Subgroups — #31795820

Order 17536.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578266	C₂	1	2	yes	maximal
2	C₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11764047	—	2	2	no	—
4	C₂×#14555485	C₄	1	1	yes	characteristic
5	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#11764047	—	4	2	no	—
8	C₂×#11764040	Q₈	1	1	yes	characteristic
9	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
10	#14555485	—	2	1	no	—
11	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₃₇	C₈	1	2	yes	complemented
13	#11764047	—	8	2	no	—
14	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
15	C₂×Dic₁₃₇	Q₁₆	1	1	yes	characteristic
16	#11764040	—	2	1	no	—
17	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₃₇	#107	1	1	yes	characteristic
21	C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₁₃₇	#108	1	1	yes	characteristic
23	Dic₁₃₇	—	2	1	no	—
24	Dic₁₃₇	D₁₆	1	1	yes	characteristic
25	Dic₁₃₇	Q₃₂	1	1	yes	characteristic
26	C₂×C₁₃₇	#335	1	1	yes	characteristic
27	C₂×C₁₃₇	#366	1	1	yes	characteristic
28	C₂×C₁₃₇	#368	1	1	yes	characteristic
29	C₁₃₇	#1371	1	1	yes	characteristic, complemented, Sylow
30	#1371	—	137	1	no	maximal, Sylow
31	C₄×C₁₆	—	137	1	no	—
32	#335	—	137	2	no	—
33	C₂×C₁₆	—	137	1	no	—
34	C₂×C₁₆	#10587739	1	1	yes	characteristic
35	C₄⋊C₈	—	274	2	no	—
36	C₄×C₈	—	137	1	no	—
37	C₂×C₈	—	548	2	no	—
38	C₂×C₈	—	137	1	no	—
39	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
40	C₄²	—	137	1	no	—
41	C₁₆	—	274	1	no	—
42	C₁₆	#11764047	1	2	yes	complemented
43	C₂×C₄	—	137	1	no	—
44	C₂×C₄	—	137	1	no	—
45	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
46	C₈	—	274	1	no	—
47	C₈	—	1096	2	no	—
48	C₈	C₂×#11764047	1	1	yes	characteristic
49	C₈	#14555520	1	1	yes	characteristic
50	C₂²	#17984157	1	1	yes	characteristic
51	C₄	—	274	1	no	—
52	C₄	—	137	1	no	—
53	C₄	—	137	1	no	—
54	C₄	C₄⋊#11764047	1	1	yes	characteristic
55	C₄	#17984159	1	1	yes	characteristic
56	C₂	#22578266	1	1	yes	characteristic
57	C₂	#22578329	1	1	yes	characteristic
58	C₂	#22578331	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984157
commutator	C₈×C₁₃₇	C₂×C₈
Frattini	C₂×C₈	C₂×#10587739
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	#107
cosocle	C₈×Dic₁₃₇	C₂²
hypercenter	C₂×C₁₆	#10587739
hypocenter	C₁₃₇	#1371
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₃₇	C₈	semidirect
29. C₁₃₇	#1371	semidirect
42. C₁₆	#11764047	semidirect

Maximal subgroups

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

subgroup	index
#22578266	2
C₁₆×Dic₁₃₇	2
#1371	137