Group #31884931 — subgroups

Subgroups — #31884931

Order 18176.

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

normal subgroups only

This group has 110 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	#22590248	C₂	1	1	yes	maximal, characteristic
2	#22586474	C₂	1	1	yes	maximal, characteristic
3	C₇₁×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₄₂	C₄	1	1	yes	characteristic
5	C₂×C₇₁⋊C₃₂	—	2	1	no	—
6	C₄⋊(C₇₁⋊C₁₆)	C₄	1	1	yes	characteristic
7	Dic₇₁⋊C₁₆	—	2	1	no	—
8	#17985774	—	2	1	no	—
9	C₂×C₃₂×C₇₁	—	2	1	no	—
10	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₄₂	C₈	1	1	yes	characteristic
12	C₂×C₇₁⋊C₁₆	—	2	1	no	—
13	C₄⋊(C₇₁⋊C₈)	C₈	1	1	yes	characteristic
14	Dic₇₁⋊C₈	—	2	1	no	—
15	C₈×Dic₇₁	—	2	1	no	—
16	C₇₁⋊C₃₂	—	2	2	no	—
17	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
19	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₃₂×C₇₁	—	4	1	no	—
21	C₂×Dic₁₄₂	C₁₆	1	1	yes	characteristic
22	C₂×C₇₁⋊C₈	—	2	1	no	—
23	C₄⋊Dic₇₁	C₁₆	1	1	yes	characteristic
24	Dic₇₁⋊C₄	—	2	1	no	—
25	C₄×Dic₇₁	—	2	1	no	—
26	C₇₁⋊C₁₆	—	4	1	no	—
27	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
29	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
30	C₁₆×C₇₁	—	2	1	no	—
31	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
32	C₁₆×C₇₁	Q₁₆	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
35	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
36	C₂×Dic₇₁	—	2	1	no	—
37	C₂×Dic₇₁	—	2	1	no	—
38	Dic₁₄₂	—	4	1	no	—
39	C₈×C₇₁	—	2	1	no	—
40	C₈×C₇₁	#103	1	1	yes	characteristic
41	C₈×C₇₁	#104	1	1	yes	characteristic
42	C₇₁⋊C₈	—	4	1	no	—
43	C₂²×C₇₁	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₄×C₇₁	—	2	1	no	—
45	C₄×C₇₁	—	2	1	no	—
46	C₄×C₇₁	Q₈⋊C₈	1	1	yes	characteristic
47	C₄×C₇₁	#350	1	1	yes	characteristic
48	Dic₇₁	—	4	1	no	—
49	Dic₇₁	—	4	1	no	—
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	—	71	1	no	—
55	C₄⋊C₃₂	D₇₁	1	1	yes	characteristic
56	#1397	—	71	1	no	—
57	C₇₁	#7393	1	1	yes	characteristic, complemented, Sylow
58	C₈×Q₈	—	71	1	no	—
59	C₂×C₃₂	—	2	1	no	—
60	C₂×C₃₂	—	142	1	no	—
61	C₄⋊C₁₆	—	71	1	no	—
62	C₄⋊C₁₆	—	142	1	no	—
63	#346	—	142	1	no	—
64	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
65	C₄×Q₈	—	71	1	no	—
66	C₂×C₁₆	—	142	1	no	—
67	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
68	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
69	C₄⋊C₈	—	71	1	no	—
70	C₄⋊C₈	—	142	1	no	—
71	C₄×C₈	—	142	1	no	—
72	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
73	C₃₂	—	4	1	no	—
74	C₃₂	—	142	2	no	—
75	C₂×Q₈	—	71	1	no	—
76	C₂×C₈	—	142	1	no	—
77	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
78	C₂×C₈	#11765848	1	1	yes	characteristic
79	C₄⋊C₄	—	71	1	no	—
80	C₄⋊C₄	—	142	1	no	—
81	C₄²	—	142	1	no	—
82	C₄²	C₈×D₇₁	1	1	yes	characteristic
83	C₁₆	—	2	1	no	—
84	C₁₆	—	284	1	no	—
85	C₁₆	#11765853	1	1	yes	characteristic
86	C₁₆	#11765855	1	1	yes	characteristic
87	Q₈	—	284	1	no	—
88	C₂×C₄	—	142	1	no	—
89	C₂×C₄	—	142	1	no	—
90	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
91	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
92	C₂×C₄	#14556067	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	284	1	no	—
95	C₈	#14556051	1	1	yes	characteristic
96	C₈	#14556054	1	1	yes	characteristic
97	C₂²	(C₂×D₇₁)⋊C₁₆	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	284	1	no	—
101	C₄	—	284	1	no	—
102	C₄	#17985729	1	1	yes	characteristic
103	C₄	#17985784	1	1	yes	characteristic
104	C₂	#22586318	1	1	yes	characteristic
105	C₂	#22586479	1	1	yes	characteristic
106	C₂	#22586481	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590248

C₂

yes

maximal, characteristic

#22586474

C₂

yes

maximal, characteristic

C₇₁×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₈×Dic₁₄₂

C₄

yes

characteristic

C₂×C₇₁⋊C₃₂

—

C₄⋊(C₇₁⋊C₁₆)

C₄

yes

characteristic

Dic₇₁⋊C₁₆

—

#17985774

—

C₂×C₃₂×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₄×Dic₁₄₂

C₈

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

C₈

yes

characteristic

Dic₇₁⋊C₈

—

C₈×Dic₇₁

—

C₇₁⋊C₃₂

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₃₂×C₇₁

—

C₂×Dic₁₄₂

C₁₆

yes

characteristic

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

C₁₆

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

M₁₆

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

SD₁₆

yes

characteristic

C₁₆×C₇₁

Q₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

M₃₂

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

Q₈⋊C₈

yes

characteristic

C₄×C₇₁

#350

yes

characteristic

Dic₇₁

—

Dic₇₁

—

#7393

—

maximal, Sylow

C₂×C₇₁

Q₈⋊C₁₆

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₄⋊C₃₂

D₇₁

yes

characteristic

#1397

—

C₇₁

#7393

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

142

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

#346

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

142

—

C₂×C₁₆

D₂₈₄

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄×C₈

—

142

—

C₄×C₈

C₄×D₇₁

yes

characteristic

C₃₂

—

C₃₂

—

142

—

C₂×Q₈

—

C₂×C₈

—

142

—

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₈

#11765848

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄²

—

142

—

C₄²

C₈×D₇₁

yes

characteristic

C₁₆

—

C₁₆

—

284

—

C₁₆

yes

C₁₆

yes

Q₈

—

284

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

C₂×C₄

C₁₆×D₇₁

yes

characteristic

C₂×C₄

#14556067

yes

characteristic

C₈

—

C₈

—

284

—

C₈

yes

C₈

yes

C₂²

(C₂×D₇₁)⋊C₁₆

yes

characteristic

C₄

—

C₄

—

100

C₄

—

284

—

101

C₄

—

284

—

102

C₄

#17985729

yes

characteristic

103

C₄

#17985784

yes

characteristic

104

C₂

#22586318

yes

characteristic

105

C₂

#22586479

yes

characteristic

106

C₂

#22586481

yes

characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₇₁	C₂²⋊C₁₆
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	C₄⋊C₃₂	D₇₁
hypocenter	C₇₁	#7393
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center

C₂×C₈

(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₂×C₁₆

Frattini

C₄×C₁₆

C₂×D₇₁

Fitting

C₇₁×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₇₁

C₂²⋊C₁₆

cosocle

C₄×C₁₆×C₇₁

C₂²

hypercenter

C₄⋊C₃₂

D₇₁

hypocenter

C₇₁

#7393

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

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
#22590248	2
#22586474	2
C₇₁×C₄⋊C₃₂	2
#7393	71

subgroup

index

#22590248

#22586474

C₇₁×C₄⋊C₃₂

#7393

Subgroups — #31884931

Conjugacy classes of subgroups

Complements

Maximal subgroups