Group #31780690 — subgroups

Subgroups — #31780690

Order 17536.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#17984023	C₂	1	2	yes	maximal
2	M₁₆×Dic₁₃₇	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#22577168	C₂	1	1	yes	maximal, characteristic
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	#17984023	C₄	1	8	yes	complemented
10	#17983998	—	2	2	no	—
11	C₈×Dic₁₃₇	—	2	2	no	—
12	C₈×Dic₁₃₇	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₃₇	Q₈	1	2	yes	—
16	#14555494	—	2	2	no	—
17	#14555494	C₂×C₄	1	4	yes	—
18	C₂×#11764040	—	2	2	no	—
19	C₂×#11764040	C₂×C₄	1	4	yes	—
20	C₂×#11764040	C₂×C₄	1	2	yes	—
21	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
23	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₃₇	—	2	2	no	—
26	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₁₃₇	—	2	2	no	—
28	C₂×C₈×C₁₃₇	—	2	2	no	—
29	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₃₇	—	2	4	no	—
31	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
32	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
33	#11764040	—	2	4	no	—
34	#11764040	C₄²	1	4	yes	—
35	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	#41	1	2	yes	—
40	C₈×C₁₃₇	—	2	2	no	—
41	C₈×C₁₃₇	—	2	4	no	—
42	C₂²×C₁₃₇	—	2	2	no	—
43	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₃₇	C₄×Q₈	1	2	yes	—
45	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₁₃₇	#117	1	1	yes	characteristic
47	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
48	Dic₁₃₇	—	2	2	no	—
49	Dic₁₃₇	#108	1	4	yes	complemented
50	C₂×C₁₃₇	—	2	2	no	—
51	C₂×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₃₇	C₂×#108	1	2	yes	—
53	C₁₃₇	C₄×#108	1	1	yes	characteristic, complemented, Sylow
54	C₄×#108	—	137	1	no	maximal, Sylow
55	C₂×#108	—	137	2	no	—
56	C₄×M₁₆	—	137	2	no	—
57	C₂×C₄×C₈	—	137	1	no	—
58	#341	—	137	1	no	—
59	#341	D₁₃₇	1	1	yes	characteristic
60	C₂×M₁₆	—	137	2	no	—
61	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
62	C₂²×C₈	—	137	1	no	—
63	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
64	C₂×C₄²	—	137	1	no	—
65	#108	—	137	8	no	—
66	#97	—	274	2	no	—
67	C₄×C₈	—	274	2	no	—
68	C₄×C₈	—	137	4	no	—
69	C₂²×C₄	—	137	2	no	—
70	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	274	2	no	—
73	M₁₆	—	137	4	no	—
74	M₁₆	C₄×D₁₃₇	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	274	2	no	—
78	C₂×C₈	—	137	2	no	—
79	C₂×C₈	—	137	4	no	—
80	C₂×C₈	C₄×D₁₃₇	1	2	yes	—
81	C₄²	—	137	2	no	—
82	C₄²	—	137	1	no	—
83	C₄²	—	137	1	no	—
84	C₂³	Q₈×D₁₃₇	1	1	yes	characteristic
85	C₂×C₄	—	274	4	no	—
86	C₂×C₄	—	137	2	no	—
87	C₂×C₄	—	137	2	no	—
88	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
90	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
91	C₂×C₄	#14555509	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	274	4	no	—
95	C₈	—	137	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
98	C₂²	#17984053	1	2	yes	—
99	C₄	—	274	2	no	—
100	C₄	—	137	4	no	—
101	C₄	C₄²×D₁₃₇	1	1	yes	characteristic
102	C₄	#17984044	1	1	yes	characteristic
103	C₄	#17984048	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#22577254	1	1	yes	characteristic
106	C₂	#108×D₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984023

C₂

yes

maximal

M₁₆×Dic₁₃₇

C₂

yes

maximal

C₂×C₈×Dic₁₃₇

C₂

yes

maximal, characteristic

#22577168

C₂

yes

maximal, characteristic

#341×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

C₂²×#11764040

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984023

C₄

yes

complemented

#17983998

—

C₈×Dic₁₃₇

—

C₈×Dic₁₃₇

C₄

yes

—

C₂×M₁₆×C₁₃₇

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂²×Dic₁₃₇

Q₈

yes

—

#14555494

—

#14555494

C₂×C₄

yes

—

C₂×#11764040

—

C₂×#11764040

C₂×C₄

yes

—

C₂×#11764040

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

D₄

yes

characteristic

C₄×Dic₁₃₇

D₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

C₂×C₄

yes

—

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

C₄²

yes

—

C₂³×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₇

C₄×Q₈

yes

—

C₄×C₁₃₇

C₂×C₄²

yes

characteristic

C₄×C₁₃₇

#117

yes

characteristic

C₄×C₁₃₇

C₄×D₄

yes

—

Dic₁₃₇

—

Dic₁₃₇

#108

yes

complemented

C₂×C₁₃₇

—

C₂×C₁₃₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₃₇

C₂×#108

yes

—

C₁₃₇

C₄×#108

yes

characteristic, complemented, Sylow

C₄×#108

—

137

maximal, Sylow

C₂×#108

—

137

—

C₄×M₁₆

—

137

—

C₂×C₄×C₈

—

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

—

#108

—

137

—

#97

—

274

—

C₄×C₈

—

274

—

C₄×C₈

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

M₁₆

—

274

—

M₁₆

—

137

—

M₁₆

C₄×D₁₃₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

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

C₂×C₄×D₁₃₇

yes

—

C₂×C₄

D₄×D₁₃₇

yes

characteristic

C₂×C₄

#14555509

yes

—

C₈

—

C₈

—

C₈

—

274

—

C₈

—

137

—

C₂²

—

C₂²

C₄⋊C₄×D₁₃₇

yes

characteristic

C₂²

#17984053

yes

—

C₄

—

274

—

100

C₄

—

137

—

101

C₄

C₄²×D₁₃₇

yes

characteristic

102

C₄

#17984044

yes

characteristic

103

C₄

#17984048

yes

—

104

C₂

—

105

C₂

#22577254

yes

characteristic

106

C₂

#108×D₁₃₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂×C₄²
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	C₄×#108
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₇

commutator

C₄×C₁₃₇

C₂×C₄²

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 ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #17984023	C₄	semidirect
49. Dic₁₃₇	#108	semidirect
53. C₁₃₇	C₄×#108	semidirect

H ≅ G/N

type

9. #17984023

C₄

semidirect

49. Dic₁₃₇

#108

semidirect

53. C₁₃₇

C₄×#108

semidirect

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

subgroup	index
C₂×#17984023	2
M₁₆×Dic₁₃₇	2
C₂×C₈×Dic₁₃₇	2
#22577168	2
#341×C₁₃₇	2
C₄×#108	137

subgroup

index

C₂×#17984023

M₁₆×Dic₁₃₇

C₂×C₈×Dic₁₃₇

#22577168

#341×C₁₃₇

C₄×#108

137

Subgroups — #31780690

Conjugacy classes of subgroups

Complements

Maximal subgroups