Group #31796990 — subgroups

Subgroups — #31796990

Order 17536.

200 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 110 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×#17984161	C₂	1	2	yes	maximal
2	#22578292	C₂	1	2	yes	maximal, complemented
3	#22578291	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#17984161	C₄	1	8	yes	complemented
6	C₂×#14555513	C₂²	1	4	yes	—
7	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
8	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
9	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
10	C₄×#11764040	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₃₇	C₄	1	1	yes	characteristic
12	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
14	#14555513	C₂×C₄	1	8	yes	complemented
15	C₂×C₄×D₁₃₇	C₈	1	2	yes	—
16	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
18	C₂×#11764040	C₂³	1	1	yes	characteristic
19	C₈×D₁₃₇	C₈	1	4	yes	—
20	C₈×D₁₃₇	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₃₇	C₈	1	2	yes	—
22	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×D₁₃₇	C₂×C₈	1	1	yes	characteristic
27	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
28	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
29	C₄×D₁₃₇	—	2	4	no	—
30	C₄×D₁₃₇	C₄²	1	2	yes	—
31	C₄×D₁₃₇	C₂×C₈	1	2	yes	—
32	#11764040	C₂×C₈	1	2	yes	—
33	#11764040	C₂²×C₄	1	2	yes	—
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	C₂×C₈	1	2	yes	—
38	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
39	C₂×D₁₃₇	C₄×C₈	1	2	yes	—
40	C₂×D₁₃₇	M₃₂	1	4	yes	—
41	C₄×C₁₃₇	—	2	2	no	—
42	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
43	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
44	Dic₁₃₇	—	2	2	no	—
45	Dic₁₃₇	C₄×C₈	1	2	yes	—
46	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
47	C₂×C₁₃₇	C₂×M₃₂	1	2	yes	—
48	D₁₃₇	#346	1	4	yes	complemented
49	C₁₃₇	C₂×#346	1	1	yes	characteristic, complemented, Sylow
50	C₂×#346	—	137	1	no	maximal, Sylow
51	C₂²×C₁₆	—	137	2	no	—
52	C₂×C₄×C₈	—	137	1	no	—
53	#346	—	137	2	no	—
54	#346	—	137	2	no	—
55	C₂²×C₈	—	137	1	no	—
56	C₂²×C₈	—	137	1	no	—
57	C₂×C₄²	—	137	1	no	—
58	C₂×C₁₆	—	137	4	no	—
59	C₂×C₁₆	—	137	8	no	—
60	C₄×C₈	—	137	1	no	—
61	C₄×C₈	—	137	1	no	—
62	C₄×C₈	—	137	1	no	—
63	C₄×C₈	#10587739	1	1	yes	characteristic
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	4	no	—
69	C₂×C₈	—	137	4	no	—
70	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
71	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
72	C₄²	—	137	2	no	—
73	C₄²	—	137	1	no	—
74	C₄²	C₂×#10587739	1	1	yes	characteristic
75	C₁₆	—	137	8	no	—
76	C₂³	—	137	1	no	—
77	C₂×C₄	—	274	4	no	—
78	C₂×C₄	—	137	2	no	—
79	C₂×C₄	—	137	1	no	—
80	C₂×C₄	—	137	2	no	—
81	C₂×C₄	—	137	2	no	—
82	C₂×C₄	#14555518	1	2	yes	—
83	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
84	C₈	—	137	2	no	—
85	C₈	—	137	2	no	—
86	C₈	C₄×#10587739	1	2	yes	—
87	C₈	#14555518	1	2	yes	—
88	C₂²	—	137	4	no	—
89	C₂²	—	137	2	no	—
90	C₂²	C₂×#14555518	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	274	2	no	—
93	C₄	—	137	2	no	—
94	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
95	C₄	C₂×#14555518	1	1	yes	characteristic
96	C₂	—	137	4	no	—
97	C₂	C₄×#14555518	1	1	yes	characteristic
98	C₂	#22578459	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984161

C₂

yes

maximal

#22578292

C₂

yes

maximal, complemented

#22578291

C₂

yes

maximal, complemented

C₄×C₈×D₁₃₇

C₂

yes

maximal, characteristic

#17984161

C₄

yes

complemented

C₂×#14555513

C₂²

yes

—

C₂×C₈×D₁₃₇

C₄

yes

characteristic

C₂×C₈×D₁₃₇

C₂²

yes

characteristic

C₄²×D₁₃₇

C₄

yes

characteristic

C₄×#11764040

C₂²

yes

characteristic

C₈×Dic₁₃₇

C₄

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₄

yes

characteristic

#14555513

C₂×C₄

yes

complemented

C₂×C₄×D₁₃₇

C₈

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

C₂³

yes

characteristic

C₈×D₁₃₇

C₈

yes

—

C₈×D₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₈

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×D₁₃₇

C₂×C₈

yes

characteristic

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

C₂×C₈

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

C₄²

yes

—

C₄×D₁₃₇

C₂×C₈

yes

—

#11764040

C₂×C₈

yes

—

#11764040

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

C₄²

yes

—

C₈×C₁₃₇

C₂×C₈

yes

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₂×D₁₃₇

C₄×C₈

yes

—

C₂×D₁₃₇

M₃₂

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×C₄²

yes

characteristic

C₄×C₁₃₇

C₂²×C₈

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

C₄×C₈

yes

—

C₂×C₁₃₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₃₇

C₂×M₃₂

yes

—

D₁₃₇

#346

yes

complemented

C₁₃₇

C₂×#346

yes

characteristic, complemented, Sylow

C₂×#346

—

137

maximal, Sylow

C₂²×C₁₆

—

137

—

C₂×C₄×C₈

—

137

—

#346

—

137

—

#346

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

#10587739

yes

characteristic

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₁₆

—

137

—

C₂³

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

#14555518

yes

—

C₂×C₄

C₂²×#10587739

yes

characteristic

C₈

—

137

—

C₈

—

137

—

C₈

C₄×#10587739

yes

—

C₈

#14555518

yes

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

C₂×#14555518

yes

characteristic

C₄

—

C₄

—

274

—

C₄

—

137

—

C₄

yes

C₄

yes

C₂

—

137

—

C₂

C₄×#14555518

yes

characteristic

C₂

#22578459

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×#10587739
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²×C₈
prime radical	C₂²×D₁₃₇	C₂×C₈
cosocle	C₂×#11764040	C₂³
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	C₂×#346
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂×#10587739

commutator

C₂×C₁₃₇

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#10587739

Fitting

C₄×C₈×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂²×C₈

prime radical

C₂²×D₁₃₇

C₂×C₈

C₂³

C₄×C₈

C₁₃₇

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
2. #22578292	C₂	semidirect
3. #22578291	C₂	semidirect
5. #17984161	C₄	semidirect
14. #14555513	C₂×C₄	semidirect
48. D₁₃₇	#346	semidirect
49. C₁₃₇	C₂×#346	semidirect

H ≅ G/N

type

2. #22578292

C₂

semidirect

3. #22578291

C₂

semidirect

5. #17984161

C₄

semidirect

14. #14555513

C₂×C₄

semidirect

48. D₁₃₇

#346

semidirect

49. C₁₃₇

C₂×#346

semidirect

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

subgroup	index
C₂×#17984161	2
#22578292	2
#22578291	2
C₄×C₈×D₁₃₇	2
C₂×#346	137

subgroup

index

C₄×C₈×D₁₃₇

137

Subgroups — #31796990

Conjugacy classes of subgroups

Complements

Maximal subgroups