Subgroups — #31778709

Order 17536.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577179	C₂	1	1	yes	maximal, characteristic
2	#22577124	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555485	C₄	1	2	yes	—
5	C₄×#11764040	C₄	1	2	yes	—
6	C₂×C₁₆×C₁₃₇	C₄	1	2	yes	—
7	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×#11764040	—	2	2	no	—
9	#14555485	—	2	2	no	—
10	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₃₇	C₈	1	4	yes	complemented
14	#11764040	—	4	2	no	—
15	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
17	C₈×C₁₃₇	C₂×C₈	1	2	yes	—
18	C₈×C₁₃₇	M₁₆	1	2	yes	—
19	C₂²×C₁₃₇	#97	1	1	yes	characteristic
20	C₄×C₁₃₇	—	2	2	no	—
21	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
22	C₄×C₁₃₇	#97	1	1	yes	characteristic
23	C₂×C₁₃₇	#322	1	1	yes	characteristic
24	C₂×C₁₃₇	#347	1	2	yes	—
25	C₁₃₇	#1313	1	1	yes	characteristic, complemented, Sylow
26	#1313	—	137	1	no	maximal, Sylow
27	#346	—	137	1	no	—
28	#346	D₁₃₇	1	1	yes	characteristic
29	#322	—	137	1	no	—
30	C₂×C₁₆	—	137	2	no	—
31	C₂×C₁₆	Dic₁₃₇	1	2	yes	—
32	C₄×C₈	—	137	2	no	—
33	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
34	C₂×C₈	—	274	2	no	—
35	C₂×C₈	C₄×D₁₃₇	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₁₃₇	1	1	yes	characteristic
37	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
38	C₁₆	—	274	2	no	—
39	C₁₆	#11764040	1	4	yes	complemented
40	C₂×C₄	C₄×Dic₁₃₇	1	1	yes	characteristic
41	C₂×C₄	#14555493	1	2	yes	—
42	C₈	—	548	2	no	—
43	C₈	C₂×#11764040	1	2	yes	—
44	C₈	#14555493	1	2	yes	—
45	C₂²	#17983998	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄×#11764040	1	1	yes	characteristic
48	C₄	#17983998	1	1	yes	characteristic
49	C₂	#22577124	1	1	yes	characteristic
50	C₂	#22577180	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₃₇
commutator	C₄×C₁₃₇	C₄×C₈
Frattini	C₄×C₈	C₂×D₁₃₇
Fitting	#346×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	#97
cosocle	C₄×C₈×C₁₃₇	C₂²
hypercenter	#346	D₁₃₇
hypocenter	C₁₃₇	#1313
LCS	G ▷ C₄×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #346

Complements

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

N	H ≅ G/N	type
13. C₁₆×C₁₃₇	C₈	semidirect
25. C₁₃₇	#1313	semidirect
39. C₁₆	#11764040	semidirect

Maximal subgroups

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

subgroup	index
#22577179	2
#22577124	2
#346×C₁₃₇	2
#1313	137