Subgroups — #14558338

Order 2368.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924582	C₂	1	2	yes	maximal, complemented
2	#11924564	C₂	1	6	yes	maximal, complemented
3	#11924558	C₂	1	3	yes	maximal, complemented
4	C₂×D₂₉₆	C₂	1	3	yes	maximal, complemented
5	#131×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₃₇	C₂²	1	1	yes	characteristic
7	#10596966	—	2	2	no	—
8	D₄×D₃₇	—	2	6	no	—
9	#10596962	C₂²	1	6	yes	complemented
10	C₂×D₁₄₈	C₂²	1	6	yes	complemented
11	M₁₆×C₃₇	C₂²	1	3	yes	complemented
12	C₂×C₈×C₃₇	C₂²	1	3	yes	complemented
13	Dic₁₄₈	C₂²	1	1	yes	characteristic, complemented
14	D₂₉₆	C₂²	1	3	yes	complemented
15	D₂₉₆	C₂²	1	6	yes	complemented
16	#10596930	C₂²	1	6	yes	complemented
17	C₂²×D₃₇	—	2	6	no	—
18	Q₈×C₃₇	D₄	1	1	yes	characteristic
19	D₄×C₃₇	D₄	1	3	yes	—
20	C₂×C₄×C₃₇	C₂³	1	3	yes	—
21	C₂²⋊D₃₇	—	2	6	no	—
22	D₁₄₈	—	2	6	no	—
23	D₁₄₈	C₂³	1	6	yes	—
24	C₄×D₃₇	—	2	6	no	—
25	Dic₇₄	C₂³	1	2	yes	—
26	C₈×C₃₇	C₂³	1	3	yes	—
27	C₈×C₃₇	C₂³	1	1	yes	characteristic
28	C₂²×C₃₇	C₂×D₄	1	3	yes	—
29	C₂×D₃₇	—	2	6	no	—
30	C₂×D₃₇	—	8	6	no	—
31	C₄×C₃₇	C₂×D₄	1	3	yes	—
32	C₄×C₃₇	C₂⁴	1	1	yes	characteristic
33	Dic₃₇	—	2	2	no	—
34	C₂×C₃₇	—	2	3	no	—
35	C₂×C₃₇	C₂²×D₄	1	1	yes	characteristic
36	D₃₇	—	4	6	no	—
37	#576	—	37	1	no	maximal, Sylow
38	C₃₇	#576	1	1	yes	characteristic, complemented, Sylow
39	H(5,𝔽₂)	—	37	2	no	—
40	AGL(1,ℤ/8)	—	37	6	no	—
41	#135	—	37	3	no	—
42	C₂×D₈	—	37	3	no	—
43	#131	D₃₇	1	1	yes	characteristic, complemented
44	#41	—	37	6	no	—
45	#41	—	74	2	no	—
46	#41	C₂×D₃₇	1	1	yes	characteristic
47	C₂×D₄	—	37	6	no	—
48	C₂×D₄	—	74	6	no	—
49	Q₁₆	—	37	1	no	—
50	SD₁₆	—	37	6	no	—
51	D₈	—	37	3	no	—
52	D₈	—	37	6	no	—
53	M₁₆	C₂×D₃₇	1	3	yes	complemented
54	C₂×C₈	C₂×D₃₇	1	3	yes	complemented
55	C₂³	—	74	6	no	—
56	Q₈	—	37	2	no	—
57	Q₈	D₁₄₈	1	1	yes	characteristic
58	D₄	—	37	6	no	—
59	D₄	—	74	6	no	—
60	D₄	—	74	6	no	—
61	D₄	D₁₄₈	1	3	yes	—
62	C₂×C₄	—	74	6	no	—
63	C₂×C₄	C₂²×D₃₇	1	3	yes	—
64	C₈	C₂²×D₃₇	1	1	yes	characteristic
65	C₈	C₂²×D₃₇	1	3	yes	—
66	C₂²	—	296	6	no	—
67	C₂²	—	74	6	no	—
68	C₂²	C₂×D₁₄₈	1	3	yes	—
69	C₄	—	74	2	no	—
70	C₄	C₂×D₁₄₈	1	3	yes	—
71	C₄	C₂³×D₃₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	148	6	no	—
74	C₂	C₂²×D₁₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₄₈
commutator, cosocle	C₄×C₃₇	C₂⁴
Frattini	C₄	C₂³×D₃₇
Fitting	#131×C₃₇	C₂
socle	C₂×C₃₇	C₂²×D₄
hypercenter	#131	D₃₇
hypocenter	C₃₇	#576
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #11924582	C₂	semidirect
2. #11924564	C₂	semidirect
3. #11924558	C₂	semidirect
4. C₂×D₂₉₆	C₂	semidirect
5. #131×C₃₇	C₂	semidirect
9. #10596962	C₂²	semidirect
10. C₂×D₁₄₈	C₂²	semidirect
11. M₁₆×C₃₇	C₂²	semidirect
12. C₂×C₈×C₃₇	C₂²	semidirect
13. Dic₁₄₈	C₂²	semidirect
14. D₂₉₆	C₂²	semidirect
15. D₂₉₆	C₂²	semidirect
16. #10596930	C₂²	semidirect
38. C₃₇	#576	semidirect
43. #131	D₃₇	semidirect
53. M₁₆	C₂×D₃₇	semidirect
54. C₂×C₈	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#11924582	2
#11924564	2
#11924558	2
C₂×D₂₉₆	2
#131×C₃₇	2
#576	37