Subgroups — #31778992

Order 17536.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17983994	C₂	1	1	yes	maximal, characteristic
2	#22577233	C₂	1	1	yes	maximal, characteristic
3	#368×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#17984024	—	2	1	no	—
5	#17983994	C₄	1	2	yes	—
6	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
7	#108×C₁₃₇	—	2	1	no	—
8	#14555494	—	4	1	no	—
9	M₁₆×C₁₃₇	—	4	1	no	—
10	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
11	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₃₇	Q₈	1	1	yes	characteristic
13	#11764040	—	8	1	no	—
14	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
15	C₈×C₁₃₇	—	8	1	no	—
16	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₃₇	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₃₇	Q₃₂	1	1	yes	characteristic
19	C₄×C₁₃₇	#107	1	1	yes	characteristic
20	C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
21	C₂×C₁₃₇	—	2	1	no	—
22	C₂×C₁₃₇	#366	1	1	yes	characteristic
23	C₁₃₇	#1425	1	1	yes	characteristic, complemented, Sylow
24	#1425	—	137	1	no	maximal, Sylow
25	C₂×C₃₂	—	137	1	no	—
26	#368	—	137	1	no	—
27	#368	D₁₃₇	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
29	#108	—	2	1	no	—
30	#108	—	274	1	no	—
31	C₃₂	—	137	2	no	—
32	M₁₆	—	4	1	no	—
33	M₁₆	—	548	1	no	—
34	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
35	C₁₆	C₄×D₁₃₇	1	1	yes	characteristic
36	C₁₆	Dic₂₇₄	1	1	yes	characteristic
37	C₂×C₄	#14555495	1	1	yes	characteristic
38	C₈	—	8	1	no	—
39	C₈	—	1096	1	no	—
40	C₈	#14555489	1	1	yes	characteristic
41	C₈	#14555500	1	1	yes	characteristic
42	C₂²	#17984034	1	1	yes	characteristic
43	C₄	#17984021	1	1	yes	characteristic
44	C₄	#17984029	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#22577228	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984029
commutator	C₁₆×C₁₃₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₃₇
Fitting	#368×C₁₃₇	C₂
socle	C₂×C₁₃₇	#366
prime radical	C₂²×C₁₃₇	Q₃₂
cosocle	C₂×C₁₆×C₁₃₇	C₂²
hypercenter	#368	D₁₃₇
hypocenter	C₁₃₇	#1425
LCS	G ▷ C₁₆×C₁₃₇ ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
23. C₁₃₇	#1425	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17983994	2
#22577233	2
#368×C₁₃₇	2
#1425	137