Subgroups — #30660366

Order 16736.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₅₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	Dic₅₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₅₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊Dic₅₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²⋊Dic₅₂₃	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₅₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×C₅₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₂₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₂₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₅₂₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₅₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₅₂₃	—	2	1	no	—
16	C₂²×C₅₂₃	—	2	1	no	—
17	C₂²×C₅₂₃	—	2	1	no	—
18	C₂²×C₅₂₃	C₂³	1	1	yes	characteristic
19	C₄×C₅₂₃	—	2	1	no	—
20	C₄×C₅₂₃	—	2	1	no	—
21	Dic₅₂₃	—	2	1	no	—
22	Dic₅₂₃	—	2	1	no	—
23	Dic₅₂₃	—	2	1	no	—
24	Dic₅₂₃	—	2	1	no	—
25	C₂×C₅₂₃	—	4	1	no	—
26	C₂×C₅₂₃	#41	1	1	yes	characteristic
27	C₂×C₅₂₃	#41	1	1	yes	characteristic
28	C₂×C₅₂₃	#41	1	1	yes	characteristic
29	C₅₂₃	#126	1	1	yes	characteristic, complemented, Sylow
30	#126	—	523	1	no	maximal, Sylow
31	C₄⋊C₄	—	523	1	no	—
32	C₄⋊C₄	—	523	1	no	—
33	C₄⋊C₄	—	523	1	no	—
34	C₂²⋊C₄	—	523	1	no	—
35	C₂²⋊C₄	—	523	1	no	—
36	C₂²⋊C₄	D₅₂₃	1	1	yes	characteristic
37	C₄²	—	523	1	no	—
38	C₂³	C₂×D₅₂₃	1	1	yes	characteristic
39	C₂×C₄	—	523	1	no	—
40	C₂×C₄	—	523	1	no	—
41	C₂×C₄	—	523	1	no	—
42	C₂×C₄	—	523	1	no	—
43	C₂×C₄	C₂×D₅₂₃	1	1	yes	characteristic
44	C₂×C₄	C₂×D₅₂₃	1	1	yes	characteristic
45	C₂²	—	2	1	no	—
46	C₂²	—	2	1	no	—
47	C₂²	—	2	1	no	—
48	C₂²	C₂²×D₅₂₃	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	1046	1	no	—
52	C₄	—	1046	1	no	—
53	C₄	—	1046	1	no	—
54	C₄	—	1046	1	no	—
55	C₂	—	4	1	no	—
56	C₂	#22552640	1	1	yes	characteristic
57	C₂	#22552640	1	1	yes	characteristic
58	C₂	#22552641	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅₂₃
commutator, socle, cosocle	C₂²×C₅₂₃	C₂³
Fitting	C₂²⋊C₄×C₅₂₃	C₂
prime radical	C₂³×C₅₂₃	C₂²
hypercenter	C₂²⋊C₄	D₅₂₃
hypocenter	C₅₂₃	#126
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
1. C₄⋊Dic₅₂₃	C₂	semidirect
2. Dic₅₂₃⋊C₄	C₂	semidirect
3. Dic₅₂₃⋊C₄	C₂	semidirect
6. C₄×Dic₅₂₃	C₂	semidirect
29. C₅₂₃	#126	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Dic₅₂₃	2
Dic₅₂₃⋊C₄	2
C₂²⋊Dic₅₂₃	2
C₄×Dic₅₂₃	2
C₂²⋊C₄×C₅₂₃	2
#126	523