Subgroups — #31780376

Order 17536.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577801	C₂	1	3	yes	maximal, complemented
2	#22577776	C₂	1	3	yes	maximal
3	C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14555499	—	2	6	no	—
5	#17984088	C₂²	1	3	yes	—
6	#17984085	C₂²	1	1	yes	characteristic
7	#17984020	—	2	6	no	—
8	C₄×C₈×C₁₃₇	C₂²	1	3	yes	—
9	C₂×Dic₂₇₄	—	2	3	no	—
10	C₂×D₅₄₈	—	2	3	no	—
11	#14555499	—	4	12	no	—
12	C₄⋊Dic₁₃₇	—	2	6	no	—
13	C₂×C₈×C₁₃₇	D₄	1	6	yes	—
14	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
15	C₂²×D₁₃₇	—	4	1	no	—
16	Dic₂₇₄	—	4	6	no	—
17	D₅₄₈	—	4	6	no	—
18	C₂×Dic₁₃₇	—	4	3	no	—
19	C₂×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
20	C₈×C₁₃₇	SD₁₆	1	12	yes	complemented
21	C₂²×C₁₃₇	#127	1	1	yes	characteristic
22	C₂×D₁₃₇	—	8	3	no	—
23	C₄×C₁₃₇	C₂×SD₁₆	1	6	yes	—
24	Dic₁₃₇	—	8	3	no	—
25	C₂×C₁₃₇	#492	1	3	yes	—
26	D₁₃₇	—	16	1	no	—
27	C₁₃₇	#1705	1	1	yes	characteristic, complemented, Sylow
28	#1705	—	137	1	no	maximal, Sylow
29	#498	—	137	3	no	—
30	#492	—	137	3	no	—
31	C₈²	D₁₃₇	1	1	yes	characteristic, complemented
32	C₂×SD₁₆	—	274	6	no	—
33	C₄⋊Q₈	—	137	3	no	—
34	#127	—	137	1	no	—
35	#106	—	274	6	no	—
36	C₄×C₈	C₂×D₁₃₇	1	3	yes	—
37	C₂×Q₈	—	274	3	no	—
38	C₂×D₄	—	274	3	no	—
39	SD₁₆	—	548	12	no	—
40	C₂×C₈	D₅₄₈	1	6	yes	—
41	C₄⋊C₄	—	274	6	no	—
42	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
43	C₂³	—	548	1	no	—
44	Q₈	—	548	6	no	—
45	D₄	—	548	6	no	—
46	C₂×C₄	—	548	3	no	—
47	C₂×C₄	C₂×D₅₄₈	1	3	yes	—
48	C₈	#14555499	1	12	yes	complemented
49	C₂²	—	1096	3	no	—
50	C₂²	#17984085	1	1	yes	characteristic
51	C₄	—	1096	3	no	—
52	C₄	C₂×#14555499	1	6	yes	—
53	C₂	—	2192	1	no	—
54	C₂	#22577776	1	3	yes	—

	subgroup	G/H
center	C₂²	#17984085
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	C₈²×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#127
prime radical	#17984085	C₂²
hypercenter	C₈²	D₁₃₇
hypocenter	C₁₃₇	#1705
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #22577801	C₂	semidirect
3. C₈²×C₁₃₇	C₂	semidirect
20. C₈×C₁₃₇	SD₁₆	semidirect
27. C₁₃₇	#1705	semidirect
31. C₈²	D₁₃₇	semidirect
48. C₈	#14555499	semidirect

Maximal subgroups

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

subgroup	index
#22577801	2
#22577776	2
C₈²×C₁₃₇	2
#1705	137