Subgroups — #43248199

Order 28352.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20296788	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₄₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₇₇₂	C₂	1	3	yes	maximal, complemented
4	#28381848	C₂	1	3	yes	maximal, complemented
5	#28381843	C₂	1	3	yes	maximal, complemented
6	C₄⋊C₄×D₄₄₃	C₂	1	3	yes	maximal, complemented
7	C₄×Q₈×C₄₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#20296788	C₄	1	8	yes	complemented
9	C₂×D₁₇₇₂	C₂²	1	3	yes	—
10	C₂×C₄×D₄₄₃	C₂²	1	6	yes	—
11	C₂×C₄×D₄₄₃	C₂²	1	3	yes	—
12	C₄⋊Dic₄₄₃	C₂²	1	3	yes	—
13	Dic₄₄₃⋊C₄	C₂²	1	3	yes	—
14	Dic₄₄₃⋊C₄	C₂²	1	1	yes	characteristic
15	(C₂×D₄₄₃)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₄₄₃)⋊C₄	C₂²	1	3	yes	—
17	C₄×Dic₄₄₃	C₂²	1	3	yes	—
18	C₂×Q₈×C₄₄₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₄₄₃	C₂²	1	3	yes	—
20	C₄²×C₄₄₃	C₂²	1	3	yes	—
21	C₂²×D₄₄₃	C₂³	1	3	yes	—
22	D₁₇₇₂	C₂×C₄	1	12	yes	complemented
23	C₂×Dic₄₄₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₄₄₃	C₂³	1	3	yes	—
25	C₂×Dic₄₄₃	C₂³	1	1	yes	characteristic
26	C₄×D₄₄₃	—	4	6	no	—
27	C₄×D₄₄₃	C₂×C₄	1	12	yes	complemented
28	Q₈×C₄₄₃	C₂×C₄	1	4	yes	complemented
29	C₂×C₄×C₄₄₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₄₄₃	C₂³	1	3	yes	—
31	C₂×C₄×C₄₄₃	C₂³	1	3	yes	—
32	C₂²×C₄₄₃	C₂⁴	1	1	yes	characteristic
33	C₂×D₄₄₃	—	2	3	no	—
34	C₂×D₄₄₃	—	2	3	no	—
35	C₂×D₄₄₃	C₂²×C₄	1	6	yes	—
36	C₄×C₄₄₃	—	2	3	no	—
37	C₄×C₄₄₃	—	2	1	no	—
38	C₄×C₄₄₃	C₂²×C₄	1	6	yes	—
39	Dic₄₄₃	—	2	3	no	—
40	Dic₄₄₃	—	2	1	no	—
41	Dic₄₄₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄₄₃	C₂³×C₄	1	1	yes	characteristic
43	C₂×C₄₄₃	H(5,𝔽₂)	1	1	yes	characteristic
44	C₂×C₄₄₃	2₋¹⁺⁴	1	1	yes	characteristic
45	D₄₄₃	—	2	6	no	—
46	C₄₄₃	#520	1	1	yes	characteristic, complemented, Sylow
47	#520	—	443	1	no	maximal, Sylow
48	C₂×#41	—	443	1	no	—
49	C₄×Q₈	—	443	1	no	—
50	C₄×Q₈	D₄₄₃	1	1	yes	characteristic, complemented
51	C₄×D₄	—	443	3	no	—
52	C₄×D₄	—	443	3	no	—
53	#117	—	443	3	no	—
54	C₂×C₄⋊C₄	—	443	3	no	—
55	#41	—	443	8	no	—
56	C₂×Q₈	C₂×D₄₄₃	1	1	yes	characteristic
57	C₂×D₄	—	443	3	no	—
58	C₂²×C₄	—	443	3	no	—
59	C₂²×C₄	—	443	6	no	—
60	C₄⋊C₄	—	443	3	no	—
61	C₄⋊C₄	—	443	1	no	—
62	C₄⋊C₄	—	443	3	no	—
63	C₄⋊C₄	C₂×D₄₄₃	1	3	yes	—
64	C₂²⋊C₄	—	443	3	no	—
65	C₂²⋊C₄	—	443	3	no	—
66	C₄²	—	443	3	no	—
67	C₄²	C₂×D₄₄₃	1	3	yes	—
68	C₂³	—	443	3	no	—
69	Q₈	C₄×D₄₄₃	1	4	yes	complemented
70	D₄	—	443	12	no	—
71	C₂×C₄	—	443	1	no	—
72	C₂×C₄	—	443	3	no	—
73	C₂×C₄	—	443	1	no	—
74	C₂×C₄	—	443	12	no	—
75	C₂×C₄	—	1772	6	no	—
76	C₂×C₄	C₂²×D₄₄₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₄₄₃	1	3	yes	—
78	C₂×C₄	C₂²×D₄₄₃	1	3	yes	—
79	C₂²	—	886	3	no	—
80	C₂²	—	886	3	no	—
81	C₂²	—	443	6	no	—
82	C₂²	C₂³×D₄₄₃	1	1	yes	characteristic
83	C₄	—	2	3	no	—
84	C₄	—	2	1	no	—
85	C₄	—	886	3	no	—
86	C₄	—	886	1	no	—
87	C₄	—	443	2	no	—
88	C₄	C₂×C₄×D₄₄₃	1	6	yes	—
89	C₂	—	886	6	no	—
90	C₂	C₂²×C₄×D₄₄₃	1	1	yes	characteristic
91	C₂	#28381932	1	1	yes	characteristic
92	C₂	#28381934	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₄₃
commutator	C₂×C₄₄₃	C₂³×C₄
Fitting	C₄×Q₈×C₄₄₃	C₂
socle, cosocle	C₂²×C₄₄₃	C₂⁴
prime radical	C₂×#20296788	C₂
hypercenter	C₄×Q₈	D₄₄₃
hypocenter	C₄₄₃	#520
LCS	G ▷ C₂×C₄₄₃ ▷ C₄₄₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₄₄₃	C₂	semidirect
3. C₄×D₁₇₇₂	C₂	semidirect
4. #28381848	C₂	semidirect
5. #28381843	C₂	semidirect
6. C₄⋊C₄×D₄₄₃	C₂	semidirect
7. C₄×Q₈×C₄₄₃	C₂	semidirect
8. #20296788	C₄	semidirect
22. D₁₇₇₂	C₂×C₄	semidirect
27. C₄×D₄₄₃	C₂×C₄	semidirect
28. Q₈×C₄₄₃	C₂×C₄	semidirect
46. C₄₄₃	#520	semidirect
50. C₄×Q₈	D₄₄₃	semidirect
69. Q₈	C₄×D₄₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20296788	2
Q₈×Dic₄₄₃	2
C₄×D₁₇₇₂	2
#28381848	2
#28381843	2
C₄⋊C₄×D₄₄₃	2
C₄×Q₈×C₄₄₃	2
#520	443