Group #31891106 — subgroups

Subgroups — #31891106

Order 18176.

262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22586185	C₂	1	2	yes	maximal
3	#22586169	C₂	1	2	yes	maximal
4	C₇₁×#1746	C₂	1	1	yes	maximal, characteristic
5	#17986052	C₂²	1	4	yes	—
6	#17985752	—	2	2	no	—
7	#17985750	—	2	2	no	—
8	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	#376×C₇₁	C₂²	1	1	yes	characteristic
11	#336×C₇₁	C₄	1	4	yes	—
12	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
13	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
14	C₄×C₇₁⋊C₈	—	2	4	no	—
15	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
19	#97×C₇₁	—	2	2	no	—
20	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
21	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
22	C₄×C₈×C₇₁	Q₈	1	2	yes	—
23	#95×C₇₁	—	2	2	no	—
24	C₂×C₇₁⋊C₈	—	4	4	no	—
25	C₂²×C₄×C₇₁	—	2	2	no	—
26	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	#41	1	2	yes	—
28	C₂×C₈×C₇₁	—	2	4	no	—
29	C₂×C₈×C₇₁	—	2	2	no	—
30	C₂×C₈×C₇₁	—	2	4	no	—
31	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
33	C₄²×C₇₁	—	2	4	no	—
34	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₇₁	#41	1	2	yes	—
37	C₂³×C₇₁	#117	1	1	yes	characteristic
38	C₂×C₄×C₇₁	—	4	2	no	—
39	C₂×C₄×C₇₁	—	4	4	no	—
40	C₂×C₄×C₇₁	—	4	2	no	—
41	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#124	1	2	yes	—
44	C₂×C₄×C₇₁	#125	1	2	yes	—
45	C₂×C₄×C₇₁	#131	1	4	yes	—
46	C₈×C₇₁	—	2	2	no	—
47	C₈×C₇₁	—	2	2	no	—
48	C₈×C₇₁	—	4	2	no	—
49	C₇₁⋊C₈	—	8	4	no	—
50	C₂²×C₇₁	—	2	2	no	—
51	C₂²×C₇₁	#382	1	1	yes	characteristic
52	C₂²×C₇₁	#433	1	2	yes	—
53	C₄×C₇₁	—	2	2	no	—
54	C₄×C₇₁	—	2	2	no	—
55	C₄×C₇₁	—	8	2	no	—
56	C₄×C₇₁	#424	1	2	yes	—
57	C₄×C₇₁	#433	1	2	yes	—
58	#8782	—	71	1	no	maximal, Sylow
59	C₂×C₇₁	—	2	2	no	—
60	C₂×C₇₁	#1832	1	1	yes	characteristic
61	C₂×C₇₁	#1833	1	2	yes	—
62	#1746	D₇₁	1	1	yes	characteristic
63	#1564	—	71	2	no	—
64	#1281	—	71	2	no	—
65	#1274	—	71	2	no	—
66	C₇₁	#8782	1	1	yes	characteristic, complemented, Sylow
67	#431	—	71	4	no	—
68	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
70	#376	C₂×D₇₁	1	1	yes	characteristic
71	#336	Dic₇₁	1	4	yes	—
72	#335	—	142	2	no	—
73	#334	—	142	2	no	—
74	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
75	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
76	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
77	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
78	C₄⋊C₈	—	142	4	no	—
79	C₂²⋊C₈	—	142	4	no	—
80	#97	—	2	2	no	—
81	C₄×C₈	—	142	4	no	—
82	C₄×C₈	Dic₁₄₂	1	2	yes	—
83	C₄×C₈	D₂₈₄	1	1	yes	characteristic
84	C₄×C₈	D₂₈₄	1	1	yes	characteristic
85	#95	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
88	C₂²×C₄	#11765865	1	2	yes	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	284	4	no	—
93	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
94	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
95	C₄²	—	2	4	no	—
96	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
97	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
98	C₄²	#11765865	1	2	yes	—
99	C₂³	#14556087	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
104	C₂×C₄	#14556087	1	1	yes	characteristic
105	C₂×C₄	#14556116	1	2	yes	—
106	C₂×C₄	#14556120	1	2	yes	—
107	C₂×C₄	#14556135	1	4	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	2	2	no	—
110	C₈	—	4	2	no	—
111	C₈	—	568	4	no	—
112	C₂²	—	2	2	no	—
113	C₂²	#17985865	1	1	yes	characteristic
114	C₂²	#17986060	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	8	2	no	—
118	C₄	#17986026	1	2	yes	—
119	C₄	#17986060	1	2	yes	—
120	C₂	—	2	2	no	—
121	C₂	#22588460	1	1	yes	characteristic
122	C₂	#22588464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22587088

C₂

yes

maximal

#22586185

C₂

yes

maximal

#22586169

C₂

yes

maximal

C₇₁×#1746

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

#17985752

—

#17985750

—

C₂×#97×C₇₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#376×C₇₁

C₂²

yes

characteristic

#336×C₇₁

C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#97×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

—

#95×C₇₁

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#125

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#382

yes

characteristic

C₂²×C₇₁

#433

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#424

yes

—

C₄×C₇₁

#433

yes

—

#8782

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1832

yes

characteristic

C₂×C₇₁

#1833

yes

—

#1746

D₇₁

yes

—

—

—

C₇₁

#8782

yes

characteristic, complemented, Sylow

#431

—

C₂×#97

C₂×D₇₁

yes

characteristic

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#376

C₂×D₇₁

yes

characteristic

#336

Dic₇₁

yes

—

#335

—

142

—

#334

—

142

—

C₂²×C₈

C₂×Dic₇₁

yes

—

C₂²×C₈

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

142

—

#97

—

C₄×C₈

—

142

—

C₄×C₈

Dic₁₄₂

yes

—

C₄×C₈

D₂₈₄

yes

characteristic

C₄×C₈

D₂₈₄

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

C₂²×C₄

#11765865

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

C₄⋊Dic₇₁

yes

—

C₂×C₈

C₄⋊Dic₇₁

yes

—

C₄²

—

C₄²

C₂×D₂₈₄

yes

characteristic

C₄²

C₂×Dic₁₄₂

yes

characteristic

C₄²

#11765865

yes

—

C₂³

#14556087

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

104

C₂×C₄

#14556087

yes

characteristic

105

C₂×C₄

#14556116

yes

—

106

C₂×C₄

#14556120

yes

—

107

C₂×C₄

#14556135

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

568

—

112

C₂²

—

113

C₂²

#17985865

yes

characteristic

114

C₂²

#17986060

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#17986026

yes

—

119

C₄

#17986060

yes

—

120

C₂

—

121

C₂

#22588460

yes

characteristic

122

C₂

#22588464

yes

—

	subgroup	G/H
center	C₂×C₄	#14556087
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1746	C₂
socle	C₂²×C₇₁	#382
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1746	D₇₁
hypocenter	C₇₁	#8782
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1746

subgroup

G/H

center

C₂×C₄

#14556087

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1746

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1746

N	H ≅ G/N	type
66. C₇₁	#8782	semidirect

H ≅ G/N

type

66. C₇₁

#8782

semidirect

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

subgroup	index
#22587088	2
#22586185	2
#22586169	2
C₇₁×#1746	2
#8782	71

subgroup

index

C₇₁×#1746

Subgroups — #31891106

Conjugacy classes of subgroups

Complements

Maximal subgroups