Group #31782385 — subgroups

Subgroups — #31782385

Order 17536.

184 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 125 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×#17984028	C₂	1	1	yes	maximal, characteristic
2	#22577358	C₂	1	1	yes	maximal, characteristic
3	#22577178	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₁₃₇	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₄	1	1	yes	characteristic
7	#17984047	C₄	1	1	yes	characteristic
8	#17984028	C₄	1	4	yes	complemented
9	C₂×#14555485	C₂²	1	2	yes	—
10	#17983998	C₄	1	2	yes	—
11	C₈×Dic₁₃₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₃₇	C₄	1	4	yes	complemented
14	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
16	#14555494	C₈	1	4	yes	—
17	C₂×#11764040	C₂×C₄	1	2	yes	—
18	C₄⋊Dic₁₃₇	C₈	1	2	yes	—
19	C₂²⋊Dic₁₃₇	C₈	1	2	yes	—
20	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
21	#14555485	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₃₇	C₂×C₈	1	4	yes	—
30	#11764040	C₂×C₈	1	4	yes	—
31	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
33	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₃₇	C₄²	1	2	yes	—
37	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₃₇	—	2	2	no	—
40	C₂²×C₁₃₇	C₄×C₈	1	2	yes	—
41	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₃₇	C₄×C₈	1	2	yes	—
43	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
45	Dic₁₃₇	—	2	4	no	—
46	C₂×C₁₃₇	—	2	2	no	—
47	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂×C₁₃₇	#504	1	2	yes	—
49	C₁₃₇	#2108	1	1	yes	characteristic, complemented, Sylow
50	#2108	—	137	1	no	maximal, Sylow
51	C₂×M₃₂	—	137	1	no	—
52	C₂²×C₁₆	D₁₃₇	1	1	yes	characteristic
53	#405	—	137	1	no	—
54	#346	—	137	2	no	—
55	C₄×C₁₆	—	137	2	no	—
56	C₂×M₁₆	—	137	1	no	—
57	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
58	#117	—	137	1	no	—
59	M₃₂	—	137	4	no	—
60	C₂×C₁₆	—	137	2	no	—
61	C₂×C₁₆	Dic₁₃₇	1	4	yes	complemented
62	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
63	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
64	#97	—	137	2	no	—
65	C₄×C₈	—	137	2	no	—
66	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
67	M₁₆	—	137	4	no	—
68	C₂×C₈	—	137	2	no	—
69	C₂×C₈	C₄×D₁₃₇	1	1	yes	characteristic
70	C₂×C₈	C₄×D₁₃₇	1	2	yes	—
71	C₂×C₈	C₂×Dic₁₃₇	1	2	yes	—
72	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
73	C₄⋊C₄	—	137	2	no	—
74	C₂²⋊C₄	—	137	2	no	—
75	C₄²	—	137	2	no	—
76	C₁₆	—	137	4	no	—
77	C₁₆	C₂×Dic₁₃₇	1	2	yes	—
78	C₁₆	C₂×Dic₁₃₇	1	2	yes	—
79	C₂³	C₈×D₁₃₇	1	1	yes	characteristic
80	C₂×C₄	—	137	4	no	—
81	C₂×C₄	C₄×Dic₁₃₇	1	2	yes	—
82	C₂×C₄	C₈×D₁₃₇	1	2	yes	—
83	C₂×C₄	C₈×D₁₃₇	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
85	C₈	—	137	4	no	—
86	C₈	C₄×Dic₁₃₇	1	2	yes	—
87	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
88	C₈	C₂²×Dic₁₃₇	1	1	yes	characteristic
89	C₂²	—	2	2	no	—
90	C₂²	C₈×Dic₁₃₇	1	2	yes	—
91	C₂²	C₂×C₈×D₁₃₇	1	1	yes	characteristic
92	C₄	—	274	4	no	—
93	C₄	C₈×Dic₁₃₇	1	2	yes	—
94	C₄	C₂×C₄×Dic₁₃₇	1	1	yes	characteristic
95	C₄	C₂×C₈×D₁₃₇	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₈×Dic₁₃₇	1	1	yes	characteristic
98	C₂	#22577819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984028

C₂

yes

maximal, characteristic

#22577358

C₂

yes

maximal, characteristic

#22577178

C₂

yes

maximal, complemented

C₁₆×Dic₁₃₇

C₂

yes

maximal, complemented

C₂²×C₁₆×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₄

yes

characteristic

#17984047

C₄

yes

characteristic

#17984028

C₄

yes

complemented

C₂×#14555485

C₂²

yes

—

#17983998

C₄

yes

—

C₈×Dic₁₃₇

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₃₇

C₄

yes

complemented

C₂×C₁₆×C₁₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₃₇

C₂²

yes

characteristic

#14555494

C₈

yes

—

C₂×#11764040

C₂×C₄

yes

—

C₄⋊Dic₁₃₇

C₈

yes

—

C₂²⋊Dic₁₃₇

C₈

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

#14555485

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×C₈×C₁₃₇

C₂³

yes

characteristic

C₁₆×C₁₃₇

C₂×C₄

yes

—

C₁₆×C₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

C₂×C₈

yes

—

#11764040

C₂×C₈

yes

—

C₂³×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₄²

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

C₄²

yes

—

C₈×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄×C₈

yes

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₄×C₁₃₇

C₄×C₈

yes

—

C₄×C₁₃₇

C₂×C₄²

yes

characteristic

C₄×C₁₃₇

C₂²×C₈

yes

characteristic

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₃₇

#504

yes

—

C₁₃₇

#2108

yes

characteristic, complemented, Sylow

#2108

—

137

maximal, Sylow

C₂×M₃₂

—

137

—

C₂²×C₁₆

D₁₃₇

yes

characteristic

#405

—

137

—

#346

—

137

—

C₄×C₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

#117

—

137

—

M₃₂

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

Dic₁₃₇

yes

complemented

C₂×C₁₆

C₂×D₁₃₇

yes

characteristic

C₂×C₁₆

C₂×D₁₃₇

yes

characteristic

#97

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

C₄×D₁₃₇

yes

characteristic

M₁₆

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₄×D₁₃₇

yes

characteristic

C₂×C₈

C₄×D₁₃₇

yes

—

C₂×C₈

C₂×Dic₁₃₇

yes

—

C₂×C₈

C₂²×D₁₃₇

yes

characteristic

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₁₆

—

137

—

C₁₆

C₂×Dic₁₃₇

yes

—

C₁₆

C₂×Dic₁₃₇

yes

—

C₂³

C₈×D₁₃₇

yes

characteristic

C₂×C₄

—

137

—

C₂×C₄

C₄×Dic₁₃₇

yes

—

C₂×C₄

C₈×D₁₃₇

yes

—

C₂×C₄

C₈×D₁₃₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

C₈

—

137

—

C₈

C₄×Dic₁₃₇

yes

—

C₈

C₂×C₄×D₁₃₇

yes

characteristic

C₈

C₂²×Dic₁₃₇

yes

characteristic

C₂²

—

C₂²

C₈×Dic₁₃₇

yes

—

C₂²

C₂×C₈×D₁₃₇

yes

characteristic

C₄

—

274

—

C₄

C₈×Dic₁₃₇

yes

—

C₄

C₂×C₄×Dic₁₃₇

yes

characteristic

C₄

C₂×C₈×D₁₃₇

yes

characteristic

C₂

—

C₂

C₂×C₈×Dic₁₃₇

yes

characteristic

C₂

#22577819

yes

—

	subgroup	G/H
center	C₂×C₁₆	C₂×D₁₃₇
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	C₂²×C₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×C₈
prime radical	C₂³×C₁₃₇	C₂×C₈
cosocle	C₂×C₈×C₁₃₇	C₂³
hypercenter	C₂²×C₁₆	D₁₃₇
hypocenter	C₁₃₇	#2108
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₁₆

C₂×D₁₃₇

commutator

C₂×C₁₃₇

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×D₁₃₇

Fitting

C₂²×C₁₆×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂²×C₈

prime radical

C₂³×C₁₃₇

C₂×C₈

cosocle

C₂×C₈×C₁₃₇

C₂³

hypercenter

C₂²×C₁₆

D₁₃₇

C₁₃₇

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. #22577178	C₂	semidirect
4. C₁₆×Dic₁₃₇	C₂	semidirect
8. #17984028	C₄	semidirect
13. C₂×C₁₆×C₁₃₇	C₄	semidirect
49. C₁₃₇	#2108	semidirect
61. C₂×C₁₆	Dic₁₃₇	semidirect

H ≅ G/N

type

3. #22577178

C₂

semidirect

4. C₁₆×Dic₁₃₇

C₂

semidirect

8. #17984028

C₄

semidirect

13. C₂×C₁₆×C₁₃₇

C₄

semidirect

49. C₁₃₇

#2108

semidirect

61. C₂×C₁₆

Dic₁₃₇

semidirect

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

subgroup	index
C₂×#17984028	2
#22577358	2
#22577178	2
C₁₆×Dic₁₃₇	2
C₂²×C₁₆×C₁₃₇	2
#2108	137

subgroup

index

C₂×#17984028

#22577358

#22577178

C₁₆×Dic₁₃₇

C₂²×C₁₆×C₁₃₇

#2108

137

Subgroups — #31782385

Conjugacy classes of subgroups

Complements

Maximal subgroups