Subgroups — #31779088

Order 17536.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577454	C₂	1	3	yes	maximal
2	#22577125	C₂	1	3	yes	maximal, complemented
3	C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
5	#17984017	C₄	1	6	yes	—
6	#17984000	C₄	1	6	yes	—
7	C₄×#11764040	C₂²	1	3	yes	—
8	C₄×C₈×C₁₃₇	C₂²	1	3	yes	—
9	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
10	C₂×#11764040	C₂×C₄	1	6	yes	—
11	C₄×Dic₁₃₇	C₂×C₄	1	3	yes	—
12	C₂×C₈×C₁₃₇	C₂×C₄	1	6	yes	—
13	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
14	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
15	C₂×Dic₁₃₇	C₄²	1	3	yes	—
16	C₄×D₁₃₇	—	2	6	no	—
17	#11764040	—	2	6	no	—
18	C₂×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
19	C₈×C₁₃₇	—	2	6	no	—
20	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
21	C₂×D₁₃₇	—	2	3	no	—
22	C₄×C₁₃₇	#131	1	6	yes	—
23	Dic₁₃₇	—	2	3	no	—
24	C₂×C₁₃₇	#405	1	3	yes	—
25	D₁₃₇	—	4	1	no	—
26	C₁₃₇	#1454	1	1	yes	characteristic, complemented, Sylow
27	#1454	—	137	1	no	maximal, Sylow
28	#431	—	137	3	no	—
29	#322	—	137	3	no	—
30	C₈²	D₁₃₇	1	1	yes	characteristic, complemented
31	C₂×C₄²	—	137	1	no	—
32	C₄⋊C₈	—	137	6	no	—
33	C₂²⋊C₈	—	137	6	no	—
34	C₄×C₈	—	137	3	no	—
35	C₄×C₈	C₂×D₁₃₇	1	3	yes	—
36	C₂²×C₄	—	137	3	no	—
37	C₂×C₈	—	137	6	no	—
38	C₂×C₈	C₄×D₁₃₇	1	6	yes	—
39	C₄²	—	137	3	no	—
40	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
41	C₂³	—	137	1	no	—
42	C₂×C₄	—	274	6	no	—
43	C₂×C₄	—	137	3	no	—
44	C₂×C₄	C₂×C₄×D₁₃₇	1	3	yes	—
45	C₈	—	2	6	no	—
46	C₈	—	274	6	no	—
47	C₂²	—	274	3	no	—
48	C₂²	C₄²×D₁₃₇	1	1	yes	characteristic
49	C₄	—	274	3	no	—
50	C₄	#17984094	1	6	yes	—
51	C₂	—	548	1	no	—
52	C₂	#22577357	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂×C₄²
Fitting	C₈²×C₁₃₇	C₂
prime radical	C₂²×D₁₃₇	C₄²
cosocle	C₄²×C₁₃₇	C₂³
hypercenter	C₈²	D₁₃₇
hypocenter	C₁₃₇	#1454
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #22577125	C₂	semidirect
3. C₈²×C₁₃₇	C₂	semidirect
26. C₁₃₇	#1454	semidirect
30. C₈²	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577454	2
#22577125	2
C₈²×C₁₃₇	2
#1454	137