Group #151741 — subgroups

Subgroups — #151741

Order 512.

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

normal subgroups only

This group has 173 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	#13733	C₂	1	2	yes	maximal, complemented
2	#13719	C₂	1	2	yes	maximal
3	#12521	C₂	1	1	yes	maximal, characteristic, complemented
4	#12503	C₂	1	1	yes	maximal, characteristic
5	#7331	C₂	1	1	yes	maximal, characteristic, complemented
6	#2252	—	2	1	no	—
7	#2252	C₂²	1	2	yes	—
8	#2252	C₂²	1	1	yes	characteristic
9	#2246	C₂²	1	1	yes	characteristic
10	#2240	—	2	1	no	—
11	#1713	—	2	1	no	—
12	#1706	—	2	1	no	—
13	#1424	—	2	4	no	—
14	#1415	—	2	4	no	—
15	C₄×C₃₂	C₂²	1	2	yes	—
16	C₈×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×D₁₆	—	2	2	no	—
18	#500	—	2	1	no	—
19	#500	—	2	2	no	—
20	#500	—	4	1	no	—
21	#494	—	4	1	no	—
22	#493	—	2	1	no	—
23	#492	—	4	1	no	—
24	#486	—	4	1	no	—
25	C₂×C₃₂	D₄	1	4	yes	—
26	#366	—	2	2	no	—
27	#366	—	2	2	no	—
28	#366	—	2	2	no	—
29	#366	—	4	2	no	—
30	#357	—	4	2	no	—
31	C₄×C₁₆	D₄	1	1	yes	characteristic
32	C₄×C₁₆	C₂³	1	1	yes	characteristic
33	C₈²	D₄	1	1	yes	characteristic
34	C₂×Q₁₆	—	8	2	no	—
35	C₂×SD₁₆	—	8	2	no	—
36	C₂×D₈	—	4	1	no	—
37	C₂×D₈	—	4	1	no	—
38	C₄⋊Q₈	—	4	1	no	—
39	C₄⋊Q₈	—	4	2	no	—
40	#127	—	4	1	no	—
41	D₁₆	—	4	4	no	—
42	C₂×C₁₆	D₈	1	2	yes	—
43	C₂×C₁₆	C₂×D₄	1	2	yes	—
44	#107	—	4	2	no	—
45	#107	—	4	1	no	—
46	#107	—	4	1	no	—
47	#107	—	4	2	no	—
48	#107	—	8	2	no	—
49	#102	—	8	2	no	—
50	C₄×C₈	D₈	1	1	yes	characteristic
51	C₄×C₈	D₈	1	1	yes	characteristic
52	C₄×C₈	C₂×D₄	1	1	yes	characteristic
53	C₃₂	—	2	4	no	—
54	C₂×Q₈	—	8	1	no	—
55	C₂×Q₈	—	8	2	no	—
56	C₂×D₄	—	8	1	no	—
57	C₂×D₄	—	8	1	no	—
58	C₂×D₄	—	8	1	no	—
59	Q₁₆	—	16	4	no	—
60	SD₁₆	—	16	4	no	—
61	D₈	—	8	2	no	—
62	D₈	—	8	2	no	—
63	C₂×C₈	D₁₆	1	2	yes	—
64	C₂×C₈	D₁₆	1	2	yes	—
65	C₂×C₈	#127	1	1	yes	characteristic
66	C₂×C₈	C₂×D₈	1	1	yes	characteristic
67	C₄⋊C₄	—	8	1	no	—
68	C₄⋊C₄	—	8	2	no	—
69	C₄⋊C₄	—	8	1	no	—
70	C₄⋊C₄	—	8	2	no	—
71	C₄²	C₂×D₈	1	1	yes	characteristic
72	C₁₆	—	2	2	no	—
73	C₁₆	#137	1	4	yes	—
74	C₂³	—	16	1	no	—
75	Q₈	—	16	2	no	—
76	Q₈	—	16	4	no	—
77	D₄	—	16	1	no	—
78	D₄	—	16	1	no	—
79	D₄	—	16	2	no	—
80	D₄	—	16	2	no	—
81	C₂×C₄	—	16	1	no	—
82	C₂×C₄	—	16	2	no	—
83	C₂×C₄	#493	1	1	yes	characteristic
84	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
85	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	SD₆₄	1	4	yes	—
88	C₈	#497	1	2	yes	—
89	C₈	#510	1	2	yes	—
90	C₂²	—	32	1	no	—
91	C₂²	—	32	1	no	—
92	C₂²	—	32	1	no	—
93	C₂²	#2246	1	1	yes	characteristic
94	C₄	—	32	1	no	—
95	C₄	—	32	2	no	—
96	C₄	#1722	1	1	yes	characteristic
97	C₄	#2251	1	1	yes	characteristic
98	C₄	C₂×SD₆₄	1	2	yes	—
99	C₄	#2263	1	2	yes	—
100	C₂	—	64	1	no	—
101	C₂	#12531	1	1	yes	characteristic
102	C₂	#13727	1	1	yes	characteristic
103	C₂	#13729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13733

C₂

yes

maximal, complemented

#13719

C₂

yes

maximal

#12521

C₂

yes

maximal, characteristic, complemented

#12503

C₂

yes

maximal, characteristic

#7331

C₂

yes

maximal, characteristic, complemented

#2252

—

#2252

C₂²

yes

—

#2252

C₂²

yes

characteristic

#2246

C₂²

yes

—

—

—

—

—

C₄×C₃₂

C₂²

yes

—

C₈×C₁₆

C₂²

yes

characteristic

C₂×D₁₆

—

—

—

—

—

—

—

—

C₂×C₃₂

D₄

yes

—

—

—

—

—

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

C₂³

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

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×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

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

—

C₄²

C₂×D₈

yes

characteristic

C₁₆

—

C₁₆

#137

yes

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#493

yes

characteristic

C₂×C₄

C₂×D₁₆

yes

characteristic

C₂×C₄

C₂×D₁₆

yes

characteristic

C₈

—

C₈

SD₆₄

yes

—

C₈

#497

yes

—

C₈

#510

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#2246

yes

characteristic

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

C₂×SD₆₄

yes

—

C₄

#2263

yes

—

100

C₂

—

101

C₂

#12531

yes

characteristic

102

C₂

#13727

yes

characteristic

103

C₂

#13729

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#2246	C₂²
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

C₂²

#2246

commutator, Frattini, cosocle

C₄×C₁₆

C₂³

prime radical

#2246

C₂²

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
1. #13733	C₂	semidirect
3. #12521	C₂	semidirect
5. #7331	C₂	semidirect

H ≅ G/N

type

1. #13733

C₂

semidirect

3. #12521

C₂

semidirect

5. #7331

C₂

semidirect

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

subgroup	index
#13733	2
#13719	2
#12521	2
#12503	2
#7331	2

subgroup

index

Subgroups — #151741

Conjugacy classes of subgroups

Complements

Maximal subgroups