Group #31781577 — subgroups

Subgroups — #31781577

Order 17536.

184 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 85 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	#22577451	C₂	1	2	yes	maximal
2	C₂×#17984024	C₂	1	1	yes	maximal, characteristic
3	#22577358	C₂	1	1	yes	maximal, characteristic
4	#22577174	C₂	1	2	yes	maximal
5	C₂×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₂²	1	2	yes	—
7	C₂×#14555494	C₂²	1	1	yes	characteristic
8	#17984047	C₂²	1	1	yes	characteristic
9	#17984024	C₄	1	4	yes	complemented
10	#17984023	—	2	4	no	—
11	#17983998	—	2	1	no	—
12	C₈×Dic₁₃₇	—	2	1	no	—
13	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	#108×C₁₃₇	—	2	1	no	—
16	#108×C₁₃₇	—	2	1	no	—
17	#14555494	—	2	1	no	—
18	#14555494	—	2	2	no	—
19	#14555494	C₂×C₄	1	4	yes	—
20	#14555494	D₄	1	2	yes	—
21	C₂×#11764040	—	2	2	no	—
22	C₂×#11764040	—	2	1	no	—
23	C₄⋊Dic₁₃₇	—	2	1	no	—
24	C₂²⋊Dic₁₃₇	Q₈	1	2	yes	—
25	C₄×Dic₁₃₇	—	2	1	no	—
26	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₃₇	—	2	2	no	—
28	M₁₆×C₁₃₇	—	2	2	no	—
29	C₂×C₈×C₁₃₇	—	2	2	no	—
30	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃₇	Q₈	1	2	yes	—
34	C₂×Dic₁₃₇	—	2	2	no	—
35	#11764040	—	2	4	no	—
36	#11764040	—	2	2	no	—
37	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
44	C₈×C₁₃₇	—	4	2	no	—
45	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₃₇	—	4	1	no	—
48	C₂²×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
53	C₄×C₁₃₇	#121	1	1	yes	characteristic
54	C₄×C₁₃₇	#125	1	1	yes	characteristic
55	Dic₁₃₇	—	4	2	no	—
56	C₂×C₁₃₇	—	2	1	no	—
57	C₂×C₁₃₇	—	2	1	no	—
58	C₂×C₁₃₇	—	2	1	no	—
59	C₂×C₁₃₇	#389	1	1	yes	characteristic
60	C₁₃₇	#1953	1	1	yes	characteristic, complemented, Sylow
61	#1953	—	137	1	no	maximal, Sylow
62	#430	—	137	2	no	—
63	C₂×#108	—	137	1	no	—
64	C₂×#108	D₁₃₇	1	1	yes	characteristic
65	#405	—	137	1	no	—
66	#344	—	137	2	no	—
67	C₂×M₁₆	—	137	2	no	—
68	C₂×M₁₆	—	137	1	no	—
69	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
70	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
71	#117	—	137	1	no	—
72	#108	—	2	1	no	—
73	#108	—	2	1	no	—
74	#108	—	274	4	no	—
75	#108	—	137	4	no	—
76	#97	—	274	1	no	—
77	C₄×C₈	—	274	1	no	—
78	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	274	1	no	—
82	M₁₆	—	274	2	no	—
83	M₁₆	—	137	4	no	—
84	M₁₆	—	137	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	274	2	no	—
87	C₂×C₈	—	274	1	no	—
88	C₂×C₈	C₄×D₁₃₇	1	2	yes	—
89	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
91	C₂×C₈	Dic₂₇₄	1	2	yes	—
92	C₄⋊C₄	—	274	1	no	—
93	C₂²⋊C₄	—	137	2	no	—
94	C₄²	—	274	1	no	—
95	C₂³	Q₈×D₁₃₇	1	1	yes	characteristic
96	C₂×C₄	—	274	2	no	—
97	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
98	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
101	C₂×C₄	#14555509	1	1	yes	characteristic
102	C₂×C₄	#14555511	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	274	4	no	—
105	C₈	—	274	2	no	—
106	C₈	#14555489	1	2	yes	—
107	C₈	#14555489	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#17984053	1	1	yes	characteristic
110	C₂²	#17984064	1	1	yes	characteristic
111	C₂²	#17984087	1	1	yes	characteristic
112	C₄	—	548	2	no	—
113	C₄	C₂×#14555489	1	1	yes	characteristic
114	C₄	#17984050	1	1	yes	characteristic
115	C₄	#17984059	1	1	yes	characteristic
116	C₄	#17984079	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#22577299	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577451

C₂

yes

maximal

C₂×#17984024

C₂

yes

maximal, characteristic

#22577358

C₂

yes

maximal, characteristic

#22577174

C₂

yes

maximal

C₂×#108×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

C₂×#14555494

C₂²

yes

characteristic

#17984047

C₂²

yes

characteristic

#17984024

C₄

yes

complemented

#17984023

—

#17983998

—

C₈×Dic₁₃₇

—

C₂×M₁₆×C₁₃₇

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

#108×C₁₃₇

—

#108×C₁₃₇

—

#14555494

—

#14555494

—

#14555494

C₂×C₄

yes

—

#14555494

D₄

yes

—

C₂×#11764040

—

C₂×#11764040

—

C₄⋊Dic₁₃₇

—

C₂²⋊Dic₁₃₇

Q₈

yes

—

C₄×Dic₁₃₇

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×C₈×C₁₃₇

D₄

yes

characteristic

C₂×C₈×C₁₃₇

D₄

yes

characteristic

C₂×C₈×C₁₃₇

Q₈

yes

—

C₂×Dic₁₃₇

—

#11764040

—

#11764040

—

C₂³×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

C₂×C₄×C₁₃₇

yes

C₂×C₄×C₁₃₇

yes

C₈×C₁₃₇

—

C₈×C₁₃₇

C₄⋊C₄

yes

—

C₈×C₁₃₇

C₄⋊C₄

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄×Q₈

yes

characteristic

C₂²×C₁₃₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₇

C₄×D₄

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#389

yes

characteristic

C₁₃₇

#1953

yes

characteristic, complemented, Sylow

#1953

—

137

maximal, Sylow

#430

—

137

—

C₂×#108

—

137

—

C₂×#108

D₁₃₇

yes

characteristic

#405

—

137

—

#344

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

C₂×D₁₃₇

yes

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

#117

—

137

—

#108

—

#108

—

#108

—

274

—

#108

—

137

—

#97

—

274

—

C₄×C₈

—

274

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

137

—

M₁₆

—

137

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

C₄×D₁₃₇

yes

—

C₂×C₈

C₂²⋊D₁₃₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₇

yes

characteristic

C₂×C₈

Dic₂₇₄

yes

—

C₄⋊C₄

—

274

—

C₂²⋊C₄

—

137

—

C₄²

—

274

—

C₂³

Q₈×D₁₃₇

yes

characteristic

C₂×C₄

—

274

—

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

C₂×C₄

D₄×D₁₃₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

100

C₂×C₄

C₂×Dic₂₇₄

yes

characteristic

101

C₂×C₄

#14555509

yes

characteristic

102

C₂×C₄

#14555511

yes

characteristic

103

C₈

—

104

C₈

—

274

—

105

C₈

—

274

—

106

C₈

#14555489

yes

—

107

C₈

#14555489

yes

—

108

C₂²

—

109

C₂²

#17984053

yes

characteristic

110

C₂²

#17984064

yes

characteristic

111

C₂²

#17984087

yes

characteristic

112

C₄

—

548

—

113

C₄

C₂×#14555489

yes

characteristic

114

C₄

#17984050

yes

characteristic

115

C₄

#17984059

yes

characteristic

116

C₄

#17984079

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#22577299

yes

characteristic

	subgroup	G/H
center	C₄	#17984059
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×#108×C₁₃₇	C₂
socle	C₂×C₁₃₇	#389
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×#108	D₁₃₇
hypocenter	C₁₃₇	#1953
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#17984059

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

C₂×#108×C₁₃₇

C₂

socle

C₂×C₁₃₇

#389

prime radical

C₂³×C₁₃₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #17984024	C₄	semidirect
60. C₁₃₇	#1953	semidirect

H ≅ G/N

type

9. #17984024

C₄

semidirect

60. C₁₃₇

#1953

semidirect

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

subgroup	index
#22577451	2
C₂×#17984024	2
#22577358	2
#22577174	2
C₂×#108×C₁₃₇	2
#1953	137

subgroup

index

C₂×#108×C₁₃₇

137

Subgroups — #31781577

Conjugacy classes of subgroups

Complements

Maximal subgroups