Subgroups — #17994205

Order 4736.

168 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	#14557885	C₂	1	2	yes	maximal
3	#14557883	C₂	1	2	yes	maximal
4	C₈×#64436	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#500	C₂	1	1	yes	maximal, characteristic
6	#11924536	C₂²	1	2	yes	—
7	#11924470	—	2	2	no	—
8	#11924460	—	2	4	no	—
9	#11924459	—	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	C₄⋊Dic₃₇	—	2	2	no	—
22	C₄⋊Dic₃₇	—	2	2	no	—
23	C₂×#64436	D₄	1	2	yes	—
24	C₂×#64436	D₄	1	2	yes	—
25	Q₈×C₃₇	—	4	4	no	—
26	C₂×C₄×C₃₇	—	4	2	no	—
27	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₃₇	—	4	2	no	—
31	Dic₇₄	—	4	4	no	—
32	C₈×C₃₇	Q₁₆	1	4	yes	—
33	#64436	—	2	2	no	—
34	#64436	—	2	2	no	—
35	C₂²×C₃₇	#127	1	1	yes	characteristic
36	C₄×C₃₇	—	8	2	no	—
37	C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
38	C₄×C₃₇	#135	1	2	yes	—
39	C₄×C₃₇	#135	1	2	yes	—
40	Dic₃₇	—	8	2	no	—
41	#1710	—	37	1	no	maximal, Sylow
42	C₂×C₃₇	#494	1	1	yes	characteristic
43	C₂×C₃₇	#495	1	1	yes	characteristic
44	C₂×C₃₇	#495	1	1	yes	characteristic
45	#500	—	37	1	no	—
46	#500	D₃₇	1	1	yes	characteristic
47	#487	—	37	2	no	—
48	#487	—	37	2	no	—
49	C₈²	—	37	1	no	—
50	C₃₇	#1710	1	1	yes	characteristic, complemented, Sylow
51	C₄⋊Q₈	—	37	2	no	—
52	C₄⋊Q₈	C₂×D₃₇	1	2	yes	—
53	#107	—	2	2	no	—
54	#107	—	74	2	no	—
55	#103	—	74	4	no	—
56	#103	—	74	4	no	—
57	C₄×C₈	—	37	1	no	—
58	C₄×C₈	—	37	1	no	—
59	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	74	2	no	—
62	C₂×C₈	—	37	2	no	—
63	C₂×C₈	—	37	2	no	—
64	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	74	2	no	—
68	C₄⋊C₄	—	74	2	no	—
69	C₄²	C₂²×D₃₇	1	1	yes	characteristic
70	Q₈	—	4	4	no	—
71	Q₈	—	148	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	148	2	no	—
74	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
75	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
76	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
77	C₈	—	74	2	no	—
78	C₈	—	74	2	no	—
79	C₈	#10596942	1	4	yes	—
80	C₂²	#11924532	1	1	yes	characteristic
81	C₄	—	8	2	no	—
82	C₄	—	296	2	no	—
83	C₄	C₂×#10596942	1	2	yes	—
84	C₄	#11924555	1	2	yes	—
85	C₄	#11924559	1	2	yes	—
86	C₂	#14557912	1	1	yes	characteristic
87	C₂	#14557916	1	1	yes	characteristic
88	C₂	#14557918	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	#1710
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
50. C₃₇	#1710	semidirect

Maximal subgroups

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

subgroup	index
#14557938	2
#14557885	2
#14557883	2
C₈×#64436	2
C₃₇×#500	2
#1710	37