Group #31796509 — subgroups

Subgroups — #31796509

Order 17536.

240 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₄×#14555518	C₂	1	1	yes	maximal, characteristic
2	#22578339	C₂	1	1	yes	maximal, characteristic
3	#22578268	C₂	1	4	yes	maximal
4	#97×D₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555518	C₂²	1	2	yes	—
6	C₂×C₄×#10587739	C₂²	1	2	yes	—
7	C₄×#11764047	C₄	1	4	yes	complemented
8	(C₂×#10587739)⋊C₄	C₄	1	4	yes	—
9	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
10	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
11	#17983999	C₄	1	1	yes	characteristic, complemented
12	#17983998	C₄	1	2	yes	complemented
13	#97×C₁₃₇	C₄	1	1	yes	characteristic, complemented
14	C₂²×#10587739	C₂×C₄	1	4	yes	—
15	#14555518	—	2	4	no	—
16	C₂×#11764047	C₂×C₄	1	4	yes	—
17	C₄×#10587739	—	2	4	no	—
18	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
20	C₂×#11764040	C₂×C₄	1	2	yes	—
21	C₈×D₁₃₇	—	2	4	no	—
22	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂×#10587739	—	2	4	no	—
27	C₂×#10587739	—	2	8	no	—
28	#11764047	—	2	4	no	—
29	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₁₃₇	C₄²	1	2	yes	—
31	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×D₁₃₇	—	2	4	no	—
33	C₄×D₁₃₇	#41	1	4	yes	—
34	#11764040	—	2	2	no	—
35	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
36	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₁₃₇	—	2	2	no	—
38	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
39	C₂×D₁₃₇	#117	1	2	yes	—
40	C₂×D₁₃₇	#131	1	4	yes	—
41	#10587739	—	4	4	no	—
42	C₄×C₁₃₇	—	2	2	no	—
43	C₄×C₁₃₇	#117	1	2	yes	—
44	Dic₁₃₇	—	2	2	no	—
45	Dic₁₃₇	#117	1	2	yes	—
46	C₂×C₁₃₇	#376	1	1	yes	characteristic
47	C₂×C₁₃₇	#405	1	2	yes	—
48	D₁₃₇	#433	1	4	yes	—
49	C₁₃₇	#1746	1	1	yes	characteristic, complemented, Sylow
50	#1746	—	137	1	no	maximal, Sylow
51	C₂×#97	—	137	1	no	—
52	C₂×C₄×C₈	—	137	1	no	—
53	#376	—	137	1	no	—
54	#336	—	137	4	no	—
55	C₂²×C₈	—	137	2	no	—
56	C₂²×C₈	—	137	2	no	—
57	C₂×C₄²	—	137	2	no	—
58	C₂×C₄²	—	137	1	no	—
59	#97	—	137	2	no	—
60	#97	—	137	1	no	—
61	#97	#10587739	1	1	yes	characteristic, complemented
62	C₄×C₈	—	137	4	no	—
63	#95	—	137	4	no	—
64	C₂²×C₄	—	137	4	no	—
65	C₂²×C₄	—	137	2	no	—
66	C₂²×C₄	—	137	1	no	—
67	C₂×C₈	—	274	4	no	—
68	C₂×C₈	—	274	4	no	—
69	C₂×C₈	—	137	4	no	—
70	C₂×C₈	—	137	2	no	—
71	C₂×C₈	C₂×#10587739	1	2	yes	—
72	C₄²	—	274	4	no	—
73	C₄²	—	137	2	no	—
74	C₄²	—	137	1	no	—
75	C₄²	C₂×#10587739	1	1	yes	characteristic
76	C₂³	—	137	1	no	—
77	C₂×C₄	—	274	4	no	—
78	C₂×C₄	—	274	4	no	—
79	C₂×C₄	—	274	8	no	—
80	C₂×C₄	—	137	2	no	—
81	C₂×C₄	—	137	1	no	—
82	C₂×C₄	—	137	4	no	—
83	C₂×C₄	C₄×#10587739	1	2	yes	—
84	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	274	4	no	—
87	C₈	—	274	2	no	—
88	C₂²	—	137	2	no	—
89	C₂²	—	137	4	no	—
90	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	274	2	no	—
93	C₄	—	548	4	no	—
94	C₄	—	137	2	no	—
95	C₄	#17984167	1	2	yes	—
96	C₂	—	137	4	no	—
97	C₂	#22578339	1	1	yes	characteristic
98	C₂	#22578369	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14555518

C₂

yes

maximal, characteristic

#22578339

C₂

yes

maximal, characteristic

#22578268

C₂

yes

maximal

#97×D₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555518

C₂²

yes

—

C₂×C₄×#10587739

C₂²

yes

—

C₄×#11764047

C₄

yes

complemented

(C₂×#10587739)⋊C₄

C₄

yes

—

C₂×C₈×D₁₃₇

C₂²

yes

—

C₄²×D₁₃₇

C₂²

yes

characteristic

#17983999

C₄

yes

characteristic, complemented

#17983998

C₄

yes

complemented

#97×C₁₃₇

C₄

yes

characteristic, complemented

C₂²×#10587739

C₂×C₄

yes

—

#14555518

—

C₂×#11764047

C₂×C₄

yes

—

C₄×#10587739

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

characteristic

C₂×#11764040

C₂×C₄

yes

—

C₈×D₁₃₇

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

—

—

C₂²×D₁₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₃₇

C₄²

yes

—

C₂×Dic₁₃₇

C₂²×C₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

#41

yes

—

#11764040

—

C₂×C₄×C₁₃₇

C₄²

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄²

yes

characteristic

C₂×D₁₃₇

#117

yes

—

C₂×D₁₃₇

#131

yes

—

#10587739

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#117

yes

—

Dic₁₃₇

—

Dic₁₃₇

#117

yes

—

C₂×C₁₃₇

#376

yes

characteristic

C₂×C₁₃₇

#405

yes

—

D₁₃₇

#433

yes

—

C₁₃₇

#1746

yes

characteristic, complemented, Sylow

#1746

—

137

maximal, Sylow

C₂×#97

—

137

—

C₂×C₄×C₈

—

137

—

#376

—

137

—

#336

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

#97

—

137

—

#97

—

137

—

#97

#10587739

yes

characteristic, complemented

C₄×C₈

—

137

—

#95

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₂×#10587739

yes

—

C₄²

—

274

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₄×#10587739

yes

—

C₂×C₄

C₂²×#10587739

yes

characteristic

C₈

—

C₈

—

274

—

C₈

—

274

—

C₂²

—

137

—

C₂²

—

137

—

C₂²

C₂×C₄×#10587739

yes

characteristic

C₄

—

C₄

—

274

—

C₄

—

548

—

C₄

—

137

—

C₄

#17984167

yes

—

C₂

—

137

—

C₂

#22578339

yes

characteristic

C₂

#22578369

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂×C₄²
Fitting	#97×C₁₃₇	C₄
prime radical	C₂²×D₁₃₇	C₂²×C₄
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	#97	#10587739
hypocenter	C₁₃₇	#1746
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10587739

commutator, socle

C₂²×C₁₃₇

C₂×C₄²

Fitting

#97×C₁₃₇

C₄

prime radical

C₂²×D₁₃₇

C₂²×C₄

cosocle

C₂×C₄×D₁₃₇

C₂³

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
7. C₄×#11764047	C₄	semidirect
11. #17983999	C₄	semidirect
12. #17983998	C₄	semidirect
13. #97×C₁₃₇	C₄	semidirect
49. C₁₃₇	#1746	semidirect
61. #97	#10587739	semidirect

H ≅ G/N

type

7. C₄×#11764047

C₄

semidirect

11. #17983999

C₄

semidirect

12. #17983998

C₄

semidirect

13. #97×C₁₃₇

C₄

semidirect

49. C₁₃₇

#1746

semidirect

61. #97

#10587739

semidirect

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

subgroup	index
C₄×#14555518	2
#22578339	2
#22578268	2
#97×D₁₃₇	2
#1746	137

subgroup

index

#97×D₁₃₇

137

Subgroups — #31796509

Conjugacy classes of subgroups

Complements

Maximal subgroups