Group #31780159 — subgroups

Subgroups — #31780159

Order 17536.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#22577783	C₂	1	1	yes	maximal, characteristic
2	#22577688	C₂	1	2	yes	maximal
3	#22577156	C₂	1	1	yes	maximal, characteristic
4	#22577150	C₂	1	2	yes	maximal
5	C₈×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₄₈	—	2	1	no	—
7	C₂×Dic₅₄₈	—	2	1	no	—
8	#17984088	C₂²	1	2	yes	—
9	#17984019	—	2	2	no	—
10	C₄⋊#11764040	C₂²	1	2	yes	—
11	#17984012	—	2	2	no	—
12	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₃₇	—	2	1	no	—
14	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₃₇	—	2	1	no	—
16	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₇₄	—	2	2	no	—
18	Dic₅₄₈	—	4	2	no	—
19	Dic₅₄₈	—	4	2	no	—
20	C₂×#11764040	—	2	2	no	—
21	C₄⋊Dic₁₃₇	—	2	2	no	—
22	C₄⋊Dic₁₃₇	—	2	2	no	—
23	C₂×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃₇	—	2	1	no	—
25	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₁₃₇	—	2	1	no	—
28	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
29	C₄²×C₁₃₇	—	2	1	no	—
30	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
31	Dic₂₇₄	—	4	2	no	—
32	Dic₂₇₄	—	4	2	no	—
33	C₂×Dic₁₃₇	—	4	2	no	—
34	#11764040	—	4	2	no	—
35	Q₈×C₁₃₇	—	2	1	no	—
36	Q₈×C₁₃₇	Q₁₆	1	2	yes	complemented
37	C₂×C₄×C₁₃₇	—	2	1	no	—
38	C₂×C₄×C₁₃₇	—	2	1	no	—
39	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
42	C₈×C₁₃₇	—	2	1	no	—
43	C₈×C₁₃₇	—	4	1	no	—
44	C₈×C₁₃₇	SD₁₆	1	2	yes	—
45	C₂²×C₁₃₇	#121	1	1	yes	characteristic
46	C₄×C₁₃₇	—	2	2	no	—
47	C₄×C₁₃₇	—	2	1	no	—
48	C₄×C₁₃₇	—	4	1	no	—
49	C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₁₃₇	#135	1	1	yes	characteristic
52	C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
53	Dic₁₃₇	—	8	2	no	—
54	C₂×C₁₃₇	#460	1	1	yes	characteristic
55	C₂×C₁₃₇	#467	1	1	yes	characteristic
56	C₂×C₁₃₇	#473	1	1	yes	characteristic
57	C₁₃₇	#1670	1	1	yes	characteristic, complemented, Sylow
58	#1670	—	137	1	no	maximal, Sylow
59	#494	—	137	1	no	—
60	#475	—	137	2	no	—
61	C₈×Q₈	D₁₃₇	1	1	yes	characteristic
62	#335	—	137	1	no	—
63	#332	—	137	2	no	—
64	C₂×Q₁₆	—	274	1	no	—
65	C₂×Q₁₆	—	274	1	no	—
66	C₄⋊Q₈	—	137	2	no	—
67	C₄×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
68	#106	—	274	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	137	2	no	—
71	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
72	#103	—	274	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
75	C₂×Q₈	—	274	2	no	—
76	C₂×Q₈	D₅₄₈	1	1	yes	characteristic
77	Q₁₆	—	548	2	no	—
78	Q₁₆	—	548	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	274	2	no	—
81	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	274	2	no	—
85	C₄⋊C₄	—	274	2	no	—
86	C₄⋊C₄	D₅₄₈	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	548	2	no	—
91	Q₈	—	548	2	no	—
92	Q₈	Dic₅₄₈	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	548	2	no	—
96	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
97	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
98	C₂×C₄	#14555510	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	548	2	no	—
102	C₈	#14555497	1	2	yes	—
103	C₂²	#17984062	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	—	4	1	no	—
107	C₄	—	1096	2	no	—
108	C₄	C₂×#14555497	1	1	yes	characteristic
109	C₄	C₂×Dic₅₄₈	1	1	yes	characteristic
110	C₄	#17984110	1	1	yes	characteristic
111	C₄	#17984113	1	1	yes	characteristic
112	C₂	#22577603	1	1	yes	characteristic
113	C₂	#22577644	1	1	yes	characteristic
114	C₂	#22577678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577783

C₂

yes

maximal, characteristic

#22577688

C₂

yes

maximal

#22577156

C₂

yes

maximal, characteristic

#22577150

C₂

yes

maximal

C₈×Q₈×C₁₃₇

C₂

yes

maximal, characteristic

C₂×Dic₅₄₈

—

C₂×Dic₅₄₈

—

#17984088

C₂²

yes

—

#17984019

—

C₄⋊#11764040

C₂²

yes

—

#17984012

—

C₄×Q₈×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

—

C₄⋊C₈×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

—

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

Dic₅₄₈

—

Dic₅₄₈

—

C₂×#11764040

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂×Q₈×C₁₃₇

D₄

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

D₄

yes

characteristic

C₂×C₈×C₁₃₇

D₄

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

D₄

yes

characteristic

C₄²×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

#11764040

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

Q₁₆

yes

complemented

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

SD₁₆

yes

—

C₂²×C₁₃₇

#121

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₃₇

C₂×Q₁₆

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

Dic₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1670

—

137

maximal, Sylow

#494

—

137

—

#475

—

137

—

C₈×Q₈

D₁₃₇

yes

characteristic

#335

—

137

—

#332

—

137

—

C₂×Q₁₆

—

274

—

C₂×Q₁₆

—

274

—

C₄⋊Q₈

—

137

—

C₄×Q₈

C₂×D₁₃₇

yes

characteristic

#106

—

274

—

C₄⋊C₈

—

C₄⋊C₈

—

137

—

C₄⋊C₈

C₂×D₁₃₇

yes

characteristic

#103

—

274

—

C₄×C₈

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×Q₈

—

274

—

C₂×Q₈

D₅₄₈

yes

characteristic

Q₁₆

—

548

—

Q₁₆

—

548

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

C₂²⋊D₁₃₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

D₅₄₈

yes

characteristic

C₄²

—

C₄²

C₂²×D₁₃₇

yes

characteristic

Q₈

—

Q₈

—

548

—

Q₈

—

548

—

Q₈

Dic₅₄₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

548

—

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

C₂×C₄

C₂×D₅₄₈

yes

characteristic

C₂×C₄

#14555510

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

548

—

102

C₈

#14555497

yes

—

103

C₂²

#17984062

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

1096

—

108

C₄

C₂×#14555497

yes

characteristic

109

C₄

C₂×Dic₅₄₈

yes

characteristic

110

C₄

#17984110

yes

characteristic

111

C₄

#17984113

yes

characteristic

112

C₂

#22577603

yes

characteristic

113

C₂

#22577644

yes

characteristic

114

C₂

#22577678

yes

characteristic

	subgroup	G/H
center	C₂²	#17984062
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	C₈×Q₈×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	#121
hypercenter	C₈×Q₈	D₁₃₇
hypocenter	C₁₃₇	#1670
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#17984062

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

C₈×Q₈×C₁₃₇

C₂

socle, prime radical

C₂²×C₁₃₇

#121

hypercenter

C₈×Q₈

D₁₃₇

hypocenter

C₁₃₇

#1670

LCS

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
36. Q₈×C₁₃₇	Q₁₆	semidirect
57. C₁₃₇	#1670	semidirect
92. Q₈	Dic₅₄₈	semidirect

H ≅ G/N

type

36. Q₈×C₁₃₇

Q₁₆

semidirect

57. C₁₃₇

#1670

semidirect

92. Q₈

Dic₅₄₈

semidirect

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

subgroup	index
#22577783	2
#22577688	2
#22577156	2
#22577150	2
C₈×Q₈×C₁₃₇	2
#1670	137

subgroup

index

C₈×Q₈×C₁₃₇

137

Subgroups — #31780159

Conjugacy classes of subgroups

Complements

Maximal subgroups