Group #31909472 — subgroups

Subgroups — #31909472

Order 18176.

198 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 77 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	#22587249	C₂	1	2	yes	maximal
2	#22586406	C₂	1	1	yes	maximal, characteristic
3	#22586310	C₂	1	2	yes	maximal
4	#22586260	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#2176	C₂	1	1	yes	maximal, characteristic
6	#17986130	—	2	2	no	—
7	#17985775	C₂²	1	1	yes	characteristic
8	#17985750	C₂²	1	2	yes	—
9	Q₈⋊(C₇₁⋊C₈)	—	2	2	no	—
10	#17985720	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₇₁	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₇₁	C₂²	1	2	yes	—
13	#367×C₇₁	C₄	1	1	yes	characteristic
14	#358×C₇₁	C₄	1	2	yes	—
15	#346×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₇₁⋊C₁₆	—	2	1	no	—
17	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
19	#14556039	—	4	2	no	—
20	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
21	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
22	C₄×Q₈×C₇₁	—	2	2	no	—
23	Q₃₂×C₇₁	—	2	2	no	—
24	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
26	#107×C₇₁	C₂×C₄	1	2	yes	—
27	#103×C₇₁	—	2	2	no	—
28	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×C₇₁⋊C₈	—	2	2	no	—
30	C₂×C₇₁⋊C₈	—	4	2	no	—
31	C₇₁⋊C₁₆	—	4	1	no	—
32	C₂×Q₈×C₇₁	—	2	2	no	—
33	Q₁₆×C₇₁	—	2	2	no	—
34	Q₁₆×C₇₁	—	2	2	no	—
35	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₇₁	—	2	2	no	—
38	C₄⋊C₄×C₇₁	—	4	2	no	—
39	C₄²×C₇₁	—	4	2	no	—
40	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₁	—	2	1	no	—
42	C₁₆×C₇₁	M₁₆	1	2	yes	—
43	Q₈×C₇₁	—	4	2	no	—
44	Q₈×C₇₁	—	4	2	no	—
45	C₂×C₄×C₇₁	—	4	2	no	—
46	C₂×C₄×C₇₁	—	4	2	no	—
47	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
49	C₈×C₇₁	—	2	1	no	—
50	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
51	C₈×C₇₁	#131	1	1	yes	characteristic
52	C₇₁⋊C₈	—	4	2	no	—
53	C₇₁⋊C₈	—	8	2	no	—
54	C₂²×C₇₁	#442	1	1	yes	characteristic
55	C₄×C₇₁	—	2	2	no	—
56	C₄×C₇₁	—	8	2	no	—
57	C₄×C₇₁	—	8	2	no	—
58	C₄×C₇₁	#436	1	1	yes	characteristic
59	C₄×C₇₁	#444	1	1	yes	characteristic
60	#12088	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	#1594	1	1	yes	characteristic
62	C₂×C₇₁	#2180	1	2	yes	—
63	#2176	D₇₁	1	1	yes	characteristic
64	#1593	—	71	2	no	—
65	#1368	—	71	1	no	—
66	#1334	—	71	2	no	—
67	#1313	—	71	1	no	—
68	C₇₁	#12088	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	Dic₇₁	1	1	yes	characteristic
70	#446	—	142	2	no	—
71	C₄×Q₁₆	C₂×D₇₁	1	2	yes	—
72	#367	Dic₇₁	1	1	yes	characteristic
73	#358	Dic₇₁	1	2	yes	—
74	#346	—	71	1	no	—
75	#346	C₂×D₇₁	1	1	yes	characteristic
76	#334	—	71	2	no	—
77	Q₈⋊C₈	—	142	2	no	—
78	#322	—	71	1	no	—
79	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	—
80	C₄×Q₈	—	2	2	no	—
81	Q₃₂	—	2	2	no	—
82	C₂×C₁₆	—	142	1	no	—
83	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
84	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
85	#107	C₂×Dic₇₁	1	2	yes	—
86	C₄⋊C₈	—	284	2	no	—
87	C₄⋊C₈	—	142	2	no	—
88	#103	—	2	2	no	—
89	#97	—	284	2	no	—
90	C₄×C₈	—	71	2	no	—
91	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	2	2	no	—
95	C₂×C₈	—	284	2	no	—
96	C₂×C₈	—	142	2	no	—
97	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
98	C₂×C₈	#11765864	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	D₄×D₇₁	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	284	1	no	—
105	C₁₆	#11765849	1	2	yes	—
106	Q₈	—	4	2	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
111	C₂×C₄	#14556152	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	284	2	no	—
114	C₈	—	568	2	no	—
115	C₈	C₂×#11765849	1	1	yes	characteristic
116	C₈	#14556135	1	1	yes	characteristic
117	C₂²	#17986111	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	8	2	no	—
120	C₄	—	8	2	no	—
121	C₄	#17986077	1	1	yes	characteristic
122	C₄	#17986121	1	1	yes	characteristic
123	C₂	#22587254	1	1	yes	characteristic
124	C₂	#22590232	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22587249

C₂

yes

maximal

#22586406

C₂

yes

maximal, characteristic

#22586310

C₂

yes

maximal

#22586260

C₂

yes

maximal, characteristic

C₇₁×#2176

C₂

yes

maximal, characteristic

#17986130

—

#17985775

C₂²

yes

characteristic

#17985750

C₂²

yes

—

Q₈⋊(C₇₁⋊C₈)

—

#17985720

C₂²

yes

characteristic

C₂×Q₃₂×C₇₁

C₄

yes

characteristic

C₄×Q₁₆×C₇₁

C₂²

yes

—

#367×C₇₁

C₄

yes

characteristic

#358×C₇₁

C₄

yes

—

#346×C₇₁

C₂²

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

#14556039

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂×Q₁₆×C₇₁

C₂×C₄

yes

—

C₄×Q₈×C₇₁

—

Q₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

#107×C₇₁

C₂×C₄

yes

—

#103×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

M₁₆

yes

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#442

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

#2176

D₇₁

yes

—

—

—

—

C₇₁

#12088

yes

characteristic, complemented, Sylow

C₂×Q₃₂

Dic₇₁

yes

characteristic

#446

—

142

—

C₄×Q₁₆

C₂×D₇₁

yes

—

#367

Dic₇₁

yes

characteristic

#358

Dic₇₁

yes

—

#346

—

#346

C₂×D₇₁

yes

characteristic

#334

—

Q₈⋊C₈

—

142

—

#322

—

C₂×Q₁₆

C₂×Dic₇₁

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

#107

C₂×Dic₇₁

yes

—

C₄⋊C₈

—

284

—

C₄⋊C₈

—

142

—

#103

—

#97

—

284

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

C₂×C₈

#11765864

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

D₄×D₇₁

yes

characteristic

103

C₁₆

—

104

C₁₆

—

284

—

105

C₁₆

#11765849

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

D₄×Dic₇₁

yes

characteristic

111

C₂×C₄

#14556152

yes

—

112

C₈

—

113

C₈

—

284

—

114

C₈

—

568

—

115

C₈

C₂×#11765849

yes

characteristic

116

C₈

#14556135

yes

characteristic

117

C₂²

#17986111

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#17986077

yes

characteristic

122

C₄

#17986121

yes

characteristic

123

C₂

#22587254

yes

characteristic

124

C₂

#22590232

yes

—

	subgroup	G/H
center	C₂²	#17986111
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2176	C₂
socle, prime radical	C₂²×C₇₁	#442
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#2176	D₇₁
hypocenter	C₇₁	#12088
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

subgroup

G/H

center

C₂²

#17986111

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#2176

C₂

socle, prime radical

C₂²×C₇₁

#442

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

N	H ≅ G/N	type
68. C₇₁	#12088	semidirect

H ≅ G/N

type

68. C₇₁

#12088

semidirect

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

subgroup	index
#22587249	2
#22586406	2
#22586310	2
#22586260	2
C₇₁×#2176	2
#12088	71

subgroup

index

C₇₁×#2176

Subgroups — #31909472

Conjugacy classes of subgroups

Complements

Maximal subgroups