Subgroups — C₂×#27165736 #38786657

Order 25664.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₈₀₂	C₂	1	2	yes	maximal
2	#27165736	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₄₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#20264345	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₄₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₄₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₈₀₂	—	2	8	no	—
8	C₂×Dic₈₀₂	C₂²	1	8	yes	—
9	C₂²×Dic₄₀₁	C₂²	1	2	yes	—
10	C₂²×Dic₄₀₁	C₂²	1	1	yes	characteristic
11	C₂²×Dic₄₀₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₀₁	C₂²	1	4	yes	—
13	#20264345	C₂²	1	8	yes	—
14	C₄×Dic₄₀₁	C₂²	1	4	yes	—
15	C₂²×C₄×C₄₀₁	C₂²	1	2	yes	—
16	C₂²×C₄×C₄₀₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₀₁	C₂²	1	4	yes	—
18	Dic₈₀₂	—	2	16	no	—
19	C₂×Dic₄₀₁	—	2	4	no	—
20	C₂×Dic₄₀₁	D₄	1	4	yes	—
21	C₂×Dic₄₀₁	Q₈	1	4	yes	complemented
22	C₂×Dic₄₀₁	C₂³	1	4	yes	—
23	C₂×Dic₄₀₁	C₂³	1	2	yes	—
24	C₂×Dic₄₀₁	C₂³	1	2	yes	—
25	C₂³×C₄₀₁	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₄₀₁	—	2	4	no	—
27	C₂×C₄×C₄₀₁	Q₈	1	4	yes	complemented
28	C₂×C₄×C₄₀₁	C₂³	1	4	yes	—
29	C₂×C₄×C₄₀₁	C₂³	1	2	yes	—
30	C₂²×C₄₀₁	C₂×D₄	1	2	yes	—
31	C₂²×C₄₀₁	C₂×Q₈	1	2	yes	—
32	C₂²×C₄₀₁	C₂×Q₈	1	2	yes	—
33	C₂²×C₄₀₁	C₂⁴	1	1	yes	characteristic
34	C₄×C₄₀₁	—	2	4	no	—
35	C₄×C₄₀₁	C₂×Q₈	1	4	yes	complemented
36	Dic₄₀₁	—	2	4	no	—
37	Dic₄₀₁	C₂×D₄	1	4	yes	—
38	Dic₄₀₁	C₂×Q₈	1	4	yes	complemented
39	C₂×C₄₀₁	C₄⋊Q₈	1	4	yes	complemented
40	C₂×C₄₀₁	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₄₀₁	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₄₀₁	C₂²×Q₈	1	1	yes	characteristic
43	C₄₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄⋊Q₈	—	401	1	no	maximal, Sylow
45	C₂²×Q₈	—	401	2	no	—
46	C₄⋊Q₈	—	401	8	no	—
47	C₂×C₄⋊C₄	—	401	2	no	—
48	C₂×C₄⋊C₄	—	401	1	no	—
49	C₂×C₄⋊C₄	D₄₀₁	1	1	yes	characteristic
50	C₂×C₄²	—	401	1	no	—
51	C₂×Q₈	—	802	8	no	—
52	C₂×Q₈	—	401	8	no	—
53	C₂²×C₄	—	401	2	no	—
54	C₂²×C₄	—	401	1	no	—
55	C₂²×C₄	—	401	1	no	—
56	C₂²×C₄	C₂×D₄₀₁	1	2	yes	—
57	C₂²×C₄	C₂×D₄₀₁	1	1	yes	characteristic
58	C₄⋊C₄	—	401	8	no	—
59	C₄⋊C₄	—	401	4	no	—
60	C₄⋊C₄	C₂×D₄₀₁	1	4	yes	—
61	C₄²	—	401	4	no	—
62	C₂³	C₂²×D₄₀₁	1	1	yes	characteristic
63	Q₈	—	802	16	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	802	4	no	—
66	C₂×C₄	—	401	4	no	—
67	C₂×C₄	—	401	4	no	—
68	C₂×C₄	—	401	4	no	—
69	C₂×C₄	—	401	2	no	—
70	C₂×C₄	—	401	2	no	—
71	C₂×C₄	Dic₈₀₂	1	4	yes	complemented
72	C₂×C₄	C₂²×D₄₀₁	1	4	yes	—
73	C₂×C₄	C₂²×D₄₀₁	1	2	yes	—
74	C₂²	D₄×D₄₀₁	1	2	yes	—
75	C₂²	Q₈×D₄₀₁	1	2	yes	—
76	C₂²	C₂×Dic₈₀₂	1	2	yes	—
77	C₂²	C₂³×D₄₀₁	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	802	4	no	—
80	C₄	—	401	4	no	—
81	C₄	—	401	4	no	—
82	C₄	C₂×Dic₈₀₂	1	4	yes	complemented
83	C₂	#27165736	1	4	yes	direct product factor
84	C₂	C₂×D₄×D₄₀₁	1	1	yes	characteristic
85	C₂	C₂×Q₈×D₄₀₁	1	1	yes	characteristic
86	C₂	C₂²×Dic₈₀₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₄₀₁
commutator, cosocle	C₂²×C₄₀₁	C₂⁴
Frattini	C₂²	C₂³×D₄₀₁
Fitting	C₂×C₄⋊C₄×C₄₀₁	C₂
socle, prime radical	C₂³×C₄₀₁	C₂³
hypercenter	C₂×C₄⋊C₄	D₄₀₁
hypocenter	C₄₀₁	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₄₀₁ ▷ C₄₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #27165736	C₂	direct (N × H)
21. C₂×Dic₄₀₁	Q₈	semidirect
27. C₂×C₄×C₄₀₁	Q₈	semidirect
35. C₄×C₄₀₁	C₂×Q₈	semidirect
38. Dic₄₀₁	C₂×Q₈	semidirect
39. C₂×C₄₀₁	C₄⋊Q₈	semidirect
43. C₄₀₁	C₂×C₄⋊Q₈	semidirect
71. C₂×C₄	Dic₈₀₂	semidirect
82. C₄	C₂×Dic₈₀₂	semidirect
83. C₂	#27165736	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×Dic₈₀₂	2
#27165736	2
C₂×C₄⋊Dic₄₀₁	2
C₂×#20264345	2
C₂×C₄×Dic₄₀₁	2
C₂×C₄⋊C₄×C₄₀₁	2
C₂×C₄⋊Q₈	401

Subgroups — C2×#27165736 #38786657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27165736 #38786657