Group #31797946 — subgroups

Subgroups — C₂×#22578498 #31797946

Order 17536.

200 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 103 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×#17984183	C₂	1	2	yes	maximal
2	#22578498	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×#10587739	C₂	1	1	yes	maximal, characteristic
4	#17984183	C₄	1	4	yes	complemented
5	#17984183	C₂²	1	4	yes	—
6	C₂×#14555522	C₄	1	4	yes	complemented
7	C₂×#14555518	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10587739	C₄	1	1	yes	characteristic
9	C₈×#10587739	C₄	1	2	yes	—
10	C₈×#10587739	C₂²	1	2	yes	—
11	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
12	#14555522	C₂×C₄	1	8	yes	complemented
13	C₂²×#10587739	C₈	1	2	yes	—
14	#14555518	C₂×C₄	1	2	yes	—
15	#14555518	C₂×C₄	1	1	yes	characteristic
16	#14555518	C₂³	1	1	yes	characteristic
17	C₂×#11764047	C₂×C₄	1	2	yes	—
18	C₄×#10587739	C₈	1	2	yes	—
19	C₄×#10587739	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×#11764040	C₈	1	1	yes	characteristic
22	C₈×D₁₃₇	C₈	1	2	yes	—
23	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₃₇	C₈	1	1	yes	characteristic
25	C₂×#10587739	—	2	4	no	—
26	C₂×#10587739	C₂×C₈	1	4	yes	—
27	#11764047	C₄²	1	2	yes	—
28	#11764047	C₂²×C₄	1	2	yes	—
29	C₂²×D₁₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
31	C₄×D₁₃₇	C₄²	1	1	yes	characteristic
32	C₄×D₁₃₇	C₂×C₈	1	1	yes	characteristic
33	C₄×D₁₃₇	C₂×C₈	1	1	yes	characteristic
34	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
35	#11764040	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
37	C₈×C₁₃₇	C₂×C₈	1	2	yes	—
38	C₂²×C₁₃₇	C₄×C₈	1	1	yes	characteristic
39	C₂×D₁₃₇	C₄×C₈	1	1	yes	characteristic
40	C₂×D₁₃₇	M₃₂	1	4	yes	—
41	C₂×D₁₃₇	C₂²×C₈	1	1	yes	characteristic
42	#10587739	—	2	4	no	—
43	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
44	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
45	Dic₁₃₇	C₄×C₈	1	1	yes	characteristic
46	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₁₃₇	#346	1	2	yes	complemented
48	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
49	D₁₃₇	#346	1	2	yes	complemented
50	D₁₃₇	C₂×M₃₂	1	2	yes	—
51	C₁₃₇	C₂×#346	1	1	yes	characteristic, complemented, Sylow
52	C₂×#346	—	137	1	no	maximal, Sylow
53	C₂²×C₁₆	—	137	2	no	—
54	C₂×C₄×C₈	—	137	1	no	—
55	#346	—	137	4	no	—
56	C₂²×C₈	—	137	1	no	—
57	C₂²×C₈	—	137	1	no	—
58	C₂×C₄²	—	137	1	no	—
59	C₂×C₁₆	—	137	4	no	—
60	C₂×C₁₆	—	137	4	no	—
61	C₂×C₁₆	—	137	4	no	—
62	C₄×C₈	—	137	2	no	—
63	C₄×C₈	—	137	2	no	—
64	C₂²×C₄	—	137	2	no	—
65	C₂²×C₄	—	137	1	no	—
66	C₂×C₈	—	137	1	no	—
67	C₂×C₈	—	137	1	no	—
68	C₂×C₈	—	137	2	no	—
69	C₂×C₈	—	137	2	no	—
70	C₂×C₈	—	137	1	no	—
71	C₂×C₈	—	137	2	no	—
72	C₂×C₈	—	137	2	no	—
73	C₂×C₈	#11764049	1	1	yes	characteristic
74	C₄²	—	137	2	no	—
75	C₄²	—	137	2	no	—
76	C₁₆	—	137	8	no	—
77	C₂³	—	137	1	no	—
78	C₂×C₄	—	274	4	no	—
79	C₂×C₄	—	137	4	no	—
80	C₂×C₄	—	137	1	no	—
81	C₂×C₄	—	137	1	no	—
82	C₂×C₄	—	137	1	no	—
83	C₂×C₄	—	137	1	no	—
84	C₂×C₄	—	137	1	no	—
85	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
86	C₈	—	137	2	no	—
87	C₈	—	137	2	no	—
88	C₈	—	137	2	no	—
89	C₈	C₂×#11764049	1	2	yes	—
90	C₂²	—	137	4	no	—
91	C₂²	—	137	1	no	—
92	C₂²	—	137	1	no	—
93	C₂²	C₄×#11764049	1	1	yes	characteristic
94	C₄	—	274	4	no	—
95	C₄	—	137	1	no	—
96	C₄	—	137	1	no	—
97	C₄	C₄×#11764049	1	1	yes	characteristic
98	C₄	C₂²×#11764049	1	1	yes	characteristic
99	C₂	—	137	2	no	—
100	C₂	—	137	2	no	—
101	C₂	#22578498	1	2	yes	direct product factor
102	C₂	C₂×C₄×#11764049	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984183

C₂

yes

maximal

#22578498

C₂

yes

maximal, direct product factor

C₂×C₈×#10587739

C₂

yes

maximal, characteristic

#17984183

C₄

yes

complemented

#17984183

C₂²

yes

—

C₂×#14555522

C₄

yes

complemented

C₂×#14555518

C₂²

yes

characteristic

C₂×C₄×#10587739

C₄

yes

characteristic

C₈×#10587739

C₄

yes

—

C₈×#10587739

C₂²

yes

—

C₂×C₈×D₁₃₇

C₄

yes

characteristic

#14555522

C₂×C₄

yes

complemented

C₂²×#10587739

C₈

yes

—

#14555518

C₂×C₄

yes

—

#14555518

C₂×C₄

yes

characteristic

#14555518

C₂³

yes

characteristic

C₂×#11764047

C₂×C₄

yes

—

C₄×#10587739

C₈

yes

—

C₄×#10587739

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×#11764040

C₈

yes

characteristic

C₈×D₁₃₇

C₈

yes

—

C₈×D₁₃₇

C₂×C₄

yes

—

C₂×C₈×C₁₃₇

C₈

yes

characteristic

C₂×#10587739

—

C₂×#10587739

C₂×C₈

yes

—

#11764047

C₄²

yes

—

#11764047

C₂²×C₄

yes

—

C₂²×D₁₃₇

C₂×C₈

yes

characteristic

C₂×Dic₁₃₇

C₄²

yes

characteristic

C₄×D₁₃₇

C₄²

yes

characteristic

C₄×D₁₃₇

C₂×C₈

yes

characteristic

C₄×D₁₃₇

C₂×C₈

yes

characteristic

C₄×D₁₃₇

C₂²×C₄

yes

characteristic

#11764040

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₈×C₁₃₇

C₂×C₈

yes

—

C₂²×C₁₃₇

C₄×C₈

yes

characteristic

C₂×D₁₃₇

C₄×C₈

yes

characteristic

C₂×D₁₃₇

M₃₂

yes

—

C₂×D₁₃₇

C₂²×C₈

yes

characteristic

#10587739

—

C₄×C₁₃₇

C₄×C₈

yes

characteristic

C₄×C₁₃₇

C₂²×C₈

yes

characteristic

Dic₁₃₇

C₄×C₈

yes

characteristic

Dic₁₃₇

C₂×C₄²

yes

characteristic

C₂×C₁₃₇

#346

yes

complemented

C₂×C₁₃₇

C₂×C₄×C₈

yes

characteristic

D₁₃₇

#346

yes

complemented

D₁₃₇

C₂×M₃₂

yes

—

C₁₃₇

C₂×#346

yes

characteristic, complemented, Sylow

C₂×#346

—

137

maximal, Sylow

C₂²×C₁₆

—

137

—

C₂×C₄×C₈

—

137

—

#346

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

#11764049

yes

characteristic

C₄²

—

137

—

C₄²

—

137

—

C₁₆

—

137

—

C₂³

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂×#11764049

yes

characteristic

C₈

—

137

—

C₈

—

137

—

C₈

—

137

—

C₈

C₂×#11764049

yes

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

C₄×#11764049

yes

characteristic

C₄

—

274

—

C₄

—

137

—

C₄

—

137

—

C₄

yes

C₄

yes

C₂

—

137

—

100

C₂

—

137

—

101

C₂

#22578498

yes

direct product factor

102

C₂

C₂×C₄×#11764049

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11764049
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Frattini	C₄	C₂²×#11764049
Fitting	C₂×C₈×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₄×C₈
prime radical	C₂²×D₁₃₇	C₂×C₈
cosocle	#14555518	C₂³
hypercenter	C₂×C₈	#11764049
hypocenter	C₁₃₇	C₂×#346
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂×C₄

C₂×#11764049

commutator

C₂×C₁₃₇

C₂×C₄×C₈

Frattini

C₄

C₂²×#11764049

Fitting

C₂×C₈×C₁₃₇

C₈

socle

C₂²×C₁₃₇

C₄×C₈

prime radical

C₂²×D₁₃₇

C₂×C₈

C₂³

C₂×C₈

C₁₃₇

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
2. #22578498	C₂	direct (N × H)
4. #17984183	C₄	semidirect
6. C₂×#14555522	C₄	semidirect
12. #14555522	C₂×C₄	semidirect
47. C₂×C₁₃₇	#346	semidirect
49. D₁₃₇	#346	semidirect
51. C₁₃₇	C₂×#346	semidirect
101. C₂	#22578498	direct (N × H)

H ≅ G/N

type

2. #22578498

C₂

direct (N × H)

4. #17984183

C₄

semidirect

6. C₂×#14555522

C₄

semidirect

12. #14555522

C₂×C₄

semidirect

47. C₂×C₁₃₇

#346

semidirect

49. D₁₃₇

#346

semidirect

51. C₁₃₇

C₂×#346

semidirect

101. C₂

#22578498

direct (N × H)

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

subgroup	index
C₂×#17984183	2
#22578498	2
C₂×C₈×#10587739	2
C₂×#346	137

subgroup

index

137

Subgroups — C₂×#22578498 #31797946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22578498 #31797946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578498 #31797946