Group #31884324 — subgroups

Subgroups — #31884324

Order 18176.

222 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22587088	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	#17986052	—	2	2	no	—
4	#17985752	C₄	1	4	yes	—
5	#17985750	—	2	2	no	—
6	C₂²×C₁₆×C₇₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₇₁	—	2	2	no	—
9	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
10	C₂²⋊(C₇₁⋊C₈)	—	4	2	no	—
11	C₄×C₇₁⋊C₈	—	4	2	no	—
12	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₇₁	—	2	2	no	—
16	C₂×C₁₆×C₇₁	—	2	2	no	—
17	C₂×C₁₆×C₇₁	C₈	1	4	yes	complemented
18	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₇₁	D₄	1	2	yes	—
20	C₂×C₇₁⋊C₈	—	4	4	no	—
21	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
23	C₂×C₈×C₇₁	—	2	2	no	—
24	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
29	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₇₁	D₈	1	1	yes	characteristic
31	C₄²×C₇₁	Q₁₆	1	1	yes	characteristic
32	C₁₆×C₇₁	—	2	4	no	—
33	C₂³×C₇₁	#106	1	1	yes	characteristic
34	C₂×C₄×C₇₁	—	2	4	no	—
35	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
36	C₂×C₄×C₇₁	#102	1	2	yes	—
37	C₂×C₄×C₇₁	#103	1	2	yes	—
38	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
39	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
40	C₂×C₄×C₇₁	#108	1	2	yes	—
41	C₈×C₇₁	—	2	2	no	—
42	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
43	C₈×C₇₁	#97	1	1	yes	characteristic
44	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
45	C₇₁⋊C₈	—	8	4	no	—
46	C₂²×C₇₁	—	2	2	no	—
47	C₂²×C₇₁	#334	1	2	yes	—
48	C₂²×C₇₁	#340	1	1	yes	characteristic
49	C₄×C₇₁	—	2	2	no	—
50	C₄×C₇₁	#335	1	2	yes	—
51	C₄×C₇₁	#336	1	1	yes	characteristic
52	C₄×C₇₁	#341	1	1	yes	characteristic
53	C₄×C₇₁	#359	1	4	yes	—
54	#7155	—	71	1	no	maximal, Sylow
55	C₂×C₇₁	—	2	2	no	—
56	C₂×C₇₁	#1277	1	1	yes	characteristic
57	C₂×C₇₁	#1382	1	2	yes	—
58	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
59	#1564	—	71	2	no	—
60	C₇₁	#7155	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₁₆	Dic₇₁	1	2	yes	—
62	#431	—	142	2	no	—
63	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
64	C₄×C₁₆	—	2	2	no	—
65	#335	—	71	4	no	—
66	#334	—	142	2	no	—
67	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
68	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
69	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₇₁⋊C₈	1	4	yes	complemented
73	C₄⋊C₈	—	142	4	no	—
74	C₂²⋊C₈	—	284	2	no	—
75	C₄×C₈	—	284	2	no	—
76	C₄×C₈	C₄×D₇₁	1	2	yes	—
77	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
78	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
79	C₂²×C₄	#11765854	1	2	yes	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	284	4	no	—
82	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
84	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
85	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
86	C₂×C₈	#11765849	1	2	yes	—
87	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
88	C₄²	D₅₆₈	1	1	yes	characteristic
89	C₄²	Dic₂₈₄	1	1	yes	characteristic
90	C₁₆	—	2	4	no	—
91	C₂³	#14556060	1	1	yes	characteristic
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
94	C₂×C₄	#14556050	1	2	yes	—
95	C₂×C₄	#14556053	1	2	yes	—
96	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
97	C₂×C₄	#14556062	1	1	yes	characteristic
98	C₂×C₄	#14556064	1	2	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	568	4	no	—
101	C₈	C₄×C₇₁⋊C₈	1	1	yes	characteristic
102	C₈	#14556039	1	1	yes	characteristic
103	C₈	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	#17985750	1	2	yes	—
106	C₂²	#17985762	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	#17985752	1	2	yes	—
109	C₄	#17985754	1	1	yes	characteristic
110	C₄	#17985764	1	1	yes	characteristic
111	C₄	#17985807	1	4	yes	—
112	C₂	—	2	2	no	—
113	C₂	#22586175	1	1	yes	characteristic
114	C₂	#22586436	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22587088

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₁

C₂

yes

maximal, characteristic

#17986052

—

#17985752

C₄

yes

—

#17985750

—

C₂²×C₁₆×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₈

yes

complemented

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

Q₁₆

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

#106

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#95

yes

characteristic

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

#103

yes

—

C₂×C₄×C₇₁

C₄⋊C₈

yes

—

C₂×C₄×C₇₁

#107

yes

characteristic

C₂×C₄×C₇₁

#108

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₄×C₈

yes

characteristic

C₈×C₇₁

#97

yes

characteristic

C₈×C₇₁

C₂²⋊C₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#334

yes

—

C₂²×C₇₁

#340

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#335

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#7155

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1277

yes

characteristic

C₂×C₇₁

#1382

yes

—

C₂×C₄×C₁₆

D₇₁

yes

characteristic

#1564

—

C₇₁

#7155

yes

characteristic, complemented, Sylow

C₂²×C₁₆

Dic₇₁

yes

—

#431

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄×C₁₆

—

#335

—

#334

—

142

—

C₂²×C₈

Dic₁₄₂

yes

characteristic

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₄²

D₂₈₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₇₁⋊C₈

yes

complemented

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

284

—

C₄×C₈

—

284

—

C₄×C₈

C₄×D₇₁

yes

—

C₄×C₈

C₂²⋊D₇₁

yes

—

C₂²×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂²×C₄

#11765854

yes

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

C₄×Dic₇₁

yes

characteristic

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₈

Dic₇₁⋊C₄

yes

—

C₂×C₈

C₂×C₇₁⋊C₈

yes

—

C₂×C₈

#11765849

yes

—

C₄²

(C₂×D₇₁)⋊C₄

yes

—

C₄²

D₅₆₈

yes

characteristic

C₄²

Dic₂₈₄

yes

characteristic

C₁₆

—

C₂³

#14556060

yes

characteristic

C₂×C₄

—

C₂×C₄

C₇₁⋊#95

yes

characteristic

C₂×C₄

#14556050

yes

—

C₂×C₄

#14556053

yes

—

C₂×C₄

C₄⋊(C₇₁⋊C₈)

yes

—

C₂×C₄

#14556062

yes

characteristic

C₂×C₄

#14556064

yes

—

C₈

—

100

C₈

—

568

—

101

C₈

C₄×C₇₁⋊C₈

yes

characteristic

102

C₈

#14556039

yes

characteristic

103

C₈

C₂²⋊(C₇₁⋊C₈)

yes

—

104

C₂²

—

105

C₂²

#17985750

yes

—

106

C₂²

#17985762

yes

characteristic

107

C₄

—

108

C₄

#17985752

yes

—

109

C₄

#17985754

yes

characteristic

110

C₄

#17985764

yes

characteristic

111

C₄

#17985807

yes

—

112

C₂

—

113

C₂

#22586175

yes

characteristic

114

C₂

#22586436

yes

—

	subgroup	G/H
center	C₄²	D₅₆₈
commutator	C₈×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#340
prime radical	C₂³×C₇₁	#106
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#7155
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₄²

D₅₆₈

commutator

C₈×C₇₁

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₁

Fitting

C₂×C₄×C₁₆×C₇₁

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄×C₈×C₇₁

C₂²

hypercenter

C₂×C₄×C₁₆

D₇₁

hypocenter

C₇₁

#7155

LCS

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

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

N	H ≅ G/N	type
17. C₂×C₁₆×C₇₁	C₈	semidirect
60. C₇₁	#7155	semidirect
72. C₂×C₁₆	C₇₁⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₁₆×C₇₁

C₈

semidirect

60. C₇₁

#7155

semidirect

72. C₂×C₁₆

C₇₁⋊C₈

semidirect

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

subgroup	index
#22587088	2
C₂×C₄×C₁₆×C₇₁	2
#7155	71

subgroup

index

#22587088

C₂×C₄×C₁₆×C₇₁

#7155

Subgroups — #31884324

Conjugacy classes of subgroups

Complements

Maximal subgroups