Group #31918020 — subgroups

Subgroups — #31918020

Order 18176.

184 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 81 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	#22590635	C₂	1	2	yes	maximal, complemented
2	#22590178	C₂	1	1	yes	maximal, characteristic
3	#22586483	C₂	1	2	yes	maximal, complemented
4	#22586474	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1422	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986415	C₂²	1	2	yes	—
7	#17986117	C₄	1	1	yes	characteristic
8	#17985832	—	2	2	no	—
9	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
11	C₃₂×D₇₁	—	2	2	no	—
12	#17985820	C₄	1	1	yes	characteristic
13	#17985782	C₄	1	2	yes	—
14	M₆₄×C₇₁	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	#14556150	C₈	1	1	yes	characteristic
17	#14556134	C₂×C₄	1	2	yes	—
18	#14556068	C₂×C₄	1	2	yes	—
19	#14556067	—	2	2	no	—
20	C₁₆×D₇₁	—	2	2	no	—
21	#14556064	C₈	1	1	yes	characteristic
22	#14556056	C₈	1	2	yes	—
23	C₇₁⋊C₃₂	—	2	1	no	—
24	C₇₁⋊C₃₂	D₄	1	2	yes	—
25	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₃₂×C₇₁	—	2	2	no	—
29	#11765865	C₂×C₈	1	2	yes	—
30	Dic₂₈₄	C₁₆	1	1	yes	characteristic
31	#11765854	C₁₆	1	2	yes	—
32	D₅₆₈	C₁₆	1	1	yes	characteristic
33	#11765849	C₂×C₈	1	2	yes	—
34	#11765848	—	2	2	no	—
35	C₈×D₇₁	—	2	2	no	—
36	C₇₁⋊C₁₆	—	2	2	no	—
37	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
40	C₁₆×C₇₁	—	2	1	no	—
41	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₁	#41	1	1	yes	characteristic
43	C₂×C₄×C₇₁	—	2	1	no	—
44	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
45	C₂²⋊D₇₁	—	2	2	no	—
46	D₂₈₄	C₂×C₁₆	1	2	yes	—
47	C₄×D₇₁	—	2	2	no	—
48	Dic₁₄₂	C₂×C₁₆	1	2	yes	—
49	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
50	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
51	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
52	C₈×C₇₁	#131	1	1	yes	characteristic
53	C₇₁⋊C₈	—	2	2	no	—
54	C₂²×C₇₁	#504	1	1	yes	characteristic
55	C₂×D₇₁	—	2	2	no	—
56	C₄×C₇₁	—	4	1	no	—
57	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
58	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
59	Dic₇₁	—	2	2	no	—
60	#13648	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	—	2	1	no	—
62	C₂×C₇₁	D₄×C₁₆	1	1	yes	characteristic
63	D₇₁	—	4	2	no	—
64	#2258	—	71	2	no	—
65	#2170	—	71	1	no	—
66	#1422	D₇₁	1	1	yes	characteristic, complemented
67	#1401	—	71	2	no	—
68	#1397	—	71	1	no	—
69	C₇₁	#13648	1	1	yes	characteristic, complemented, Sylow
70	#504	—	71	2	no	—
71	#443	—	71	1	no	—
72	M₆₄	—	142	2	no	—
73	M₆₄	C₂×D₇₁	1	2	yes	—
74	C₂×C₃₂	—	71	1	no	—
75	C₂×C₃₂	—	71	1	no	—
76	C₂×C₃₂	—	142	2	no	—
77	#364	—	71	1	no	—
78	#350	—	71	2	no	—
79	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
80	#135	—	71	1	no	—
81	#131	—	71	2	no	—
82	M₃₂	—	71	2	no	—
83	M₃₂	—	142	2	no	—
84	C₂×C₁₆	—	142	2	no	—
85	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
86	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
87	#108	—	71	1	no	—
88	#104	—	71	2	no	—
89	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
90	C₃₂	—	2	2	no	—
91	C₃₂	—	71	2	no	—
92	C₃₂	—	142	1	no	—
93	#41	—	71	2	no	—
94	Q₁₆	—	71	1	no	—
95	SD₁₆	—	71	2	no	—
96	D₈	—	71	1	no	—
97	M₁₆	—	71	2	no	—
98	M₁₆	—	142	2	no	—
99	C₂×C₈	—	142	2	no	—
100	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
101	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
102	C₄²	C₈×D₇₁	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	142	2	no	—
105	C₁₆	D₄×D₇₁	1	1	yes	characteristic
106	C₁₆	#11765866	1	1	yes	characteristic
107	Q₈	—	71	2	no	—
108	D₄	—	71	2	no	—
109	D₄	—	142	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	142	2	no	—
112	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
113	C₈	—	142	2	no	—
114	C₈	C₁₆×D₇₁	1	1	yes	characteristic
115	C₈	C₁₆×D₇₁	1	1	yes	characteristic
116	C₈	#14556090	1	1	yes	characteristic
117	C₈	#14556136	1	1	yes	characteristic
118	C₂²	—	142	2	no	—
119	C₂²	#17986417	1	1	yes	characteristic
120	C₄	—	4	1	no	—
121	C₄	—	142	2	no	—
122	C₄	#17986064	1	1	yes	characteristic
123	C₄	C₂×C₁₆×D₇₁	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	284	2	no	—
126	C₂	#22590159	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590635

C₂

yes

maximal, complemented

#22590178

C₂

yes

maximal, characteristic

#22586483

C₂

yes

maximal, complemented

#22586474

C₂

yes

maximal, characteristic, complemented

C₇₁×#1422

C₂

yes

maximal, characteristic, complemented

#17986415

C₂²

yes

—

#17986117

C₄

yes

characteristic

#17985832

—

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

C₃₂×D₇₁

—

#17985820

C₄

yes

characteristic

#17985782

C₄

yes

—

M₆₄×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#14556150

C₈

yes

characteristic

#14556134

C₂×C₄

yes

—

#14556068

C₂×C₄

yes

—

#14556067

—

C₁₆×D₇₁

—

#14556064

C₈

yes

characteristic

#14556056

C₈

yes

—

C₇₁⋊C₃₂

—

C₇₁⋊C₃₂

D₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₃₂×C₇₁

—

#11765865

C₂×C₈

yes

—

Dic₂₈₄

C₁₆

yes

characteristic

#11765854

C₁₆

yes

—

D₅₆₈

C₁₆

yes

characteristic

#11765849

C₂×C₈

yes

—

#11765848

—

C₈×D₇₁

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂²⋊D₇₁

—

D₂₈₄

C₂×C₁₆

yes

—

C₄×D₇₁

—

Dic₁₄₂

C₂×C₁₆

yes

—

C₈×C₇₁

C₂×C₁₆

yes

characteristic

C₈×C₇₁

C₂×C₁₆

yes

characteristic

C₈×C₇₁

C₄×D₄

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

#504

yes

characteristic

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₈×D₄

yes

characteristic

C₄×C₇₁

C₂²×C₁₆

yes

characteristic

Dic₇₁

—

#13648

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

D₄×C₁₆

yes

characteristic

D₇₁

—

#2258

—

#2170

—

#1422

D₇₁

yes

characteristic, complemented

#1401

—

#1397

—

C₇₁

#13648

yes

characteristic, complemented, Sylow

#504

—

#443

—

M₆₄

—

142

—

M₆₄

C₂×D₇₁

yes

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

142

—

#364

—

#350

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#135

—

#131

—

M₃₂

—

M₃₂

—

142

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#108

—

#104

—

C₄×C₈

C₄×D₇₁

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

142

—

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

142

—

C₂×C₈

—

142

—

100

C₂×C₈

C₈×D₇₁

yes

characteristic

101

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

102

C₄²

C₈×D₇₁

yes

characteristic

103

C₁₆

—

104

C₁₆

—

142

—

105

C₁₆

D₄×D₇₁

yes

characteristic

106

C₁₆

#11765866

yes

characteristic

107

Q₈

—

108

D₄

—

109

D₄

—

142

—

110

C₂×C₄

—

111

C₂×C₄

—

142

—

112

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

113

C₈

—

142

—

114

C₈

C₁₆×D₇₁

yes

characteristic

115

C₈

C₁₆×D₇₁

yes

characteristic

116

C₈

#14556090

yes

characteristic

117

C₈

#14556136

yes

characteristic

118

C₂²

—

142

—

119

C₂²

#17986417

yes

characteristic

120

C₄

—

121

C₄

—

142

—

122

C₄

#17986064

yes

characteristic

123

C₄

C₂×C₁₆×D₇₁

yes

characteristic

124

C₂

—

125

C₂

—

284

—

126

C₂

#22590159

yes

characteristic

	subgroup	G/H
center	C₁₆	D₄×D₇₁
commutator	C₄×C₇₁	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1422	C₂
socle	C₂×C₇₁	D₄×C₁₆
prime radical	#14556150	C₈
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	#1422	D₇₁
hypocenter	C₇₁	#13648
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

subgroup

G/H

center

C₁₆

D₄×D₇₁

commutator

C₄×C₇₁

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#1422

C₂

socle

C₂×C₇₁

D₄×C₁₆

prime radical

#14556150

C₈

cosocle

C₂×C₁₆×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

N	H ≅ G/N	type
1. #22590635	C₂	semidirect
3. #22586483	C₂	semidirect
4. #22586474	C₂	semidirect
5. C₇₁×#1422	C₂	semidirect
66. #1422	D₇₁	semidirect
69. C₇₁	#13648	semidirect

H ≅ G/N

type

1. #22590635

C₂

semidirect

3. #22586483

C₂

semidirect

4. #22586474

C₂

semidirect

5. C₇₁×#1422

C₂

semidirect

66. #1422

D₇₁

semidirect

69. C₇₁

#13648

semidirect

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

subgroup	index
#22590635	2
#22590178	2
#22586483	2
#22586474	2
C₇₁×#1422	2
#13648	71

subgroup

index

C₇₁×#1422

Subgroups — #31918020

Conjugacy classes of subgroups

Complements

Maximal subgroups