Subgroups — #31795939

Order 17536.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578334	C₂	1	1	yes	maximal, characteristic
2	#22578329	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#17984157	—	2	1	no	—
5	#17984138	—	2	1	no	—
6	Q₁₆×D₁₃₇	—	2	1	no	—
7	Dic₁₀₉₆	C₄	1	1	yes	characteristic, complemented
8	#17984034	—	2	1	no	—
9	C₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
10	Q₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#10587739	—	4	1	no	—
12	Q₈×D₁₃₇	—	4	1	no	—
13	Dic₅₄₈	—	2	1	no	—
14	#14555500	—	4	1	no	—
15	C₈×D₁₃₇	D₄	1	1	yes	characteristic
16	#14555485	D₄	1	1	yes	characteristic
17	Q₁₆×C₁₃₇	—	2	1	no	—
18	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×#10587739	—	8	1	no	—
20	Dic₂₇₄	—	4	1	no	—
21	Dic₂₇₄	—	8	1	no	—
22	C₄×D₁₃₇	—	8	1	no	—
23	C₄×D₁₃₇	D₈	1	1	yes	characteristic
24	#11764040	SD₁₆	1	1	yes	characteristic
25	Q₈×C₁₃₇	—	4	1	no	—
26	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₁₃₇	D₁₆	1	1	yes	characteristic
28	#10587739	—	16	1	no	—
29	C₄×C₁₃₇	—	8	1	no	—
30	C₄×C₁₃₇	#102	1	1	yes	characteristic
31	Dic₁₃₇	—	8	1	no	—
32	Dic₁₃₇	SD₃₂	1	1	yes	characteristic
33	C₂×C₁₃₇	#357	1	1	yes	characteristic
34	D₁₃₇	SD₆₄	1	1	yes	characteristic
35	D₁₃₇	Q₆₄	1	1	yes	characteristic
36	C₁₃₇	#1416	1	1	yes	characteristic, complemented, Sylow
37	#1416	—	137	1	no	maximal, Sylow
38	C₂×Q₃₂	—	137	1	no	—
39	C₂×C₃₂	—	137	1	no	—
40	#366	—	137	1	no	—
41	C₂×Q₁₆	—	274	1	no	—
42	Q₃₂	—	274	1	no	—
43	Q₃₂	—	137	1	no	—
44	Q₃₂	#10587739	1	1	yes	characteristic, complemented
45	C₂×C₁₆	—	137	1	no	—
46	#107	—	274	1	no	—
47	C₃₂	—	274	1	no	—
48	C₂×Q₈	—	548	1	no	—
49	Q₁₆	—	2	1	no	—
50	Q₁₆	—	274	1	no	—
51	Q₁₆	—	548	1	no	—
52	C₂×C₈	—	137	1	no	—
53	C₄⋊C₄	—	548	1	no	—
54	C₁₆	—	137	1	no	—
55	C₁₆	C₂×#10587739	1	1	yes	characteristic
56	Q₈	—	4	1	no	—
57	Q₈	—	548	1	no	—
58	Q₈	—	1096	1	no	—
59	C₂×C₄	—	1096	1	no	—
60	C₂×C₄	—	1096	1	no	—
61	C₂×C₄	—	137	1	no	—
62	C₈	—	137	1	no	—
63	C₈	C₂²⋊#10587739	1	1	yes	characteristic
64	C₂²	—	137	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	1096	1	no	—
67	C₄	—	137	1	no	—
68	C₄	—	2192	1	no	—
69	C₄	#17984150	1	1	yes	characteristic
70	C₂	—	137	1	no	—
71	C₂	—	137	1	no	—
72	C₂	#22578315	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578315
commutator	C₁₆×C₁₃₇	C₂×C₄
Frattini	C₁₆	C₂×#10587739
Fitting	Q₃₂×C₁₃₇	C₄
socle	C₂×C₁₃₇	#357
prime radical	C₂×D₁₃₇	D₁₆
cosocle	C₁₆×D₁₃₇	C₂²
hypercenter	Q₃₂	#10587739
hypocenter	C₁₃₇	#1416
LCS	G ▷ C₁₆×C₁₃₇ ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
7. Dic₁₀₉₆	C₄	semidirect
10. Q₃₂×C₁₃₇	C₄	semidirect
36. C₁₃₇	#1416	semidirect
44. Q₃₂	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578334	2
#22578329	2
Q₃₂×D₁₃₇	2
#1416	137