Subgroups — #31797805

Order 17536.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14555522	C₂	1	2	yes	maximal
2	C₈×#11764047	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555522	C₄	1	4	yes	—
4	C₄×#11764047	C₄	1	1	yes	characteristic
5	C₄×#11764047	C₂²	1	1	yes	characteristic
6	C₈×Dic₁₃₇	C₄	1	1	yes	characteristic
7	#14555522	C₈	1	8	yes	complemented
8	C₂×#11764047	C₈	1	2	yes	—
9	C₂×#11764047	C₂×C₄	1	2	yes	—
10	C₂×#11764040	C₈	1	1	yes	characteristic
11	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₁₃₇	C₈	1	1	yes	characteristic
13	#11764047	—	2	2	no	—
14	#11764047	C₂×C₈	1	4	yes	—
15	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
16	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
17	#11764040	—	2	1	no	—
18	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₃₇	—	2	1	no	—
20	C₂²×C₁₃₇	C₄×C₈	1	1	yes	characteristic
21	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
22	C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
23	Dic₁₃₇	C₄×C₈	1	2	yes	—
24	Dic₁₃₇	M₃₂	1	2	yes	—
25	C₂×C₁₃₇	C₈²	1	1	yes	characteristic
26	C₂×C₁₃₇	#346	1	1	yes	characteristic
27	C₂×C₁₃₇	#346	1	1	yes	characteristic
28	C₁₃₇	#1311	1	1	yes	characteristic, complemented, Sylow
29	#1311	—	137	1	no	maximal, Sylow
30	C₄×C₁₆	—	137	2	no	—
31	C₈²	—	137	1	no	—
32	C₂×C₁₆	—	137	4	no	—
33	C₄×C₈	—	137	1	no	—
34	C₄×C₈	—	137	1	no	—
35	C₄×C₈	—	137	1	no	—
36	C₂×C₈	—	137	2	no	—
37	C₂×C₈	—	137	1	no	—
38	C₂×C₈	—	137	2	no	—
39	C₂×C₈	#11764049	1	1	yes	characteristic
40	C₄²	—	137	1	no	—
41	C₁₆	—	137	8	no	—
42	C₂×C₄	—	137	1	no	—
43	C₂×C₄	—	137	1	no	—
44	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	274	2	no	—
47	C₈	—	274	1	no	—
48	C₈	—	137	4	no	—
49	C₂²	C₄×#11764049	1	1	yes	characteristic
50	C₄	—	137	2	no	—
51	C₄	—	137	2	no	—
52	C₄	#17984184	1	1	yes	characteristic
53	C₄	#17984185	1	1	yes	characteristic
54	C₂	C₈×#11764049	1	1	yes	characteristic
55	C₂	#22578497	1	1	yes	characteristic
56	C₂	#22578498	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#11764049
commutator	C₂×C₁₃₇	C₈²
Fitting	C₂×C₈×C₁₃₇	C₈
socle, prime radical	C₂²×C₁₃₇	C₄×C₈
cosocle	C₄×#11764047	C₂²
hypercenter	C₂×C₈	#11764049
hypocenter	C₁₃₇	#1311
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
7. #14555522	C₈	semidirect
28. C₁₃₇	#1311	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14555522	2
C₈×#11764047	2
#1311	137