Group #72322983 — subgroups

Subgroups — #72322983

Order 1024.

179 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 73 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	#151084	C₂	1	1	yes	maximal, characteristic
2	#122962	C₂	1	2	yes	maximal
3	#93877	C₂	1	1	yes	maximal, characteristic
4	#93758	C₂	1	2	yes	maximal
5	#93668	C₂	1	1	yes	maximal, characteristic
6	#13653	C₄	1	1	yes	characteristic
7	#12059	—	2	2	no	—
8	#12044	C₂²	1	2	yes	—
9	#7463	C₄	1	1	yes	characteristic
10	#7386	C₄	1	2	yes	—
11	#7332	C₂²	1	1	yes	characteristic
12	#7330	C₂²	1	1	yes	characteristic
13	#7099	C₂²	1	2	yes	—
14	#7071	—	2	2	no	—
15	#7053	C₂²	1	1	yes	characteristic
16	C₂×Q₆₄	C₈	1	1	yes	characteristic
17	Q₈×C₁₆	—	4	2	no	—
18	C₄×Q₃₂	C₂×C₄	1	2	yes	—
19	C₈×Q₁₆	—	2	2	no	—
20	#1424	C₈	1	1	yes	characteristic
21	#1416	C₈	1	2	yes	—
22	#1397	—	2	1	no	—
23	#1397	C₂×C₄	1	1	yes	characteristic
24	#1397	C₂×C₄	1	1	yes	characteristic
25	#1372	C₂×C₄	1	2	yes	—
26	#1367	—	2	2	no	—
27	Q₈⋊C₁₆	—	4	2	no	—
28	#1332	—	2	2	no	—
29	#1312	D₄	1	2	yes	—
30	#1311	C₂³	1	1	yes	characteristic
31	C₂×Q₃₂	C₂×C₈	1	2	yes	—
32	C₈×Q₈	—	4	2	no	—
33	C₄×Q₁₆	—	2	2	no	—
34	Q₆₄	C₁₆	1	4	yes	complemented
35	C₂×C₃₂	—	2	1	no	—
36	C₂×C₃₂	—	4	1	no	—
37	C₂×C₃₂	C₂×C₈	1	1	yes	characteristic
38	C₂×C₃₂	C₂×C₈	1	1	yes	characteristic
39	#366	C₂×C₈	1	2	yes	—
40	C₄⋊C₁₆	—	4	2	no	—
41	C₄⋊C₁₆	—	8	2	no	—
42	#358	—	2	2	no	—
43	C₄×C₁₆	—	2	2	no	—
44	C₄×C₁₆	—	8	2	no	—
45	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
46	C₄×C₁₆	#41	1	1	yes	characteristic
47	#335	—	2	2	no	—
48	Q₈⋊C₈	—	4	2	no	—
49	C₈²	C₂×D₄	1	1	yes	characteristic
50	C₂×Q₁₆	—	2	2	no	—
51	C₄×Q₈	—	4	2	no	—
52	Q₃₂	—	2	2	no	—
53	Q₃₂	C₂×C₁₆	1	4	yes	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₂×C₁₆	—	8	2	no	—
57	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₁₆	#131	1	1	yes	characteristic
59	#107	—	2	2	no	—
60	C₄⋊C₈	—	4	2	no	—
61	C₄⋊C₈	—	8	2	no	—
62	#103	—	4	2	no	—
63	C₄×C₈	—	8	2	no	—
64	C₄×C₈	C₄×D₄	1	1	yes	characteristic
65	C₄×C₈	AGL(1,ℤ/8)	1	2	yes	—
66	C₃₂	—	2	1	no	—
67	C₃₂	—	4	1	no	—
68	C₃₂	—	8	1	no	—
69	C₃₂	C₂×C₁₆	1	2	yes	—
70	C₂×Q₈	—	4	2	no	—
71	Q₁₆	—	2	4	no	—
72	Q₁₆	—	4	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	8	2	no	—
75	C₂×C₈	C₈×D₄	1	1	yes	characteristic
76	C₂×C₈	#444	1	1	yes	characteristic
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	8	2	no	—
79	C₄²	—	8	2	no	—
80	C₄²	#442	1	1	yes	characteristic
81	C₁₆	—	2	1	no	—
82	C₁₆	—	4	1	no	—
83	C₁₆	—	8	2	no	—
84	C₁₆	—	16	2	no	—
85	C₁₆	C₂²×C₁₆	1	1	yes	characteristic
86	C₁₆	#504	1	1	yes	characteristic
87	Q₈	—	4	4	no	—
88	Q₈	—	8	2	no	—
89	C₂×C₄	—	8	2	no	—
90	C₂×C₄	—	8	2	no	—
91	C₂×C₄	#1580	1	1	yes	characteristic
92	C₂×C₄	AGL(1,ℤ/16)	1	2	yes	—
93	C₈	—	2	1	no	—
94	C₈	—	4	2	no	—
95	C₈	—	16	2	no	—
96	C₈	D₄×C₁₆	1	1	yes	characteristic
97	C₈	#2171	1	1	yes	characteristic
98	C₂²	#12067	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	8	4	no	—
101	C₄	—	16	2	no	—
102	C₄	#12062	1	1	yes	characteristic
103	C₄	#12068	1	1	yes	characteristic
104	C₂	#122963	1	1	yes	characteristic
105	C₂	#151105	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#151084

C₂

yes

maximal, characteristic

#122962

C₂

yes

maximal

#93877

C₂

yes

maximal, characteristic

#93758

C₂

yes

maximal

#93668

C₂

yes

maximal, characteristic

#13653

C₄

yes

characteristic

#12059

—

#12044

C₂²

yes

—

#7463

C₄

yes

characteristic

#7386

C₄

yes

—

#7332

C₂²

yes

characteristic

#7330

C₂²

yes

characteristic

#7099

C₂²

yes

—

#7071

—

#7053

C₂²

yes

characteristic

C₂×Q₆₄

C₈

yes

characteristic

Q₈×C₁₆

—

C₄×Q₃₂

C₂×C₄

yes

—

C₈×Q₁₆

—

#1424

C₈

yes

characteristic

#1416

C₈

yes

—

#1397

—

#1397

C₂×C₄

yes

characteristic

#1397

C₂×C₄

yes

characteristic

#1372

C₂×C₄

yes

—

#1367

—

Q₈⋊C₁₆

—

#1332

—

#1312

D₄

yes

—

#1311

C₂³

yes

characteristic

C₂×Q₃₂

C₂×C₈

yes

—

C₈×Q₈

—

C₄×Q₁₆

—

Q₆₄

C₁₆

yes

complemented

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×C₈

yes

characteristic

C₂×C₃₂

C₂×C₈

yes

characteristic

#366

C₂×C₈

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂²×C₄

yes

C₄×C₁₆

yes

—

Q₈⋊C₈

—

C₈²

C₂×D₄

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

C₂×C₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₈

yes

C₂×C₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₄×D₄

yes

characteristic

C₄×C₈

AGL(1,ℤ/8)

yes

—

C₃₂

—

C₃₂

—

C₃₂

—

C₃₂

C₂×C₁₆

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₄

yes

characteristic

C₂×C₈

#444

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

#442

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

C₂²×C₁₆

yes

characteristic

C₁₆

#504

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#1580

yes

characteristic

C₂×C₄

AGL(1,ℤ/16)

yes

—

C₈

—

C₈

—

C₈

—

C₈

D₄×C₁₆

yes

C₈

yes

C₂²

yes

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#12062

yes

characteristic

103

C₄

#12068

yes

characteristic

104

C₂

#122963

yes

characteristic

105

C₂

#151105

yes

—

	subgroup	G/H
center, socle, prime radical	C₂²	#12067
commutator	C₁₆	C₂²×C₁₆
Frattini, cosocle	#1311	C₂³
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #7053 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

#12067

commutator

C₁₆

C₂²×C₁₆

Frattini, cosocle

#1311

C₂³

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #7053 ◁ G

LECS

G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
34. Q₆₄	C₁₆	semidirect

H ≅ G/N

type

34. Q₆₄

C₁₆

semidirect

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

subgroup	index
#151084	2
#122962	2
#93877	2
#93758	2
#93668	2

subgroup

index

Subgroups — #72322983

Conjugacy classes of subgroups

Complements

Maximal subgroups