Subgroups — #17994213

Order 4736.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14557938	C₂	1	1	yes	maximal, characteristic
2	#14557936	C₂	1	2	yes	maximal
3	#14557912	C₂	1	2	yes	maximal
4	C₈×#64436	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596942	—	2	4	no	—
7	#11924536	C₂²	1	2	yes	—
8	#11924470	—	2	2	no	—
9	#11924469	—	2	4	no	—
10	C₄×#64436	C₂²	1	1	yes	characteristic
11	C₄×#64436	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
13	#107×C₃₇	—	2	2	no	—
14	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₃₇	—	2	2	no	—
16	C₂×Dic₇₄	—	2	2	no	—
17	C₂×C₈×C₃₇	D₄	1	2	yes	—
18	C₄⋊C₄×C₃₇	—	2	2	no	—
19	C₄⋊C₄×C₃₇	—	2	2	no	—
20	C₄²×C₃₇	C₂³	1	1	yes	characteristic
21	#10596942	—	4	8	no	—
22	C₄⋊Dic₃₇	—	2	2	no	—
23	C₄⋊Dic₃₇	—	2	2	no	—
24	C₂×#64436	D₄	1	2	yes	—
25	C₂×#64436	D₄	1	2	yes	—
26	Q₈×C₃₇	—	4	4	no	—
27	C₂×C₄×C₃₇	—	4	2	no	—
28	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₃₇	—	4	2	no	—
32	Dic₇₄	—	4	4	no	—
33	C₈×C₃₇	Q₁₆	1	4	yes	complemented
34	#64436	D₈	1	4	yes	—
35	#64436	Q₁₆	1	4	yes	complemented
36	C₂²×C₃₇	#127	1	1	yes	characteristic
37	C₄×C₃₇	—	8	2	no	—
38	C₄×C₃₇	C₂×D₈	1	2	yes	—
39	C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
40	C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
41	Dic₃₇	—	8	2	no	—
42	#1713	—	37	1	no	maximal, Sylow
43	C₂×C₃₇	#493	1	1	yes	characteristic
44	C₂×C₃₇	#494	1	1	yes	characteristic
45	C₂×C₃₇	#494	1	1	yes	characteristic
46	#500	—	37	2	no	—
47	#500	—	37	1	no	—
48	#500	D₃₇	1	1	yes	characteristic
49	#494	—	37	2	no	—
50	C₈²	—	37	1	no	—
51	C₃₇	#1713	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₁₆	—	74	4	no	—
53	C₄⋊Q₈	—	37	2	no	—
54	C₄⋊Q₈	C₂×D₃₇	1	2	yes	—
55	#107	—	2	2	no	—
56	#107	—	74	4	no	—
57	#107	—	74	2	no	—
58	C₄×C₈	—	37	1	no	—
59	C₄×C₈	—	37	1	no	—
60	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	74	2	no	—
63	Q₁₆	—	148	8	no	—
64	C₂×C₈	—	37	2	no	—
65	C₂×C₈	—	37	2	no	—
66	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	74	2	no	—
70	C₄⋊C₄	—	74	2	no	—
71	C₄²	C₂²×D₃₇	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	148	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	148	2	no	—
76	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
77	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
79	C₈	—	37	4	no	—
80	C₈	—	37	4	no	—
81	C₈	#10596942	1	4	yes	complemented
82	C₂²	#11924532	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	296	2	no	—
85	C₄	D₈×D₃₇	1	2	yes	—
86	C₄	Q₁₆×D₃₇	1	2	yes	—
87	C₄	C₂×#10596942	1	2	yes	—
88	C₂	#14557908	1	1	yes	characteristic
89	C₂	#14557911	1	1	yes	characteristic
90	C₂	#14557912	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11924532
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#500	C₂
socle, prime radical	C₂²×C₃₇	#127
hypercenter	#500	D₃₇
hypocenter	C₃₇	#1713
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
33. C₈×C₃₇	Q₁₆	semidirect
35. #64436	Q₁₆	semidirect
51. C₃₇	#1713	semidirect
81. C₈	#10596942	semidirect

Maximal subgroups

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

subgroup	index
#14557938	2
#14557936	2
#14557912	2
C₈×#64436	2
C₃₇×#500	2
#1713	37