Group #135757 — subgroups

Subgroups — #135757

Order 512.

218 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 102 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	#12901	C₂	1	1	yes	maximal, characteristic
2	#12880	C₂	1	1	yes	maximal, characteristic
3	#12005	C₂	1	2	yes	maximal
4	#11997	C₂	1	2	yes	maximal
5	#8138	C₂	1	1	yes	maximal, characteristic
6	C₂×#364	—	2	2	no	—
7	#2150	C₂²	1	2	yes	—
8	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	#1726	C₂²	1	1	yes	characteristic
11	#1450	C₂²	1	2	yes	—
12	#1374	—	2	2	no	—
13	#1370	—	2	4	no	—
14	#1367	—	2	4	no	—
15	#1275	C₄	1	2	yes	—
16	#1274	C₄	1	2	yes	—
17	C₂×M₃₂	—	2	2	no	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₂²×C₁₆	—	2	1	no	—
20	#431	C₂×C₄	1	4	yes	—
21	C₂×#97	C₂×C₄	1	1	yes	characteristic
22	C₂×#97	Q₈	1	2	yes	—
23	C₂×C₄×C₈	D₄	1	2	yes	—
24	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
25	#376	C₂×C₄	1	1	yes	characteristic
26	#364	—	4	4	no	—
27	C₄⋊C₁₆	—	2	2	no	—
28	C₄⋊C₁₆	—	2	2	no	—
29	C₄⋊C₁₆	—	2	2	no	—
30	C₄⋊C₁₆	—	2	2	no	—
31	#322	Q₈	1	4	yes	—
32	C₈²	D₄	1	4	yes	—
33	C₂²×C₈	C₄⋊C₄	1	2	yes	—
34	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄²	C₄⋊C₄	1	2	yes	—
37	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
38	M₃₂	—	4	4	no	—
39	C₂×C₁₆	—	4	1	no	—
40	C₂×C₁₆	—	4	1	no	—
41	C₂×C₁₆	—	4	2	no	—
42	C₂×C₁₆	—	4	2	no	—
43	C₂×C₁₆	—	4	2	no	—
44	C₄⋊C₈	—	2	4	no	—
45	C₂²⋊C₈	—	2	4	no	—
46	#97	—	2	4	no	—
47	#97	—	2	2	no	—
48	C₄×C₈	—	2	2	no	—
49	C₄×C₈	—	2	2	no	—
50	C₄×C₈	C₄⋊C₄	1	8	yes	—
51	C₄×C₈	C₂×D₄	1	2	yes	—
52	C₄×C₈	C₂×Q₈	1	2	yes	—
53	#95	—	2	2	no	—
54	C₂²×C₄	—	2	2	no	—
55	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄	#131	1	2	yes	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	8	no	—
61	C₂×C₈	#108	1	4	yes	—
62	C₂×C₈	#127	1	1	yes	characteristic
63	C₂×C₈	C₄⋊Q₈	1	2	yes	—
64	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
66	C₂×C₈	#137	1	2	yes	—
67	C₄²	—	2	2	no	—
68	C₄²	—	2	2	no	—
69	C₄²	C₂×C₄⋊C₄	1	2	yes	—
70	C₄²	#131	1	2	yes	—
71	C₁₆	—	8	1	no	—
72	C₁₆	—	8	2	no	—
73	C₁₆	—	8	1	no	—
74	C₂³	#424	1	1	yes	characteristic
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	4	4	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	#384	1	1	yes	characteristic
79	C₂×C₄	#424	1	4	yes	—
80	C₂×C₄	#424	1	1	yes	characteristic
81	C₂×C₄	#428	1	2	yes	—
82	C₂×C₄	C₂×#108	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	—	2	2	no	—
85	C₈	—	4	8	no	—
86	C₈	#496	1	1	yes	characteristic
87	C₈	#497	1	1	yes	characteristic
88	C₈	#501	1	2	yes	—
89	C₂²	—	2	2	no	—
90	C₂²	#1574	1	2	yes	—
91	C₂²	#1843	1	1	yes	characteristic
92	C₄	—	2	4	no	—
93	C₄	—	8	2	no	—
94	C₄	#1573	1	2	yes	—
95	C₄	#1847	1	1	yes	characteristic
96	C₄	#1848	1	1	yes	characteristic
97	C₂	—	2	2	no	—
98	C₂	#8812	1	1	yes	characteristic
99	C₂	#12902	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12901

C₂

yes

maximal, characteristic

#12880

C₂

yes

maximal, characteristic

#12005

C₂

yes

maximal

#11997

C₂

yes

maximal

#8138

C₂

yes

maximal, characteristic

C₂×#364

—

#2150

C₂²

yes

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

#1726

C₂²

yes

characteristic

#1450

C₂²

yes

—

—

—

—

C₄

yes

—

#1274

C₄

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

#431

C₂×C₄

yes

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

Q₈

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#376

C₂×C₄

yes

characteristic

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#322

Q₈

yes

—

C₈²

D₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×Q₈

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#131

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#108

yes

—

C₂×C₈

#127

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄⋊C₄

yes

—

C₄²

#131

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#424

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#384

yes

characteristic

C₂×C₄

#424

yes

—

C₂×C₄

#424

yes

characteristic

C₂×C₄

#428

yes

—

C₂×C₄

C₂×#108

yes

—

C₈

—

C₈

—

C₈

—

C₈

yes

C₈

yes

C₈

yes

—

C₂²

—

C₂²

#1574

yes

—

C₂²

#1843

yes

characteristic

C₄

—

C₄

—

C₄

#1573

yes

—

C₄

yes

C₄

yes

C₂

—

C₂

#8812

yes

characteristic

C₂

#12902

yes

—

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1843
prime radical	C₂³	#424
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#127

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12901	2
#12880	2
#12005	2
#11997	2
#8138	2

subgroup

index

Subgroups — #135757

Conjugacy classes of subgroups

Maximal subgroups