Group #31918425 — subgroups

Subgroups — #31918425

Order 18176.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22590615	C₂	1	1	yes	maximal, characteristic
2	#22590593	C₂	1	1	yes	maximal, characteristic
3	#22586522	C₂	1	4	yes	maximal
4	C₄×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₆₈	—	2	1	no	—
6	C₂×Dic₅₆₈	C₂²	1	2	yes	—
7	#17986404	—	2	1	no	—
8	#17986379	—	2	1	no	—
9	#17985825	—	2	1	no	—
10	#17985825	C₂²	1	2	yes	—
11	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
12	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₈₄	—	2	2	no	—
14	C₂×Dic₂₈₄	—	4	1	no	—
15	#14556128	—	4	1	no	—
16	#14556128	—	4	1	no	—
17	Dic₅₆₈	—	2	4	no	—
18	Dic₅₆₈	—	4	2	no	—
19	#14556062	—	2	2	no	—
20	#14556062	—	4	1	no	—
21	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₃₂×C₇₁	—	2	2	no	—
25	C₂×Dic₁₄₂	—	4	2	no	—
26	C₂×Dic₁₄₂	—	8	1	no	—
27	Dic₂₈₄	—	4	2	no	—
28	Dic₂₈₄	—	4	2	no	—
29	Dic₂₈₄	—	8	2	no	—
30	C₄⋊Dic₇₁	—	4	2	no	—
31	C₄⋊Dic₇₁	—	8	1	no	—
32	C₄⋊Dic₇₁	—	8	1	no	—
33	C₄⋊Dic₇₁	—	8	1	no	—
34	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₇₁	D₈	1	1	yes	characteristic
37	C₁₆×C₇₁	SD₁₆	1	2	yes	—
38	C₁₆×C₇₁	#41	1	2	yes	—
39	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
40	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
42	C₂×Dic₇₁	—	8	2	no	—
43	C₂×Dic₇₁	—	8	2	no	—
44	Dic₁₄₂	—	8	2	no	—
45	Dic₁₄₂	—	8	2	no	—
46	Dic₁₄₂	—	16	2	no	—
47	C₈×C₇₁	SD₃₂	1	2	yes	—
48	C₈×C₇₁	#124	1	1	yes	characteristic
49	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
50	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
51	C₄×C₇₁	SD₆₄	1	2	yes	—
52	C₄×C₇₁	Q₆₄	1	2	yes	—
53	C₄×C₇₁	#486	1	1	yes	characteristic
54	C₄×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
55	Dic₇₁	—	16	2	no	—
56	Dic₇₁	—	16	2	no	—
57	#13720	—	71	1	no	maximal, Sylow
58	C₂×C₇₁	#2240	1	1	yes	characteristic
59	C₂×C₇₁	C₂×SD₆₄	1	1	yes	characteristic
60	C₂×C₇₁	C₂×Q₆₄	1	1	yes	characteristic
61	#2252	—	71	1	no	—
62	#2247	—	71	1	no	—
63	#1416	—	71	4	no	—
64	C₄×C₃₂	D₇₁	1	1	yes	characteristic
65	C₇₁	#13720	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₃₂	—	71	2	no	—
67	C₂×Q₃₂	—	142	1	no	—
68	#500	—	142	1	no	—
69	#494	—	142	1	no	—
70	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
71	#366	—	71	2	no	—
72	#366	—	142	1	no	—
73	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
74	C₂×Q₁₆	—	284	1	no	—
75	C₂×Q₁₆	—	142	2	no	—
76	C₄⋊Q₈	—	284	1	no	—
77	C₄⋊Q₈	—	284	1	no	—
78	Q₃₂	—	284	2	no	—
79	Q₃₂	—	142	4	no	—
80	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
82	#107	—	284	1	no	—
83	#107	—	142	2	no	—
84	C₄×C₈	D₂₈₄	1	1	yes	characteristic
85	C₃₂	—	2	2	no	—
86	C₂×Q₈	—	284	2	no	—
87	C₂×Q₈	—	568	1	no	—
88	Q₁₆	—	284	2	no	—
89	Q₁₆	—	284	2	no	—
90	Q₁₆	—	568	2	no	—
91	C₂×C₈	D₅₆₈	1	1	yes	characteristic
92	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
93	C₄⋊C₄	—	284	2	no	—
94	C₄⋊C₄	—	568	1	no	—
95	C₄⋊C₄	—	568	1	no	—
96	C₄⋊C₄	—	568	1	no	—
97	C₄²	D₅₆₈	1	1	yes	characteristic
98	C₁₆	#11765854	1	2	yes	—
99	C₁₆	#11765865	1	2	yes	—
100	Q₈	—	568	2	no	—
101	Q₈	—	568	2	no	—
102	Q₈	—	1136	2	no	—
103	C₂×C₄	—	568	2	no	—
104	C₂×C₄	—	568	2	no	—
105	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
106	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
107	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
108	C₈	#14556073	1	2	yes	—
109	C₈	#14556116	1	1	yes	characteristic
110	C₈	C₂×#11765854	1	1	yes	characteristic
111	C₂²	C₂×D₁₁₃₆	1	1	yes	characteristic
112	C₄	—	1136	2	no	—
113	C₄	—	1136	2	no	—
114	C₄	#17985838	1	2	yes	—
115	C₄	Dic₁₁₃₆	1	2	yes	—
116	C₄	#17986346	1	1	yes	characteristic
117	C₄	C₂×#14556073	1	1	yes	characteristic
118	C₂	#22590564	1	1	yes	characteristic
119	C₂	C₂×#17985838	1	1	yes	characteristic
120	C₂	C₂×Dic₁₁₃₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590615

C₂

yes

maximal, characteristic

#22590593

C₂

yes

maximal, characteristic

#22586522

C₂

yes

maximal

C₄×C₃₂×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

—

C₂×Dic₅₆₈

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₃₂×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

#14556128

—

Dic₅₆₈

—

Dic₅₆₈

—

#14556062

—

#14556062

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₃₂×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₁₆×C₇₁

SD₁₆

yes

—

C₁₆×C₇₁

#41

yes

—

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

SD₃₂

yes

—

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

SD₆₄

yes

—

C₄×C₇₁

Q₆₄

yes

—

C₄×C₇₁

#486

yes

characteristic

C₄×C₇₁

C₂×SD₃₂

yes

characteristic

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

C₂×SD₆₄

yes

characteristic

C₂×C₇₁

C₂×Q₆₄

yes

—

—

—

C₄×C₃₂

D₇₁

yes

characteristic

C₇₁

#13720

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

142

—

#500

—

142

—

#494

—

142

—

C₂×C₃₂

C₂×D₇₁

yes

—

#366

—

#366

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

284

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

284

—

C₄⋊Q₈

—

284

—

Q₃₂

—

284

—

Q₃₂

—

142

—

C₂×C₁₆

D₂₈₄

yes

characteristic

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#107

—

284

—

#107

—

142

—

C₄×C₈

D₂₈₄

yes

characteristic

C₃₂

—

C₂×Q₈

—

284

—

C₂×Q₈

—

568

—

Q₁₆

—

284

—

Q₁₆

—

284

—

Q₁₆

—

568

—

C₂×C₈

D₅₆₈

yes

characteristic

C₂×C₈

C₂×D₂₈₄

yes

characteristic

C₄⋊C₄

—

284

—

C₄⋊C₄

—

568

—

C₄⋊C₄

—

568

—

C₄⋊C₄

—

568

—

C₄²

D₅₆₈

yes

characteristic

C₁₆

#11765854

yes

—

C₁₆

#11765865

yes

—

100

Q₈

—

568

—

101

Q₈

—

568

—

102

Q₈

—

1136

—

103

C₂×C₄

—

568

—

104

C₂×C₄

—

568

—

105

C₂×C₄

D₁₁₃₆

yes

characteristic

106

C₂×C₄

D₁₁₃₆

yes

characteristic

107

C₂×C₄

C₂×D₅₆₈

yes

characteristic

108

C₈

#14556073

yes

—

109

C₈

#14556116

yes

characteristic

110

C₈

C₂×#11765854

yes

characteristic

111

C₂²

C₂×D₁₁₃₆

yes

characteristic

112

C₄

—

1136

—

113

C₄

—

1136

—

114

C₄

#17985838

yes

—

115

C₄

Dic₁₁₃₆

yes

—

116

C₄

#17986346

yes

characteristic

117

C₄

C₂×#14556073

yes

characteristic

118

C₂

#22590564

yes

characteristic

119

C₂

C₂×#17985838

yes

characteristic

120

C₂

C₂×Dic₁₁₃₆

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₁₃₆
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₄×C₃₂×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	C₄×C₃₂	D₇₁
hypocenter	C₇₁	#13720
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂²

C₂×D₁₁₃₆

commutator, cosocle

C₂×C₁₆×C₇₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₄×C₃₂×C₇₁

C₂

socle, prime radical

C₂²×C₇₁

C₂×D₁₆

hypercenter

C₄×C₃₂

D₇₁

hypocenter

C₇₁

#13720

LCS

G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
65. C₇₁	#13720	semidirect

H ≅ G/N

type

65. C₇₁

#13720

semidirect

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

subgroup	index
#22590615	2
#22590593	2
#22586522	2
C₄×C₃₂×C₇₁	2
#13720	71

subgroup

index

C₄×C₃₂×C₇₁

Subgroups — #31918425

Conjugacy classes of subgroups

Complements

Maximal subgroups