Subgroups — #37559207

Order 23872.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26045798	C₂	1	1	yes	maximal, characteristic
2	C₂×#20238585	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20238610	C₄	1	1	yes	characteristic
5	#20238594	C₄	1	1	yes	characteristic
6	#20238593	—	2	1	no	—
7	C₈×D₃₇₃	—	2	1	no	—
8	#20238585	—	2	1	no	—
9	C₂×C₈×C₃₇₃	C₂²	1	1	yes	characteristic
10	C₁₆×C₃₇₃	—	2	1	no	—
11	Dic₇₄₆	C₈	1	1	yes	characteristic
12	D₁₄₉₂	C₈	1	1	yes	characteristic
13	C₂²⋊D₃₇₃	—	2	1	no	—
14	C₄×D₃₇₃	—	2	1	no	—
15	#15688412	—	2	1	no	—
16	C₂×C₄×C₃₇₃	C₂×C₄	1	1	yes	characteristic
17	C₈×C₃₇₃	D₄	1	1	yes	characteristic
18	C₈×C₃₇₃	D₄	1	1	yes	characteristic
19	C₂²×C₃₇₃	M₁₆	1	1	yes	characteristic
20	C₂×D₃₇₃	—	2	1	no	—
21	C₄×C₃₇₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₃₇₃	C₂×C₈	1	1	yes	characteristic
23	Dic₃₇₃	—	2	1	no	—
24	C₂×C₃₇₃	—	2	1	no	—
25	C₂×C₃₇₃	C₂²⋊C₈	1	1	yes	characteristic
26	D₃₇₃	—	4	1	no	—
27	C₃₇₃	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	373	1	no	maximal, Sylow
29	#131	—	373	1	no	—
30	M₃₂	D₃₇₃	1	1	yes	characteristic, complemented
31	C₂×C₁₆	—	373	1	no	—
32	#41	—	373	1	no	—
33	M₁₆	—	373	1	no	—
34	M₁₆	—	746	1	no	—
35	C₂×C₈	—	746	1	no	—
36	C₂×C₈	C₂×D₃₇₃	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	746	1	no	—
39	Q₈	—	373	1	no	—
40	D₄	—	373	1	no	—
41	D₄	—	746	1	no	—
42	C₂×C₄	—	746	1	no	—
43	C₂×C₄	C₄×D₃₇₃	1	1	yes	characteristic
44	C₈	—	746	1	no	—
45	C₈	C₂²⋊D₃₇₃	1	1	yes	characteristic
46	C₈	D₁₄₉₂	1	1	yes	characteristic
47	C₂²	—	746	1	no	—
48	C₂²	#20238593	1	1	yes	characteristic
49	C₄	—	746	1	no	—
50	C₄	#20238587	1	1	yes	characteristic
51	C₄	C₈×D₃₇₃	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	1492	1	no	—
54	C₂	#26045704	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₃₇₃
commutator	C₄×C₃₇₃	C₂×C₈
Frattini	C₂×C₈	C₂×D₃₇₃
Fitting	M₃₂×C₃₇₃	C₂
socle	C₂×C₃₇₃	C₂²⋊C₈
prime radical	#20238610	C₄
cosocle	C₂×C₈×C₃₇₃	C₂²
hypercenter	M₃₂	D₃₇₃
hypocenter	C₃₇₃	#350
LCS	G ▷ C₄×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#20238585	C₂	semidirect
3. M₃₂×C₃₇₃	C₂	semidirect
27. C₃₇₃	#350	semidirect
30. M₃₂	D₃₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26045798	2
C₂×#20238585	2
M₃₂×C₃₇₃	2
#350	373