Group #31796505 — subgroups

Subgroups — #31796505

Order 17536.

272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 98 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	#22578362	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10587739	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	C₄	1	4	yes	—
7	C₂×C₄×#10587739	C₂²	1	2	yes	—
8	#17984143	C₄	1	4	yes	complemented
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	8	no	—
18	C₄×#10587739	—	2	4	no	—
19	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
21	C₂×#11764040	C₂×C₄	1	2	yes	—
22	C₈×D₁₃₇	—	2	4	no	—
23	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×#10587739	—	2	4	no	—
28	C₂×#10587739	—	2	4	no	—
29	C₂×#10587739	M₁₆	1	8	yes	—
30	#11764047	—	2	4	no	—
31	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₃₇	C₄²	1	2	yes	—
33	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₄×D₁₃₇	—	2	4	no	—
35	C₄×D₁₃₇	#41	1	4	yes	—
36	#11764040	—	2	2	no	—
37	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
38	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₈×C₁₃₇	—	2	2	no	—
40	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
41	C₂×D₁₃₇	#117	1	2	yes	—
42	C₂×D₁₃₇	C₂×M₁₆	1	4	yes	—
43	#10587739	—	2	8	no	—
44	C₄×C₁₃₇	—	2	2	no	—
45	C₄×C₁₃₇	#117	1	2	yes	—
46	Dic₁₃₇	—	2	2	no	—
47	Dic₁₃₇	#117	1	2	yes	—
48	C₂×C₁₃₇	#376	1	1	yes	characteristic
49	C₂×C₁₃₇	C₄×M₁₆	1	2	yes	—
50	D₁₃₇	#432	1	4	yes	—
51	C₁₃₇	#1745	1	1	yes	characteristic, complemented, Sylow
52	#1745	—	137	1	no	maximal, Sylow
53	C₂×#97	—	137	1	no	—
54	C₂×#97	—	137	1	no	—
55	C₄³	—	137	1	no	—
56	#336	—	137	4	no	—
57	C₂²×C₈	—	137	2	no	—
58	C₂²×C₈	—	137	2	no	—
59	C₂×C₄²	—	137	2	no	—
60	C₂×C₄²	—	137	4	no	—
61	C₂×C₄²	—	137	1	no	—
62	#97	—	137	4	no	—
63	#97	—	137	2	no	—
64	#97	—	137	1	no	—
65	#97	#10587739	1	1	yes	characteristic, complemented
66	C₂²×C₄	—	137	4	no	—
67	C₂²×C₄	—	137	2	no	—
68	C₂²×C₄	—	137	1	no	—
69	C₂×C₈	—	274	4	no	—
70	C₂×C₈	—	274	4	no	—
71	C₂×C₈	—	137	4	no	—
72	C₂×C₈	—	137	2	no	—
73	C₂×C₈	C₂×#10587739	1	2	yes	—
74	C₄²	—	274	8	no	—
75	C₄²	—	274	4	no	—
76	C₄²	—	137	2	no	—
77	C₄²	—	137	1	no	—
78	C₄²	C₂×#10587739	1	1	yes	characteristic
79	C₂³	—	137	1	no	—
80	C₂×C₄	—	274	4	no	—
81	C₂×C₄	—	274	4	no	—
82	C₂×C₄	—	274	4	no	—
83	C₂×C₄	—	137	8	no	—
84	C₂×C₄	—	137	2	no	—
85	C₂×C₄	—	137	1	no	—
86	C₂×C₄	—	137	4	no	—
87	C₂×C₄	C₄×#10587739	1	2	yes	—
88	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	274	4	no	—
91	C₈	—	274	2	no	—
92	C₂²	—	137	2	no	—
93	C₂²	—	137	4	no	—
94	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	274	8	no	—
97	C₄	—	274	2	no	—
98	C₄	—	137	2	no	—
99	C₄	#17984167	1	2	yes	—
100	C₂	—	137	4	no	—
101	C₂	#22578339	1	1	yes	characteristic
102	C₂	M₁₆×#10587739	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22578362

C₂

yes

maximal, characteristic

C₄²×#10587739

C₂

yes

maximal, characteristic

#22578268

C₂

yes

maximal

#97×D₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555518

C₂²

yes

—

C₂×C₄×#10587739

C₄

yes

—

C₂×C₄×#10587739

C₂²

yes

—

#17984143

C₄

yes

complemented

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

—

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

—

—

M₁₆

yes

—

#11764047

—

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

C₂×M₁₆

yes

—

#10587739

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#117

yes

—

Dic₁₃₇

—

Dic₁₃₇

#117

yes

—

C₂×C₁₃₇

#376

yes

characteristic

C₂×C₁₃₇

C₄×M₁₆

yes

—

D₁₃₇

#432

yes

—

C₁₃₇

#1745

yes

characteristic, complemented, Sylow

#1745

—

137

maximal, Sylow

C₂×#97

—

137

—

C₂×#97

—

137

—

C₄³

—

137

—

#336

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

#97

—

137

—

#97

—

137

—

#97

—

137

—

#97

#10587739

yes

characteristic, complemented

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

—

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

—

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

—

274

—

C₄

—

137

—

C₄

#17984167

yes

—

100

C₂

—

137

—

101

C₂

#22578339

yes

characteristic

102

C₂

M₁₆×#10587739

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	#1745
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
8. #17984143	C₄	semidirect
11. #17983999	C₄	semidirect
12. #17983998	C₄	semidirect
13. #97×C₁₃₇	C₄	semidirect
51. C₁₃₇	#1745	semidirect
65. #97	#10587739	semidirect

H ≅ G/N

type

8. #17984143

C₄

semidirect

11. #17983999

C₄

semidirect

12. #17983998

C₄

semidirect

13. #97×C₁₃₇

C₄

semidirect

51. C₁₃₇

#1745

semidirect

65. #97

#10587739

semidirect

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

subgroup	index
#22578362	2
C₄²×#10587739	2
#22578268	2
#97×D₁₃₇	2
#1745	137

subgroup

index

#97×D₁₃₇

137

Subgroups — #31796505

Conjugacy classes of subgroups

Complements

Maximal subgroups