Group #151737 — subgroups

Subgroups — #151737

Order 512.

186 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 62 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	#13726	C₂	1	2	yes	maximal
2	#13720	C₂	1	2	yes	maximal
3	#12516	C₂	1	1	yes	maximal, characteristic
4	#12504	C₂	1	1	yes	maximal, characteristic
5	#7331	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₆₄	—	2	4	no	—
7	#2252	C₂²	1	1	yes	characteristic
8	#2247	—	2	1	no	—
9	#2247	C₂²	1	2	yes	—
10	#2247	C₂²	1	1	yes	characteristic
11	#2241	—	2	1	no	—
12	#1713	—	2	1	no	—
13	#1707	—	2	1	no	—
14	#1416	—	2	4	no	—
15	C₄×C₃₂	C₂²	1	2	yes	—
16	C₈×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂	—	2	2	no	—
18	C₂×Q₃₂	—	2	2	no	—
19	C₂×Q₃₂	—	2	2	no	—
20	C₂×Q₃₂	—	4	2	no	—
21	#500	—	2	1	no	—
22	#500	—	4	1	no	—
23	#500	—	4	1	no	—
24	#498	—	4	1	no	—
25	#494	—	2	1	no	—
26	#494	—	2	2	no	—
27	#487	—	4	1	no	—
28	Q₆₄	—	4	8	no	—
29	C₂×C₃₂	D₄	1	4	yes	—
30	#366	—	2	2	no	—
31	#358	—	4	2	no	—
32	C₄×C₁₆	D₄	1	1	yes	characteristic
33	C₄×C₁₆	C₂³	1	1	yes	characteristic
34	C₈²	D₄	1	1	yes	characteristic
35	C₂×Q₁₆	—	4	1	no	—
36	C₂×Q₁₆	—	4	2	no	—
37	C₂×Q₁₆	—	4	1	no	—
38	C₂×Q₁₆	—	4	2	no	—
39	C₄⋊Q₈	—	4	1	no	—
40	C₄⋊Q₈	—	4	1	no	—
41	C₄⋊Q₈	—	4	2	no	—
42	Q₃₂	—	4	4	no	—
43	Q₃₂	—	4	4	no	—
44	Q₃₂	—	4	4	no	—
45	Q₃₂	—	8	4	no	—
46	C₂×C₁₆	D₈	1	2	yes	—
47	C₂×C₁₆	C₂×D₄	1	2	yes	—
48	#107	—	4	1	no	—
49	#107	—	4	1	no	—
50	#107	—	8	2	no	—
51	#107	—	8	2	no	—
52	#106	—	8	2	no	—
53	#103	—	8	2	no	—
54	C₄×C₈	D₈	1	1	yes	characteristic
55	C₄×C₈	D₈	1	1	yes	characteristic
56	C₄×C₈	C₂×D₄	1	1	yes	characteristic
57	C₃₂	—	2	4	no	—
58	C₂×Q₈	—	8	1	no	—
59	C₂×Q₈	—	8	2	no	—
60	C₂×Q₈	—	8	1	no	—
61	Q₁₆	—	8	2	no	—
62	Q₁₆	—	8	2	no	—
63	Q₁₆	—	8	2	no	—
64	Q₁₆	—	8	4	no	—
65	Q₁₆	—	8	2	no	—
66	C₂×C₈	D₁₆	1	2	yes	—
67	C₂×C₈	D₁₆	1	2	yes	—
68	C₂×C₈	#127	1	1	yes	characteristic
69	C₂×C₈	C₂×D₈	1	1	yes	characteristic
70	C₄⋊C₄	—	8	1	no	—
71	C₄⋊C₄	—	8	1	no	—
72	C₄⋊C₄	—	8	1	no	—
73	C₄⋊C₄	—	8	1	no	—
74	C₄⋊C₄	—	8	2	no	—
75	C₄⋊C₄	—	8	2	no	—
76	C₄²	C₂×D₈	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	AGL(1,ℤ/8)	1	4	yes	—
79	Q₈	—	16	1	no	—
80	Q₈	—	16	2	no	—
81	Q₈	—	16	2	no	—
82	Q₈	—	16	1	no	—
83	Q₈	—	16	2	no	—
84	C₂×C₄	—	16	1	no	—
85	C₂×C₄	—	16	1	no	—
86	C₂×C₄	—	16	2	no	—
87	C₂×C₄	#493	1	1	yes	characteristic
88	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
89	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	Q₆₄	1	4	yes	complemented
92	C₈	#496	1	2	yes	—
93	C₈	#509	1	2	yes	—
94	C₂²	#2246	1	1	yes	characteristic
95	C₄	—	32	1	no	—
96	C₄	—	32	1	no	—
97	C₄	—	32	2	no	—
98	C₄	#1721	1	1	yes	characteristic
99	C₄	#2250	1	1	yes	characteristic
100	C₄	C₂×Q₆₄	1	2	yes	—
101	C₄	#2264	1	2	yes	—
102	C₂	#12525	1	1	yes	characteristic
103	C₂	#13726	1	1	yes	characteristic
104	C₂	#13730	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13726

C₂

yes

maximal

#13720

C₂

yes

maximal

#12516

C₂

yes

maximal, characteristic

#12504

C₂

yes

maximal, characteristic

#7331

C₂

yes

maximal, characteristic

C₂×Q₆₄

—

#2252

C₂²

yes

characteristic

#2247

—

#2247

C₂²

yes

—

#2247

C₂²

yes

—

—

—

—

C₄×C₃₂

C₂²

yes

—

C₈×C₁₆

C₂²

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

—

—

—

—

—

—

—

Q₆₄

—

C₂×C₃₂

D₄

yes

—

#366

—

#358

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

C₂³

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

C₂×D₄

yes

—

—

—

—

—

—

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

#127

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₈

yes

characteristic

C₁₆

—

C₁₆

AGL(1,ℤ/8)

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#493

yes

characteristic

C₂×C₄

C₂×D₁₆

yes

characteristic

C₂×C₄

C₂×D₁₆

yes

characteristic

C₈

—

C₈

Q₆₄

yes

complemented

C₈

#496

yes

—

C₈

#509

yes

—

C₂²

#2246

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

100

C₄

C₂×Q₆₄

yes

—

101

C₄

#2264

yes

—

102

C₂

#12525

yes

characteristic

103

C₂

#13726

yes

characteristic

104

C₂

#13730

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

#2246

commutator, Frattini, cosocle

C₄×C₁₆

C₂³

LCS

G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G

LECS

G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
91. C₈	Q₆₄	semidirect

H ≅ G/N

type

91. C₈

Q₆₄

semidirect

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

subgroup	index
#13726	2
#13720	2
#12516	2
#12504	2
#7331	2

subgroup

index

Subgroups — #151737

Conjugacy classes of subgroups

Complements

Maximal subgroups