Subgroups — #31797855

Order 17536.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578493	C₂	1	2	yes	maximal
2	C₁₆×#10587739	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11764049	—	2	2	no	—
4	#17984161	C₄	1	1	yes	characteristic
5	C₈×#10587739	C₂²	1	1	yes	characteristic
6	C₁₆×D₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#11764049	—	4	2	no	—
8	#14555518	Q₈	1	1	yes	characteristic
9	C₄×#10587739	D₄	1	1	yes	characteristic
10	#14555513	—	2	1	no	—
11	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
12	#14555485	C₈	1	1	yes	characteristic, complemented
13	C₁₆×C₁₃₇	C₈	1	1	yes	characteristic, complemented
14	#11764049	—	8	2	no	—
15	C₂×#10587739	D₈	1	1	yes	characteristic
16	C₂×#10587739	Q₁₆	1	1	yes	characteristic
17	#11764047	—	2	1	no	—
18	C₄×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
19	#11764040	M₁₆	1	1	yes	characteristic
20	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
21	C₂×D₁₃₇	#107	1	1	yes	characteristic
22	#10587739	—	2	1	no	—
23	#10587739	D₁₆	1	1	yes	characteristic
24	#10587739	Q₃₂	1	1	yes	characteristic
25	C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
26	Dic₁₃₇	#108	1	1	yes	characteristic
27	C₂×C₁₃₇	#335	1	1	yes	characteristic
28	D₁₃₇	#366	1	1	yes	characteristic
29	D₁₃₇	#368	1	1	yes	characteristic
30	C₁₃₇	#1371	1	1	yes	characteristic, complemented, Sylow
31	#1371	—	137	1	no	maximal, Sylow
32	C₄×C₁₆	—	137	1	no	—
33	#335	—	137	2	no	—
34	C₂×C₁₆	—	137	1	no	—
35	C₂×C₁₆	—	137	1	no	—
36	C₄⋊C₈	—	274	2	no	—
37	C₄×C₈	—	137	1	no	—
38	C₂×C₈	—	548	2	no	—
39	C₂×C₈	—	137	1	no	—
40	C₂×C₈	—	137	1	no	—
41	C₄²	—	137	1	no	—
42	C₁₆	—	274	1	no	—
43	C₁₆	—	137	1	no	—
44	C₁₆	#11764049	1	1	yes	characteristic, complemented
45	C₂×C₄	—	137	1	no	—
46	C₂×C₄	—	137	1	no	—
47	C₂×C₄	—	137	1	no	—
48	C₈	—	274	1	no	—
49	C₈	—	1096	2	no	—
50	C₈	—	137	1	no	—
51	C₈	C₂×#11764049	1	1	yes	characteristic
52	C₂²	—	137	1	no	—
53	C₄	—	274	1	no	—
54	C₄	—	137	1	no	—
55	C₄	—	137	1	no	—
56	C₄	—	137	1	no	—
57	C₄	C₄⋊#11764049	1	1	yes	characteristic
58	C₂	—	137	1	no	—
59	C₂	—	137	1	no	—
60	C₂	#22578493	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578493
commutator	C₈×C₁₃₇	C₂×C₈
Frattini	C₈	C₂×#11764049
Fitting	C₁₆×C₁₃₇	C₈
socle	C₂×C₁₃₇	#335
prime radical	C₂×D₁₃₇	#107
cosocle	C₈×#10587739	C₂²
hypercenter	C₁₆	#11764049
hypocenter	C₁₃₇	#1371
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
12. #14555485	C₈	semidirect
13. C₁₆×C₁₃₇	C₈	semidirect
30. C₁₃₇	#1371	semidirect
44. C₁₆	#11764049	semidirect

Maximal subgroups

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

subgroup	index
#22578493	2
C₁₆×#10587739	2
#1371	137