Group #153370 — subgroups

Subgroups — #153370

Order 512.

185 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 185 conjugacy classes of subgroups — too many to draw. Only its 64 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×#1424	C₂	1	1	yes	maximal, characteristic, complemented
2	#13554	C₂	1	1	yes	maximal, characteristic
3	#12842	C₂	1	2	yes	maximal
4	#7478	C₂	1	2	yes	maximal, complemented
5	#7472	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#504	C₂²	1	1	yes	characteristic
7	C₂×M₆₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃₂	C₂²	1	1	yes	characteristic
9	C₂×#368	C₂²	1	2	yes	—
10	C₂×#366	C₂²	1	2	yes	—
11	#1816	—	2	2	no	—
12	#1424	—	2	2	no	—
13	C₂²⋊C₃₂	—	2	2	no	—
14	#1387	—	2	2	no	—
15	C₂×#131	D₄	1	1	yes	characteristic
16	#504	—	2	2	no	—
17	#504	—	2	2	no	—
18	C₂×M₃₂	—	2	1	no	—
19	C₂×M₃₂	D₄	1	1	yes	characteristic
20	C₂²×C₁₆	—	2	1	no	—
21	C₂²×C₁₆	C₂³	1	1	yes	characteristic
22	C₂×#108	—	2	2	no	—
23	C₂×#107	—	2	2	no	—
24	#417	—	4	2	no	—
25	M₆₄	—	2	2	no	—
26	C₂×C₃₂	—	2	2	no	—
27	C₂×C₃₂	—	2	1	no	—
28	C₂×C₃₂	D₄	1	2	yes	—
29	#368	—	2	2	no	—
30	#368	—	2	2	no	—
31	#366	—	2	2	no	—
32	#366	—	2	2	no	—
33	#343	—	4	2	no	—
34	C₂×#41	D₈	1	1	yes	characteristic
35	#131	—	2	2	no	—
36	#131	—	2	2	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂×M₁₆	—	4	2	no	—
39	C₂×M₁₆	D₈	1	1	yes	characteristic
40	C₂²×C₈	—	2	1	no	—
41	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄	—	4	2	no	—
43	M₃₂	—	2	2	no	—
44	M₃₂	—	4	2	no	—
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	—	4	1	no	—
47	C₂×C₁₆	—	4	1	no	—
48	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
49	C₂×C₁₆	#41	1	2	yes	—
50	C₂×C₁₆	#41	1	2	yes	—
51	C₂×C₁₆	#41	1	1	yes	characteristic
52	#108	—	4	2	no	—
53	#108	—	4	2	no	—
54	#107	—	4	2	no	—
55	#107	—	4	2	no	—
56	#103	—	8	2	no	—
57	#102	—	8	2	no	—
58	C₃₂	—	2	2	no	—
59	C₃₂	—	4	1	no	—
60	#41	—	2	2	no	—
61	#41	—	2	2	no	—
62	C₂×Q₈	D₁₆	1	1	yes	characteristic
63	C₂×D₄	—	2	1	no	—
64	C₂×D₄	D₁₆	1	1	yes	characteristic
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	—	8	2	no	—
67	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	4	2	no	—
70	M₁₆	—	8	2	no	—
71	M₁₆	—	8	2	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	—	8	2	no	—
76	C₂×C₈	#121	1	1	yes	characteristic
77	C₂×C₈	#123	1	2	yes	—
78	C₂×C₈	#135	1	1	yes	characteristic
79	C₂×C₈	#137	1	2	yes	—
80	C₄⋊C₄	—	8	2	no	—
81	C₄⋊C₄	—	8	2	no	—
82	C₁₆	—	4	1	no	—
83	C₁₆	#123	1	1	yes	characteristic
84	C₁₆	#124	1	1	yes	characteristic
85	C₁₆	#126	1	2	yes	—
86	C₂³	—	2	1	no	—
87	C₂³	#508	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	D₄	—	2	2	no	—
90	D₄	—	4	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	16	2	no	—
95	C₂×C₄	—	32	2	no	—
96	C₂×C₄	#466	1	1	yes	characteristic
97	C₂×C₄	#480	1	2	yes	—
98	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
99	C₂×C₄	#510	1	2	yes	—
100	C₈	—	4	1	no	—
101	C₈	—	16	2	no	—
102	C₈	#397	1	1	yes	characteristic
103	C₈	#470	1	1	yes	characteristic
104	C₈	#485	1	2	yes	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	2	no	—
107	C₂²	—	4	1	no	—
108	C₂²	#2215	1	1	yes	characteristic
109	C₂²	#2237	1	2	yes	—
110	C₄	—	4	1	no	—
111	C₄	—	32	2	no	—
112	C₄	#2075	1	1	yes	characteristic
113	C₄	#2219	1	1	yes	characteristic
114	C₄	#2232	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	8	1	no	—
117	C₂	#13492	1	1	yes	characteristic
118	C₂	#13702	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1424

C₂

yes

maximal, characteristic, complemented

#13554

C₂

yes

maximal, characteristic

#12842

C₂

yes

maximal

#7478

C₂

yes

maximal, complemented

#7472

C₂

yes

maximal, characteristic, complemented

C₂×#504

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

C₂×#366

C₂²

yes

—

#1816

—

#1424

—

C₂²⋊C₃₂

—

#1387

—

C₂×#131

D₄

yes

characteristic

#504

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

C₂³

yes

—

—

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

—

—

—

—

—

C₂×#41

D₈

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

#41

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

—

—

C₃₂

—

C₃₂

—

#41

—

#41

—

C₂×Q₈

D₁₆

yes

characteristic

C₂×D₄

—

C₂×D₄

D₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#121

yes

characteristic

C₂×C₈

#123

yes

—

C₂×C₈

#135

yes

characteristic

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₁₆

—

C₁₆

yes

C₁₆

yes

C₁₆

yes

—

C₂³

—

C₂³

#508

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#466

yes

characteristic

C₂×C₄

#480

yes

—

C₂×C₄

C₂×D₁₆

yes

characteristic

C₂×C₄

#510

yes

—

100

C₈

—

101

C₈

—

102

C₈

#397

yes

characteristic

103

C₈

#470

yes

characteristic

104

C₈

#485

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#2215

yes

characteristic

109

C₂²

#2237

yes

—

110

C₄

—

111

C₄

—

112

C₄

#2075

yes

characteristic

113

C₄

#2219

yes

characteristic

114

C₄

#2232

yes

—

115

C₂

—

116

C₂

—

117

C₂

#13492

yes

characteristic

118

C₂

#13702

yes

—

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2215

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂×#41

D₈

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#1424	C₂	semidirect
4. #7478	C₂	semidirect
5. #7472	C₂	semidirect

H ≅ G/N

type

1. C₂×#1424

C₂

semidirect

4. #7478

C₂

semidirect

5. #7472

C₂

semidirect

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

subgroup	index
C₂×#1424	2
#13554	2
#12842	2
#7478	2
#7472	2

subgroup

index

Subgroups — #153370

Conjugacy classes of subgroups

Complements

Maximal subgroups