Group #31885087 — subgroups

Subgroups — #31885087

Order 18176.

132 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22590593	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₇₁⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1371	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₆₈	—	2	1	no	—
5	C₂×Dic₅₆₈	C₄	1	2	yes	—
6	#17986379	—	2	1	no	—
7	C₂×C₇₁⋊C₃₂	—	2	1	no	—
8	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	#335×C₇₁	—	2	1	no	—
10	C₂×Dic₂₈₄	—	2	2	no	—
11	C₂×Dic₂₈₄	—	4	1	no	—
12	#14556128	—	4	1	no	—
13	Dic₅₆₈	—	4	2	no	—
14	Dic₅₆₈	—	4	1	no	—
15	Dic₅₆₈	—	4	1	no	—
16	C₇₁⋊C₃₂	—	4	1	no	—
17	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
19	C₄⋊C₈×C₇₁	—	4	1	no	—
20	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
21	C₂×Dic₁₄₂	—	4	2	no	—
22	Dic₂₈₄	—	4	2	no	—
23	Dic₂₈₄	—	4	2	no	—
24	Dic₂₈₄	—	8	2	no	—
25	C₄⋊Dic₇₁	—	8	1	no	—
26	C₄⋊Dic₇₁	—	8	1	no	—
27	C₂×C₈×C₇₁	—	8	1	no	—
28	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
30	C₄²×C₇₁	D₈	1	1	yes	characteristic
31	C₁₆×C₇₁	—	2	1	no	—
32	C₁₆×C₇₁	D₈	1	1	yes	characteristic
33	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
35	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
36	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
37	C₂×Dic₇₁	—	8	2	no	—
38	Dic₁₄₂	—	8	2	no	—
39	Dic₁₄₂	—	8	2	no	—
40	C₈×C₇₁	—	2	1	no	—
41	C₈×C₇₁	—	16	1	no	—
42	C₈×C₇₁	#100	1	1	yes	characteristic
43	C₈×C₇₁	#102	1	1	yes	characteristic
44	C₂²×C₇₁	#357	1	1	yes	characteristic
45	C₄×C₇₁	—	2	1	no	—
46	C₄×C₇₁	#331	1	1	yes	characteristic
47	C₄×C₇₁	#361	1	1	yes	characteristic
48	C₄×C₇₁	SD₆₄	1	1	yes	characteristic
49	C₄×C₇₁	Q₆₄	1	1	yes	characteristic
50	Dic₇₁	—	16	2	no	—
51	#7445	—	71	1	no	maximal, Sylow
52	C₂×C₇₁	#1361	1	1	yes	characteristic
53	C₂×C₇₁	#1416	1	1	yes	characteristic
54	C₂×C₇₁	#1420	1	1	yes	characteristic
55	#2247	—	71	1	no	—
56	C₄⋊C₃₂	—	71	1	no	—
57	#1371	D₇₁	1	1	yes	characteristic
58	C₇₁	#7445	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₃₂	—	71	2	no	—
60	C₂×Q₃₂	—	142	1	no	—
61	#494	—	142	1	no	—
62	C₂×C₃₂	—	142	1	no	—
63	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
64	#335	—	2	1	no	—
65	C₂×Q₁₆	—	284	1	no	—
66	C₂×Q₁₆	—	142	2	no	—
67	C₄⋊Q₈	—	284	1	no	—
68	Q₃₂	—	284	2	no	—
69	Q₃₂	—	284	1	no	—
70	Q₃₂	—	284	1	no	—
71	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
72	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
73	C₄⋊C₈	—	4	1	no	—
74	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
75	C₃₂	—	284	1	no	—
76	C₂×Q₈	—	284	2	no	—
77	Q₁₆	—	284	2	no	—
78	Q₁₆	—	284	2	no	—
79	Q₁₆	—	568	2	no	—
80	C₂×C₈	—	8	1	no	—
81	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
82	C₂×C₈	#11765852	1	1	yes	characteristic
83	C₄⋊C₄	—	568	1	no	—
84	C₄⋊C₄	—	568	1	no	—
85	C₄²	#11765850	1	1	yes	characteristic
86	C₁₆	—	2	1	no	—
87	C₁₆	D₅₆₈	1	1	yes	characteristic
88	C₁₆	#11765854	1	1	yes	characteristic
89	Q₈	—	568	2	no	—
90	Q₈	—	568	2	no	—
91	C₂×C₄	—	568	2	no	—
92	C₂×C₄	#14556048	1	1	yes	characteristic
93	C₂×C₄	#14556069	1	1	yes	characteristic
94	C₂×C₄	#14556071	1	1	yes	characteristic
95	C₈	—	2	1	no	—
96	C₈	—	16	1	no	—
97	C₈	#14556044	1	1	yes	characteristic
98	C₈	#14556050	1	1	yes	characteristic
99	C₂²	#17985799	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	1136	2	no	—
102	C₄	#17985742	1	1	yes	characteristic
103	C₄	#17985811	1	1	yes	characteristic
104	C₄	#17985837	1	1	yes	characteristic
105	C₄	#17985839	1	1	yes	characteristic
106	C₂	#22586385	1	1	yes	characteristic
107	C₂	#22586520	1	1	yes	characteristic
108	C₂	#22586532	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590593

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₃₂)

C₂

yes

maximal, characteristic

C₇₁×#1371

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

—

C₂×Dic₅₆₈

C₄

yes

—

#17986379

—

C₂×C₇₁⋊C₃₂

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#335×C₇₁

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

Dic₅₆₈

—

Dic₅₆₈

—

Dic₅₆₈

—

C₇₁⋊C₃₂

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

SD₁₆

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

D₈

yes

characteristic

C₁₆×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

#102

yes

characteristic

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₂×C₄×C₇₁

SD₃₂

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

SD₆₄

yes

characteristic

C₄×C₇₁

Q₆₄

yes

characteristic

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₄⋊C₃₂

—

#1371

D₇₁

yes

characteristic

C₇₁

#7445

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

142

—

#494

—

142

—

C₂×C₃₂

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

C₂×Q₁₆

—

284

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

284

—

Q₃₂

—

284

—

Q₃₂

—

284

—

Q₃₂

—

284

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

D₂₈₄

yes

characteristic

C₄⋊C₈

—

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

C₃₂

—

284

—

C₂×Q₈

—

284

—

Q₁₆

—

284

—

Q₁₆

—

284

—

Q₁₆

—

568

—

C₂×C₈

—

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₈

#11765852

yes

characteristic

C₄⋊C₄

—

568

—

C₄⋊C₄

—

568

—

C₄²

#11765850

yes

characteristic

C₁₆

—

C₁₆

D₅₆₈

yes

characteristic

C₁₆

#11765854

yes

characteristic

Q₈

—

568

—

Q₈

—

568

—

C₂×C₄

—

568

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

yes

100

C₄

—

101

C₄

—

1136

—

102

C₄

#17985742

yes

characteristic

103

C₄

#17985811

yes

characteristic

104

C₄

#17985837

yes

characteristic

105

C₄

#17985839

yes

characteristic

106

C₂

#22586385

yes

characteristic

107

C₂

#22586520

yes

characteristic

108

C₂

#22586532

yes

characteristic

	subgroup	G/H
center	C₂²	#17985799
commutator	C₂×C₁₆×C₇₁	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×#1371	C₂
socle, prime radical	C₂²×C₇₁	#357
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	#1371	D₇₁
hypocenter	C₇₁	#7445
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1371

subgroup

G/H

center

C₂²

#17985799

commutator

C₂×C₁₆×C₇₁

C₂×C₄

Frattini

C₄×C₁₆

C₂×D₇₁

Fitting

C₇₁×#1371

C₂

socle, prime radical

C₂²×C₇₁

#357

cosocle

C₄×C₁₆×C₇₁

C₂²

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1371

N	H ≅ G/N	type
58. C₇₁	#7445	semidirect

H ≅ G/N

type

58. C₇₁

#7445

semidirect

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

subgroup	index
#22590593	2
C₄⋊(C₇₁⋊C₃₂)	2
C₇₁×#1371	2
#7445	71

subgroup

index

#22590593

C₄⋊(C₇₁⋊C₃₂)

C₇₁×#1371

#7445

Subgroups — #31885087

Conjugacy classes of subgroups

Complements

Maximal subgroups