Subgroups — #37560327

Order 23872.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26045851	C₂	1	2	yes	maximal, complemented
2	#26045789	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#13093637	—	2	2	no	—
4	C₂×D₁₄₉₂	—	2	2	no	—
5	C₂×D₁₄₉₂	C₄	1	1	yes	characteristic, complemented
6	C₂×D₁₄₉₂	C₂²	1	1	yes	characteristic
7	C₄²×C₃₇₃	C₄	1	1	yes	characteristic, complemented
8	C₂×#13093637	—	4	2	no	—
9	C₂²×D₃₇₃	—	2	1	no	—
10	C₂²×D₃₇₃	D₄	1	2	yes	—
11	D₁₄₉₂	—	4	1	no	—
12	D₁₄₉₂	—	4	4	no	—
13	D₁₄₉₂	—	4	1	no	—
14	C₂×C₄×C₃₇₃	—	2	1	no	—
15	C₂×C₄×C₃₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃₇₃	C₂²⋊C₄	1	1	yes	characteristic
17	C₂×D₃₇₃	—	2	2	no	—
18	C₂×D₃₇₃	—	4	2	no	—
19	C₂×D₃₇₃	—	4	2	no	—
20	C₂×D₃₇₃	—	4	1	no	—
21	#13093637	—	8	2	no	—
22	C₄×C₃₇₃	—	2	2	no	—
23	C₄×C₃₇₃	—	2	1	no	—
24	C₂×C₃₇₃	—	2	1	no	—
25	C₂×C₃₇₃	#99	1	1	yes	characteristic
26	D₃₇₃	—	4	2	no	—
27	D₃₇₃	—	8	1	no	—
28	C₃₇₃	#353	1	1	yes	characteristic, complemented, Sylow
29	#353	—	373	1	no	maximal, Sylow
30	#127	—	373	1	no	—
31	#99	—	373	2	no	—
32	C₂×D₄	—	373	1	no	—
33	C₂×D₄	—	373	1	no	—
34	C₂×D₄	—	746	2	no	—
35	C₂²⋊C₄	—	746	2	no	—
36	C₄²	#13093637	1	1	yes	characteristic, complemented
37	C₂³	—	373	2	no	—
38	C₂³	—	746	1	no	—
39	D₄	—	1492	1	no	—
40	D₄	—	1492	4	no	—
41	D₄	—	1492	1	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	1492	2	no	—
44	C₂×C₄	C₂×#13093637	1	1	yes	characteristic
45	C₂²	—	1492	2	no	—
46	C₂²	—	1492	2	no	—
47	C₂²	—	1492	1	no	—
48	C₂²	—	746	2	no	—
49	C₂²	C₂²⋊#13093637	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2984	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	2984	1	no	—
55	C₂	—	1492	2	no	—
56	C₂	#26045851	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26045851
commutator	C₂×C₄×C₃₇₃	C₂×C₄
Frattini	C₂×C₄	C₂×#13093637
Fitting	C₄²×C₃₇₃	C₄
socle	C₂×C₃₇₃	#99
prime radical	#26045789	C₂
cosocle	C₂×D₁₄₉₂	C₂²
hypercenter	C₄²	#13093637
hypocenter	C₃₇₃	#353
LCS	G ▷ C₂×C₄×C₃₇₃ ▷ C₂²×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. #26045851	C₂	semidirect
5. C₂×D₁₄₉₂	C₄	semidirect
7. C₄²×C₃₇₃	C₄	semidirect
28. C₃₇₃	#353	semidirect
36. C₄²	#13093637	semidirect

Maximal subgroups

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

subgroup	index
#26045851	2
#26045789	2
#353	373