Group #151876 — subgroups

Subgroups — #151876

Order 512.

266 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 194 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×C₄×C₃₂	C₂	1	2	yes	maximal
2	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#7465	C₂	1	4	yes	maximal
4	C₂²×C₃₂	C₂²	1	4	yes	—
5	C₂×C₄×C₁₆	C₄	1	2	yes	—
6	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₄²×C₈	C₄	1	1	yes	characteristic
10	C₄×C₃₂	C₄	1	8	yes	complemented
11	C₂²×C₁₆	C₂×C₄	1	2	yes	—
12	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₁₆	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₈	C₈	1	2	yes	—
15	C₂×C₄×C₈	C₈	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₄³	C₈	1	1	yes	characteristic
19	C₂×C₃₂	—	2	8	no	—
20	C₂×C₃₂	C₂×C₄	1	8	yes	—
21	C₄×C₁₆	—	2	4	no	—
22	C₄×C₁₆	—	2	4	no	—
23	C₄×C₁₆	C₈	1	4	yes	—
24	C₄×C₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈	C₂×C₈	1	2	yes	—
26	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄²	C₁₆	1	2	yes	—
29	C₂×C₄²	C₁₆	1	2	yes	—
30	C₂×C₄²	C₂×C₈	1	2	yes	—
31	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
32	C₂×C₁₆	—	2	2	no	—
33	C₂×C₁₆	—	2	2	no	—
34	C₂×C₁₆	C₄²	1	2	yes	—
35	C₂×C₁₆	C₂×C₈	1	4	yes	—
36	C₂×C₁₆	M₁₆	1	4	yes	—
37	C₂×C₁₆	C₂²×C₄	1	2	yes	—
38	C₂×C₁₆	#41	1	4	yes	—
39	C₄×C₈	—	2	4	no	—
40	C₄×C₈	—	2	2	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	C₁₆	1	4	yes	—
43	C₄×C₈	C₄²	1	2	yes	—
44	C₄×C₈	C₂×C₈	1	2	yes	—
45	C₃₂	—	2	8	no	—
46	C₂²×C₄	C₂×C₁₆	1	4	yes	—
47	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
48	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
49	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	C₄×C₈	1	2	yes	—
53	C₂×C₈	C₂×C₁₆	1	4	yes	—
54	C₂×C₈	M₃₂	1	4	yes	—
55	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₈	#117	1	2	yes	—
57	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₈	C₂×M₁₆	1	2	yes	—
59	C₄²	—	2	4	no	—
60	C₄²	—	2	2	no	—
61	C₄²	—	2	2	no	—
62	C₄²	C₃₂	1	8	yes	complemented
63	C₄²	C₄×C₈	1	2	yes	—
64	C₄²	C₂×C₁₆	1	2	yes	—
65	C₁₆	—	2	4	no	—
66	C₁₆	#97	1	4	yes	—
67	C₁₆	#117	1	4	yes	—
68	C₂³	C₂²×C₁₆	1	1	yes	characteristic
69	C₂×C₄	—	2	4	no	—
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	C₄×C₁₆	1	2	yes	—
72	C₂×C₄	C₄×C₁₆	1	2	yes	—
73	C₂×C₄	C₂×C₃₂	1	8	yes	—
74	C₂×C₄	C₂×C₃₂	1	4	yes	—
75	C₂×C₄	M₆₄	1	4	yes	—
76	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
77	C₂×C₄	#431	1	2	yes	—
78	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
79	C₂×C₄	C₂×M₃₂	1	2	yes	—
80	C₈	—	2	4	no	—
81	C₈	#346	1	4	yes	—
82	C₈	#376	1	1	yes	characteristic
83	C₈	C₂×#97	1	1	yes	characteristic
84	C₈	#431	1	2	yes	—
85	C₂²	C₂×C₄×C₁₆	1	1	yes	characteristic
86	C₂²	#2162	1	2	yes	—
87	C₂²	C₂²×C₃₂	1	2	yes	—
88	C₂²	C₂×M₆₄	1	2	yes	—
89	C₄	—	2	8	no	—
90	C₄	C₄×C₃₂	1	4	yes	—
91	C₄	#1397	1	4	yes	—
92	C₄	#1744	1	1	yes	characteristic
93	C₄	C₂×#346	1	1	yes	characteristic
94	C₄	#2162	1	2	yes	—
95	C₂	#12590	1	1	yes	characteristic
96	C₂	C₂×C₄×C₃₂	1	1	yes	characteristic
97	C₂	C₂×#1397	1	1	yes	characteristic
98	C₂	#13639	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₃₂

C₂

yes

maximal

C₄²×C₁₆

C₂

yes

maximal, characteristic

#7465

C₂

yes

maximal

C₂²×C₃₂

C₂²

yes

—

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄²×C₈

C₄

yes

characteristic

C₄×C₃₂

C₄

yes

complemented

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

C₂×C₄×C₈

C₈

yes

—

C₂×C₄×C₈

C₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₄³

C₈

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂×C₄²

C₁₆

yes

—

C₂×C₄²

C₁₆

yes

—

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

#41

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₁₆

yes

—

C₄×C₈

C₄²

yes

—

C₄×C₈

C₂×C₈

yes

—

C₃₂

—

C₂²×C₄

C₂×C₁₆

yes

—

C₂²×C₄

C₂×C₁₆

yes

characteristic

C₂²×C₄

C₂×C₁₆

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

#117

yes

—

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₃₂

yes

complemented

C₄²

C₄×C₈

yes

—

C₄²

C₂×C₁₆

yes

—

C₁₆

—

C₁₆

#97

yes

—

C₁₆

#117

yes

—

C₂³

C₂²×C₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₁₆

yes

—

C₂×C₄

C₄×C₁₆

yes

—

C₂×C₄

C₂×C₃₂

yes

—

C₂×C₄

C₂×C₃₂

yes

—

C₂×C₄

M₆₄

yes

—

C₂×C₄

C₂×C₄×C₈

yes

characteristic

C₂×C₄

#431

yes

—

C₂×C₄

C₂²×C₁₆

yes

characteristic

C₂×C₄

C₂×M₃₂

yes

—

C₈

—

C₈

#346

yes

—

C₈

#376

yes

characteristic

C₈

C₂×#97

yes

characteristic

C₈

#431

yes

—

C₂²

C₂×C₄×C₁₆

yes

characteristic

C₂²

#2162

yes

—

C₂²

C₂²×C₃₂

yes

—

C₂²

C₂×M₆₄

yes

—

C₄

—

C₄

C₄×C₃₂

yes

—

C₄

#1397

yes

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

#12590

yes

characteristic

C₂

C₂×C₄×C₃₂

yes

C₂

yes

C₂

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₆	C₂²
commutator	C₂	C₂×C₄×C₃₂
Frattini, cosocle	C₂²×C₁₆	C₂³
socle, prime radical	C₂³	C₂²×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄×C₁₆

C₂²

commutator

C₂

C₂×C₄×C₃₂

Frattini, cosocle

C₂²×C₁₆

C₂³

socle, prime radical

C₂³

C₂²×C₁₆

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
10. C₄×C₃₂	C₄	semidirect
62. C₄²	C₃₂	semidirect

H ≅ G/N

type

10. C₄×C₃₂

C₄

semidirect

62. C₄²

C₃₂

semidirect

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

subgroup	index
C₂×C₄×C₃₂	2
C₄²×C₁₆	2
#7465	2

subgroup

index

C₂×C₄×C₃₂

C₄²×C₁₆

#7465

Subgroups — #151876

Conjugacy classes of subgroups

Complements

Maximal subgroups