Subgroups — #17996505

Order 4736.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#10596925	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#10596925	C₂	1	2	yes	maximal
3	#14557309	C₂	1	2	yes	maximal
4	C₄×#10596925	C₂	1	1	yes	maximal, characteristic
5	C₈×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596925	C₂²	1	1	yes	characteristic
7	C₂×#10596925	C₂²	1	2	yes	—
8	C₂×#10596925	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₃₇	C₄	1	1	yes	characteristic
10	C₄⋊C₈×C₃₇	C₄	1	2	yes	—
11	C₄⋊C₈×C₃₇	C₄	1	1	yes	characteristic
12	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₃₇	C₂²	1	2	yes	—
14	C₂×Q₈×C₃₇	C₈	1	1	yes	characteristic
15	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₃₇	C₈	1	2	yes	—
19	C₄⋊C₄×C₃₇	C₈	1	1	yes	characteristic
20	C₄²×C₃₇	C₂×C₄	1	2	yes	—
21	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
22	#10596925	—	2	1	no	—
23	#10596925	—	2	2	no	—
24	#10596925	Q₈	1	2	yes	complemented
25	Q₈×C₃₇	—	2	2	no	—
26	C₂×C₄×C₃₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₃₇	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₃₇	—	2	1	no	—
32	C₈×C₃₇	—	2	2	no	—
33	C₈×C₃₇	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₃₇	#41	1	1	yes	characteristic
35	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₃₇	—	2	2	no	—
37	C₄×C₃₇	—	2	2	no	—
38	C₄×C₃₇	—	2	1	no	—
39	C₄×C₃₇	M₃₂	1	2	yes	—
40	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
41	C₄×C₃₇	#131	1	1	yes	characteristic
42	#2183	—	37	1	no	maximal, Sylow
43	C₂×C₃₇	C₈×Q₈	1	1	yes	characteristic
44	C₂×C₃₇	C₂×M₃₂	1	1	yes	characteristic
45	C₂×C₃₇	#504	1	1	yes	characteristic
46	C₈×Q₈	D₃₇	1	1	yes	characteristic
47	C₄⋊C₁₆	—	37	1	no	—
48	C₄⋊C₁₆	—	37	2	no	—
49	#346	—	37	2	no	—
50	C₄×C₁₆	—	37	1	no	—
51	C₃₇	#2183	1	1	yes	characteristic, complemented, Sylow
52	C₄×Q₈	Dic₃₇	1	1	yes	characteristic
53	C₂×C₁₆	—	37	1	no	—
54	C₂×C₁₆	—	37	1	no	—
55	C₂×C₁₆	—	37	2	no	—
56	C₄⋊C₈	Dic₃₇	1	2	yes	—
57	C₄⋊C₈	Dic₃₇	1	1	yes	characteristic
58	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
59	C₄×C₈	C₂×D₃₇	1	2	yes	—
60	C₂×Q₈	#64436	1	1	yes	characteristic
61	C₂×C₈	C₂×Dic₃₇	1	1	yes	characteristic
62	C₂×C₈	C₂×Dic₃₇	1	2	yes	—
63	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
64	C₄⋊C₄	#64436	1	2	yes	—
65	C₄⋊C₄	#64436	1	1	yes	characteristic
66	C₄²	C₂×Dic₃₇	1	2	yes	—
67	C₄²	C₂×Dic₃₇	1	1	yes	characteristic
68	C₁₆	—	37	2	no	—
69	C₁₆	—	74	1	no	—
70	C₁₆	—	74	2	no	—
71	Q₈	—	2	2	no	—
72	C₂×C₄	C₂×#64436	1	2	yes	—
73	C₂×C₄	C₂×#64436	1	1	yes	characteristic
74	C₂×C₄	C₂×#64436	1	2	yes	—
75	C₂×C₄	C₂×#64436	1	1	yes	characteristic
76	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	2	no	—
79	C₈	Q₈×D₃₇	1	1	yes	characteristic
80	C₈	#10596966	1	1	yes	characteristic
81	C₂²	C₂²×#64436	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	#11924476	1	2	yes	—
86	C₄	Q₈×Dic₃₇	1	1	yes	characteristic
87	C₄	#11924543	1	1	yes	characteristic
88	C₂	Q₈×#64436	1	1	yes	characteristic
89	C₂	C₂×#11924476	1	1	yes	characteristic
90	C₂	#14557950	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃₇
commutator, socle, prime radical	C₂²×C₃₇	C₂²×C₈
Fitting	C₈×Q₈×C₃₇	C₂
cosocle	C₂×C₈×C₃₇	C₂³
hypercenter	C₈×Q₈	D₃₇
hypocenter	C₃₇	#2183
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
24. #10596925	Q₈	semidirect
51. C₃₇	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#10596925	2
#14557309	2
C₄×#10596925	2
C₈×Q₈×C₃₇	2
#2183	37