Group #31913842 — subgroups

Subgroups — #31913842

Order 18176.

282 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 155 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	#22588512	C₂	1	2	yes	maximal
2	#22586434	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556064	—	2	2	no	—
5	C₂×#14556064	C₂²	1	4	yes	—
6	#17986023	—	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	2	yes	—
10	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
11	C₂×#11765849	—	2	4	no	—
12	#14556064	—	2	8	no	—
13	#14556064	—	4	4	no	—
14	C₄⋊(C₇₁⋊C₈)	—	4	4	no	—
15	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	—	2	4	no	—
19	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
23	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
24	#11765849	—	4	4	no	—
25	#11765849	—	4	4	no	—
26	C₂×C₇₁⋊C₈	—	4	4	no	—
27	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
28	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
32	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₇₁	#41	1	4	yes	—
35	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₇₁	D₈	1	2	yes	—
38	C₁₆×C₇₁	—	2	4	no	—
39	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
40	C₂×C₄×C₇₁	—	2	4	no	—
41	C₂×C₄×C₇₁	#107	1	2	yes	—
42	C₂×C₄×C₇₁	#107	1	2	yes	—
43	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	#124	1	2	yes	—
45	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
47	C₈×C₇₁	#106	1	2	yes	—
48	C₈×C₇₁	#106	1	2	yes	—
49	C₈×C₇₁	#117	1	2	yes	—
50	C₈×C₇₁	#125	1	2	yes	—
51	C₇₁⋊C₈	—	8	4	no	—
52	C₂²×C₇₁	—	2	2	no	—
53	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
54	C₂²×C₇₁	#487	1	2	yes	—
55	C₄×C₇₁	—	2	2	no	—
56	C₄×C₇₁	#368	1	4	yes	—
57	C₄×C₇₁	#382	1	1	yes	characteristic
58	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
59	C₄×C₇₁	#486	1	2	yes	—
60	#12891	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	—	2	2	no	—
62	C₂×C₇₁	#1834	1	1	yes	characteristic
63	C₂×C₇₁	C₂×#368	1	2	yes	—
64	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
65	#1848	—	71	2	no	—
66	#1381	—	71	4	no	—
67	C₇₁	#12891	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
69	C₂×#108	—	71	4	no	—
70	C₂×#108	—	142	2	no	—
71	#423	—	142	2	no	—
72	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
73	C₄×C₁₆	Dic₇₁	1	2	yes	—
74	C₄×C₁₆	Dic₇₁	1	2	yes	—
75	C₂×M₁₆	—	142	4	no	—
76	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
78	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
79	C₂×C₁₆	—	2	4	no	—
80	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
81	#108	—	284	4	no	—
82	#108	—	142	8	no	—
83	C₄⋊C₈	—	284	4	no	—
84	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
85	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
87	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
88	C₂²×C₄	Dic₂₈₄	1	2	yes	—
89	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
90	M₁₆	—	284	4	no	—
91	M₁₆	—	284	4	no	—
92	C₂×C₈	—	284	4	no	—
93	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
95	C₂×C₈	#11765854	1	4	yes	—
96	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
97	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
98	C₂×C₈	#11765865	1	4	yes	—
99	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
100	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
101	C₄²	D₅₆₈	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₂³	C₂×Dic₂₈₄	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	#14556062	1	2	yes	—
106	C₂×C₄	#14556062	1	2	yes	—
107	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
108	C₂×C₄	#14556116	1	2	yes	—
109	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
110	C₂×C₄	C₂×#11765854	1	2	yes	—
111	C₈	—	568	4	no	—
112	C₈	#14556060	1	2	yes	—
113	C₈	#14556060	1	2	yes	—
114	C₈	#14556087	1	2	yes	—
115	C₈	#14556120	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#14556062	1	1	yes	characteristic
118	C₂²	#17986350	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	#17985829	1	4	yes	—
121	C₄	#17985865	1	1	yes	characteristic
122	C₄	C₂×#14556060	1	1	yes	characteristic
123	C₄	#17986346	1	2	yes	—
124	C₂	—	2	2	no	—
125	C₂	#22588468	1	1	yes	characteristic
126	C₂	C₂×#17985829	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588512

C₂

yes

maximal

#22586434

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556064

—

C₂×#14556064

C₂²

yes

—

#17986023

—

C₂²×C₁₆×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₄

yes

—

C₄×C₁₆×C₇₁

C₄

yes

—

C₂×#11765849

—

#14556064

—

#14556064

—

C₄⋊(C₇₁⋊C₈)

—

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

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

characteristic

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

Q₁₆

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

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

characteristic

C₄²×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

D₈

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#124

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₈×C₇₁

#106

yes

—

C₈×C₇₁

#106

yes

—

C₈×C₇₁

#117

yes

—

C₈×C₇₁

#125

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

#487

yes

—

C₄×C₇₁

—

C₄×C₇₁

#368

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#12891

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1834

yes

characteristic

C₂×C₇₁

C₂×#368

yes

—

C₂×C₄×C₁₆

D₇₁

yes

characteristic

#1848

—

#1381

—

C₇₁

#12891

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₁

yes

—

C₂×#108

—

C₂×#108

—

142

—

#423

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄×C₁₆

Dic₇₁

yes

—

C₄×C₁₆

Dic₇₁

yes

—

C₂×M₁₆

—

142

—

C₂²×C₈

D₂₈₄

yes

characteristic

C₂²×C₈

C₂²×D₇₁

yes

characteristic

C₂×C₄²

D₂₈₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₁

yes

—

#108

—

284

—

#108

—

142

—

C₄⋊C₈

—

284

—

C₄×C₈

Dic₁₄₂

yes

characteristic

C₄×C₈

Dic₁₄₂

yes

characteristic

C₄×C₈

C₂×Dic₇₁

yes

characteristic

C₄×C₈

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

Dic₂₈₄

yes

—

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

M₁₆

—

284

—

M₁₆

—

284

—

C₂×C₈

—

284

—

C₂×C₈

C₄⋊Dic₇₁

yes

characteristic

C₂×C₈

C₄⋊Dic₇₁

yes

characteristic

C₂×C₈

#11765854

yes

—

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

C₂×C₈

#11765865

yes

—

C₄²

C₄⋊Dic₇₁

yes

characteristic

100

C₄²

C₄⋊Dic₇₁

yes

characteristic

101

C₄²

D₅₆₈

yes

—

102

C₁₆

—

103

C₂³

C₂×Dic₂₈₄

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#14556062

yes

—

106

C₂×C₄

#14556062

yes

—

107

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

108

C₂×C₄

#14556116

yes

—

109

C₂×C₄

C₂×D₅₆₈

yes

characteristic

110

C₂×C₄

C₂×#11765854

yes

—

111

C₈

—

568

—

112

C₈

#14556060

yes

—

113

C₈

#14556060

yes

—

114

C₈

#14556087

yes

—

115

C₈

#14556120

yes

—

116

C₂²

—

117

C₂²

C₂×#14556062

yes

characteristic

118

C₂²

#17986350

yes

—

119

C₄

—

120

C₄

#17985829

yes

—

121

C₄

#17985865

yes

characteristic

122

C₄

C₂×#14556060

yes

characteristic

123

C₄

#17986346

yes

—

124

C₂

—

125

C₂

#22588468

yes

characteristic

126

C₂

C₂×#17985829

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#12891
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₅₆₈

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×C₄×C₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×#107

prime radical

C₂³×C₇₁

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

hypercenter

C₂×C₄×C₁₆

D₇₁

hypocenter

C₇₁

#12891

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

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

N	H ≅ G/N	type
67. C₇₁	#12891	semidirect

H ≅ G/N

type

67. C₇₁

#12891

semidirect

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

subgroup	index
#22588512	2
#22586434	2
C₂×C₄×C₁₆×C₇₁	2
#12891	71

subgroup

index

#22588512

#22586434

C₂×C₄×C₁₆×C₇₁

#12891

Subgroups — #31913842

Conjugacy classes of subgroups

Complements

Maximal subgroups