Subgroups — #31780930

Order 17536.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577427	C₂	1	2	yes	maximal
2	#22577168	C₂	1	4	yes	maximal
3	C₂×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555494	C₂²	1	4	yes	—
5	C₄⋊#11764040	—	2	4	no	—
6	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
8	#97×C₁₃₇	C₄	1	4	yes	—
9	#14555494	—	2	8	no	—
10	C₂×#11764040	—	2	4	no	—
11	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
12	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
13	C₂×C₈×C₁₃₇	—	2	4	no	—
14	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
15	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
16	C₄²×C₁₃₇	D₄	1	2	yes	—
17	#11764040	—	4	4	no	—
18	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
19	C₂×C₄×C₁₃₇	—	4	2	no	—
20	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
21	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	#41	1	4	yes	—
24	C₈×C₁₃₇	—	2	4	no	—
25	C₂²×C₁₃₇	—	2	2	no	—
26	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₃₇	#125	1	2	yes	—
28	C₄×C₁₃₇	—	2	4	no	—
29	C₄×C₁₃₇	#117	1	2	yes	—
30	C₄×C₁₃₇	#124	1	2	yes	—
31	C₂×C₁₃₇	—	2	2	no	—
32	C₂×C₁₃₇	#382	1	1	yes	characteristic
33	C₂×C₁₃₇	#430	1	2	yes	—
34	C₁₃₇	#1838	1	1	yes	characteristic, complemented, Sylow
35	#1838	—	137	1	no	maximal, Sylow
36	#423	—	137	2	no	—
37	C₂×#97	D₁₃₇	1	1	yes	characteristic
38	#341	—	137	4	no	—
39	C₂×M₁₆	—	137	4	no	—
40	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
41	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
42	C₄⋊C₈	—	274	4	no	—
43	#97	Dic₁₃₇	1	4	yes	—
44	C₂²×C₄	Dic₂₇₄	1	2	yes	—
45	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
46	M₁₆	—	274	8	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	274	4	no	—
49	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
50	C₄²	C₂×Dic₁₃₇	1	2	yes	—
51	C₄²	D₅₄₈	1	2	yes	—
52	C₂³	C₂×Dic₂₇₄	1	1	yes	characteristic
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	C₄⋊Dic₁₃₇	1	4	yes	—
55	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
56	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
57	C₂×C₄	#14555510	1	4	yes	—
58	C₈	—	2	4	no	—
59	C₈	—	548	4	no	—
60	C₂²	—	2	2	no	—
61	C₂²	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
62	C₂²	#17984080	1	2	yes	—
63	C₄	—	2	4	no	—
64	C₄	#17984047	1	2	yes	—
65	C₄	#17984076	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	#22577269	1	1	yes	characteristic
68	C₂	#22577452	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×#97×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×#97	D₁₃₇
hypocenter	C₁₃₇	#1838
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
34. C₁₃₇	#1838	semidirect

Maximal subgroups

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

subgroup	index
#22577427	2
#22577168	2
C₂×#97×C₁₃₇	2
#1838	137