Subgroups — #31797889

Order 17536.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578511	C₂	1	2	yes	maximal
2	#22578289	C₂	1	1	yes	maximal, characteristic
3	#17984178	—	2	2	no	—
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	—	2	1	no	—
8	#14555513	D₄	1	1	yes	characteristic
9	#14555513	Q₈	1	1	yes	characteristic
10	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
11	C₄×Dic₁₃₇	C₈	1	1	yes	characteristic
12	C₂×C₈×C₁₃₇	C₈	1	1	yes	characteristic
13	C₂×Dic₁₃₇	—	2	1	no	—
14	#11764040	C₄⋊C₄	1	1	yes	characteristic
15	#11764040	M₁₆	1	1	yes	characteristic
16	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
17	C₈×C₁₃₇	C₁₆	1	2	yes	—
18	C₂²×C₁₃₇	M₃₂	1	1	yes	characteristic
19	C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
20	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
21	Dic₁₃₇	—	2	2	no	—
22	C₂×C₁₃₇	—	2	1	no	—
23	C₂×C₁₃₇	C₄⋊C₁₆	1	1	yes	characteristic
24	C₁₃₇	#1422	1	1	yes	characteristic, complemented, Sylow
25	#1422	—	137	1	no	maximal, Sylow
26	M₆₄	—	137	2	no	—
27	C₄×C₁₆	—	137	1	no	—
28	C₂×C₁₆	—	137	1	no	—
29	C₂×C₁₆	—	137	1	no	—
30	C₄×C₈	—	137	1	no	—
31	C₃₂	—	274	2	no	—
32	C₂×C₈	—	137	1	no	—
33	C₂×C₈	#11764049	1	1	yes	characteristic
34	C₄²	—	137	1	no	—
35	C₁₆	—	274	1	no	—
36	C₁₆	—	137	1	no	—
37	C₁₆	—	137	1	no	—
38	C₂×C₄	—	274	1	no	—
39	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
40	C₈	—	137	1	no	—
41	C₈	—	137	1	no	—
42	C₈	#14555522	1	2	yes	—
43	C₂²	#17984185	1	1	yes	characteristic
44	C₄	—	274	2	no	—
45	C₄	C₄⋊#11764049	1	1	yes	characteristic
46	C₄	C₂×#14555522	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	C₄⋊#14555522	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#11764049
commutator	C₄×C₁₃₇	C₂×C₁₆
Frattini	C₂×C₄	C₂×#11764049
Fitting	C₂×C₈×C₁₃₇	C₈
socle	C₂×C₁₃₇	C₄⋊C₁₆
prime radical	C₂²×C₁₃₇	M₃₂
cosocle	C₂×#14555513	C₂²
hypercenter	C₂×C₈	#11764049
hypocenter	C₁₃₇	#1422
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
24. C₁₃₇	#1422	semidirect

Maximal subgroups

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

subgroup	index
#22578511	2
#22578289	2
#1422	137