Subgroups — #30660356

Order 16736.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₅₂₃	C₂	1	1	yes	maximal, characteristic
2	Dic₅₂₃⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Dic₅₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₅₂₃)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₅₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₂₃	C₂²	1	1	yes	characteristic
7	C₂²⋊D₅₂₃	—	2	2	no	—
8	C₂×Dic₅₂₃	C₂²	1	2	yes	—
9	C₂×Dic₅₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₅₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅₂₃	—	2	2	no	—
12	C₂×C₄×C₅₂₃	C₂²	1	2	yes	—
13	C₂²×C₅₂₃	—	2	2	no	—
14	C₂²×C₅₂₃	D₄	1	2	yes	—
15	C₂²×C₅₂₃	C₂³	1	1	yes	characteristic
16	C₂×D₅₂₃	—	2	1	no	—
17	C₂×D₅₂₃	—	2	2	no	—
18	C₄×C₅₂₃	—	2	2	no	—
19	Dic₅₂₃	—	2	1	no	—
20	Dic₅₂₃	—	2	2	no	—
21	C₂×C₅₂₃	—	2	2	no	—
22	C₂×C₅₂₃	C₂×D₄	1	1	yes	characteristic
23	C₂×C₅₂₃	#41	1	2	yes	—
24	D₅₂₃	—	4	1	no	—
25	C₅₂₃	#123	1	1	yes	characteristic, complemented, Sylow
26	#123	—	523	1	no	maximal, Sylow
27	C₂×D₄	—	523	1	no	—
28	C₂²×C₄	D₅₂₃	1	1	yes	characteristic, complemented
29	C₄⋊C₄	—	523	2	no	—
30	C₂²⋊C₄	—	523	1	no	—
31	C₂²⋊C₄	—	523	2	no	—
32	C₂³	—	523	1	no	—
33	C₂³	C₂×D₅₂₃	1	1	yes	characteristic
34	D₄	—	1046	2	no	—
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	—	523	2	no	—
37	C₂×C₄	—	523	1	no	—
38	C₂×C₄	C₂×D₅₂₃	1	2	yes	—
39	C₂²	—	2	2	no	—
40	C₂²	—	1046	1	no	—
41	C₂²	—	1046	2	no	—
42	C₂²	C₂²⋊D₅₂₃	1	2	yes	—
43	C₂²	C₂²×D₅₂₃	1	1	yes	characteristic
44	C₄	—	2	2	no	—
45	C₄	—	1046	1	no	—
46	C₄	—	1046	2	no	—
47	C₂	—	2	2	no	—
48	C₂	—	2092	1	no	—
49	C₂	C₂×C₂²⋊D₅₂₃	1	1	yes	characteristic
50	C₂	#22552641	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅₂₃
commutator, socle, cosocle	C₂²×C₅₂₃	C₂³
Fitting	C₂²×C₄×C₅₂₃	C₂
prime radical	C₂×C₂²⋊D₅₂₃	C₂
hypercenter	C₂²×C₄	D₅₂₃
hypocenter	C₅₂₃	#123
LCS	G ▷ C₂²×C₅₂₃ ▷ C₅₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. Dic₅₂₃⋊C₄	C₂	semidirect
3. C₂²⋊Dic₅₂₃	C₂	semidirect
4. (C₂×D₅₂₃)⋊C₄	C₂	semidirect
5. C₂²×C₄×C₅₂₃	C₂	semidirect
25. C₅₂₃	#123	semidirect
28. C₂²×C₄	D₅₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₅₂₃	2
Dic₅₂₃⋊C₄	2
C₂²⋊Dic₅₂₃	2
(C₂×D₅₂₃)⋊C₄	2
C₂²×C₄×C₅₂₃	2
#123	523