Group #31884932 — subgroups

Subgroups — #31884932

Order 18176.

110 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 59 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	C₄⋊(C₇₁⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
2	#22586474	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2183	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₃₂	—	2	1	no	—
5	C₂×C₇₁⋊C₃₂	—	2	1	no	—
6	C₈×Q₈×C₇₁	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆×C₇₁	—	2	1	no	—
8	C₄⋊C₁₆×C₇₁	C₄	1	1	yes	characteristic
9	#346×C₇₁	—	2	1	no	—
10	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
11	C₇₁⋊C₃₂	—	2	2	no	—
12	C₇₁⋊C₃₂	—	4	1	no	—
13	C₄×Q₈×C₇₁	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₇₁	—	2	1	no	—
15	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
17	C₄⋊C₈×C₇₁	—	2	1	no	—
18	C₄⋊C₈×C₇₁	C₈	1	1	yes	characteristic
19	C₄×C₈×C₇₁	—	2	1	no	—
20	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×Q₈×C₇₁	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₇₁	—	2	1	no	—
23	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
25	C₄⋊C₄×C₇₁	—	2	1	no	—
26	C₄⋊C₄×C₇₁	C₁₆	1	1	yes	characteristic
27	C₄²×C₇₁	—	2	1	no	—
28	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
29	C₁₆×C₇₁	—	2	1	no	—
30	C₁₆×C₇₁	—	4	1	no	—
31	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
32	C₁₆×C₇₁	Q₁₆	1	1	yes	characteristic
33	Q₈×C₇₁	—	4	1	no	—
34	C₂×C₄×C₇₁	—	2	1	no	—
35	C₂×C₄×C₇₁	—	2	1	no	—
36	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
39	C₈×C₇₁	—	2	1	no	—
40	C₈×C₇₁	—	4	1	no	—
41	C₈×C₇₁	#103	1	1	yes	characteristic
42	C₈×C₇₁	#104	1	1	yes	characteristic
43	C₂²×C₇₁	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₄×C₇₁	—	2	1	no	—
45	C₄×C₇₁	—	2	1	no	—
46	C₄×C₇₁	—	4	1	no	—
47	C₄×C₇₁	—	4	1	no	—
48	C₄×C₇₁	Q₈⋊C₈	1	1	yes	characteristic
49	C₄×C₇₁	#350	1	1	yes	characteristic
50	#7393	—	71	1	no	maximal, Sylow
51	C₂×C₇₁	Q₈⋊C₁₆	1	1	yes	characteristic
52	C₂×C₇₁	#1400	1	1	yes	characteristic
53	C₂×C₇₁	#1401	1	1	yes	characteristic
54	#2183	D₇₁	1	1	yes	characteristic
55	C₄⋊C₃₂	—	71	1	no	—
56	#1397	—	71	1	no	—
57	C₇₁	#7393	1	1	yes	characteristic, complemented, Sylow
58	C₈×Q₈	Dic₇₁	1	1	yes	characteristic
59	C₂×C₃₂	—	142	1	no	—
60	C₂×C₃₂	—	142	1	no	—
61	C₄⋊C₁₆	—	2	1	no	—
62	C₄⋊C₁₆	Dic₇₁	1	1	yes	characteristic
63	#346	—	2	1	no	—
64	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
65	C₄×Q₈	C₇₁⋊C₈	1	1	yes	characteristic
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
68	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	C₇₁⋊C₈	1	1	yes	characteristic
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
73	C₃₂	—	284	1	no	—
74	C₃₂	—	142	2	no	—
75	C₂×Q₈	C₇₁⋊C₁₆	1	1	yes	characteristic
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
78	C₂×C₈	#11765849	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	C₇₁⋊C₁₆	1	1	yes	characteristic
81	C₄²	—	2	1	no	—
82	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
83	C₁₆	—	2	1	no	—
84	C₁₆	—	4	1	no	—
85	C₁₆	#11765852	1	1	yes	characteristic
86	C₁₆	#11765855	1	1	yes	characteristic
87	Q₈	—	4	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
91	C₂×C₄	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
92	C₂×C₄	#14556068	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	4	1	no	—
95	C₈	#14556052	1	1	yes	characteristic
96	C₈	#14556055	1	1	yes	characteristic
97	C₂²	C₂²⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	Q₈⋊(C₇₁⋊C₈)	1	1	yes	characteristic
103	C₄	#17985783	1	1	yes	characteristic
104	C₂	Q₈⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
105	C₂	#22586480	1	1	yes	characteristic
106	C₂	#22586482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₇₁⋊C₃₂)

C₂

yes

maximal, characteristic

#22586474

C₂

yes

maximal, characteristic

C₇₁×#2183

C₂

yes

maximal, characteristic

C₂×C₇₁⋊C₃₂

—

C₂×C₇₁⋊C₃₂

—

C₈×Q₈×C₇₁

C₄

yes

characteristic

C₄⋊C₁₆×C₇₁

—

C₄⋊C₁₆×C₇₁

C₄

yes

characteristic

#346×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₇₁⋊C₃₂

—

C₇₁⋊C₃₂

—

C₄×Q₈×C₇₁

C₈

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

C₈

yes

characteristic

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₇₁

C₁₆

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

M₁₆

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₁₆

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

SD₁₆

yes

characteristic

C₁₆×C₇₁

Q₁₆

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

M₃₂

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₂²×C₇₁

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

Q₈⋊C₈

yes

C₄×C₇₁

yes

—

C₂×C₇₁

Q₈⋊C₁₆

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

yes

characteristic

C₄⋊C₃₂

—

#1397

—

C₇₁

#7393

yes

characteristic, complemented, Sylow

C₈×Q₈

Dic₇₁

yes

characteristic

C₂×C₃₂

—

142

—

C₂×C₃₂

—

142

—

C₄⋊C₁₆

—

C₄⋊C₁₆

Dic₇₁

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₄×Q₈

C₇₁⋊C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₇₁⋊C₈

yes

characteristic

C₄×C₈

—

C₄×C₈

C₂×Dic₇₁

yes

characteristic

C₃₂

—

284

—

C₃₂

—

142

—

C₂×Q₈

C₇₁⋊C₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₈

#11765849

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₇₁⋊C₁₆

yes

characteristic

C₄²

—

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

yes

C₁₆

yes

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

C₂×C₄

C₂×C₇₁⋊C₁₆

yes

characteristic

C₂×C₄

#14556068

yes

characteristic

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

C₂²⋊(C₇₁⋊C₁₆)

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

Q₈⋊(C₇₁⋊C₈)

yes

characteristic

103

C₄

#17985783

yes

characteristic

104

C₂

Q₈⋊(C₇₁⋊C₁₆)

yes

characteristic

105

C₂

#22586480

yes

characteristic

106

C₂

#22586482

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇₁
commutator	C₂×C₄×C₇₁	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×#2183	C₂
socle, prime radical	C₂²×C₇₁	C₂²⋊C₁₆
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	#2183	D₇₁
hypocenter	C₇₁	#7393
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #2183

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₇₁

commutator

C₂×C₄×C₇₁

C₂×C₁₆

Frattini

C₄×C₁₆

C₂×D₇₁

Fitting

C₇₁×#2183

C₂

socle, prime radical

C₂²×C₇₁

C₂²⋊C₁₆

cosocle

C₄×C₁₆×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #2183

N	H ≅ G/N	type
57. C₇₁	#7393	semidirect

H ≅ G/N

type

57. C₇₁

#7393

semidirect

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

subgroup	index
C₄⋊(C₇₁⋊C₃₂)	2
#22586474	2
C₇₁×#2183	2
#7393	71

subgroup

index

C₄⋊(C₇₁⋊C₃₂)

#22586474

C₇₁×#2183

#7393

Subgroups — #31884932

Conjugacy classes of subgroups

Complements

Maximal subgroups