Subgroups — #31780943

Order 17536.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11764040	C₂	1	6	yes	maximal
2	#376×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11764040	C₂²	1	4	yes	—
4	C₄⋊#11764040	—	2	12	no	—
5	C₂×C₄²×C₁₃₇	C₂²	1	3	yes	—
6	#95×C₁₃₇	C₄	1	4	yes	—
7	C₂×#11764040	—	2	12	no	—
8	C₂²×C₄×C₁₃₇	C₂×C₄	1	6	yes	—
9	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
10	C₄²×C₁₃₇	D₄	1	6	yes	—
11	C₄²×C₁₃₇	Q₈	1	6	yes	—
12	#11764040	—	4	4	no	—
13	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₃₇	—	2	12	no	—
15	C₂×C₄×C₁₃₇	C₄⋊C₄	1	12	yes	—
16	C₂×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
17	C₂×C₄×C₁₃₇	C₂×Q₈	1	3	yes	—
18	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	3	yes	—
19	C₂²×C₁₃₇	#131	1	4	yes	—
20	C₄×C₁₃₇	—	2	12	no	—
21	C₄×C₁₃₇	#127	1	1	yes	characteristic
22	C₄×C₁₃₇	C₄⋊Q₈	1	3	yes	—
23	C₂×C₁₃₇	#384	1	1	yes	characteristic
24	C₂×C₁₃₇	#424	1	6	yes	—
25	C₁₃₇	#1843	1	1	yes	characteristic, complemented, Sylow
26	#1843	—	137	1	no	maximal, Sylow
27	C₂×C₄⋊C₈	—	137	6	no	—
28	#376	D₁₃₇	1	1	yes	characteristic
29	C₂²×C₈	—	137	4	no	—
30	C₂×C₄²	C₂×D₁₃₇	1	3	yes	—
31	C₄⋊C₈	—	274	12	no	—
32	#95	Dic₁₃₇	1	4	yes	—
33	C₂²×C₄	C₂×Dic₁₃₇	1	6	yes	—
34	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
35	C₂×C₈	—	274	12	no	—
36	C₄²	D₅₄₈	1	6	yes	—
37	C₄²	Dic₂₇₄	1	6	yes	—
38	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
39	C₂×C₄	—	2	12	no	—
40	C₂×C₄	C₄⋊Dic₁₃₇	1	12	yes	—
41	C₂×C₄	C₂×D₅₄₈	1	3	yes	—
42	C₂×C₄	C₂×Dic₂₇₄	1	3	yes	—
43	C₈	—	548	4	no	—
44	C₂²	C₂×C₄⋊Dic₁₃₇	1	3	yes	—
45	C₂²	#17984095	1	4	yes	—
46	C₄	—	2	12	no	—
47	C₄	#17984085	1	1	yes	characteristic
48	C₄	#17984088	1	3	yes	—
49	C₂	#22577273	1	1	yes	characteristic
50	C₂	#22577430	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₇
commutator, socle, prime radical	C₂³×C₁₃₇	C₂²×C₄
Fitting	#376×C₁₃₇	C₂
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#376	D₁₃₇
hypocenter	C₁₃₇	#1843
LCS	G ▷ C₂³×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
25. C₁₃₇	#1843	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11764040	2
#376×C₁₃₇	2
#1843	137