Subgroups — #31797872

Order 17536.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984178	C₂	1	2	yes	maximal
2	#22578289	C₂	1	1	yes	maximal, characteristic
3	#17984178	C₄	1	4	yes	complemented
4	C₂×#14555513	C₄	1	1	yes	characteristic
5	C₂×#14555513	C₂²	1	1	yes	characteristic
6	C₈×Dic₁₃₇	C₄	1	1	yes	characteristic
7	#14555513	C₈	1	2	yes	—
8	#14555513	C₂×C₄	1	2	yes	—
9	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
10	C₄×Dic₁₃₇	C₈	1	1	yes	characteristic
11	C₂×C₈×C₁₃₇	C₈	1	1	yes	characteristic
12	C₂×Dic₁₃₇	C₁₆	1	2	yes	—
13	#11764040	C₄²	1	1	yes	characteristic
14	#11764040	C₂×C₈	1	1	yes	characteristic
15	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
16	C₈×C₁₃₇	C₁₆	1	2	yes	—
17	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
18	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
19	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
20	Dic₁₃₇	C₃₂	1	4	yes	complemented
21	C₂×C₁₃₇	C₄×C₁₆	1	1	yes	characteristic
22	C₂×C₁₃₇	C₂×C₃₂	1	2	yes	—
23	C₁₃₇	C₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
24	C₄×C₃₂	—	137	1	no	maximal, Sylow
25	C₂×C₃₂	—	137	2	no	—
26	C₄×C₁₆	—	137	1	no	—
27	C₂×C₁₆	—	137	1	no	—
28	C₂×C₁₆	—	137	1	no	—
29	C₄×C₈	—	137	1	no	—
30	C₃₂	—	137	4	no	—
31	C₂×C₈	—	137	1	no	—
32	C₂×C₈	#11764049	1	1	yes	characteristic
33	C₄²	—	137	1	no	—
34	C₁₆	—	137	2	no	—
35	C₁₆	—	137	2	no	—
36	C₂×C₄	—	137	2	no	—
37	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
38	C₈	—	137	1	no	—
39	C₈	—	137	1	no	—
40	C₈	#14555522	1	2	yes	—
41	C₂²	#17984183	1	1	yes	characteristic
42	C₄	—	137	4	no	—
43	C₄	C₄×#11764049	1	1	yes	characteristic
44	C₄	C₂×#14555522	1	1	yes	characteristic
45	C₂	C₄×#14555522	1	1	yes	characteristic
46	C₂	#22578510	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₈	#11764049
commutator, hypocenter	C₁₃₇	C₄×C₃₂
Frattini	C₂×C₄	C₂×#11764049
Fitting	C₂×C₈×C₁₃₇	C₈
socle, prime radical	C₂²×C₁₃₇	C₂×C₁₆
cosocle	C₂×#14555513	C₂²
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
3. #17984178	C₄	semidirect
20. Dic₁₃₇	C₃₂	semidirect
23. C₁₃₇	C₄×C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984178	2
#22578289	2
C₄×C₃₂	137