Group #46819348 — subgroups

Subgroups — #46819348

Order 34432.

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

normal subgroups only

This group has 716 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×#22575501	C₂	1	1	yes	maximal, characteristic
2	#31771083	C₂	1	1	yes	maximal, characteristic, complemented
3	#31770578	C₂	1	12	yes	maximal, complemented
4	#519×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575501	—	2	4	no	—
6	#22575501	—	2	4	no	—
7	C₂×#16864403	C₄	1	4	yes	complemented
8	C₂×#16864402	C₂²	1	6	yes	—
9	C₂×Q₈×D₂₆₉	C₄	1	4	yes	complemented
10	Q₈×Dic₂₆₉	—	2	4	no	—
11	#22575412	C₂²	1	6	yes	—
12	#22575367	C₂²	1	16	yes	—
13	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₂₆₉	—	2	4	no	—
15	#117×C₂₆₉	C₂²	1	6	yes	—
16	#16864403	—	2	16	no	—
17	#16864402	—	2	12	no	—
18	#16864402	—	2	12	no	—
19	C₂×Dic₅₃₈	C₂×C₄	1	6	yes	—
20	Q₈×D₂₆₉	—	2	16	no	—
21	C₂×D₁₀₇₆	C₂×C₄	1	6	yes	—
22	C₂×C₂²⋊D₂₆₉	C₂³	1	4	yes	—
23	C₂×C₄×D₂₆₉	C₂×C₄	1	12	yes	—
24	C₄⋊Dic₂₆₉	—	2	6	no	—
25	C₂²⋊Dic₂₆₉	C₂³	1	4	yes	—
26	C₄×Dic₂₆₉	—	2	6	no	—
27	C₂×Q₈×C₂₆₉	C₂×C₄	1	4	yes	—
28	C₂×Q₈×C₂₆₉	D₄	1	4	yes	—
29	C₂×Q₈×C₂₆₉	D₄	1	4	yes	—
30	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
31	C₄⋊C₄×C₂₆₉	—	2	6	no	—
32	C₂²⋊C₄×C₂₆₉	C₂³	1	4	yes	—
33	C₄²×C₂₆₉	—	2	6	no	—
34	C₂²×D₂₆₉	C₂²×C₄	1	4	yes	—
35	Dic₅₃₈	—	2	12	no	—
36	D₁₀₇₆	—	2	12	no	—
37	C₂²⋊D₂₆₉	—	2	4	no	—
38	C₂²⋊D₂₆₉	—	2	4	no	—
39	C₂×Dic₂₆₉	—	2	4	no	—
40	C₂×Dic₂₆₉	C₂²×C₄	1	4	yes	—
41	C₄×D₂₆₉	—	2	24	no	—
42	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
43	Q₈×C₂₆₉	C₂²⋊C₄	1	16	yes	—
44	C₂×C₄×C₂₆₉	—	2	4	no	—
45	C₂×C₄×C₂₆₉	C₂²×C₄	1	6	yes	—
46	C₂×C₄×C₂₆₉	C₂×D₄	1	6	yes	—
47	C₂×C₄×C₂₆₉	C₂×D₄	1	6	yes	—
48	C₂²×C₂₆₉	—	4	1	no	—
49	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
50	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₂₆₉	—	2	4	no	—
53	C₂×D₂₆₉	—	4	4	no	—
54	C₄×C₂₆₉	—	4	4	no	—
55	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	12	yes	—
56	Dic₂₆₉	—	2	4	no	—
57	Dic₂₆₉	—	4	4	no	—
58	C₂×C₂₆₉	—	2	1	no	—
59	C₂×C₂₆₉	—	2	1	no	—
60	C₂×C₂₆₉	—	2	1	no	—
61	C₂×C₂₆₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
62	D₂₆₉	—	4	4	no	—
63	C₂₆₉	#2884	1	1	yes	characteristic, complemented, Sylow
64	#2884	—	269	1	no	maximal, Sylow
65	C₂×2₋¹⁺⁴	—	269	1	no	—
66	#519	—	269	1	no	—
67	#519	D₂₆₉	1	1	yes	characteristic, complemented
68	#410	—	269	12	no	—
69	2₋¹⁺⁴	—	538	4	no	—
70	2₋¹⁺⁴	—	538	4	no	—
71	C₂×#41	—	269	4	no	—
72	C₂×#41	—	269	6	no	—
73	C₂²×Q₈	—	269	4	no	—
74	C₂²×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
75	C₄×Q₈	—	2	4	no	—
76	C₄×Q₈	—	538	4	no	—
77	#117	—	269	6	no	—
78	#117	C₂×D₂₆₉	1	6	yes	—
79	#99	—	269	16	no	—
80	#41	—	538	16	no	—
81	#41	—	538	12	no	—
82	#41	—	538	12	no	—
83	C₂×Q₈	—	269	6	no	—
84	C₂×Q₈	—	538	16	no	—
85	C₂×Q₈	C₄×D₂₆₉	1	4	yes	—
86	C₂×Q₈	C₂²⋊D₂₆₉	1	4	yes	—
87	C₂×Q₈	D₁₀₇₆	1	4	yes	—
88	C₂×D₄	—	269	4	no	—
89	C₂×D₄	—	269	6	no	—
90	C₂²×C₄	—	269	12	no	—
91	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
92	C₄⋊C₄	—	2	6	no	—
93	C₄⋊C₄	—	538	6	no	—
94	C₂²⋊C₄	—	269	4	no	—
95	C₂²⋊C₄	C₂²×D₂₆₉	1	4	yes	—
96	C₄²	—	2	6	no	—
97	C₄²	—	538	6	no	—
98	C₂³	—	269	4	no	—
99	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
100	Q₈	—	538	12	no	—
101	Q₈	#16864379	1	16	yes	—
102	D₄	—	538	4	no	—
103	D₄	—	538	4	no	—
104	D₄	—	538	12	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	269	4	no	—
107	C₂×C₄	—	538	24	no	—
108	C₂×C₄	—	538	4	no	—
109	C₂×C₄	C₂×C₄×D₂₆₉	1	6	yes	—
110	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	6	yes	—
111	C₂×C₄	C₂×D₁₀₇₆	1	6	yes	—
112	C₂²	—	4	1	no	—
113	C₂²	—	538	4	no	—
114	C₂²	—	1076	4	no	—
115	C₂²	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
116	C₂²	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic
117	C₂²	C₂²×D₁₀₇₆	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	538	4	no	—
120	C₄	—	1076	4	no	—
121	C₄	C₂×#16864379	1	12	yes	—
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	1076	4	no	—
126	C₂	C₂²×#16864379	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575501

C₂

yes

maximal, characteristic

#31771083

C₂

yes

maximal, characteristic, complemented

#31770578

C₂

yes

maximal, complemented

#519×C₂₆₉

C₂

yes

maximal, characteristic, complemented

#22575501

—

#22575501

—

C₂×#16864403

C₄

yes

complemented

C₂×#16864402

C₂²

yes

—

C₂×Q₈×D₂₆₉

C₄

yes

complemented

Q₈×Dic₂₆₉

—

#22575412

C₂²

yes

—

#22575367

C₂²

yes

—

C₂²×Q₈×C₂₆₉

C₂²

yes

characteristic

C₄×Q₈×C₂₆₉

—

#117×C₂₆₉

C₂²

yes

—

#16864403

—

#16864402

—

#16864402

—

C₂×Dic₅₃₈

C₂×C₄

yes

—

Q₈×D₂₆₉

—

C₂×D₁₀₇₆

C₂×C₄

yes

—

C₂×C₂²⋊D₂₆₉

C₂³

yes

—

C₂×C₄×D₂₆₉

C₂×C₄

yes

—

C₄⋊Dic₂₆₉

—

C₂²⋊Dic₂₆₉

C₂³

yes

—

C₄×Dic₂₆₉

—

C₂×Q₈×C₂₆₉

C₂×C₄

yes

—

C₂×Q₈×C₂₆₉

D₄

yes

—

C₂×Q₈×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

—

C₄⋊C₄×C₂₆₉

—

C₂²⋊C₄×C₂₆₉

C₂³

yes

—

C₄²×C₂₆₉

—

C₂²×D₂₆₉

C₂²×C₄

yes

—

Dic₅₃₈

—

D₁₀₇₆

—

C₂²⋊D₂₆₉

—

C₂²⋊D₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₂²×C₄

yes

—

C₄×D₂₆₉

—

C₂³×C₂₆₉

C₂⁴

yes

characteristic

Q₈×C₂₆₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂³×C₄

yes

characteristic

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×C₂²⋊C₄

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂²×C₂²⋊C₄

yes

characteristic

D₂₆₉

—

C₂₆₉

#2884

yes

characteristic, complemented, Sylow

#2884

—

269

maximal, Sylow

C₂×2₋¹⁺⁴

—

269

—

#519

—

269

—

#519

D₂₆₉

yes

characteristic, complemented

#410

—

269

—

2₋¹⁺⁴

—

538

—

2₋¹⁺⁴

—

538

—

C₂×#41

—

269

—

C₂×#41

—

269

—

C₂²×Q₈

—

269

—

C₂²×Q₈

C₂×D₂₆₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

538

—

#117

—

269

—

#117

C₂×D₂₆₉

yes

—

#99

—

269

—

#41

—

538

—

#41

—

538

—

#41

—

538

—

C₂×Q₈

—

269

—

C₂×Q₈

—

538

—

C₂×Q₈

C₄×D₂₆₉

yes

—

C₂×Q₈

C₂²⋊D₂₆₉

yes

—

C₂×Q₈

D₁₀₇₆

yes

—

C₂×D₄

—

269

—

C₂×D₄

—

269

—

C₂²×C₄

—

269

—

C₂²×C₄

C₂²×D₂₆₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

538

—

C₂²⋊C₄

—

269

—

C₂²⋊C₄

C₂²×D₂₆₉

yes

—

C₄²

—

C₄²

—

538

—

C₂³

—

269

—

C₂³

C₂³×D₂₆₉

yes

characteristic

100

Q₈

—

538

—

101

Q₈

#16864379

yes

—

102

D₄

—

538

—

103

D₄

—

538

—

104

D₄

—

538

—

105

C₂×C₄

—

106

C₂×C₄

—

269

—

107

C₂×C₄

—

538

—

108

C₂×C₄

—

538

—

109

C₂×C₄

C₂×C₄×D₂₆₉

yes

—

110

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

111

C₂×C₄

C₂×D₁₀₇₆

yes

—

112

C₂²

—

113

C₂²

—

538

—

114

C₂²

—

1076

—

115

C₂²

C₂²×C₄×D₂₆₉

yes

characteristic

116

C₂²

C₂²×C₂²⋊D₂₆₉

yes

characteristic

117

C₂²

C₂²×D₁₀₇₆

yes

characteristic

118

C₄

—

119

C₄

—

538

—

120

C₄

—

1076

—

121

C₄

C₂×#16864379

yes

—

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

—

1076

—

126

C₂

C₂²×#16864379

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#16864379
commutator	C₂²×C₂₆₉	C₂³×C₄
Frattini	C₂³	C₂³×D₂₆₉
Fitting	#519×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₂²×C₂²⋊C₄
prime radical	C₂×#22575501	C₂
cosocle	C₂³×C₂₆₉	C₂⁴
hypercenter	#519	D₂₆₉
hypocenter	C₂₆₉	#2884
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

subgroup

G/H

center

C₂

C₂²×#16864379

commutator

C₂²×C₂₆₉

C₂³×C₄

Frattini

C₂³

C₂³×D₂₆₉

Fitting

#519×C₂₆₉

C₂

socle

C₂×C₂₆₉

C₂²×C₂²⋊C₄

prime radical

C₂×#22575501

C₂

cosocle

C₂³×C₂₆₉

C₂⁴

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

N	H ≅ G/N	type
2. #31771083	C₂	semidirect
3. #31770578	C₂	semidirect
4. #519×C₂₆₉	C₂	semidirect
7. C₂×#16864403	C₄	semidirect
9. C₂×Q₈×D₂₆₉	C₄	semidirect
63. C₂₆₉	#2884	semidirect
67. #519	D₂₆₉	semidirect

H ≅ G/N

type

2. #31771083

C₂

semidirect

3. #31770578

C₂

semidirect

4. #519×C₂₆₉

C₂

semidirect

7. C₂×#16864403

C₄

semidirect

9. C₂×Q₈×D₂₆₉

C₄

semidirect

63. C₂₆₉

#2884

semidirect

67. #519

D₂₆₉

semidirect

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

subgroup	index
C₂×#22575501	2
#31771083	2
#31770578	2
#519×C₂₆₉	2
#2884	269

subgroup

index

#519×C₂₆₉

269

Subgroups — #46819348

Conjugacy classes of subgroups

Complements

Maximal subgroups