Group #31780925 — subgroups

Subgroups — #31780925

Order 17536.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#22577426	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#22577168	C₂	1	1	yes	maximal, characteristic
4	#22577167	C₂	1	2	yes	maximal
5	#341×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₂²	1	2	yes	—
7	C₂²×#11764040	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
9	#17984017	—	2	4	no	—
10	C₈×Dic₁₃₇	C₄	1	2	yes	—
11	C₈×Dic₁₃₇	C₄	1	2	yes	—
12	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₃₇	Q₈	1	2	yes	—
15	#14555494	—	2	2	no	—
16	#14555494	—	2	2	no	—
17	C₂×#11764040	—	2	2	no	—
18	C₂×#11764040	—	2	2	no	—
19	C₂×#11764040	C₂×C₄	1	2	yes	—
20	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₁₃₇	D₄	1	2	yes	—
23	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₃₇	—	2	2	no	—
25	M₁₆×C₁₃₇	—	2	2	no	—
26	C₂×C₈×C₁₃₇	—	2	2	no	—
27	C₂×C₈×C₁₃₇	—	2	2	no	—
28	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₃₇	—	2	4	no	—
30	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
32	#11764040	—	2	2	no	—
33	#11764040	—	4	2	no	—
34	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	#41	1	2	yes	—
38	C₂×C₄×C₁₃₇	#41	1	2	yes	—
39	C₈×C₁₃₇	—	2	2	no	—
40	C₈×C₁₃₇	—	4	2	no	—
41	C₂²×C₁₃₇	—	2	2	no	—
42	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₃₇	#125	1	2	yes	—
44	C₄×C₁₃₇	#117	1	1	yes	characteristic
45	C₄×C₁₃₇	#117	1	1	yes	characteristic
46	C₄×C₁₃₇	#124	1	2	yes	—
47	Dic₁₃₇	—	2	2	no	—
48	Dic₁₃₇	#108	1	4	yes	—
49	C₂×C₁₃₇	—	2	2	no	—
50	C₂×C₁₃₇	#382	1	1	yes	characteristic
51	C₂×C₁₃₇	C₂×#108	1	2	yes	—
52	C₁₃₇	#1837	1	1	yes	characteristic, complemented, Sylow
53	#1837	—	137	1	no	maximal, Sylow
54	#423	—	137	2	no	—
55	C₂×C₄×C₈	—	137	1	no	—
56	#341	—	137	2	no	—
57	#341	—	137	1	no	—
58	#341	D₁₃₇	1	1	yes	characteristic
59	C₂×M₁₆	—	137	2	no	—
60	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
61	C₂²×C₈	—	137	1	no	—
62	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
63	C₂×C₄²	—	137	1	no	—
64	C₄⋊C₈	—	274	4	no	—
65	C₄×C₈	—	137	2	no	—
66	C₄×C₈	—	137	2	no	—
67	C₂²×C₄	—	137	2	no	—
68	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	274	2	no	—
72	M₁₆	—	274	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	274	2	no	—
76	C₂×C₈	—	274	2	no	—
77	C₂×C₈	—	137	2	no	—
78	C₂×C₈	C₄×D₁₃₇	1	2	yes	—
79	C₄²	—	137	1	no	—
80	C₄²	—	137	1	no	—
81	C₄²	—	137	2	no	—
82	C₂³	Q₈×D₁₃₇	1	1	yes	characteristic
83	C₂×C₄	—	274	4	no	—
84	C₂×C₄	—	137	2	no	—
85	C₂×C₄	—	137	2	no	—
86	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
87	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
88	C₂×C₄	#14555509	1	2	yes	—
89	C₂×C₄	#14555510	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	274	2	no	—
93	C₈	—	548	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
96	C₂²	#17984078	1	2	yes	—
97	C₄	—	274	2	no	—
98	C₄	—	137	4	no	—
99	C₄	#17984044	1	1	yes	characteristic
100	C₄	#17984045	1	1	yes	characteristic
101	C₄	#17984074	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#22577267	1	1	yes	characteristic
104	C₂	#108×D₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22577426

C₂

yes

maximal

C₂×C₈×Dic₁₃₇

C₂

yes

maximal, characteristic

#22577168

C₂

yes

maximal, characteristic

#22577167

C₂

yes

maximal

#341×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

C₂²×#11764040

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984017

—

C₈×Dic₁₃₇

C₄

yes

—

C₈×Dic₁₃₇

C₄

yes

—

C₂×M₁₆×C₁₃₇

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂²×Dic₁₃₇

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

#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

#41

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₇

#125

yes

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

Dic₁₃₇

—

Dic₁₃₇

#108

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#382

yes

characteristic

C₂×C₁₃₇

C₂×#108

yes

—

C₁₃₇

#1837

yes

characteristic, complemented, Sylow

#1837

—

137

maximal, Sylow

#423

—

137

—

C₂×C₄×C₈

—

137

—

#341

—

137

—

#341

—

137

—

#341

D₁₃₇

yes

characteristic

C₂×M₁₆

—

137

—

C₂×M₁₆

C₂×D₁₃₇

yes

—

C₂²×C₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

C₂×C₄²

—

137

—

C₄⋊C₈

—

274

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

C₄×D₁₃₇

yes

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₂³

Q₈×D₁₃₇

yes

characteristic

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

C₂×C₄

D₄×D₁₃₇

yes

characteristic

C₂×C₄

#14555509

yes

—

C₂×C₄

#14555510

yes

—

C₈

—

C₈

—

C₈

—

274

—

C₈

—

548

—

C₂²

—

C₂²

C₄⋊C₄×D₁₃₇

yes

characteristic

C₂²

#17984078

yes

—

C₄

—

274

—

C₄

—

137

—

C₄

#17984044

yes

characteristic

100

C₄

#17984045

yes

characteristic

101

C₄

#17984074

yes

—

102

C₂

—

103

C₂

#22577267

yes

characteristic

104

C₂

#108×D₁₃₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#341×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#341	D₁₃₇
hypocenter	C₁₃₇	#1837
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₇

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

#341×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₃₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
52. C₁₃₇	#1837	semidirect

H ≅ G/N

type

52. C₁₃₇

#1837

semidirect

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

subgroup	index
#22577426	2
C₂×C₈×Dic₁₃₇	2
#22577168	2
#22577167	2
#341×C₁₃₇	2
#1837	137

subgroup

index

C₂×C₈×Dic₁₃₇

#341×C₁₃₇

137

Subgroups — #31780925

Conjugacy classes of subgroups

Complements

Maximal subgroups