Subgroups — #38786296

Order 25664.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27165785	C₂	1	1	yes	maximal, characteristic
2	#27165768	C₂	1	2	yes	maximal, complemented
3	#27165665	C₂	1	2	yes	maximal, complemented
4	#27165656	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₄₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20264367	—	2	2	no	—
7	#20264366	—	2	2	no	—
8	#20264366	C₂²	1	2	yes	—
9	Q₈×D₄₀₁	—	2	2	no	—
10	#20264356	—	2	2	no	—
11	#20264353	—	2	2	no	—
12	#20264350	C₂²	1	2	yes	—
13	C₂×Q₈×C₄₀₁	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₄₀₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₄₀₁	—	2	2	no	—
16	C₄²×C₄₀₁	C₂²	1	1	yes	characteristic
17	Dic₈₀₂	—	2	2	no	—
18	Dic₈₀₂	D₄	1	2	yes	—
19	D₁₆₀₄	—	2	2	no	—
20	D₁₆₀₄	D₄	1	2	yes	—
21	C₂²⋊D₄₀₁	—	2	2	no	—
22	C₄×D₄₀₁	—	2	4	no	—
23	C₄×D₄₀₁	—	2	2	no	—
24	#15691999	—	2	2	no	—
25	Q₈×C₄₀₁	—	2	2	no	—
26	Q₈×C₄₀₁	—	2	2	no	—
27	C₂×C₄×C₄₀₁	—	2	1	no	—
28	C₂×C₄×C₄₀₁	—	2	1	no	—
29	C₂×C₄×C₄₀₁	D₄	1	2	yes	—
30	C₂×C₄×C₄₀₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄₀₁	C₂×D₄	1	1	yes	characteristic
32	C₂×D₄₀₁	—	2	2	no	—
33	C₄×C₄₀₁	—	2	2	no	—
34	C₄×C₄₀₁	—	2	2	no	—
35	C₄×C₄₀₁	—	4	1	no	—
36	C₄×C₄₀₁	C₂×D₄	1	2	yes	—
37	Dic₄₀₁	—	2	2	no	—
38	C₂×C₄₀₁	—	2	1	no	—
39	C₂×C₄₀₁	#120	1	1	yes	characteristic
40	D₄₀₁	—	4	2	no	—
41	C₄₀₁	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	401	1	no	maximal, Sylow
43	2₋¹⁺⁴	—	401	1	no	—
44	#137	—	401	2	no	—
45	C₄⋊Q₈	D₄₀₁	1	1	yes	characteristic, complemented
46	#104	—	401	2	no	—
47	#101	—	401	1	no	—
48	#41	—	802	2	no	—
49	#41	—	802	2	no	—
50	#41	—	401	2	no	—
51	C₂×Q₈	—	802	2	no	—
52	C₂×Q₈	C₂×D₄₀₁	1	1	yes	characteristic
53	C₂×Q₈	C₂×D₄₀₁	1	1	yes	characteristic
54	Q₁₆	—	802	2	no	—
55	SD₁₆	—	802	2	no	—
56	M₁₆	—	401	2	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄²	C₂×D₄₀₁	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	802	2	no	—
62	Q₈	—	401	2	no	—
63	D₄	—	802	2	no	—
64	D₄	—	802	2	no	—
65	D₄	—	401	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	802	4	no	—
69	C₂×C₄	—	802	2	no	—
70	C₂×C₄	C₂²⋊D₄₀₁	1	2	yes	—
71	C₂×C₄	C₂²×D₄₀₁	1	1	yes	characteristic
72	C₈	—	802	2	no	—
73	C₂²	—	802	2	no	—
74	C₂²	C₂×C₂²⋊D₄₀₁	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	802	2	no	—
79	C₄	D₄×D₄₀₁	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	1604	2	no	—
82	C₂	#27165705	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27165705
commutator, cosocle	C₂×C₄×C₄₀₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₀₁
Fitting	C₄⋊Q₈×C₄₀₁	C₂
socle	C₂×C₄₀₁	#120
prime radical	#27165785	C₂
hypercenter	C₄⋊Q₈	D₄₀₁
hypocenter	C₄₀₁	#456
LCS	G ▷ C₂×C₄×C₄₀₁ ▷ C₂×C₄₀₁ ▷ C₄₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #27165768	C₂	semidirect
3. #27165665	C₂	semidirect
4. #27165656	C₂	semidirect
5. C₄⋊Q₈×C₄₀₁	C₂	semidirect
41. C₄₀₁	#456	semidirect
45. C₄⋊Q₈	D₄₀₁	semidirect

Maximal subgroups

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

subgroup	index
#27165785	2
#27165768	2
#27165665	2
#27165656	2
C₄⋊Q₈×C₄₀₁	2
#456	401