Subgroups — #31795708

Order 17536.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578292	C₂	1	1	yes	maximal, characteristic
2	#22578244	C₂	1	1	yes	maximal, characteristic
3	#22577178	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	C₄	1	2	yes	—
5	C₄×#11764047	C₄	1	1	yes	characteristic
6	C₄×#11764047	C₄	1	1	yes	characteristic
7	C₂×#14555485	C₄	1	1	yes	characteristic
8	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
10	C₂×#11764047	—	2	1	no	—
11	C₂×#11764047	—	2	1	no	—
12	#14555513	C₈	1	4	yes	complemented
13	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
14	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
15	#14555485	—	2	1	no	—
16	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₃₇	—	2	1	no	—
18	#11764047	—	4	1	no	—
19	#11764047	—	4	1	no	—
20	C₂×Dic₁₃₇	M₁₆	1	1	yes	characteristic
21	C₂×Dic₁₃₇	M₁₆	1	1	yes	characteristic
22	#11764040	C₂×C₈	1	2	yes	—
23	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
24	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
25	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
26	C₂²×C₁₃₇	#97	1	1	yes	characteristic
27	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
28	C₄×C₁₃₇	#97	1	1	yes	characteristic
29	Dic₁₃₇	—	2	1	no	—
30	Dic₁₃₇	—	2	1	no	—
31	C₂×C₁₃₇	#322	1	1	yes	characteristic
32	C₂×C₁₃₇	#347	1	1	yes	characteristic
33	C₂×C₁₃₇	#347	1	1	yes	characteristic
34	C₁₃₇	#1313	1	1	yes	characteristic, complemented, Sylow
35	#1313	—	137	1	no	maximal, Sylow
36	#346	—	137	1	no	—
37	#346	—	137	1	no	—
38	#322	—	137	1	no	—
39	C₂×C₁₆	—	137	1	no	—
40	C₂×C₁₆	—	137	2	no	—
41	C₂×C₁₆	#10587739	1	1	yes	characteristic
42	C₄×C₈	—	137	1	no	—
43	C₄×C₈	—	137	1	no	—
44	C₄×C₈	—	137	1	no	—
45	C₂×C₈	—	274	1	no	—
46	C₂×C₈	—	274	1	no	—
47	C₂×C₈	—	137	1	no	—
48	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
49	C₄²	—	137	1	no	—
50	C₁₆	—	2	1	no	—
51	C₁₆	—	274	1	no	—
52	C₁₆	—	137	4	no	—
53	C₂×C₄	—	137	1	no	—
54	C₂×C₄	—	137	1	no	—
55	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
56	C₈	—	548	1	no	—
57	C₈	—	548	1	no	—
58	C₈	—	137	2	no	—
59	C₈	#14555519	1	1	yes	characteristic
60	C₈	#14555520	1	1	yes	characteristic
61	C₂²	#17984142	1	1	yes	characteristic
62	C₄	—	274	1	no	—
63	C₄	—	274	1	no	—
64	C₄	C₈×#10587739	1	1	yes	characteristic
65	C₄	#17984143	1	1	yes	characteristic
66	C₂	#22578245	1	1	yes	characteristic
67	C₂	#22578293	1	1	yes	characteristic
68	C₂	#22578294	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#10587739
commutator	C₄×C₁₃₇	C₄×C₈
Frattini	C₂×C₈	C₂×#10587739
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	#97
cosocle	C₈×Dic₁₃₇	C₂²
hypercenter	C₂×C₁₆	#10587739
hypocenter	C₁₃₇	#1313
LCS	G ▷ C₄×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
12. #14555513	C₈	semidirect
34. C₁₃₇	#1313	semidirect

Maximal subgroups

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

subgroup	index
#22578292	2
#22578244	2
#22577178	2
#1313	137