Group #31918038 — subgroups

Subgroups — Q₆₄×Dic₇₁ #31918038

Order 18176.

190 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 75 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	Q₃₂×Dic₇₁	C₂	1	2	yes	maximal
2	#22586540	C₂	1	1	yes	maximal, characteristic
3	#22586521	C₂	1	2	yes	maximal
4	C₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	Q₁₆×Dic₇₁	—	2	2	no	—
7	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
8	#17985825	C₂²	1	2	yes	—
9	#17985803	—	2	2	no	—
10	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₇₁	C₂²	1	2	yes	—
12	Q₆₄×C₇₁	C₄	1	4	yes	complemented
13	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
14	Q₈×Dic₇₁	—	4	2	no	—
15	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
16	#14556062	—	2	2	no	—
17	#14556052	—	4	2	no	—
18	C₈×Dic₇₁	D₄	1	1	yes	characteristic
19	C₇₁⋊C₃₂	—	2	1	no	—
20	C₂×Q₁₆×C₇₁	—	2	2	no	—
21	Q₃₂×C₇₁	—	2	2	no	—
22	Q₃₂×C₇₁	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
24	C₃₂×C₇₁	C₂×C₄	1	2	yes	—
25	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
26	C₄⋊Dic₇₁	—	4	2	no	—
27	C₄⋊Dic₇₁	—	8	2	no	—
28	C₄×Dic₇₁	—	8	2	no	—
29	C₄×Dic₇₁	D₈	1	1	yes	characteristic
30	C₇₁⋊C₁₆	—	2	1	no	—
31	C₂×Q₈×C₇₁	—	4	2	no	—
32	Q₁₆×C₇₁	—	2	4	no	—
33	Q₁₆×C₇₁	—	4	2	no	—
34	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₁₆×C₇₁	#41	1	1	yes	characteristic
37	Q₈×C₇₁	—	4	4	no	—
38	Q₈×C₇₁	—	8	2	no	—
39	C₂×C₄×C₇₁	—	8	2	no	—
40	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
41	C₂×Dic₇₁	—	8	2	no	—
42	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
43	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
44	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
45	C₈×C₇₁	#135	1	1	yes	characteristic
46	C₇₁⋊C₈	—	2	1	no	—
47	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
48	C₄×C₇₁	—	8	4	no	—
49	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
50	C₄×C₇₁	#508	1	1	yes	characteristic
51	Dic₇₁	—	2	1	no	—
52	Dic₇₁	—	16	2	no	—
53	Dic₇₁	Q₆₄	1	2	yes	direct product factor
54	C₄×Q₆₄	—	71	1	no	maximal, Sylow
55	C₂×C₇₁	C₄×D₁₆	1	1	yes	characteristic
56	C₂×C₇₁	C₂×Q₆₄	1	1	yes	characteristic
57	C₂×C₇₁	#2262	1	1	yes	characteristic
58	C₂×Q₆₄	D₇₁	1	1	yes	characteristic
59	C₄×Q₃₂	—	71	2	no	—
60	#1423	—	71	1	no	—
61	#1416	—	71	2	no	—
62	C₄×C₃₂	—	71	1	no	—
63	C₇₁	C₄×Q₆₄	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₃₂	C₂×D₇₁	1	2	yes	—
65	C₄×Q₁₆	—	142	2	no	—
66	Q₆₄	Dic₇₁	1	4	yes	direct product factor
67	C₂×C₃₂	—	71	1	no	—
68	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
69	#366	—	71	2	no	—
70	#358	—	142	2	no	—
71	C₄×C₁₆	—	71	1	no	—
72	C₂×Q₁₆	—	2	2	no	—
73	C₄×Q₈	—	284	2	no	—
74	Q₃₂	—	2	2	no	—
75	Q₃₂	C₂×Dic₇₁	1	4	yes	—
76	C₂×C₁₆	—	71	1	no	—
77	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
78	#107	—	142	2	no	—
79	#103	—	284	2	no	—
80	C₄×C₈	—	71	1	no	—
81	C₃₂	—	142	1	no	—
82	C₃₂	C₂×Dic₇₁	1	2	yes	—
83	C₂×Q₈	—	4	2	no	—
84	Q₁₆	—	2	4	no	—
85	Q₁₆	—	4	2	no	—
86	C₂×C₈	—	71	1	no	—
87	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
88	C₄⋊C₄	—	284	2	no	—
89	C₄⋊C₄	—	568	2	no	—
90	C₄²	—	568	2	no	—
91	C₄²	—	71	1	no	—
92	C₁₆	—	142	1	no	—
93	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
94	C₁₆	#11765864	1	1	yes	characteristic
95	Q₈	—	4	4	no	—
96	Q₈	—	8	2	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	568	2	no	—
99	C₂×C₄	—	71	1	no	—
100	C₂×C₄	—	71	1	no	—
101	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
102	C₈	—	142	1	no	—
103	C₈	D₄×Dic₇₁	1	1	yes	characteristic
104	C₈	#14556147	1	1	yes	characteristic
105	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
106	C₄	—	8	4	no	—
107	C₄	—	71	2	no	—
108	C₄	—	1136	2	no	—
109	C₄	—	142	1	no	—
110	C₄	D₈×Dic₇₁	1	1	yes	characteristic
111	C₄	#17986428	1	1	yes	characteristic
112	C₂	D₁₆×Dic₇₁	1	1	yes	characteristic
113	C₂	Q₆₄×D₇₁	1	1	yes	characteristic
114	C₂	#22590650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₃₂×Dic₇₁

C₂

yes

maximal

#22586540

C₂

yes

maximal, characteristic

#22586521

C₂

yes

maximal

C₃₂×Dic₇₁

C₂

yes

maximal, characteristic

C₂×Q₆₄×C₇₁

C₂

yes

maximal, characteristic

Q₁₆×Dic₇₁

—

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

#17985825

C₂²

yes

—

#17985803

—

C₁₆×Dic₇₁

C₂²

yes

characteristic

C₂×Q₃₂×C₇₁

C₂²

yes

—

Q₆₄×C₇₁

C₄

yes

complemented

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

Q₈×Dic₇₁

—

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

#14556062

—

#14556052

—

C₈×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₃₂

—

C₂×Q₁₆×C₇₁

—

Q₃₂×C₇₁

—

Q₃₂×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₃₂×C₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₈

yes

characteristic

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

D₁₆

yes

characteristic

C₂×Dic₇₁

D₁₆

yes

characteristic

C₈×C₇₁

C₄×D₄

yes

characteristic

C₈×C₇₁

#135

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₄×D₈

yes

characteristic

C₄×C₇₁

#508

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

Q₆₄

yes

direct product factor

C₄×Q₆₄

—

maximal, Sylow

C₂×C₇₁

C₄×D₁₆

yes

characteristic

C₂×C₇₁

C₂×Q₆₄

yes

characteristic

C₂×C₇₁

#2262

yes

characteristic

C₂×Q₆₄

D₇₁

yes

characteristic

C₄×Q₃₂

—

#1423

—

#1416

—

C₄×C₃₂

—

C₇₁

C₄×Q₆₄

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₇₁

yes

—

C₄×Q₁₆

—

142

—

Q₆₄

Dic₇₁

yes

direct product factor

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₁

yes

characteristic

#366

—

#358

—

142

—

C₄×C₁₆

—

C₂×Q₁₆

—

C₄×Q₈

—

284

—

Q₃₂

—

Q₃₂

C₂×Dic₇₁

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#107

—

142

—

#103

—

284

—

C₄×C₈

—

C₃₂

—

142

—

C₃₂

C₂×Dic₇₁

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₇₁

yes

characteristic

C₄⋊C₄

—

284

—

C₄⋊C₄

—

568

—

C₄²

—

568

—

C₄²

—

C₁₆

—

142

—

C₁₆

C₂²×Dic₇₁

yes

characteristic

C₁₆

#11765864

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

568

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

D₈×D₇₁

yes

characteristic

102

C₈

—

142

—

103

C₈

D₄×Dic₇₁

yes

characteristic

104

C₈

#14556147

yes

characteristic

105

C₂²

D₁₆×D₇₁

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

1136

—

109

C₄

—

142

—

110

C₄

D₈×Dic₇₁

yes

characteristic

111

C₄

#17986428

yes

characteristic

112

C₂

D₁₆×Dic₇₁

yes

characteristic

113

C₂

Q₆₄×D₇₁

yes

characteristic

114

C₂

#22590650

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₁
commutator	C₁₆×C₇₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₂×Q₆₄×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂×Q₆₄	D₇₁
hypocenter	C₇₁	C₄×Q₆₄
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

subgroup

G/H

center

C₂²

D₁₆×D₇₁

commutator

C₁₆×C₇₁

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₂×Q₆₄×C₇₁

C₂

socle, prime radical

C₂²×C₇₁

C₂×D₁₆

cosocle

C₂×C₁₆×C₇₁

C₂³

hypercenter

C₂×Q₆₄

D₇₁

hypocenter

C₇₁

C₄×Q₆₄

LCS

G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

N	H ≅ G/N	type
12. Q₆₄×C₇₁	C₄	semidirect
53. Dic₇₁	Q₆₄	direct (N × H)
63. C₇₁	C₄×Q₆₄	semidirect
66. Q₆₄	Dic₇₁	direct (N × H)

H ≅ G/N

type

12. Q₆₄×C₇₁

C₄

semidirect

53. Dic₇₁

Q₆₄

direct (N × H)

63. C₇₁

C₄×Q₆₄

semidirect

66. Q₆₄

Dic₇₁

direct (N × H)

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

subgroup	index
Q₃₂×Dic₇₁	2
#22586540	2
#22586521	2
C₃₂×Dic₇₁	2
C₂×Q₆₄×C₇₁	2
C₄×Q₆₄	71

subgroup

index

Q₃₂×Dic₇₁

#22586540

#22586521

C₃₂×Dic₇₁

C₂×Q₆₄×C₇₁

C₄×Q₆₄

Subgroups — Q₆₄×Dic₇₁ #31918038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q64×Dic71 #31918038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₆₄×Dic₇₁ #31918038