Group #72224639 — subgroups

Subgroups — #72224639

Order 1024.

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

normal subgroups only

This group has 199 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	#151085	C₂	1	1	yes	maximal, characteristic
2	#122963	C₂	1	2	yes	maximal, complemented
3	#93877	C₂	1	1	yes	maximal, characteristic, complemented
4	#93757	C₂	1	2	yes	maximal, complemented
5	#93668	C₂	1	1	yes	maximal, characteristic, complemented
6	#13654	C₄	1	1	yes	characteristic
7	#12062	—	2	2	no	—
8	#12045	C₂²	1	2	yes	—
9	#7463	C₄	1	1	yes	characteristic
10	#7385	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	#7070	—	2	2	no	—
15	#7053	C₂²	1	1	yes	characteristic
16	C₂×D₃₂	C₈	1	1	yes	characteristic
17	C₄×D₁₆	C₂×C₄	1	2	yes	—
18	D₄×C₁₆	—	4	2	no	—
19	C₈×D₈	—	2	2	no	—
20	#1424	C₈	1	1	yes	characteristic
21	#1415	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	#1331	—	2	2	no	—
28	D₄⋊C₁₆	—	4	2	no	—
29	#1312	D₄	1	2	yes	—
30	#1311	C₂³	1	1	yes	characteristic
31	C₂×D₁₆	C₂×C₈	1	2	yes	—
32	C₂²×C₁₆	—	8	2	no	—
33	C₄×D₈	—	2	2	no	—
34	C₈×D₄	—	4	2	no	—
35	D₃₂	C₁₆	1	4	yes	complemented
36	C₂×C₃₂	—	2	1	no	—
37	C₂×C₃₂	—	4	1	no	—
38	C₂×C₃₂	C₂×C₈	1	1	yes	characteristic
39	C₂×C₃₂	C₂×C₈	1	1	yes	characteristic
40	#366	C₂×C₈	1	2	yes	—
41	C₄⋊C₁₆	—	4	2	no	—
42	#357	—	2	2	no	—
43	C₂²⋊C₁₆	—	8	2	no	—
44	C₄×C₁₆	—	2	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	D₄⋊C₈	—	4	2	no	—
49	C₈²	C₂×D₄	1	1	yes	characteristic
50	C₂×D₈	—	2	2	no	—
51	C₂²×C₈	—	8	2	no	—
52	C₄×D₄	—	4	2	no	—
53	D₁₆	—	2	2	no	—
54	D₁₆	C₂×C₁₆	1	4	yes	complemented
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	4	2	no	—
57	C₂×C₁₆	—	8	2	no	—
58	C₂×C₁₆	—	16	4	no	—
59	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₁₆	#131	1	1	yes	characteristic
61	#107	—	2	2	no	—
62	C₄⋊C₈	—	4	2	no	—
63	#102	—	4	2	no	—
64	C₂²⋊C₈	—	8	2	no	—
65	C₄×C₈	C₄×D₄	1	1	yes	characteristic
66	C₄×C₈	AGL(1,ℤ/8)	1	2	yes	—
67	C₃₂	—	2	1	no	—
68	C₃₂	—	4	1	no	—
69	C₃₂	—	8	1	no	—
70	C₃₂	C₂×C₁₆	1	2	yes	complemented
71	C₂×D₄	—	4	2	no	—
72	C₂²×C₄	—	8	2	no	—
73	D₈	—	2	4	no	—
74	D₈	—	4	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	8	2	no	—
77	C₂×C₈	—	16	2	no	—
78	C₂×C₈	—	16	2	no	—
79	C₂×C₈	C₈×D₄	1	1	yes	characteristic
80	C₂×C₈	#444	1	1	yes	characteristic
81	C₄⋊C₄	—	4	2	no	—
82	C₂²⋊C₄	—	8	2	no	—
83	C₄²	#442	1	1	yes	characteristic
84	C₁₆	—	2	1	no	—
85	C₁₆	—	4	1	no	—
86	C₁₆	—	8	2	no	—
87	C₁₆	—	16	2	no	—
88	C₁₆	C₂²×C₁₆	1	1	yes	characteristic
89	C₁₆	#504	1	1	yes	characteristic
90	C₂³	—	8	2	no	—
91	D₄	—	4	4	no	—
92	D₄	—	8	2	no	—
93	C₂×C₄	—	8	2	no	—
94	C₂×C₄	—	16	2	no	—
95	C₂×C₄	—	16	2	no	—
96	C₂×C₄	#1580	1	1	yes	characteristic
97	C₂×C₄	AGL(1,ℤ/16)	1	2	yes	—
98	C₈	—	2	1	no	—
99	C₈	—	4	2	no	—
100	C₈	—	16	2	no	—
101	C₈	D₄×C₁₆	1	1	yes	characteristic
102	C₈	#2171	1	1	yes	characteristic
103	C₂²	—	8	4	no	—
104	C₂²	—	16	2	no	—
105	C₂²	—	16	2	no	—
106	C₂²	#12067	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	16	2	no	—
109	C₄	#12062	1	1	yes	characteristic
110	C₄	#12068	1	1	yes	characteristic
111	C₂	—	16	4	no	—
112	C₂	#122963	1	1	yes	characteristic
113	C₂	AGL(1,ℤ/32)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#151085

C₂

yes

maximal, characteristic

#122963

C₂

yes

maximal, complemented

#93877

C₂

yes

maximal, characteristic, complemented

#93757

C₂

yes

maximal, complemented

#93668

C₂

yes

maximal, characteristic, complemented

#13654

C₄

yes

characteristic

#12062

—

#12045

C₂²

yes

—

#7463

C₄

yes

characteristic

#7385

C₄

yes

—

#7332

C₂²

yes

characteristic

#7330

C₂²

yes

characteristic

#7099

C₂²

yes

—

#7070

—

#7053

C₂²

yes

characteristic

C₂×D₃₂

C₈

yes

characteristic

C₄×D₁₆

C₂×C₄

yes

—

D₄×C₁₆

—

C₈×D₈

—

#1424

C₈

yes

characteristic

#1415

C₈

yes

—

#1397

—

#1397

C₂×C₄

yes

characteristic

#1397

C₂×C₄

yes

characteristic

#1372

C₂×C₄

yes

—

#1367

—

#1331

—

D₄⋊C₁₆

—

#1312

D₄

yes

—

#1311

C₂³

yes

characteristic

C₂×D₁₆

C₂×C₈

yes

—

C₂²×C₁₆

—

C₄×D₈

—

C₈×D₄

—

D₃₂

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

—

#357

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂²×C₄

yes

C₄×C₁₆

yes

—

D₄⋊C₈

—

C₈²

C₂×D₄

yes

characteristic

C₂×D₈

—

C₂²×C₈

—

C₄×D₄

—

D₁₆

—

D₁₆

C₂×C₁₆

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₈

yes

C₂×C₁₆

yes

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

C₄×D₄

yes

characteristic

C₄×C₈

AGL(1,ℤ/8)

yes

—

C₃₂

—

C₃₂

—

C₃₂

—

C₃₂

C₂×C₁₆

yes

complemented

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₄

yes

characteristic

C₂×C₈

#444

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

#442

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

C₂²×C₁₆

yes

characteristic

C₁₆

#504

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#1580

yes

characteristic

C₂×C₄

AGL(1,ℤ/16)

yes

—

C₈

—

C₈

—

100

C₈

—

101

C₈

D₄×C₁₆

yes

characteristic

102

C₈

#2171

yes

characteristic

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#12067

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#12062

yes

characteristic

110

C₄

#12068

yes

characteristic

111

C₂

—

112

C₂

#122963

yes

characteristic

113

C₂

AGL(1,ℤ/32)

yes

—

	subgroup	G/H
center, socle	C₂²	#12067
commutator	C₁₆	C₂²×C₁₆
Frattini, cosocle	#1311	C₂³
prime radical	C₂×D₃₂	C₈
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

C₂²

#12067

commutator

C₁₆

C₂²×C₁₆

Frattini, cosocle

#1311

C₂³

prime radical

C₂×D₃₂

C₈

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
2. #122963	C₂	semidirect
3. #93877	C₂	semidirect
4. #93757	C₂	semidirect
5. #93668	C₂	semidirect
35. D₃₂	C₁₆	semidirect
54. D₁₆	C₂×C₁₆	semidirect
70. C₃₂	C₂×C₁₆	semidirect

H ≅ G/N

type

2. #122963

C₂

semidirect

3. #93877

C₂

semidirect

4. #93757

C₂

semidirect

5. #93668

C₂

semidirect

35. D₃₂

C₁₆

semidirect

54. D₁₆

C₂×C₁₆

semidirect

70. C₃₂

C₂×C₁₆

semidirect

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

subgroup	index
#151085	2
#122963	2
#93877	2
#93757	2
#93668	2

subgroup

index

Subgroups — #72224639

Conjugacy classes of subgroups

Complements

Maximal subgroups