Group #31913835 — subgroups

Subgroups — #31913835

Order 18176.

310 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 163 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	#22588504	C₂	1	2	yes	maximal
2	#22586432	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556062	C₂²	1	4	yes	—
5	C₂×#14556060	—	2	2	no	—
6	#17985865	—	2	2	no	—
7	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₇₁	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₇₁	—	2	4	no	—
11	#14556062	—	2	8	no	—
12	#14556060	—	4	4	no	—
13	C₇₁⋊#95	—	4	4	no	—
14	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₇₁	—	2	4	no	—
18	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
19	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₇₁	Q₈	1	2	yes	—
21	C₂²×Dic₇₁	—	4	4	no	—
22	C₄⋊Dic₇₁	—	4	4	no	—
23	C₄⋊Dic₇₁	—	4	4	no	—
24	C₂²×C₄×C₇₁	D₈	1	2	yes	—
25	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
28	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₇₁	#41	1	4	yes	—
31	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
32	C₄²×C₇₁	Q₁₆	1	2	yes	—
33	C₁₆×C₇₁	—	2	4	no	—
34	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄×C₇₁	—	2	4	no	—
36	C₂×C₄×C₇₁	#107	1	4	yes	—
37	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	#124	1	2	yes	—
39	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
40	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
41	C₂×Dic₇₁	—	8	4	no	—
42	C₂×Dic₇₁	—	8	8	no	—
43	C₈×C₇₁	#106	1	4	yes	—
44	C₈×C₇₁	#117	1	2	yes	—
45	C₈×C₇₁	#125	1	2	yes	—
46	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
47	C₂²×C₇₁	#486	1	2	yes	—
48	C₂²×C₇₁	#508	1	4	yes	—
49	C₄×C₇₁	—	2	4	no	—
50	C₄×C₇₁	#382	1	1	yes	characteristic
51	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
52	C₄×C₇₁	#487	1	2	yes	—
53	Dic₇₁	—	16	4	no	—
54	#12889	—	71	1	no	maximal, Sylow
55	C₂×C₇₁	#1834	1	1	yes	characteristic
56	C₂×C₇₁	#2158	1	2	yes	—
57	C₂×C₇₁	#2242	1	4	yes	—
58	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
59	#1846	—	71	2	no	—
60	#1380	—	71	4	no	—
61	C₇₁	#12889	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
63	C₂×#107	—	71	4	no	—
64	C₂×#106	—	142	2	no	—
65	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
66	#382	—	142	2	no	—
67	C₄×C₁₆	Dic₇₁	1	4	yes	complemented
68	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
69	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	142	4	no	—
71	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
74	#107	—	142	8	no	—
75	#106	—	284	4	no	—
76	C₄×C₈	Dic₁₄₂	1	2	yes	—
77	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
78	#95	—	284	4	no	—
79	C₂²×C₄	—	284	4	no	—
80	C₂²×C₄	D₅₆₈	1	2	yes	—
81	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
82	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
83	C₂×C₈	#11765854	1	4	yes	—
84	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
85	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
86	C₂×C₈	#11765865	1	4	yes	—
87	C₄⋊C₄	—	284	4	no	—
88	C₄⋊C₄	—	284	4	no	—
89	C₄²	C₄⋊Dic₇₁	1	2	yes	—
90	C₄²	Dic₂₈₄	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	568	4	no	—
95	C₂×C₄	—	568	8	no	—
96	C₂×C₄	#14556062	1	4	yes	—
97	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
98	C₂×C₄	#14556116	1	2	yes	—
99	C₂×C₄	C₂×#11765854	1	2	yes	—
100	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
101	C₈	#14556060	1	4	yes	—
102	C₈	#14556087	1	2	yes	—
103	C₈	#14556120	1	2	yes	—
104	C₂²	C₂×#14556062	1	1	yes	characteristic
105	C₂²	#17986346	1	2	yes	—
106	C₂²	#17986431	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	1136	4	no	—
109	C₄	#17985865	1	1	yes	characteristic
110	C₄	C₂×#14556060	1	1	yes	characteristic
111	C₄	#17986350	1	2	yes	—
112	C₂	#22588468	1	1	yes	characteristic
113	C₂	#22590124	1	2	yes	—
114	C₂	#22590572	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588504

C₂

yes

maximal

#22586432

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556062

C₂²

yes

—

C₂×#14556060

—

#17985865

—

C₂²×C₁₆×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₄

yes

complemented

C₂×C₄⋊Dic₇₁

—

#14556062

—

#14556060

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

D₈

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

Q₁₆

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#124

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

#106

yes

—

C₈×C₇₁

#117

yes

—

C₈×C₇₁

#125

yes

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

#486

yes

—

C₂²×C₇₁

#508

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2242

yes

—

C₂×C₄×C₁₆

D₇₁

yes

characteristic

#1846

—

#1380

—

C₇₁

#12889

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₁

yes

—

C₂×#107

—

C₂×#106

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#382

—

142

—

C₄×C₁₆

Dic₇₁

yes

complemented

C₂²×C₈

D₂₈₄

yes

characteristic

C₂²×C₈

C₂²×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

D₂₈₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₁

yes

—

#107

—

142

—

#106

—

284

—

C₄×C₈

Dic₁₄₂

yes

—

C₄×C₈

C₂×Dic₇₁

yes

—

#95

—

284

—

C₂²×C₄

—

284

—

C₂²×C₄

D₅₆₈

yes

—

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

C₂×C₈

C₄⋊Dic₇₁

yes

—

C₂×C₈

#11765854

yes

—

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

C₂×C₈

#11765865

yes

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄²

C₄⋊Dic₇₁

yes

—

C₄²

Dic₂₈₄

yes

—

C₁₆

—

C₂³

C₂×D₅₆₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

568

—

C₂×C₄

—

568

—

C₂×C₄

#14556062

yes

—

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

#14556116

yes

—

C₂×C₄

C₂×#11765854

yes

—

100

C₂×C₄

C₂×Dic₂₈₄

yes

characteristic

101

C₈

#14556060

yes

—

102

C₈

#14556087

yes

—

103

C₈

#14556120

yes

—

104

C₂²

C₂×#14556062

yes

characteristic

105

C₂²

#17986346

yes

—

106

C₂²

#17986431

yes

—

107

C₄

—

108

C₄

—

1136

—

109

C₄

#17985865

yes

characteristic

110

C₄

C₂×#14556060

yes

characteristic

111

C₄

#17986350

yes

—

112

C₂

#22588468

yes

characteristic

113

C₂

#22590124

yes

—

114

C₂

#22590572

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#12889
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₅₆₈

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×C₄×C₁₆×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂×D₈

cosocle

C₂²×C₈×C₇₁

C₂³

hypercenter

C₂×C₄×C₁₆

D₇₁

hypocenter

C₇₁

#12889

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
9. C₄×C₁₆×C₇₁	C₄	semidirect
61. C₇₁	#12889	semidirect
67. C₄×C₁₆	Dic₇₁	semidirect

H ≅ G/N

type

9. C₄×C₁₆×C₇₁

C₄

semidirect

61. C₇₁

#12889

semidirect

67. C₄×C₁₆

Dic₇₁

semidirect

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

subgroup	index
#22588504	2
#22586432	2
C₂×C₄×C₁₆×C₇₁	2
#12889	71

subgroup

index

#22588504

#22586432

C₂×C₄×C₁₆×C₇₁

#12889

Subgroups — #31913835

Conjugacy classes of subgroups

Complements

Maximal subgroups