Subgroups — C₂×#30660372 #45690748

Order 33472.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₀₄₆	C₂	1	2	yes	maximal
2	#30660372	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₅₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₂₃⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₅₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₅₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₁₀₄₆	—	2	8	no	—
8	C₂×Dic₁₀₄₆	C₂²	1	8	yes	—
9	C₂²×Dic₅₂₃	C₂²	1	2	yes	—
10	C₂²×Dic₅₂₃	C₂²	1	1	yes	characteristic
11	C₂²×Dic₅₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₅₂₃	C₂²	1	4	yes	—
13	Dic₅₂₃⋊C₄	C₂²	1	8	yes	—
14	C₄×Dic₅₂₃	C₂²	1	4	yes	—
15	C₂²×C₄×C₅₂₃	C₂²	1	2	yes	—
16	C₂²×C₄×C₅₂₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₅₂₃	C₂²	1	4	yes	—
18	Dic₁₀₄₆	—	2	16	no	—
19	C₂×Dic₅₂₃	—	2	4	no	—
20	C₂×Dic₅₂₃	D₄	1	4	yes	—
21	C₂×Dic₅₂₃	Q₈	1	4	yes	complemented
22	C₂×Dic₅₂₃	C₂³	1	4	yes	—
23	C₂×Dic₅₂₃	C₂³	1	2	yes	—
24	C₂×Dic₅₂₃	C₂³	1	2	yes	—
25	C₂³×C₅₂₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₅₂₃	—	2	4	no	—
27	C₂×C₄×C₅₂₃	Q₈	1	4	yes	complemented
28	C₂×C₄×C₅₂₃	C₂³	1	4	yes	—
29	C₂×C₄×C₅₂₃	C₂³	1	2	yes	—
30	C₂²×C₅₂₃	C₂×D₄	1	2	yes	—
31	C₂²×C₅₂₃	C₂×Q₈	1	2	yes	—
32	C₂²×C₅₂₃	C₂×Q₈	1	2	yes	—
33	C₂²×C₅₂₃	C₂⁴	1	1	yes	characteristic
34	C₄×C₅₂₃	—	2	4	no	—
35	C₄×C₅₂₃	C₂×Q₈	1	4	yes	complemented
36	Dic₅₂₃	—	2	4	no	—
37	Dic₅₂₃	C₂×D₄	1	4	yes	—
38	Dic₅₂₃	C₂×Q₈	1	4	yes	complemented
39	C₂×C₅₂₃	C₄⋊Q₈	1	4	yes	complemented
40	C₂×C₅₂₃	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₅₂₃	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₅₂₃	C₂²×Q₈	1	1	yes	characteristic
43	C₅₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄⋊Q₈	—	523	1	no	maximal, Sylow
45	C₂²×Q₈	—	523	2	no	—
46	C₄⋊Q₈	—	523	8	no	—
47	C₂×C₄⋊C₄	—	523	2	no	—
48	C₂×C₄⋊C₄	—	523	1	no	—
49	C₂×C₄⋊C₄	D₅₂₃	1	1	yes	characteristic
50	C₂×C₄²	—	523	1	no	—
51	C₂×Q₈	—	523	8	no	—
52	C₂×Q₈	—	1046	8	no	—
53	C₂²×C₄	—	523	1	no	—
54	C₂²×C₄	—	523	1	no	—
55	C₂²×C₄	—	523	2	no	—
56	C₂²×C₄	C₂×D₅₂₃	1	2	yes	—
57	C₂²×C₄	C₂×D₅₂₃	1	1	yes	characteristic
58	C₄⋊C₄	—	523	8	no	—
59	C₄⋊C₄	—	523	4	no	—
60	C₄⋊C₄	C₂×D₅₂₃	1	4	yes	—
61	C₄²	—	523	4	no	—
62	C₂³	C₂²×D₅₂₃	1	1	yes	characteristic
63	Q₈	—	1046	16	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	523	4	no	—
66	C₂×C₄	—	523	4	no	—
67	C₂×C₄	—	523	2	no	—
68	C₂×C₄	—	523	2	no	—
69	C₂×C₄	—	523	4	no	—
70	C₂×C₄	—	1046	4	no	—
71	C₂×C₄	Dic₁₀₄₆	1	4	yes	complemented
72	C₂×C₄	C₂²×D₅₂₃	1	4	yes	—
73	C₂×C₄	C₂²×D₅₂₃	1	2	yes	—
74	C₂²	D₄×D₅₂₃	1	2	yes	—
75	C₂²	Q₈×D₅₂₃	1	2	yes	—
76	C₂²	C₂×Dic₁₀₄₆	1	2	yes	—
77	C₂²	C₂³×D₅₂₃	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	523	4	no	—
80	C₄	—	523	4	no	—
81	C₄	—	1046	4	no	—
82	C₄	C₂×Dic₁₀₄₆	1	4	yes	complemented
83	C₂	#30660372	1	4	yes	direct product factor
84	C₂	C₂×D₄×D₅₂₃	1	1	yes	characteristic
85	C₂	C₂×Q₈×D₅₂₃	1	1	yes	characteristic
86	C₂	C₂²×Dic₁₀₄₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂₃
commutator, cosocle	C₂²×C₅₂₃	C₂⁴
Frattini	C₂²	C₂³×D₅₂₃
Fitting	C₂×C₄⋊C₄×C₅₂₃	C₂
socle, prime radical	C₂³×C₅₂₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₅₂₃
hypocenter	C₅₂₃	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₅₂₃ ▷ C₅₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #30660372	C₂	direct (N × H)
21. C₂×Dic₅₂₃	Q₈	semidirect
27. C₂×C₄×C₅₂₃	Q₈	semidirect
35. C₄×C₅₂₃	C₂×Q₈	semidirect
38. Dic₅₂₃	C₂×Q₈	semidirect
39. C₂×C₅₂₃	C₄⋊Q₈	semidirect
43. C₅₂₃	C₂×C₄⋊Q₈	semidirect
71. C₂×C₄	Dic₁₀₄₆	semidirect
82. C₄	C₂×Dic₁₀₄₆	semidirect
83. C₂	#30660372	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₀₄₆	2
#30660372	2
C₂×C₄⋊Dic₅₂₃	2
C₂×Dic₅₂₃⋊C₄	2
C₂×C₄×Dic₅₂₃	2
C₂×C₄⋊C₄×C₅₂₃	2
C₂×C₄⋊Q₈	523

Subgroups — C2×#30660372 #45690748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30660372 #45690748