Subgroups — #31797804

Order 17536.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10587739⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₈×#11764049	C₂	1	1	yes	maximal, characteristic
3	C₁₆×#10587739	C₂	1	1	yes	maximal, characteristic
4	#17984183	C₄	1	2	yes	—
5	C₄×#11764049	C₄	1	2	yes	—
6	#17984161	C₄	1	1	yes	characteristic
7	C₈×#10587739	C₂²	1	1	yes	characteristic
8	C₁₆×D₁₃₇	C₄	1	1	yes	characteristic
9	C₂×#11764049	C₈	1	4	yes	—
10	#14555522	C₈	1	4	yes	complemented
11	#14555518	C₂×C₄	1	1	yes	characteristic
12	C₄×#10587739	C₂×C₄	1	1	yes	characteristic
13	#14555513	C₈	1	2	yes	complemented
14	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
15	#14555485	C₈	1	1	yes	characteristic, complemented
16	C₁₆×C₁₃₇	C₈	1	1	yes	characteristic, complemented
17	#11764049	—	2	4	no	—
18	C₂×#10587739	C₂×C₈	1	2	yes	—
19	#11764047	C₂×C₈	1	2	yes	—
20	C₄×D₁₃₇	C₄²	1	1	yes	characteristic
21	#11764040	C₂×C₈	1	1	yes	characteristic
22	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
23	C₂×D₁₃₇	C₄×C₈	1	1	yes	characteristic
24	#10587739	M₃₂	1	4	yes	—
25	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
26	Dic₁₃₇	C₄×C₈	1	1	yes	characteristic
27	C₂×C₁₃₇	C₈²	1	1	yes	characteristic
28	D₁₃₇	#346	1	2	yes	—
29	C₁₃₇	#1311	1	1	yes	characteristic, complemented, Sylow
30	#1311	—	137	1	no	maximal, Sylow
31	C₄×C₁₆	—	137	1	no	—
32	C₄×C₁₆	—	137	1	no	—
33	C₈²	—	137	1	no	—
34	C₂×C₁₆	—	137	1	no	—
35	C₂×C₁₆	—	137	2	no	—
36	C₂×C₁₆	—	137	1	no	—
37	C₄×C₈	—	137	1	no	—
38	C₄×C₈	—	137	2	no	—
39	C₂×C₈	—	137	4	no	—
40	C₂×C₈	—	137	1	no	—
41	C₂×C₈	—	137	1	no	—
42	C₄²	—	137	1	no	—
43	C₁₆	—	137	2	no	—
44	C₁₆	—	137	4	no	—
45	C₁₆	—	137	1	no	—
46	C₁₆	#11764049	1	1	yes	characteristic, complemented
47	C₂×C₄	—	137	2	no	—
48	C₂×C₄	—	137	1	no	—
49	C₈	—	274	4	no	—
50	C₈	—	137	2	no	—
51	C₈	—	137	1	no	—
52	C₈	C₂×#11764049	1	1	yes	characteristic
53	C₂²	—	137	1	no	—
54	C₄	—	137	4	no	—
55	C₄	—	137	1	no	—
56	C₄	C₄×#11764049	1	1	yes	characteristic
57	C₂	—	137	2	no	—
58	C₂	C₈×#11764049	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂×#11764049
commutator, socle	C₂×C₁₃₇	C₈²
Fitting	C₁₆×C₁₃₇	C₈
prime radical	C₂×D₁₃₇	C₄×C₈
cosocle	C₈×#10587739	C₂²
hypercenter	C₁₆	#11764049
hypocenter	C₁₃₇	#1311
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
10. #14555522	C₈	semidirect
13. #14555513	C₈	semidirect
15. #14555485	C₈	semidirect
16. C₁₆×C₁₃₇	C₈	semidirect
29. C₁₃₇	#1311	semidirect
46. C₁₆	#11764049	semidirect

Maximal subgroups

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

subgroup	index
#10587739⋊C₁₆	2
C₈×#11764049	2
C₁₆×#10587739	2
#1311	137