Subgroups — #31780407

Order 17536.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577808	C₂	1	4	yes	maximal
2	#22577783	C₂	1	2	yes	maximal
3	C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₄₈	—	2	4	no	—
5	#17984088	C₂²	1	4	yes	—
6	#17984022	—	2	8	no	—
7	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂×Dic₂₇₄	—	2	4	no	—
10	Dic₅₄₈	—	4	8	no	—
11	C₄⋊Dic₁₃₇	—	2	8	no	—
12	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
13	C₂×C₈×C₁₃₇	D₄	1	4	yes	—
14	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
15	Dic₂₇₄	—	4	8	no	—
16	C₂×Dic₁₃₇	—	4	4	no	—
17	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
19	C₈×C₁₃₇	D₈	1	4	yes	—
20	C₈×C₁₃₇	Q₁₆	1	8	yes	complemented
21	C₂²×C₁₃₇	#127	1	1	yes	characteristic
22	C₄×C₁₃₇	C₂×D₈	1	2	yes	—
23	C₄×C₁₃₇	C₂×Q₁₆	1	4	yes	—
24	Dic₁₃₇	—	8	4	no	—
25	C₂×C₁₃₇	#493	1	1	yes	characteristic
26	C₂×C₁₃₇	#494	1	2	yes	—
27	C₁₃₇	#1713	1	1	yes	characteristic, complemented, Sylow
28	#1713	—	137	1	no	maximal, Sylow
29	#500	—	137	4	no	—
30	#494	—	137	2	no	—
31	C₈²	D₁₃₇	1	1	yes	characteristic
32	C₂×Q₁₆	—	274	4	no	—
33	C₄⋊Q₈	—	137	4	no	—
34	#107	—	274	8	no	—
35	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
36	C₄×C₈	C₂×D₁₃₇	1	2	yes	—
37	C₂×Q₈	—	274	4	no	—
38	Q₁₆	—	548	8	no	—
39	C₂×C₈	D₅₄₈	1	2	yes	—
40	C₂×C₈	D₅₄₈	1	4	yes	—
41	C₄⋊C₄	—	274	8	no	—
42	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
43	Q₈	—	548	8	no	—
44	C₂×C₄	—	548	4	no	—
45	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
46	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
47	C₈	D₁₀₉₆	1	4	yes	—
48	C₈	Dic₅₄₈	1	8	yes	complemented
49	C₂²	#17984085	1	1	yes	characteristic
50	C₄	—	1096	4	no	—
51	C₄	C₂×D₁₀₉₆	1	2	yes	—
52	C₄	C₂×Dic₅₄₈	1	4	yes	—
53	C₂	#22577780	1	1	yes	characteristic
54	C₂	#22577783	1	2	yes	—

	subgroup	G/H
center	C₂²	#17984085
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	C₈²×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	#127
hypercenter	C₈²	D₁₃₇
hypocenter	C₁₃₇	#1713
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₁₃₇	Q₁₆	semidirect
27. C₁₃₇	#1713	semidirect
48. C₈	Dic₅₄₈	semidirect

Maximal subgroups

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

subgroup	index
#22577808	2
#22577783	2
C₈²×C₁₃₇	2
#1713	137