Subgroups — #31778891

Order 17536.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577441	C₂	1	2	yes	maximal
2	C₂×M₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#17984047	—	2	2	no	—
4	#17984022	—	2	2	no	—
5	#17984020	C₄	1	4	yes	complemented
6	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
7	M₃₂×C₁₃₇	—	2	2	no	—
8	C₂×C₁₆×C₁₃₇	C₄	1	2	yes	complemented
9	C₄⋊Dic₁₃₇	—	2	4	no	—
10	C₂²⋊Dic₁₃₇	—	4	2	no	—
11	C₄×Dic₁₃₇	—	4	2	no	—
12	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
16	C₂×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
17	C₁₆×C₁₃₇	—	2	2	no	—
18	C₂×Dic₁₃₇	—	4	4	no	—
19	C₂³×C₁₃₇	D₈	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
21	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₃₇	SD₁₆	1	2	yes	—
23	C₂×C₄×C₁₃₇	Q₁₆	1	1	yes	characteristic
24	C₈×C₁₃₇	C₄²	1	1	yes	characteristic
25	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
27	C₂²×C₁₃₇	—	4	1	no	—
28	C₂²×C₁₃₇	#102	1	2	yes	—
29	C₂²×C₁₃₇	#107	1	1	yes	characteristic
30	C₄×C₁₃₇	#95	1	1	yes	characteristic
31	C₄×C₁₃₇	#103	1	2	yes	—
32	C₄×C₁₃₇	#106	1	1	yes	characteristic
33	Dic₁₃₇	—	8	4	no	—
34	C₂×C₁₃₇	—	2	2	no	—
35	C₂×C₁₃₇	—	2	1	no	—
36	C₂×C₁₃₇	#340	1	1	yes	characteristic
37	C₁₃₇	#1384	1	1	yes	characteristic, complemented, Sylow
38	#1384	—	137	1	no	maximal, Sylow
39	C₂×M₃₂	D₁₃₇	1	1	yes	characteristic
40	#427	—	137	2	no	—
41	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
42	#117	—	274	2	no	—
43	M₃₂	—	2	2	no	—
44	C₂×C₁₆	Dic₁₃₇	1	2	yes	complemented
45	#107	—	274	2	no	—
46	#106	—	137	4	no	—
47	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
48	C₂×C₈	C₄×D₁₃₇	1	2	yes	—
49	C₂×C₈	C₂×Dic₁₃₇	1	1	yes	characteristic
50	C₂×C₈	C₂²⋊D₁₃₇	1	2	yes	—
51	C₂×C₈	Dic₂₇₄	1	1	yes	characteristic
52	C₄⋊C₄	—	274	4	no	—
53	C₂²⋊C₄	—	548	2	no	—
54	C₄²	—	548	2	no	—
55	C₁₆	—	2	2	no	—
56	C₂³	D₁₀₉₆	1	1	yes	characteristic
57	C₂×C₄	—	548	4	no	—
58	C₂×C₄	#14555487	1	2	yes	—
59	C₂×C₄	C₄⋊Dic₁₃₇	1	1	yes	characteristic
60	C₂×C₄	#14555499	1	2	yes	—
61	C₂×C₄	Dic₅₄₈	1	1	yes	characteristic
62	C₈	C₄×Dic₁₃₇	1	1	yes	characteristic
63	C₈	C₂²⋊Dic₁₃₇	1	1	yes	characteristic
64	C₈	#14555489	1	2	yes	—
65	C₂²	—	4	1	no	—
66	C₂²	#17984010	1	2	yes	—
67	C₂²	#17984022	1	1	yes	characteristic
68	C₄	—	1096	4	no	—
69	C₄	#17983995	1	1	yes	characteristic
70	C₄	#17984013	1	2	yes	—
71	C₄	#17984020	1	1	yes	characteristic
72	C₂	—	2	2	no	—
73	C₂	—	2	1	no	—
74	C₂	#22577166	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984020
commutator	C₈×C₁₃₇	C₄²
Frattini	C₂²×C₈	C₂×D₁₃₇
Fitting	C₂×M₃₂×C₁₃₇	C₂
socle	C₂×C₁₃₇	#340
prime radical	C₂³×C₁₃₇	D₈
cosocle	C₂²×C₈×C₁₃₇	C₂²
hypercenter	C₂×M₃₂	D₁₃₇
hypocenter	C₁₃₇	#1384
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
5. #17984020	C₄	semidirect
8. C₂×C₁₆×C₁₃₇	C₄	semidirect
37. C₁₃₇	#1384	semidirect
44. C₂×C₁₆	Dic₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577441	2
C₂×M₃₂×C₁₃₇	2
#1384	137