Subgroups — #37559801

Order 23872.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20238600	C₂	1	4	yes	maximal
2	#26045792	C₂	1	1	yes	maximal, characteristic
3	C₄×#15688412	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₃₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₇₄₆	C₂²	1	2	yes	—
6	#20238600	—	2	8	no	—
7	C₂×#15688412	C₂²	1	2	yes	—
8	C₄⋊Dic₃₇₃	—	2	2	no	—
9	C₂×Q₈×C₃₇₃	C₂²	1	2	yes	—
10	C₄⋊C₄×C₃₇₃	—	2	2	no	—
11	C₄²×C₃₇₃	C₂²	1	1	yes	characteristic
12	Dic₇₄₆	—	2	4	no	—
13	C₂×Dic₃₇₃	—	2	2	no	—
14	#15688412	D₄	1	4	yes	—
15	Q₈×C₃₇₃	—	2	4	no	—
16	C₂×C₄×C₃₇₃	—	2	2	no	—
17	C₂×C₄×C₃₇₃	D₄	1	2	yes	—
18	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
19	C₂²×C₃₇₃	C₂×D₄	1	1	yes	characteristic
20	C₄×C₃₇₃	—	4	2	no	—
21	C₄×C₃₇₃	Q₁₆	1	4	yes	complemented
22	C₄×C₃₇₃	C₂×D₄	1	2	yes	—
23	Dic₃₇₃	—	4	2	no	—
24	C₂×C₃₇₃	#127	1	1	yes	characteristic
25	C₂×C₃₇₃	C₂×Q₁₆	1	2	yes	—
26	C₃₇₃	#494	1	1	yes	characteristic, complemented, Sylow
27	#494	—	373	1	no	maximal, Sylow
28	C₂×Q₁₆	—	373	4	no	—
29	C₄⋊Q₈	—	373	1	no	—
30	C₄⋊Q₈	D₃₇₃	1	1	yes	characteristic
31	C₄×C₈	—	373	1	no	—
32	C₂×Q₈	—	373	2	no	—
33	C₂×Q₈	C₂×D₃₇₃	1	2	yes	—
34	Q₁₆	—	746	8	no	—
35	C₂×C₈	—	373	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	746	2	no	—
38	C₄²	C₂×D₃₇₃	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	746	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	746	2	no	—
43	C₂×C₄	C₂²⋊D₃₇₃	1	2	yes	—
44	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
45	C₈	—	373	4	no	—
46	C₂²	C₂×C₂²⋊D₃₇₃	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	1492	2	no	—
49	C₄	#20238600	1	4	yes	complemented
50	C₄	D₄×D₃₇₃	1	2	yes	—
51	C₂	#26045788	1	1	yes	characteristic
52	C₂	C₂×#20238600	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₇₃
commutator, cosocle	C₂×C₄×C₃₇₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₇₃
Fitting	C₄⋊Q₈×C₃₇₃	C₂
socle, prime radical	C₂²×C₃₇₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₃₇₃
hypocenter	C₃₇₃	#494
LCS	G ▷ C₂×C₄×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×C₃₇₃	Q₁₆	semidirect
26. C₃₇₃	#494	semidirect
49. C₄	#20238600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20238600	2
#26045792	2
C₄×#15688412	2
C₄⋊Q₈×C₃₇₃	2
#494	373