Subgroups — #43252622

Order 28544.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383633	C₂	1	2	yes	maximal
2	#376×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₂₃⋊C₈)	—	2	2	no	—
4	C₂²⋊(C₂₂₃⋊C₈)	—	2	2	no	—
5	C₄×C₂₂₃⋊C₈	C₄	1	4	yes	complemented
6	C₂×C₄²×C₂₂₃	C₄	1	2	yes	—
7	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
8	#95×C₂₂₃	—	2	2	no	—
9	C₂×C₂₂₃⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₂₂₃	—	2	2	no	—
11	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	D₄	1	2	yes	—
13	C₄²×C₂₂₃	—	2	2	no	—
14	C₄²×C₂₂₃	C₈	1	4	yes	complemented
15	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
16	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
17	C₄²×C₂₂₃	Q₈	1	1	yes	characteristic
18	C₂₂₃⋊C₈	—	4	4	no	—
19	C₂³×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	—	2	4	no	—
21	C₂×C₄×C₂₂₃	—	4	2	no	—
22	C₂×C₄×C₂₂₃	—	4	4	no	—
23	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₂₂₃	M₁₆	1	2	yes	—
29	C₂²×C₂₂₃	—	2	2	no	—
30	C₂²×C₂₂₃	#95	1	1	yes	characteristic
31	C₂²×C₂₂₃	C₂²⋊C₈	1	2	yes	—
32	C₄×C₂₂₃	—	2	2	no	—
33	C₄×C₂₂₃	—	2	2	no	—
34	C₄×C₂₂₃	—	4	4	no	—
35	C₄×C₂₂₃	C₄×C₈	1	1	yes	characteristic
36	C₄×C₂₂₃	#97	1	1	yes	characteristic
37	C₄×C₂₂₃	C₄⋊C₈	1	2	yes	—
38	C₂×C₂₂₃	—	2	2	no	—
39	C₂×C₂₂₃	#336	1	1	yes	characteristic
40	C₂×C₂₂₃	H(3,ℤ/4)	1	2	yes	—
41	C₂₂₃	#1274	1	1	yes	characteristic, complemented, Sylow
42	#1274	—	223	1	no	maximal, Sylow
43	#431	—	223	2	no	—
44	#376	D₂₂₃	1	1	yes	characteristic
45	C₂×C₄²	Dic₂₂₃	1	2	yes	—
46	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
47	C₄⋊C₈	—	446	2	no	—
48	C₂²⋊C₈	—	446	2	no	—
49	C₄×C₈	—	223	4	no	—
50	#95	—	2	2	no	—
51	C₂²×C₄	—	2	2	no	—
52	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
53	C₂²×C₄	C₂²⋊D₂₂₃	1	2	yes	—
54	C₂×C₈	—	446	4	no	—
55	C₄²	—	2	2	no	—
56	C₄²	C₂₂₃⋊C₈	1	4	yes	complemented
57	C₄²	C₄×D₂₂₃	1	2	yes	—
58	C₄²	D₈₉₂	1	1	yes	characteristic
59	C₄²	Dic₄₄₆	1	1	yes	characteristic
60	C₂³	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	4	4	no	—
64	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
65	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
66	C₂×C₄	Dic₂₂₃⋊C₄	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
68	C₂×C₄	C₂×C₂₂₃⋊C₈	1	2	yes	—
69	C₂×C₄	#16813154	1	2	yes	—
70	C₈	—	892	4	no	—
71	C₂²	—	2	2	no	—
72	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
73	C₂²	C₂²⋊(C₂₂₃⋊C₈)	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	4	4	no	—
77	C₄	C₄×C₂₂₃⋊C₈	1	1	yes	characteristic
78	C₄	#20297215	1	1	yes	characteristic
79	C₄	C₄⋊(C₂₂₃⋊C₈)	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	#28383335	1	1	yes	characteristic
82	C₂	#28383336	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₂₂₃
commutator	C₄×C₂₂₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₂₃
Fitting	#376×C₂₂₃	C₂
socle	C₂²×C₂₂₃	#95
prime radical	C₂³×C₂₂₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₂₂₃	C₂²
hypercenter	#376	D₂₂₃
hypocenter	C₂₂₃	#1274
LCS	G ▷ C₄×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
5. C₄×C₂₂₃⋊C₈	C₄	semidirect
14. C₄²×C₂₂₃	C₈	semidirect
41. C₂₂₃	#1274	semidirect
56. C₄²	C₂₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#28383633	2
#376×C₂₂₃	2
#1274	223