771082

yes

characteristic

131

C₂

#31771082

yes

characteristic

132

C₂

#31771357

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₆₉
commutator	C₂²×C₂₆₉	C₂³×C₄
Fitting	#383×C₂₆₉	C₂
socle, prime radical, cosocle	C₂³×C₂₆₉	C₂⁴
hypercenter	#383	D₂₆₉
hypocenter	C₂₆₉	#2381
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₆₉

commutator

C₂²×C₂₆₉

C₂³×C₄

Fitting

#383×C₂₆₉

C₂

socle, prime radical, cosocle

C₂³×C₂₆₉

C₂⁴

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
8. #22575453	C₄	semidirect
66. C₂₆₉	#2381	semidirect

H ≅ G/N

type

8. #22575453

C₄

semidirect

66. C₂₆₉

#2381

semidirect

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

subgroup	index
C₂×#22575453	2
#31770509	2
#31770490	2
#31770488	2
#31770473	2
#383×C₂₆₉	2
#2381	269

subgroup

index

#383×C₂₆₉

269

Subgroups — #46814065

Conjugacy classes of subgroups

Complements

Maximal subgroups