Group #153393 — subgroups

Subgroups — #153393

Order 512.

163 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 163 conjugacy classes of subgroups — too many to draw. Only its 66 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	#13636	C₂	1	1	yes	maximal, characteristic
2	#13627	C₂	1	1	yes	maximal, characteristic
3	#13169	C₂	1	2	yes	maximal
4	#7480	C₂	1	2	yes	maximal
5	#7475	C₂	1	1	yes	maximal, characteristic
6	C₂×M₆₄	C₂²	1	1	yes	characteristic
7	C₂×M₆₄	C₂²	1	1	yes	characteristic
8	C₂×#368	C₂²	1	2	yes	—
9	#2158	C₂²	1	2	yes	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1945	—	2	2	no	—
12	#1425	—	2	2	no	—
13	C₄⋊C₃₂	—	2	2	no	—
14	#1382	—	2	2	no	—
15	C₂²×C₁₆	D₄	1	1	yes	characteristic
16	C₂²×C₁₆	C₂³	1	1	yes	characteristic
17	C₂×#108	—	2	2	no	—
18	#427	—	2	2	no	—
19	#424	—	4	2	no	—
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	M₆₄	—	2	2	no	—
22	M₆₄	—	2	2	no	—
23	C₂×C₃₂	—	2	1	no	—
24	C₂×C₃₂	D₄	1	2	yes	—
25	#368	—	2	2	no	—
26	#368	—	2	2	no	—
27	#367	Q₈	1	4	yes	—
28	#366	—	2	2	no	—
29	C₄×C₁₆	—	2	1	no	—
30	C₄×C₁₆	—	2	1	no	—
31	#339	—	4	2	no	—
32	C₂×M₁₆	—	4	2	no	—
33	C₂²×C₈	D₈	1	1	yes	characteristic
34	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
35	#117	—	4	2	no	—
36	C₂×C₄²	D₈	1	1	yes	characteristic
37	C₂×C₁₆	—	2	1	no	—
38	C₂×C₁₆	—	2	2	no	—
39	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
40	C₂×C₁₆	C₂×Q₈	1	2	yes	—
41	C₂×C₁₆	#41	1	2	yes	—
42	C₂×C₁₆	#41	1	1	yes	characteristic
43	#108	—	4	2	no	—
44	#108	—	4	2	no	—
45	#107	—	2	4	no	—
46	#106	—	4	2	no	—
47	C₄⋊C₈	—	8	4	no	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₃₂	—	2	2	no	—
51	C₃₂	—	4	1	no	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
54	M₁₆	—	8	2	no	—
55	M₁₆	—	8	2	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	8	2	no	—
59	C₂×C₈	#121	1	1	yes	characteristic
60	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₈	#135	1	1	yes	characteristic
62	C₂×C₈	#137	1	2	yes	—
63	C₄⋊C₄	—	4	4	no	—
64	C₂²⋊C₄	—	8	2	no	—
65	C₄²	—	2	1	no	—
66	C₄²	—	8	2	no	—
67	C₄²	Q₃₂	1	1	yes	characteristic
68	C₄²	Q₃₂	1	1	yes	characteristic
69	C₁₆	—	4	1	no	—
70	C₁₆	#123	1	1	yes	characteristic
71	C₁₆	#125	1	2	yes	—
72	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
73	C₂³	#508	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	8	4	no	—
78	C₂×C₄	#466	1	1	yes	characteristic
79	C₂×C₄	#474	1	2	yes	—
80	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
81	C₂×C₄	#509	1	2	yes	—
82	C₈	—	4	1	no	—
83	C₈	—	16	2	no	—
84	C₈	#398	1	1	yes	characteristic
85	C₈	#470	1	1	yes	characteristic
86	C₈	#479	1	2	yes	—
87	C₂²	—	4	1	no	—
88	C₂²	#2216	1	1	yes	characteristic
89	C₂²	#2228	1	2	yes	—
90	C₄	—	2	2	no	—
91	C₄	—	4	1	no	—
92	C₄	—	16	4	no	—
93	C₄	#2086	1	1	yes	characteristic
94	C₄	#2218	1	1	yes	characteristic
95	C₄	#2227	1	2	yes	—
96	C₂	—	2	1	no	—
97	C₂	—	2	2	no	—
98	C₂	#13515	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13636

C₂

yes

maximal, characteristic

#13627

C₂

yes

maximal, characteristic

#13169

C₂

yes

maximal

#7480

C₂

yes

maximal

#7475

C₂

yes

maximal, characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

#2158

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1945

—

#1425

—

C₄⋊C₃₂

—

#1382

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

#368

—

#368

—

#367

Q₈

yes

—

#366

—

C₄×C₁₆

—

C₄×C₁₆

—

#339

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#135

yes

characteristic

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

Q₃₂

yes

characteristic

C₄²

Q₃₂

yes

characteristic

C₁₆

—

C₁₆

#123

yes

characteristic

C₁₆

#125

yes

—

C₁₆

C₄⋊Q₈

yes

characteristic

C₂³

#508

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#466

yes

characteristic

C₂×C₄

#474

yes

—

C₂×C₄

C₂×Q₃₂

yes

characteristic

C₂×C₄

#509

yes

—

C₈

—

C₈

—

C₈

yes

C₈

yes

C₈

yes

—

C₂²

—

C₂²

#2216

yes

characteristic

C₂²

#2228

yes

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

—

C₂

#13515

yes

characteristic

	subgroup	G/H
center	C₄	#2218
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13515
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#2218

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

C₂

C₂³

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#13636	2
#13627	2
#13169	2
#7480	2
#7475	2

subgroup

index

Subgroups — #153393

Conjugacy classes of subgroups

Maximal subgroups