Subgroups — #31783071

Order 17536.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577808	C₂	1	2	yes	maximal
2	#22577231	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#17984088	—	2	2	no	—
5	#17984022	—	2	2	no	—
6	#17984022	C₂²	1	4	yes	—
7	C₂×C₁₆×C₁₃₇	C₂²	1	2	yes	—
8	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₇₄	—	4	2	no	—
10	C₄⋊Dic₁₃₇	—	2	4	no	—
11	C₄⋊Dic₁₃₇	—	4	2	no	—
12	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
14	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
15	C₁₆×C₁₃₇	Q₈	1	4	yes	complemented
16	Dic₂₇₄	—	8	4	no	—
17	C₂×Dic₁₃₇	—	4	4	no	—
18	C₂×C₄×C₁₃₇	D₈	1	2	yes	—
19	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
20	C₈×C₁₃₇	Q₁₆	1	2	yes	—
21	C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
22	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
23	C₄×C₁₃₇	SD₃₂	1	4	yes	—
24	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
25	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
26	Dic₁₃₇	—	8	4	no	—
27	C₂×C₁₃₇	#500	1	1	yes	characteristic
28	C₂×C₁₃₇	C₂×SD₃₂	1	2	yes	—
29	C₁₃₇	#2254	1	1	yes	characteristic, complemented, Sylow
30	#2254	—	137	1	no	maximal, Sylow
31	#500	—	137	2	no	—
32	#367	—	137	4	no	—
33	C₄×C₁₆	D₁₃₇	1	1	yes	characteristic
34	C₄⋊Q₈	—	274	2	no	—
35	C₂×C₁₆	C₂×D₁₃₇	1	2	yes	—
36	#107	—	274	2	no	—
37	#107	—	137	4	no	—
38	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
39	C₂×Q₈	—	548	2	no	—
40	C₂×C₈	D₅₄₈	1	1	yes	characteristic
41	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
42	C₄⋊C₄	—	274	4	no	—
43	C₄⋊C₄	—	548	2	no	—
44	C₄²	D₅₄₈	1	1	yes	characteristic
45	C₁₆	Dic₂₇₄	1	4	yes	complemented
46	Q₈	—	1096	4	no	—
47	C₂×C₄	—	548	4	no	—
48	C₂×C₄	D₁₀₉₆	1	2	yes	—
49	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
50	C₈	Dic₅₄₈	1	2	yes	—
51	C₈	C₂×Dic₂₇₄	1	2	yes	—
52	C₂²	C₂×D₁₀₉₆	1	1	yes	characteristic
53	C₄	—	1096	4	no	—
54	C₄	#17984033	1	4	yes	—
55	C₄	#17984088	1	1	yes	characteristic
56	C₄	C₂×Dic₅₄₈	1	1	yes	characteristic
57	C₂	#22577808	1	1	yes	characteristic
58	C₂	C₂×#17984033	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₉₆
commutator, cosocle	C₂×C₈×C₁₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	C₄×C₁₆×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	C₂×D₈
hypercenter	C₄×C₁₆	D₁₃₇
hypocenter	C₁₃₇	#2254
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₁₃₇	Q₈	semidirect
29. C₁₃₇	#2254	semidirect
45. C₁₆	Dic₂₇₄	semidirect

Maximal subgroups

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

subgroup	index
#22577808	2
#22577231	2
C₄×C₁₆×C₁₃₇	2
#2254	137