Subgroups — #31796716

Order 17536.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578269	C₂	1	6	yes	maximal
2	C₄²×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11764047	C₂²	1	4	yes	—
4	C₂×C₄×Dic₁₃₇	C₄	1	3	yes	—
5	C₂×C₄×Dic₁₃₇	C₂²	1	3	yes	—
6	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#11764047	—	2	12	no	—
8	C₂²×Dic₁₃₇	C₂×C₄	1	3	yes	—
9	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
10	C₄×Dic₁₃₇	—	2	6	no	—
11	C₄×Dic₁₃₇	—	2	6	no	—
12	C₂²×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
13	C₄²×C₁₃₇	C₈	1	4	yes	complemented
14	#11764047	—	4	4	no	—
15	C₂×Dic₁₃₇	—	2	3	no	—
16	C₂×Dic₁₃₇	—	2	3	no	—
17	C₂×Dic₁₃₇	M₁₆	1	6	yes	—
18	C₂×Dic₁₃₇	#41	1	6	yes	—
19	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	—	2	3	no	—
21	C₂×C₄×C₁₃₇	—	2	3	no	—
22	C₂×C₄×C₁₃₇	C₂×C₈	1	6	yes	—
23	C₂²×C₁₃₇	#117	1	3	yes	—
24	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
25	C₂²×C₁₃₇	C₂×M₁₆	1	3	yes	—
26	C₄×C₁₃₇	—	2	6	no	—
27	Dic₁₃₇	—	2	6	no	—
28	Dic₁₃₇	#126	1	4	yes	—
29	C₂×C₁₃₇	#383	1	1	yes	characteristic
30	C₂×C₁₃₇	#431	1	3	yes	—
31	C₂×C₁₃₇	#432	1	3	yes	—
32	C₁₃₇	#1839	1	1	yes	characteristic, complemented, Sylow
33	#1839	—	137	1	no	maximal, Sylow
34	C₄³	—	137	1	no	—
35	#336	—	137	6	no	—
36	C₂²×C₈	—	137	4	no	—
37	C₂×C₄²	—	137	3	no	—
38	C₂×C₄²	—	137	3	no	—
39	C₂×C₄²	#10587739	1	1	yes	characteristic
40	C₂²×C₄	—	137	3	no	—
41	C₂²×C₄	—	137	1	no	—
42	C₂²×C₄	C₂×#10587739	1	3	yes	—
43	C₂×C₈	—	274	12	no	—
44	C₄²	—	274	6	no	—
45	C₄²	—	274	6	no	—
46	C₄²	#11764047	1	4	yes	complemented
47	C₂³	C₂²×#10587739	1	1	yes	characteristic
48	C₂×C₄	—	2	3	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	—	274	3	no	—
51	C₂×C₄	—	274	3	no	—
52	C₂×C₄	—	137	6	no	—
53	C₂×C₄	—	137	6	no	—
54	C₂×C₄	C₂×#11764047	1	6	yes	—
55	C₈	—	548	4	no	—
56	C₂²	#17984167	1	3	yes	—
57	C₂²	C₂²×#11764047	1	1	yes	characteristic
58	C₂²	D₁₃₇⋊M₁₆	1	3	yes	—
59	C₄	—	2	6	no	—
60	C₄	—	274	6	no	—
61	C₄	—	137	4	no	—
62	C₂	#22578348	1	1	yes	characteristic
63	C₂	#22578419	1	3	yes	—
64	C₂	#22578426	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#10587739
commutator	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₂×C₄²×C₁₃₇	C₄
socle, prime radical	C₂³×C₁₃₇	C₂²×C₄
cosocle	C₂²×Dic₁₃₇	C₂³
hypercenter	C₂×C₄²	#10587739
hypocenter	C₁₃₇	#1839
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
13. C₄²×C₁₃₇	C₈	semidirect
32. C₁₃₇	#1839	semidirect
46. C₄²	#11764047	semidirect

Maximal subgroups

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

subgroup	index
#22578269	2
C₄²×Dic₁₃₇	2
#1839	137