Group #31782365 — subgroups

Subgroups — #31782365

Order 17536.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#22577452	C₂	1	1	yes	maximal, characteristic
2	#22577426	C₂	1	1	yes	maximal, characteristic
3	#22577173	C₂	1	2	yes	maximal
4	#22577162	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₂²	1	2	yes	—
7	C₂×#14555494	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
9	#17984024	—	2	2	no	—
10	#17984017	—	2	2	no	—
11	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
12	#117×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×Dic₁₃₇	—	2	1	no	—
14	#14555494	—	2	2	no	—
15	#14555494	—	2	2	no	—
16	#14555494	—	2	2	no	—
17	C₂×#11764040	—	2	1	no	—
18	C₂×#11764040	—	2	2	no	—
19	C₄×Dic₁₃₇	—	2	1	no	—
20	C₄×Dic₁₃₇	Q₈	1	1	yes	characteristic
21	C₄×Dic₁₃₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₃₇	—	2	2	no	—
24	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₃₇	—	2	2	no	—
26	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
27	C₄²×C₁₃₇	—	2	2	no	—
28	C₂×Dic₁₃₇	—	2	1	no	—
29	C₂×Dic₁₃₇	—	2	1	no	—
30	C₂×Dic₁₃₇	—	4	2	no	—
31	#11764040	—	4	1	no	—
32	#11764040	—	4	2	no	—
33	C₂³×C₁₃₇	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	—	4	2	no	—
35	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	C₂×Q₈	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	C₂²×C₁₃₇	—	4	1	no	—
41	C₂²×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₃₇	#126	1	2	yes	—
43	C₄×C₁₃₇	—	8	2	no	—
44	C₄×C₁₃₇	#123	1	1	yes	characteristic
45	C₄×C₁₃₇	#124	1	1	yes	characteristic
46	C₄×C₁₃₇	#125	1	2	yes	—
47	Dic₁₃₇	—	2	2	no	—
48	Dic₁₃₇	—	4	1	no	—
49	C₂×C₁₃₇	—	2	1	no	—
50	C₂×C₁₃₇	—	2	2	no	—
51	C₂×C₁₃₇	#400	1	1	yes	characteristic
52	C₁₃₇	#2102	1	1	yes	characteristic, complemented, Sylow
53	#2102	—	137	1	no	maximal, Sylow
54	#430	—	137	1	no	—
55	#423	—	137	1	no	—
56	#344	—	137	2	no	—
57	#339	—	137	2	no	—
58	H(3,ℤ/4)	D₁₃₇	1	1	yes	characteristic
59	C₂×M₁₆	—	137	2	no	—
60	C₂×M₁₆	—	137	1	no	—
61	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
62	#117	C₂×D₁₃₇	1	2	yes	—
63	C₂×C₄²	—	137	1	no	—
64	#108	—	274	2	no	—
65	C₄⋊C₈	—	274	2	no	—
66	C₂²×C₄	—	274	1	no	—
67	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	274	2	no	—
70	M₁₆	—	274	2	no	—
71	M₁₆	—	274	2	no	—
72	C₂×C₈	—	274	1	no	—
73	C₂×C₈	—	274	2	no	—
74	C₂×C₈	D₅₄₈	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	274	1	no	—
79	C₄²	—	137	1	no	—
80	C₄²	—	137	1	no	—
81	C₂³	#14555509	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	274	1	no	—
84	C₂×C₄	—	274	1	no	—
85	C₂×C₄	—	548	2	no	—
86	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
87	C₂×C₄	Q₈×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	—	548	1	no	—
92	C₈	—	548	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#17984068	1	1	yes	characteristic
95	C₂²	#17984081	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	274	2	no	—
98	C₄	—	548	1	no	—
99	C₄	#17984072	1	1	yes	characteristic
100	C₄	#17984076	1	1	yes	characteristic
101	C₄	#17984078	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#22577343	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577452

C₂

yes

maximal, characteristic

#22577426

C₂

yes

maximal, characteristic

#22577173

C₂

yes

maximal

#22577162

C₂

yes

maximal

H(3,ℤ/4)×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

C₂×#14555494

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984024

—

#17984017

—

C₂×M₁₆×C₁₃₇

C₂²

yes

characteristic

#117×C₁₃₇

C₂²

yes

—

C₂²×Dic₁₃₇

—

—

—

—

—

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

Q₈

yes

characteristic

C₄×Dic₁₃₇

Q₈

yes

characteristic

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₄⋊C₄×C₁₃₇

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

#11764040

—

C₂³×C₁₃₇

#41

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃₇

#126

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#400

yes

characteristic

C₁₃₇

#2102

yes

characteristic, complemented, Sylow

#2102

—

137

maximal, Sylow

#430

—

137

—

#423

—

137

—

#344

—

137

—

#339

—

137

—

H(3,ℤ/4)

D₁₃₇

yes

characteristic

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

C₂×D₁₃₇

yes

characteristic

#117

C₂×D₁₃₇

yes

—

C₂×C₄²

—

137

—

#108

—

274

—

C₄⋊C₈

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

D₅₄₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

274

—

C₄²

—

137

—

C₄²

—

137

—

C₂³

#14555509

yes

characteristic

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

548

—

C₂×C₄

C₂×D₅₄₈

yes

characteristic

C₂×C₄

Q₈×D₁₃₇

yes

characteristic

C₂×C₄

#14555509

yes

—

C₂×C₄

#14555510

yes

—

C₈

—

C₈

—

548

—

C₈

—

548

—

C₂²

—

C₂²

#17984068

yes

characteristic

C₂²

#17984081

yes

—

C₄

—

C₄

—

274

—

C₄

—

548

—

C₄

#17984072

yes

characteristic

100

C₄

#17984076

yes

characteristic

101

C₄

#17984078

yes

—

102

C₂

—

103

C₂

—

104

C₂

#22577343

yes

characteristic

	subgroup	G/H
center	C₄	#17984076
commutator, cosocle	C₂²×C₄×C₁₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	H(3,ℤ/4)×C₁₃₇	C₂
socle	C₂×C₁₃₇	#400
prime radical	C₂³×C₁₃₇	#41
hypercenter	H(3,ℤ/4)	D₁₃₇
hypocenter	C₁₃₇	#2102
LCS	G ▷ C₂²×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#17984076

commutator, cosocle

C₂²×C₄×C₁₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

H(3,ℤ/4)×C₁₃₇

C₂

C₂×C₁₃₇

C₂³×C₁₃₇

D₁₃₇

C₁₃₇

G ▷ C₂²×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
52. C₁₃₇	#2102	semidirect

H ≅ G/N

type

52. C₁₃₇

#2102

semidirect

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

subgroup	index
#22577452	2
#22577426	2
#22577173	2
#22577162	2
H(3,ℤ/4)×C₁₃₇	2
#2102	137

subgroup

index

H(3,ℤ/4)×C₁₃₇

137

Subgroups — #31782365

Conjugacy classes of subgroups

Complements

Maximal subgroups