Subgroups — #28384323

Order 14272.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297291	C₂	1	3	yes	maximal, complemented
2	#20297290	C₂	1	6	yes	maximal, complemented
3	#20297273	C₂	1	2	yes	maximal, complemented
4	#20297272	C₂	1	3	yes	maximal, complemented
5	#120×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₂₂₃	C₂²	1	1	yes	characteristic, complemented
7	C₂×Dic₄₄₆	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊D₂₂₃	C₂²	1	6	yes	—
9	C₂²⋊Dic₂₂₃	C₂²	1	3	yes	complemented
10	C₂²⋊Dic₂₂₃	C₂²	1	6	yes	complemented
11	(C₂×D₂₂₃)⋊C₄	C₂²	1	6	yes	complemented
12	C₄×Dic₂₂₃	C₂²	1	3	yes	complemented
13	C₂⁴×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
14	C₂×D₄×C₂₂₃	C₂²	1	3	yes	—
15	C₂²⋊C₄×C₂₂₃	C₂²	1	3	yes	complemented
16	C₂²×D₂₂₃	—	4	3	no	—
17	C₂²×D₂₂₃	C₂³	1	2	yes	—
18	Dic₄₄₆	—	4	3	no	—
19	C₂²⋊D₂₂₃	—	4	6	no	—
20	C₂×Dic₂₂₃	C₂³	1	6	yes	—
21	C₂³×C₂₂₃	—	4	3	no	—
22	C₂³×C₂₂₃	C₂³	1	3	yes	—
23	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
24	D₄×C₂₂₃	—	4	3	no	—
25	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
26	C₂²×C₂₂₃	—	2	3	no	—
27	C₂²×C₂₂₃	—	2	3	no	—
28	C₂²×C₂₂₃	—	2	6	no	—
29	C₂²×C₂₂₃	—	4	4	no	—
30	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
31	C₂×D₂₂₃	—	2	6	no	—
32	C₂×D₂₂₃	—	4	4	no	—
33	C₄×C₂₂₃	—	2	3	no	—
34	Dic₂₂₃	—	2	6	no	—
35	C₂×C₂₂₃	—	4	1	no	—
36	C₂×C₂₂₃	—	4	3	no	—
37	C₂×C₂₂₃	H(5,𝔽₂)	1	3	yes	—
38	D₂₂₃	—	4	2	no	—
39	C₂₂₃	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
40	H(3,𝔽₄)	—	223	1	no	maximal, Sylow
41	#124	—	223	3	no	—
42	#124	—	223	6	no	—
43	#120	—	223	2	no	—
44	#120	—	223	3	no	—
45	#120	D₂₂₃	1	1	yes	characteristic, complemented
46	C₂⁴	—	223	1	no	—
47	C₂⁴	C₂×D₂₂₃	1	1	yes	characteristic, complemented
48	C₂×Q₈	—	223	3	no	—
49	C₂×D₄	—	223	6	no	—
50	C₂×D₄	C₂×D₂₂₃	1	3	yes	—
51	C₂²⋊C₄	—	223	3	no	—
52	C₂²⋊C₄	—	223	6	no	—
53	C₂²⋊C₄	—	223	6	no	—
54	C₂²⋊C₄	C₂×D₂₂₃	1	3	yes	complemented
55	C₄²	—	223	3	no	—
56	C₂³	—	4	3	no	—
57	C₂³	—	892	3	no	—
58	C₂³	—	223	2	no	—
59	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
60	C₂³	C₂²×D₂₂₃	1	3	yes	—
61	Q₈	—	892	3	no	—
62	D₄	—	4	3	no	—
63	D₄	—	892	6	no	—
64	C₂×C₄	—	223	6	no	—
65	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	892	4	no	—
71	C₂²	—	446	6	no	—
72	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
73	C₄	—	2	3	no	—
74	C₄	—	446	6	no	—
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	892	2	no	—
78	C₂	#20297349	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	#120×C₂₂₃	C₂
hypercenter	#120	D₂₂₃
hypocenter	C₂₂₃	H(3,𝔽₄)
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. #20297291	C₂	semidirect
2. #20297290	C₂	semidirect
3. #20297273	C₂	semidirect
4. #20297272	C₂	semidirect
5. #120×C₂₂₃	C₂	semidirect
6. C₂³×D₂₂₃	C₂²	semidirect
7. C₂×Dic₄₄₆	C₂²	semidirect
9. C₂²⋊Dic₂₂₃	C₂²	semidirect
10. C₂²⋊Dic₂₂₃	C₂²	semidirect
11. (C₂×D₂₂₃)⋊C₄	C₂²	semidirect
12. C₄×Dic₂₂₃	C₂²	semidirect
13. C₂⁴×C₂₂₃	C₂²	semidirect
15. C₂²⋊C₄×C₂₂₃	C₂²	semidirect
39. C₂₂₃	H(3,𝔽₄)	semidirect
45. #120	D₂₂₃	semidirect
47. C₂⁴	C₂×D₂₂₃	semidirect
54. C₂²⋊C₄	C₂×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297291	2
#20297290	2
#20297273	2
#20297272	2
#120×C₂₂₃	2
H(3,𝔽₄)	223