Subgroups — #17996498

Order 4736.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₇₄	C₂	1	1	yes	maximal, characteristic
2	#14557351	C₂	1	2	yes	maximal
3	C₄×#10596925	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₃₇	C₂	1	2	yes	maximal
5	C₃₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₇₄	C₄	1	1	yes	characteristic
7	C₂×#10596925	C₂²	1	1	yes	characteristic
8	C₂×#10596925	C₂²	1	1	yes	characteristic
9	C₄⋊#64436	C₄	1	1	yes	characteristic
10	#11924465	C₄	1	2	yes	—
11	C₈×Dic₃₇	C₂²	1	2	yes	—
12	C₂×C₁₆×C₃₇	C₂²	1	2	yes	—
13	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₇₄	C₈	1	1	yes	characteristic
15	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
17	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₃₇	C₈	1	1	yes	characteristic
19	#10596936	C₈	1	2	yes	—
20	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
21	C₂×#64436	C₂×C₄	1	2	yes	—
22	C₁₆×C₃₇	—	2	2	no	—
23	#10596925	—	2	1	no	—
24	#10596925	Q₈	1	2	yes	complemented
25	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
29	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
30	Dic₇₄	C₁₆	1	4	yes	complemented
31	C₈×C₃₇	—	2	1	no	—
32	C₈×C₃₇	C₂×Q₈	1	1	yes	characteristic
33	C₈×C₃₇	#41	1	1	yes	characteristic
34	#64436	—	2	2	no	—
35	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₃₇	—	2	1	no	—
37	C₄×C₃₇	C₂×C₁₆	1	2	yes	—
38	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₃₇	#131	1	1	yes	characteristic
40	Dic₃₇	—	2	2	no	—
41	Dic₃₇	C₂×C₁₆	1	4	yes	—
42	Q₈×C₁₆	—	37	1	no	maximal, Sylow
43	C₂×C₃₇	C₈×Q₈	1	1	yes	characteristic
44	C₂×C₃₇	C₂²×C₁₆	1	1	yes	characteristic
45	C₂×C₃₇	#504	1	1	yes	characteristic
46	C₈×Q₈	—	37	1	no	—
47	C₄⋊C₁₆	—	37	2	no	—
48	C₄⋊C₁₆	D₃₇	1	1	yes	characteristic
49	C₄×C₁₆	—	37	2	no	—
50	C₄×C₁₆	—	37	1	no	—
51	C₃₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
52	C₄×Q₈	—	37	1	no	—
53	C₂×C₁₆	—	37	1	no	—
54	C₂×C₁₆	—	37	1	no	—
55	C₂×C₁₆	C₂×D₃₇	1	2	yes	—
56	C₄⋊C₈	—	37	2	no	—
57	C₄⋊C₈	—	37	1	no	—
58	C₄×C₈	—	37	2	no	—
59	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
60	C₂×Q₈	—	37	1	no	—
61	C₂×C₈	—	37	2	no	—
62	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
64	C₄⋊C₄	—	37	2	no	—
65	C₄⋊C₄	—	37	1	no	—
66	C₄²	—	37	2	no	—
67	C₄²	C₄×D₃₇	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	37	2	no	—
70	C₁₆	—	74	1	no	—
71	Q₈	—	37	4	no	—
72	C₂×C₄	—	37	2	no	—
73	C₂×C₄	—	37	2	no	—
74	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
75	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	74	2	no	—
79	C₈	#10596964	1	1	yes	characteristic
80	C₈	Q₈×D₃₇	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	37	4	no	—
84	C₄	—	74	2	no	—
85	C₄	C₁₆×D₃₇	1	2	yes	—
86	C₄	#11924501	1	1	yes	characteristic
87	C₄	#11924544	1	1	yes	characteristic
88	C₂	#14557657	1	1	yes	characteristic
89	C₂	C₂×C₁₆×D₃₇	1	1	yes	characteristic
90	C₂	#14557951	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃₇
commutator	C₂×C₃₇	C₂²×C₁₆
Fitting	C₃₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃₇	C₂²×C₈
cosocle	C₂×C₈×C₃₇	C₂³
hypercenter	C₄⋊C₁₆	D₃₇
hypocenter	C₃₇	Q₈×C₁₆
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
24. #10596925	Q₈	semidirect
30. Dic₇₄	C₁₆	semidirect
51. C₃₇	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₇₄	2
#14557351	2
C₄×#10596925	2
C₁₆×Dic₃₇	2
C₃₇×C₄⋊C₁₆	2
Q₈×C₁₆	37