Subgroups — #45687719

Order 33344.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₂₀₈₄	C₂	1	1	yes	maximal, characteristic
2	#30659639	C₂	1	2	yes	maximal, complemented
3	C₄⋊#16861960	C₂	1	1	yes	maximal, characteristic, complemented
4	#30659628	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀₈₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₅₂₁	—	2	1	no	—
8	C₂×#16861960	C₂²	1	2	yes	—
9	C₄⋊Dic₅₂₁	C₂²	1	1	yes	characteristic
10	#22552341	—	2	1	no	—
11	C₂×Q₈×C₅₂₁	—	2	1	no	—
12	C₄⋊C₄×C₅₂₁	—	2	1	no	—
13	C₄⋊C₄×C₅₂₁	C₂²	1	2	yes	—
14	C₄²×C₅₂₁	C₂²	1	1	yes	characteristic
15	C₂²×D₅₂₁	—	2	1	no	—
16	D₂₀₈₄	—	2	1	no	—
17	D₂₀₈₄	Q₈	1	2	yes	complemented
18	C₂×Dic₅₂₁	—	2	1	no	—
19	C₄×D₅₂₁	—	4	2	no	—
20	#16861960	—	2	2	no	—
21	Q₈×C₅₂₁	—	4	2	no	—
22	C₂×C₄×C₅₂₁	—	2	2	no	—
23	C₂×C₄×C₅₂₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅₂₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
26	C₂²×C₅₂₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₅₂₁	—	2	2	no	—
28	C₂×D₅₂₁	—	4	1	no	—
29	C₂×D₅₂₁	—	4	1	no	—
30	C₄×C₅₂₁	—	2	1	no	—
31	C₄×C₅₂₁	—	4	2	no	—
32	C₄×C₅₂₁	SD₁₆	1	2	yes	—
33	C₄×C₅₂₁	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₅₂₁	#41	1	1	yes	characteristic
35	Dic₅₂₁	—	4	1	no	—
36	C₂×C₅₂₁	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₅₂₁	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₅₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₅₂₁	—	4	2	no	—
40	C₅₂₁	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	521	1	no	maximal, Sylow
42	C₄⋊Q₈	D₅₂₁	1	1	yes	characteristic, complemented
43	C₄×D₄	—	521	1	no	—
44	#106	—	521	2	no	—
45	C₄⋊C₈	—	521	1	no	—
46	#102	—	521	2	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	521	1	no	—
49	C₂²×C₄	—	1042	1	no	—
50	C₂×C₈	—	521	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	521	1	no	—
53	C₄⋊C₄	C₂×D₅₂₁	1	2	yes	—
54	C₂²⋊C₄	—	1042	1	no	—
55	C₄²	C₂×D₅₂₁	1	1	yes	characteristic
56	C₂³	—	1042	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	521	2	no	—
59	D₄	—	1042	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	1042	1	no	—
62	C₂×C₄	—	2084	2	no	—
63	C₂×C₄	C₂²⋊D₅₂₁	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₅₂₁	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
66	C₈	—	1042	2	no	—
67	C₂²	—	1042	2	no	—
68	C₂²	—	2084	1	no	—
69	C₂²	—	2084	1	no	—
70	C₂²	C₂×C₂²⋊D₅₂₁	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	2084	1	no	—
74	C₄	#22552351	1	2	yes	—
75	C₄	Q₈×D₅₂₁	1	1	yes	characteristic
76	C₄	#22552365	1	1	yes	characteristic
77	C₂	—	2084	2	no	—
78	C₂	#30659684	1	1	yes	characteristic
79	C₂	C₂×#22552351	1	1	yes	characteristic
80	C₂	#30659734	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₂₁
commutator, cosocle	C₂×C₄×C₅₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₅₂₁
Fitting	C₄⋊Q₈×C₅₂₁	C₂
socle	C₂²×C₅₂₁	C₂×D₄
prime radical	C₂×D₂₀₈₄	C₂²
hypercenter	C₄⋊Q₈	D₅₂₁
hypocenter	C₅₂₁	#476
LCS	G ▷ C₂×C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #30659639	C₂	semidirect
3. C₄⋊#16861960	C₂	semidirect
5. C₄⋊Q₈×C₅₂₁	C₂	semidirect
17. D₂₀₈₄	Q₈	semidirect
40. C₅₂₁	#476	semidirect
42. C₄⋊Q₈	D₅₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₂₀₈₄	2
#30659639	2
C₄⋊#16861960	2
#30659628	2
C₄⋊Q₈×C₅₂₁	2
#476	521