Group #31891264 — subgroups

Subgroups — #31891264

Order 18176.

338 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 163 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	1	yes	maximal, characteristic
2	#22587098	C₂	1	4	yes	maximal
3	C₂×#17985720	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556064	—	2	2	no	—
6	#17986023	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
8	#17985751	—	2	8	no	—
9	#17985720	C₄	1	4	yes	—
10	C₂×#108×C₇₁	—	2	2	no	—
11	#423×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#11765849	—	2	2	no	—
14	C₂²×C₇₁⋊C₈	Q₈	1	4	yes	—
15	#14556064	—	4	4	no	—
16	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
17	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₇₁⋊C₈	D₄	1	4	yes	—
19	C₂×M₁₆×C₇₁	—	2	2	no	—
20	C₂²×C₈×C₇₁	D₄	1	2	yes	—
21	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
22	#108×C₇₁	—	4	4	no	—
23	C₄⋊C₈×C₇₁	—	2	4	no	—
24	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₁	Q₈	1	2	yes	—
26	#11765849	—	4	4	no	—
27	C₂×C₇₁⋊C₈	—	4	4	no	—
28	C₂×C₇₁⋊C₈	—	4	2	no	—
29	C₂×C₇₁⋊C₈	C₄⋊C₄	1	8	yes	—
30	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
32	M₁₆×C₇₁	—	4	4	no	—
33	C₂×C₈×C₇₁	—	4	2	no	—
34	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₇₁	D₈	1	4	yes	—
36	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
37	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₇₁	C₂×D₄	1	2	yes	—
39	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
40	C₂³×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇₁	—	2	4	no	—
42	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
46	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
47	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
48	C₈×C₇₁	—	8	2	no	—
49	C₈×C₇₁	#107	1	8	yes	—
50	C₇₁⋊C₈	—	2	8	no	—
51	C₇₁⋊C₈	—	8	2	no	—
52	C₂²×C₇₁	—	2	2	no	—
53	C₂²×C₇₁	#384	1	1	yes	characteristic
54	C₂²×C₇₁	#500	1	2	yes	—
55	C₄×C₇₁	—	2	2	no	—
56	C₄×C₇₁	C₂×#107	1	2	yes	—
57	C₄×C₇₁	#430	1	4	yes	—
58	C₄×C₇₁	#493	1	1	yes	characteristic
59	C₄×C₇₁	#494	1	1	yes	characteristic
60	#8826	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	—	2	2	no	—
62	C₂×C₇₁	#1845	1	1	yes	characteristic
63	C₂×C₇₁	#1849	1	2	yes	—
64	#1848	—	71	1	no	—
65	#1848	D₇₁	1	1	yes	characteristic
66	#1567	—	71	4	no	—
67	C₂×#322	—	71	1	no	—
68	C₇₁	#8826	1	1	yes	characteristic, complemented, Sylow
69	C₂×#108	—	2	2	no	—
70	C₂×#108	—	142	2	no	—
71	#423	—	71	2	no	—
72	#423	C₂×D₇₁	1	2	yes	—
73	C₂×C₄×C₈	—	71	2	no	—
74	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
75	#335	—	142	8	no	—
76	#322	—	71	4	no	—
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	142	2	no	—
79	C₂²×C₈	—	71	4	no	—
80	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
81	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
82	#108	—	4	4	no	—
83	#108	—	284	4	no	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	142	4	no	—
86	C₄×C₈	—	71	4	no	—
87	C₄×C₈	—	71	4	no	—
88	C₄×C₈	Dic₁₄₂	1	2	yes	—
89	C₄×C₈	C₄×D₇₁	1	2	yes	—
90	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
91	C₂²×C₄	Q₈×D₇₁	1	2	yes	—
92	M₁₆	—	4	4	no	—
93	M₁₆	—	284	4	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	284	2	no	—
96	C₂×C₈	—	284	4	no	—
97	C₂×C₈	—	71	8	no	—
98	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
99	C₂×C₈	#11765850	1	4	yes	—
100	C₂×C₈	#11765855	1	4	yes	—
101	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
102	C₄²	D₄×D₇₁	1	2	yes	—
103	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
104	C₂³	#14556126	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	C₄⋊C₄×D₇₁	1	2	yes	—
107	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
108	C₂×C₄	#14556124	1	1	yes	characteristic
109	C₂×C₄	#14556127	1	2	yes	—
110	C₂×C₄	C₂×#11765850	1	2	yes	—
111	C₂×C₄	C₂×#11765855	1	2	yes	—
112	C₈	—	8	2	no	—
113	C₈	—	568	2	no	—
114	C₈	—	142	8	no	—
115	C₈	#14556061	1	8	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	#17985868	1	1	yes	characteristic
118	C₂²	#17986402	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	C₂×#14556061	1	2	yes	—
121	C₄	#17986046	1	4	yes	—
122	C₄	#17986375	1	1	yes	characteristic
123	C₄	#17986378	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#22588499	1	1	yes	characteristic
126	C₂	#22588513	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588512

C₂

yes

maximal, characteristic

#22587098

C₂

yes

maximal

C₂×#17985720

C₂

yes

maximal, characteristic

C₇₁×#1848

C₂

yes

maximal, characteristic

C₂×#14556064

—

#17986023

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

#17985751

—

#17985720

C₄

yes

—

C₂×#108×C₇₁

—

#423×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

C₂²×C₇₁⋊C₈

Q₈

yes

—

#14556064

—

C₄⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#108×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

Q₁₆

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

C₂×Q₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

#107

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#384

yes

characteristic

C₂²×C₇₁

#500

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#107

yes

—

C₄×C₇₁

#430

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

#1845

yes

characteristic

C₂×C₇₁

#1849

yes

—

#1848

—

#1848

D₇₁

yes

characteristic

#1567

—

C₂×#322

—

C₇₁

#8826

yes

characteristic, complemented, Sylow

C₂×#108

—

C₂×#108

—

142

—

#423

—

#423

C₂×D₇₁

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#335

—

142

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₇₁

yes

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

#108

—

#108

—

284

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₄₂

yes

—

C₄×C₈

C₄×D₇₁

yes

—

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

Q₈×D₇₁

yes

—

M₁₆

—

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

—

284

—

C₂×C₈

—

C₂×C₈

Dic₇₁⋊C₄

yes

—

C₂×C₈

#11765850

yes

—

100

C₂×C₈

#11765855

yes

—

101

C₄²

C₂×C₄×D₇₁

yes

characteristic

102

C₄²

D₄×D₇₁

yes

—

103

C₄²

C₂×Dic₁₄₂

yes

characteristic

104

C₂³

#14556126

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

C₄⋊C₄×D₇₁

yes

—

107

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

108

C₂×C₄

#14556124

yes

characteristic

109

C₂×C₄

#14556127

yes

—

110

C₂×C₄

C₂×#11765850

yes

—

111

C₂×C₄

C₂×#11765855

yes

—

112

C₈

—

113

C₈

—

568

—

114

C₈

—

142

—

115

C₈

#14556061

yes

—

116

C₂²

—

117

C₂²

#17985868

yes

characteristic

118

C₂²

#17986402

yes

—

119

C₄

—

120

C₄

C₂×#14556061

yes

—

121

C₄

#17986046

yes

—

122

C₄

#17986375

yes

characteristic

123

C₄

#17986378

yes

characteristic

124

C₂

—

125

C₂

#22588499

yes

characteristic

126

C₂

#22588513

yes

—

	subgroup	G/H
center	C₂×C₄	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1848	C₂
socle	C₂²×C₇₁	#384
prime radical	C₂³×C₇₁	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1848	D₇₁
hypocenter	C₇₁	#8826
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#14556124

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1848

C₂

socle

C₂²×C₇₁

#384

prime radical

C₂³×C₇₁

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
68. C₇₁	#8826	semidirect

H ≅ G/N

type

68. C₇₁

#8826

semidirect

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

subgroup	index
#22588512	2
#22587098	2
C₂×#17985720	2
C₇₁×#1848	2
#8826	71

subgroup

index

C₇₁×#1848

Subgroups — #31891264

Conjugacy classes of subgroups

Complements

Maximal subgroups