Group #151663 — subgroups

Subgroups — #151663

Order 512.

135 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 135 conjugacy classes of subgroups — too many to draw. Only its 48 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	#13731	C₂	1	1	yes	maximal, characteristic, complemented
2	#13646	C₂	1	1	yes	maximal, characteristic
3	#12451	C₂	1	2	yes	maximal
4	#7461	C₂	1	1	yes	maximal, characteristic, complemented
5	#7450	C₂	1	2	yes	maximal, complemented
6	C₂×M₆₄	—	2	1	no	—
7	#2252	C₂²	1	2	yes	—
8	D₄×C₁₆	C₂²	1	1	yes	characteristic
9	#1673	—	2	2	no	—
10	#1423	—	2	1	no	—
11	#1423	—	2	1	no	—
12	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
13	C₂²⋊C₃₂	—	2	1	no	—
14	C₄×C₃₂	C₂²	1	1	yes	characteristic
15	#1371	C₂²	1	2	yes	—
16	#1364	—	2	2	no	—
17	C₂²×C₁₆	—	2	1	no	—
18	#500	—	2	2	no	—
19	#461	—	4	2	no	—
20	C₈×D₄	D₄	1	1	yes	characteristic
21	M₆₄	—	4	2	no	—
22	C₂×C₃₂	—	2	1	no	—
23	C₂×C₃₂	D₄	1	1	yes	characteristic
24	C₂×C₃₂	D₄	1	1	yes	characteristic
25	#366	—	2	2	no	—
26	#366	—	2	2	no	—
27	C₄⋊C₁₆	D₄	1	1	yes	characteristic
28	C₂²⋊C₁₆	—	2	1	no	—
29	C₄×C₁₆	C₂³	1	1	yes	characteristic
30	#335	—	2	2	no	—
31	#332	—	4	2	no	—
32	C₂×SD₁₆	—	8	2	no	—
33	C₂²×C₈	—	2	1	no	—
34	C₄⋊Q₈	—	4	2	no	—
35	C₄×D₄	D₈	1	1	yes	characteristic
36	C₂×C₁₆	—	2	1	no	—
37	C₂×C₁₆	—	4	1	no	—
38	C₂×C₁₆	—	4	1	no	—
39	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
40	C₂×C₁₆	#41	1	1	yes	characteristic
41	#107	—	4	2	no	—
42	#107	—	4	2	no	—
43	C₄⋊C₈	—	4	2	no	—
44	C₄⋊C₈	D₈	1	1	yes	characteristic
45	#103	—	8	2	no	—
46	C₂²⋊C₈	—	2	1	no	—
47	C₄×C₈	C₂×D₄	1	1	yes	characteristic
48	C₃₂	—	2	1	no	—
49	C₃₂	—	4	1	no	—
50	C₃₂	SD₁₆	1	2	yes	complemented
51	C₂×Q₈	—	8	2	no	—
52	C₂×D₄	D₁₆	1	1	yes	characteristic
53	C₂²×C₄	—	2	1	no	—
54	SD₁₆	—	16	4	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	4	1	no	—
57	C₂×C₈	—	4	1	no	—
58	C₂×C₈	—	8	2	no	—
59	C₂×C₈	#121	1	1	yes	characteristic
60	C₂×C₈	#135	1	1	yes	characteristic
61	C₄⋊C₄	—	8	2	no	—
62	C₄⋊C₄	—	8	2	no	—
63	C₄⋊C₄	D₁₆	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×D₈	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	—	4	1	no	—
68	C₁₆	C₂×SD₁₆	1	1	yes	characteristic
69	C₁₆	#137	1	1	yes	characteristic
70	C₂³	—	2	1	no	—
71	Q₈	—	16	2	no	—
72	Q₈	—	16	2	no	—
73	D₄	—	2	2	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	16	2	no	—
78	C₂×C₄	#466	1	1	yes	characteristic
79	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
80	C₂×C₄	#508	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	4	1	no	—
83	C₈	—	16	2	no	—
84	C₈	#461	1	1	yes	characteristic
85	C₈	#473	1	1	yes	characteristic
86	C₂²	—	4	1	no	—
87	C₂²	—	4	1	no	—
88	C₂²	—	4	1	no	—
89	C₂²	#2215	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	32	2	no	—
93	C₄	#1673	1	1	yes	characteristic
94	C₄	#2222	1	1	yes	characteristic
95	C₄	#2263	1	2	yes	—
96	C₂	—	8	1	no	—
97	C₂	#12451	1	1	yes	characteristic
98	C₂	#13697	1	1	yes	characteristic
99	C₂	#13701	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13731

C₂

yes

maximal, characteristic, complemented

#13646

C₂

yes

maximal, characteristic

#12451

C₂

yes

maximal

#7461

C₂

yes

maximal, characteristic, complemented

#7450

C₂

yes

maximal, complemented

C₂×M₆₄

—

#2252

C₂²

yes

—

D₄×C₁₆

C₂²

yes

—

—

—

C₄⋊C₃₂

C₂²

yes

characteristic

C₂²⋊C₃₂

—

C₄×C₃₂

C₂²

yes

characteristic

#1371

C₂²

yes

—

#1364

—

C₂²×C₁₆

—

#500

—

#461

—

C₈×D₄

D₄

yes

characteristic

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

D₄

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

D₄

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂³

yes

characteristic

#335

—

#332

—

C₂×SD₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×D₄

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

C₂×C₁₆

yes

—

—

C₄⋊C₈

—

C₄⋊C₈

D₈

yes

characteristic

#103

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

SD₁₆

yes

complemented

C₂×Q₈

—

C₂×D₄

D₁₆

yes

characteristic

C₂²×C₄

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₆

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂×D₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₂×SD₁₆

yes

characteristic

C₁₆

#137

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#466

yes

characteristic

C₂×C₄

C₂×D₁₆

yes

characteristic

C₂×C₄

#508

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

#2215

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂×D₄	D₁₆
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2215

commutator, Frattini, cosocle

C₄×C₁₆

C₂³

prime radical

C₂×D₄

D₁₆

LCS

G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G

LECS

G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #13731	C₂	semidirect
4. #7461	C₂	semidirect
5. #7450	C₂	semidirect
50. C₃₂	SD₁₆	semidirect

H ≅ G/N

type

1. #13731

C₂

semidirect

4. #7461

C₂

semidirect

5. #7450

C₂

semidirect

50. C₃₂

SD₁₆

semidirect

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

subgroup	index
#13731	2
#13646	2
#12451	2
#7461	2
#7450	2

subgroup

index

Subgroups — #151663

Conjugacy classes of subgroups

Complements

Maximal subgroups