Group #31884635 — subgroups

Subgroups — #31884635

Order 18176.

208 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂×#17985820	C₂	1	2	yes	maximal
2	C₇₁×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556068	—	2	2	no	—
4	#17985820	—	2	2	no	—
5	#17985820	C₄	1	4	yes	—
6	C₂×#108×C₇₁	—	2	2	no	—
7	#423×C₇₁	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	#14556068	—	2	4	no	—
10	#14556068	—	4	2	no	—
11	C₂×C₇₁⋊C₁₆	—	4	2	no	—
12	C₂×M₁₆×C₇₁	—	2	2	no	—
13	C₂²×C₈×C₇₁	D₄	1	2	yes	—
14	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
15	#108×C₇₁	—	4	2	no	—
16	#108×C₇₁	—	4	2	no	—
17	C₄⋊C₈×C₇₁	—	4	2	no	—
18	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
20	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
21	C₇₁⋊C₁₆	—	4	4	no	—
22	C₂²×C₄×C₇₁	—	2	1	no	—
23	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
24	M₁₆×C₇₁	—	4	4	no	—
25	C₂×C₈×C₇₁	—	4	2	no	—
26	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
27	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₇₁	D₈	1	2	yes	—
29	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
30	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
31	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
32	C₄²×C₇₁	—	2	1	no	—
33	C₄²×C₇₁	C₄²	1	1	yes	characteristic
34	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂³×C₇₁	#101	1	1	yes	characteristic
36	C₂×C₄×C₇₁	—	2	1	no	—
37	C₂×C₄×C₇₁	—	2	1	no	—
38	C₂×C₄×C₇₁	—	4	2	no	—
39	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
40	C₂×C₄×C₇₁	#100	1	1	yes	characteristic
41	C₂×C₄×C₇₁	#102	1	2	yes	—
42	C₂×C₄×C₇₁	#103	1	2	yes	—
43	C₈×C₇₁	—	8	2	no	—
44	C₈×C₇₁	#102	1	2	yes	—
45	C₈×C₇₁	#103	1	2	yes	—
46	C₈×C₇₁	#106	1	2	yes	—
47	C₈×C₇₁	#107	1	2	yes	—
48	C₂²×C₇₁	—	4	1	no	—
49	C₂²×C₇₁	#332	1	2	yes	—
50	C₂²×C₇₁	#343	1	1	yes	characteristic
51	C₄×C₇₁	—	2	2	no	—
52	C₄×C₇₁	—	4	1	no	—
53	C₄×C₇₁	#331	1	1	yes	characteristic
54	C₄×C₇₁	#333	1	1	yes	characteristic
55	C₄×C₇₁	#340	1	2	yes	—
56	#7273	—	71	1	no	maximal, Sylow
57	C₂×C₇₁	—	2	1	no	—
58	C₂×C₇₁	—	2	2	no	—
59	C₂×C₇₁	#1296	1	1	yes	characteristic
60	C₂×#364	—	71	2	no	—
61	#1848	D₇₁	1	1	yes	characteristic
62	C₇₁	#7273	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₃₂	—	142	2	no	—
64	C₂×#108	—	2	2	no	—
65	#423	Dic₇₁	1	2	yes	—
66	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
67	#364	—	71	4	no	—
68	#364	—	142	2	no	—
69	C₂×M₁₆	—	2	2	no	—
70	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
71	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
72	M₃₂	—	284	2	no	—
73	M₃₂	—	142	4	no	—
74	C₂×C₁₆	—	284	2	no	—
75	#108	—	4	2	no	—
76	#108	—	4	2	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
79	C₄×C₈	C₄×D₇₁	1	2	yes	—
80	C₄×C₈	D₂₈₄	1	1	yes	characteristic
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
83	M₁₆	—	4	4	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
86	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
87	C₂×C₈	#11765850	1	2	yes	—
88	C₂×C₈	#11765852	1	2	yes	—
89	C₂×C₈	#11765853	1	2	yes	—
90	C₂×C₈	#11765855	1	2	yes	—
91	C₄²	—	2	1	no	—
92	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
93	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
94	C₁₆	—	284	4	no	—
95	C₂³	#14556047	1	1	yes	characteristic
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
100	C₂×C₄	#14556045	1	1	yes	characteristic
101	C₂×C₄	#14556049	1	2	yes	—
102	C₂×C₄	#14556052	1	2	yes	—
103	C₈	—	8	2	no	—
104	C₈	#14556048	1	2	yes	—
105	C₈	#14556051	1	2	yes	—
106	C₈	#14556059	1	2	yes	—
107	C₈	#14556061	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#17985745	1	2	yes	—
110	C₂²	#17985768	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	4	1	no	—
113	C₄	#17985743	1	1	yes	characteristic
114	C₄	#17985748	1	1	yes	characteristic
115	C₄	#17985761	1	2	yes	—
116	C₂	—	2	1	no	—
117	C₂	—	2	2	no	—
118	C₂	#22586223	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985820

C₂

yes

maximal

C₇₁×#1848

C₂

yes

maximal, characteristic

C₂×#14556068

—

#17985820

—

#17985820

C₄

yes

—

C₂×#108×C₇₁

—

#423×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

#108×C₇₁

—

#108×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

characteristic

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

Q₁₆

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₄²

yes

characteristic

C₄²×C₇₁

C₄⋊C₄

yes

characteristic

C₂³×C₇₁

#101

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#103

yes

—

C₈×C₇₁

—

C₈×C₇₁

#102

yes

—

C₈×C₇₁

#103

yes

—

C₈×C₇₁

#106

yes

—

C₈×C₇₁

#107

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

#332

yes

—

C₂²×C₇₁

#343

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#7273

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

C₇₁

#7273

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

142

—

C₂×#108

—

#423

Dic₇₁

yes

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#364

—

#364

—

142

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

characteristic

M₃₂

—

284

—

M₃₂

—

142

—

C₂×C₁₆

—

284

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

Dic₁₄₂

yes

characteristic

C₄×C₈

C₄×D₇₁

yes

—

C₄×C₈

D₂₈₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₇₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

C₂×C₈

Dic₇₁⋊C₄

yes

—

C₂×C₈

#11765850

yes

—

C₂×C₈

#11765852

yes

—

C₂×C₈

#11765853

yes

—

C₂×C₈

#11765855

yes

—

C₄²

—

C₄²

C₄×Dic₇₁

yes

characteristic

C₄²

C₄⋊Dic₇₁

yes

characteristic

C₁₆

—

284

—

C₂³

#14556047

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₇₁⋊#95

yes

characteristic

100

C₂×C₄

#14556045

yes

characteristic

101

C₂×C₄

#14556049

yes

—

102

C₂×C₄

#14556052

yes

—

103

C₈

—

104

C₈

#14556048

yes

—

105

C₈

#14556051

yes

—

106

C₈

#14556059

yes

—

107

C₈

#14556061

yes

—

108

C₂²

—

109

C₂²

#17985745

yes

—

110

C₂²

#17985768

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

#17985743

yes

characteristic

114

C₄

#17985748

yes

characteristic

115

C₄

#17985761

yes

—

116

C₂

—

117

C₂

—

118

C₂

#22586223

yes

characteristic

	subgroup	G/H
center	C₄	#17985748
commutator	C₄²×C₇₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₇₁×#1848	C₂
socle	C₂×C₇₁	#1296
prime radical	C₂³×C₇₁	#101
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	#1848	D₇₁
hypocenter	C₇₁	#7273
LCS	G ▷ C₄²×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1848

subgroup

G/H

center

C₄

#17985748

commutator

C₄²×C₇₁

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₁

Fitting

C₇₁×#1848

C₂

C₂×C₇₁

C₂³×C₇₁

C₂×C₄×C₈×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1848

N	H ≅ G/N	type
62. C₇₁	#7273	semidirect

H ≅ G/N

type

62. C₇₁

#7273

semidirect

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

subgroup	index
C₂×#17985820	2
C₇₁×#1848	2
#7273	71

subgroup

index

C₂×#17985820

C₇₁×#1848

#7273

Subgroups — #31884635

Conjugacy classes of subgroups

Complements

Maximal subgroups