Subgroups — #18010993

Order 4736.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×#3866	C₂	1	2	yes	maximal
2	#14558460	C₂	1	1	yes	maximal, characteristic
3	#14558446	C₂	1	2	yes	maximal
4	#14558420	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×D₃₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×#3866	—	2	2	no	—
7	D₃₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#11924605	C₂²	1	2	yes	—
9	#11924599	—	2	2	no	—
10	C₈×#3866	C₂²	1	1	yes	characteristic
11	Q₁₆×D₃₇	C₂²	1	2	yes	—
12	Dic₂₉₆	C₄	1	1	yes	characteristic, complemented
13	#11924482	C₄	1	2	yes	complemented
14	C₁₆×D₃₇	C₂²	1	1	yes	characteristic
15	Q₃₂×C₃₇	C₄	1	1	yes	characteristic, complemented
16	Q₈×D₃₇	—	2	2	no	—
17	C₄⋊#3866	—	2	2	no	—
18	C₄⋊#3866	—	4	2	no	—
19	C₄×#3866	—	4	2	no	—
20	C₄×#3866	D₄	1	1	yes	characteristic
21	D₃₇⋊C₈	D₄	1	1	yes	characteristic
22	Q₁₆×C₃₇	C₂×C₄	1	2	yes	—
23	#10596942	—	2	2	no	—
24	Dic₁₄₈	C₂×C₄	1	2	yes	—
25	C₈×D₃₇	C₂³	1	1	yes	characteristic
26	#10596927	—	2	1	no	—
27	C₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
28	#10596925	C₂×C₄	1	1	yes	characteristic
29	C₂×#3866	—	4	2	no	—
30	C₂×#3866	D₈	1	2	yes	—
31	Q₈×C₃₇	—	2	2	no	—
32	C₄×D₃₇	—	4	2	no	—
33	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
34	Dic₇₄	—	2	2	no	—
35	Dic₇₄	—	4	2	no	—
36	#64438	—	2	1	no	—
37	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
38	#64436	#41	1	1	yes	characteristic
39	C₂×D₃₇	C₂×D₈	1	1	yes	characteristic
40	#3866	—	2	2	no	—
41	#3866	—	8	2	no	—
42	C₄×C₃₇	—	4	2	no	—
43	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
44	Dic₃₇	—	4	2	no	—
45	Dic₃₇	#135	1	1	yes	characteristic
46	#2176	—	37	1	no	maximal, Sylow
47	C₂×C₃₇	C₄×D₈	1	1	yes	characteristic
48	D₃₇	#510	1	2	yes	—
49	C₂×Q₃₂	—	37	1	no	—
50	C₄×Q₁₆	—	37	2	no	—
51	#367	—	37	1	no	—
52	#358	—	37	2	no	—
53	#346	—	37	1	no	—
54	C₃₇	#2176	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	37	2	no	—
56	C₄×Q₈	—	74	2	no	—
57	Q₃₂	—	37	2	no	—
58	Q₃₂	—	37	1	no	—
59	Q₃₂	#3866	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	37	1	no	—
61	C₂×C₁₆	—	37	1	no	—
62	#107	—	37	2	no	—
63	#103	—	74	2	no	—
64	C₄×C₈	—	37	1	no	—
65	C₂×Q₈	—	74	2	no	—
66	Q₁₆	—	37	2	no	—
67	Q₁₆	—	74	2	no	—
68	Q₁₆	C₂×#3866	1	2	yes	—
69	C₂×C₈	—	37	1	no	—
70	C₂×C₈	—	37	1	no	—
71	C₄⋊C₄	—	74	2	no	—
72	C₄⋊C₄	—	148	2	no	—
73	C₄²	—	37	1	no	—
74	C₄²	—	148	2	no	—
75	C₁₆	—	37	1	no	—
76	C₁₆	—	74	1	no	—
77	C₁₆	C₂×#3866	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	74	2	no	—
80	Q₈	—	148	2	no	—
81	C₂×C₄	—	37	1	no	—
82	C₂×C₄	—	37	2	no	—
83	C₂×C₄	—	148	2	no	—
84	C₂×C₄	—	148	2	no	—
85	C₈	—	37	1	no	—
86	C₈	—	74	1	no	—
87	C₈	C₂²×#3866	1	1	yes	characteristic
88	C₂²	—	37	1	no	—
89	C₄	—	4	2	no	—
90	C₄	—	37	1	no	—
91	C₄	—	296	2	no	—
92	C₄	—	74	2	no	—
93	C₄	—	148	2	no	—
94	C₄	D₄×#3866	1	1	yes	characteristic
95	C₂	—	37	2	no	—
96	C₂	D₈×#3866	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#3866
commutator	C₈×C₃₇	C₂²×C₄
Frattini	C₈	C₂²×#3866
Fitting	Q₃₂×C₃₇	C₄
socle	C₂×C₃₇	C₄×D₈
prime radical	C₂×D₃₇	C₂×D₈
cosocle	C₈×D₃₇	C₂³
hypercenter	Q₃₂	#3866
hypocenter	C₃₇	#2176
LCS	G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
12. Dic₂₉₆	C₄	semidirect
13. #11924482	C₄	semidirect
15. Q₃₂×C₃₇	C₄	semidirect
54. C₃₇	#2176	semidirect
59. Q₃₂	#3866	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×#3866	2
#14558460	2
#14558446	2
#14558420	2
Q₃₂×D₃₇	2
#2176	37