Subgroups — #31783068

Order 17536.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577805	C₂	1	2	yes	maximal
2	#22577231	C₂	1	2	yes	maximal
3	#22577229	C₂	1	2	yes	maximal
4	C₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#17984080	—	2	2	no	—
6	#17984022	C₂²	1	4	yes	—
7	#17984020	—	2	2	no	—
8	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₃₇	—	2	4	no	—
12	#14555489	—	4	4	no	—
13	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
15	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
16	C₁₆×C₁₃₇	Q₈	1	2	yes	—
17	C₁₆×C₁₃₇	Q₈	1	2	yes	—
18	C₂×Dic₁₃₇	—	4	4	no	—
19	C₂×C₄×C₁₃₇	D₈	1	2	yes	—
20	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
21	C₈×C₁₃₇	Q₁₆	1	2	yes	—
22	C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
23	C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
25	C₄×C₁₃₇	—	2	2	no	—
26	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₁₃₇	—	8	4	no	—
29	C₂×C₁₃₇	#500	1	1	yes	characteristic
30	C₂×C₁₃₇	#508	1	2	yes	—
31	C₁₃₇	#2253	1	1	yes	characteristic, complemented, Sylow
32	#2253	—	137	1	no	maximal, Sylow
33	#499	—	137	2	no	—
34	#367	—	137	2	no	—
35	#366	—	137	2	no	—
36	C₄×C₁₆	D₁₃₇	1	1	yes	characteristic
37	#125	—	274	2	no	—
38	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
39	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
40	#107	—	137	4	no	—
41	#106	—	274	2	no	—
42	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
43	C₂×C₈	D₅₄₈	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
45	C₄⋊C₄	—	274	4	no	—
46	C₄⋊C₄	—	548	4	no	—
47	C₄²	D₅₄₈	1	1	yes	characteristic
48	C₁₆	Dic₂₇₄	1	2	yes	—
49	C₁₆	Dic₂₇₄	1	2	yes	—
50	C₂×C₄	—	548	4	no	—
51	C₂×C₄	D₁₀₉₆	1	2	yes	—
52	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
53	C₈	Dic₅₄₈	1	2	yes	—
54	C₈	C₂×Dic₂₇₄	1	1	yes	characteristic
55	C₈	C₂×Dic₂₇₄	1	1	yes	characteristic
56	C₂²	C₂×D₁₀₉₆	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	1096	4	no	—
59	C₄	#17984088	1	1	yes	characteristic
60	C₄	C₂×Dic₅₄₈	1	1	yes	characteristic
61	C₂	#22577808	1	1	yes	characteristic
62	C₂	#22577835	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	#2253
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
31. C₁₃₇	#2253	semidirect

Maximal subgroups

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

subgroup	index
#22577805	2
#22577231	2
#22577229	2
C₄×C₁₆×C₁₃₇	2
#2253	137