Group #31997702 — subgroups

Subgroups — #31997702

Order 18176.

1062 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 411 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	#22588483	C₂	1	8	yes	maximal, complemented
2	#22588034	C₂	1	6	yes	maximal
3	C₂×C₄³×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11765849	C₂²	1	4	yes	—
5	#17985754	C₂²	1	24	yes	—
6	C₂²×C₄²×C₇₁	C₄	1	4	yes	—
7	C₂²×C₄²×C₇₁	C₂²	1	3	yes	—
8	C₄³×C₇₁	C₄	1	4	yes	—
9	C₄³×C₇₁	C₂²	1	4	yes	—
10	C₂×#11765849	—	2	24	no	—
11	C₂²×C₇₁⋊C₈	C₂³	1	8	yes	—
12	C₂³×C₄×C₇₁	C₂×C₄	1	6	yes	—
13	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₇₁	—	2	12	no	—
15	C₂×C₄²×C₇₁	—	2	16	no	—
16	C₂×C₄²×C₇₁	C₂×C₄	1	12	yes	—
17	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
18	C₂×C₄²×C₇₁	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₇₁	C₂³	1	6	yes	—
20	#11765849	—	4	16	no	—
21	C₂×C₇₁⋊C₈	—	2	24	no	—
22	C₂⁴×C₇₁	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	—	2	12	no	—
24	C₂²×C₄×C₇₁	—	2	24	no	—
25	C₂²×C₄×C₇₁	C₂²×C₄	1	12	yes	—
26	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	#41	1	12	yes	—
28	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
29	C₄²×C₇₁	—	2	12	no	—
30	C₄²×C₇₁	—	2	24	no	—
31	C₄²×C₇₁	—	2	12	no	—
32	C₄²×C₇₁	M₁₆	1	16	yes	complemented
33	C₂³×C₇₁	—	2	4	no	—
34	C₂³×C₇₁	#117	1	6	yes	—
35	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	—	2	48	no	—
37	C₂×C₄×C₇₁	—	2	12	no	—
38	C₂×C₄×C₇₁	—	2	12	no	—
39	C₂×C₄×C₇₁	#117	1	6	yes	—
40	C₂×C₄×C₇₁	#126	1	16	yes	—
41	C₂×C₄×C₇₁	C₂×M₁₆	1	24	yes	—
42	C₂×C₄×C₇₁	C₂×#41	1	6	yes	—
43	C₇₁⋊C₈	—	4	8	no	—
44	C₂²×C₇₁	—	2	12	no	—
45	C₂²×C₇₁	#383	1	4	yes	—
46	C₂²×C₇₁	C₂×#117	1	3	yes	—
47	C₂²×C₇₁	C₂²×M₁₆	1	4	yes	—
48	C₄×C₇₁	—	2	24	no	—
49	C₄×C₇₁	#383	1	4	yes	—
50	C₄×C₇₁	C₂×#126	1	4	yes	—
51	#20772	—	71	1	no	maximal, Sylow
52	C₂×C₇₁	—	2	4	no	—
53	C₂×C₇₁	C₂×#383	1	1	yes	characteristic
54	C₂×C₇₁	#2920	1	6	yes	—
55	C₂×C₄³	D₇₁	1	1	yes	characteristic
56	#1839	—	71	8	no	—
57	#1728	—	71	6	no	—
58	C₇₁	#20772	1	1	yes	characteristic, complemented, Sylow
59	C₂²×M₁₆	—	71	4	no	—
60	C₂²×C₄²	Dic₇₁	1	4	yes	—
61	C₂²×C₄²	C₂×D₇₁	1	3	yes	—
62	C₄³	Dic₇₁	1	4	yes	—
63	C₄³	C₂×D₇₁	1	4	yes	—
64	#336	—	71	24	no	—
65	C₂³×C₄	C₂×Dic₇₁	1	6	yes	—
66	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
67	C₂×M₁₆	—	142	24	no	—
68	C₂²×C₈	—	71	8	no	—
69	C₂×C₄²	—	2	12	no	—
70	C₂×C₄²	—	2	16	no	—
71	C₂×C₄²	C₂×Dic₇₁	1	6	yes	—
72	C₂×C₄²	C₂×Dic₇₁	1	12	yes	—
73	C₂×C₄²	C₂×Dic₇₁	1	4	yes	—
74	C₂×C₄²	C₂²×D₇₁	1	6	yes	—
75	C₂⁴	C₂²×Dic₇₁	1	1	yes	characteristic
76	C₂²×C₄	—	2	12	no	—
77	C₂²×C₄	—	2	24	no	—
78	C₂²×C₄	C₂²×Dic₇₁	1	12	yes	—
79	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
80	C₂²×C₄	#11765865	1	12	yes	—
81	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
82	M₁₆	—	284	16	no	—
83	C₂×C₈	—	142	24	no	—
84	C₄²	—	2	12	no	—
85	C₄²	—	2	12	no	—
86	C₄²	—	2	24	no	—
87	C₄²	#11765849	1	16	yes	complemented
88	C₂³	—	2	4	no	—
89	C₂³	#14556087	1	6	yes	—
90	C₂³	C₂³×Dic₇₁	1	1	yes	characteristic
91	C₂×C₄	—	2	48	no	—
92	C₂×C₄	—	2	12	no	—
93	C₂×C₄	—	2	12	no	—
94	C₂×C₄	#14556087	1	6	yes	—
95	C₂×C₄	#14556122	1	16	yes	—
96	C₂×C₄	C₂×#11765849	1	24	yes	—
97	C₂×C₄	C₂×#11765865	1	6	yes	—
98	C₈	—	284	8	no	—
99	C₂²	—	2	12	no	—
100	C₂²	#17985867	1	4	yes	—
101	C₂²	C₂×#14556087	1	3	yes	—
102	C₂²	C₂²×#11765849	1	4	yes	—
103	C₄	—	2	24	no	—
104	C₄	#17985867	1	4	yes	—
105	C₄	C₂×#14556122	1	4	yes	—
106	C₂	—	2	4	no	—
107	C₂	C₂×#17985867	1	1	yes	characteristic
108	C₂	#22596973	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588483

C₂

yes

maximal, complemented

#22588034

C₂

yes

maximal

C₂×C₄³×C₇₁

C₂

yes

maximal, characteristic

C₂²×#11765849

C₂²

yes

—

#17985754

C₂²

yes

—

C₂²×C₄²×C₇₁

C₄

yes

—

C₂²×C₄²×C₇₁

C₂²

yes

—

C₄³×C₇₁

C₄

yes

—

C₄³×C₇₁

C₂²

yes

—

C₂×#11765849

—

C₂²×C₇₁⋊C₈

C₂³

yes

—

C₂³×C₄×C₇₁

C₂×C₄

yes

—

C₂³×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂⁴×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂²×C₄×C₇₁

C₂⁴

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

M₁₆

yes

complemented

C₂³×C₇₁

—

C₂³×C₇₁

#117

yes

—

C₂³×C₇₁

C₂³×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#126

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₂×C₄×C₇₁

C₂×#41

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#383

yes

—

C₂²×C₇₁

C₂×#117

yes

—

C₂²×C₇₁

C₂²×M₁₆

yes

—

C₄×C₇₁

—

C₄×C₇₁

#383

yes

—

C₄×C₇₁

C₂×#126

yes

—

#20772

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₂×#383

yes

characteristic

C₂×C₇₁

#2920

yes

—

C₂×C₄³

D₇₁

yes

characteristic

#1839

—

#1728

—

C₇₁

#20772

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×C₄²

Dic₇₁

yes

—

C₂²×C₄²

C₂×D₇₁

yes

—

C₄³

Dic₇₁

yes

—

C₄³

C₂×D₇₁

yes

—

#336

—

C₂³×C₄

C₂×Dic₇₁

yes

—

C₂³×C₄

C₂²×D₇₁

yes

characteristic

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂²×D₇₁

yes

—

C₂⁴

C₂²×Dic₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×Dic₇₁

yes

—

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

C₂²×C₄

#11765865

yes

—

C₂²×C₄

C₂³×D₇₁

yes

characteristic

M₁₆

—

284

—

C₂×C₈

—

142

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11765849

yes

complemented

C₂³

—

C₂³

#14556087

yes

—

C₂³

C₂³×Dic₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14556087

yes

—

C₂×C₄

#14556122

yes

—

C₂×C₄

C₂×#11765849

yes

—

C₂×C₄

C₂×#11765865

yes

—

C₈

—

284

—

C₂²

—

100

C₂²

#17985867

yes

—

101

C₂²

C₂×#14556087

yes

—

102

C₂²

C₂²×#11765849

yes

—

103

C₄

—

104

C₄

#17985867

yes

—

105

C₄

C₂×#14556122

yes

—

106

C₂

—

107

C₂

C₂×#17985867

yes

characteristic

108

C₂

#22596973

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₁
commutator, socle	C₂³×C₇₁	C₂³×C₄
Fitting	C₂×C₄³×C₇₁	C₂
prime radical	C₂⁴×C₇₁	C₂²×C₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂×C₄³	D₇₁
hypocenter	C₇₁	#20772
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₇₁

commutator, socle

C₂³×C₇₁

C₂³×C₄

Fitting

C₂×C₄³×C₇₁

C₂

prime radical

C₂⁴×C₇₁

C₂²×C₄

cosocle

C₂²×C₄×C₇₁

C₂⁴

hypercenter

C₂×C₄³

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

N	H ≅ G/N	type
1. #22588483	C₂	semidirect
32. C₄²×C₇₁	M₁₆	semidirect
58. C₇₁	#20772	semidirect
87. C₄²	#11765849	semidirect

H ≅ G/N

type

1. #22588483

C₂

semidirect

32. C₄²×C₇₁

M₁₆

semidirect

58. C₇₁

#20772

semidirect

87. C₄²

#11765849

semidirect

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

subgroup	index
#22588483	2
#22588034	2
C₂×C₄³×C₇₁	2
#20772	71

subgroup

index

#22588483

#22588034

C₂×C₄³×C₇₁

#20772

Subgroups — #31997702

Conjugacy classes of subgroups

Complements

Maximal subgroups