Subgroups — #43247899

Order 28352.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28381934	C₂	1	1	yes	maximal, characteristic
2	#28381917	C₂	1	2	yes	maximal, complemented
3	#28381814	C₂	1	2	yes	maximal, complemented
4	#28381805	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₄₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#20296788	—	2	2	no	—
7	#20296787	—	2	2	no	—
8	#20296787	C₂²	1	2	yes	—
9	Q₈×D₄₄₃	—	2	2	no	—
10	#20296777	—	2	2	no	—
11	#20296774	—	2	2	no	—
12	#20296771	C₂²	1	2	yes	—
13	C₂×Q₈×C₄₄₃	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₄₄₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₄₄₃	—	2	2	no	—
16	C₄²×C₄₄₃	C₂²	1	1	yes	characteristic
17	Dic₈₈₆	—	2	2	no	—
18	Dic₈₈₆	D₄	1	2	yes	—
19	D₁₇₇₂	—	2	2	no	—
20	D₁₇₇₂	D₄	1	2	yes	—
21	C₂²⋊D₄₄₃	—	2	2	no	—
22	C₄×D₄₄₃	—	2	4	no	—
23	C₄×D₄₄₃	—	2	2	no	—
24	C₄₄₃⋊C₈	—	2	2	no	—
25	Q₈×C₄₄₃	—	2	2	no	—
26	Q₈×C₄₄₃	—	2	2	no	—
27	C₂×C₄×C₄₄₃	—	2	1	no	—
28	C₂×C₄×C₄₄₃	—	2	1	no	—
29	C₂×C₄×C₄₄₃	D₄	1	2	yes	—
30	C₂×C₄×C₄₄₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄₄₃	C₂×D₄	1	1	yes	characteristic
32	C₂×D₄₄₃	—	2	2	no	—
33	C₄×C₄₄₃	—	2	2	no	—
34	C₄×C₄₄₃	—	2	2	no	—
35	C₄×C₄₄₃	—	4	1	no	—
36	C₄×C₄₄₃	C₂×D₄	1	2	yes	—
37	Dic₄₄₃	—	2	2	no	—
38	C₂×C₄₄₃	—	2	1	no	—
39	C₂×C₄₄₃	#120	1	1	yes	characteristic
40	D₄₄₃	—	4	2	no	—
41	C₄₄₃	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	443	1	no	maximal, Sylow
43	2₋¹⁺⁴	—	443	1	no	—
44	#137	—	443	2	no	—
45	C₄⋊Q₈	D₄₄₃	1	1	yes	characteristic, complemented
46	#104	—	443	2	no	—
47	#101	—	443	1	no	—
48	#41	—	886	2	no	—
49	#41	—	886	2	no	—
50	#41	—	443	2	no	—
51	C₂×Q₈	—	886	2	no	—
52	C₂×Q₈	C₂×D₄₄₃	1	1	yes	characteristic
53	C₂×Q₈	C₂×D₄₄₃	1	1	yes	characteristic
54	Q₁₆	—	886	2	no	—
55	SD₁₆	—	886	2	no	—
56	M₁₆	—	443	2	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄²	C₂×D₄₄₃	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	886	2	no	—
62	Q₈	—	443	2	no	—
63	D₄	—	886	2	no	—
64	D₄	—	886	2	no	—
65	D₄	—	443	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	886	4	no	—
69	C₂×C₄	—	886	2	no	—
70	C₂×C₄	C₂²⋊D₄₄₃	1	2	yes	—
71	C₂×C₄	C₂²×D₄₄₃	1	1	yes	characteristic
72	C₈	—	886	2	no	—
73	C₂²	—	886	2	no	—
74	C₂²	C₂×C₂²⋊D₄₄₃	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	886	2	no	—
79	C₄	D₄×D₄₄₃	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	1772	2	no	—
82	C₂	#28381854	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28381854
commutator, cosocle	C₂×C₄×C₄₄₃	C₂³
Frattini	C₂×C₄	C₂²×D₄₄₃
Fitting	C₄⋊Q₈×C₄₄₃	C₂
socle	C₂×C₄₄₃	#120
prime radical	#28381934	C₂
hypercenter	C₄⋊Q₈	D₄₄₃
hypocenter	C₄₄₃	#456
LCS	G ▷ C₂×C₄×C₄₄₃ ▷ C₂×C₄₄₃ ▷ C₄₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #28381917	C₂	semidirect
3. #28381814	C₂	semidirect
4. #28381805	C₂	semidirect
5. C₄⋊Q₈×C₄₄₃	C₂	semidirect
41. C₄₄₃	#456	semidirect
45. C₄⋊Q₈	D₄₄₃	semidirect

Maximal subgroups

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

subgroup	index
#28381934	2
#28381917	2
#28381814	2
#28381805	2
C₄⋊Q₈×C₄₄₃	2
#456	443