Subgroups — #43252704

Order 28544.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383637	C₂	1	2	yes	maximal
2	#423×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₂₃⋊C₈)	—	2	2	no	—
4	C₂²⋊(C₂₂₃⋊C₈)	C₄	1	4	yes	—
5	#20297215	—	2	2	no	—
6	C₂×M₁₆×C₂₂₃	C₄	1	2	yes	—
7	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₂₂₃	—	2	2	no	—
9	C₂×C₂₂₃⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
11	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	M₁₆×C₂₂₃	—	4	2	no	—
13	C₂×C₈×C₂₂₃	—	2	2	no	—
14	C₄²×C₂₂₃	D₄	1	2	yes	—
15	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
16	C₄²×C₂₂₃	Q₈	1	1	yes	characteristic
17	C₂₂₃⋊C₈	—	4	4	no	—
18	C₂³×C₂₂₃	C₄²	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	—	2	1	no	—
20	C₂×C₄×C₂₂₃	—	2	1	no	—
21	C₂×C₄×C₂₂₃	—	4	2	no	—
22	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
26	C₈×C₂₂₃	—	4	2	no	—
27	C₂²×C₂₂₃	—	2	1	no	—
28	C₂²×C₂₂₃	—	2	2	no	—
29	C₂²×C₂₂₃	#95	1	1	yes	characteristic
30	C₄×C₂₂₃	—	2	1	no	—
31	C₄×C₂₂₃	—	2	2	no	—
32	C₄×C₂₂₃	—	2	2	no	—
33	C₄×C₂₂₃	#100	1	1	yes	characteristic
34	C₄×C₂₂₃	#101	1	1	yes	characteristic
35	C₂×C₂₂₃	—	4	1	no	—
36	C₂×C₂₂₃	#343	1	1	yes	characteristic
37	C₂×C₂₂₃	#344	1	2	yes	—
38	C₂₂₃	#1307	1	1	yes	characteristic, complemented, Sylow
39	#1307	—	223	1	no	maximal, Sylow
40	#432	—	223	2	no	—
41	#423	D₂₂₃	1	1	yes	characteristic
42	C₂×M₁₆	Dic₂₂₃	1	2	yes	—
43	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
44	C₄⋊C₈	—	2	2	no	—
45	C₄⋊C₈	—	446	2	no	—
46	C₂²⋊C₈	—	223	4	no	—
47	#97	—	446	2	no	—
48	C₂²×C₄	C₄×D₂₂₃	1	2	yes	—
49	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
50	M₁₆	—	4	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	446	4	no	—
53	C₄²	C₂²⋊D₂₂₃	1	2	yes	—
54	C₄²	D₈₉₂	1	1	yes	characteristic
55	C₄²	Dic₄₄₆	1	1	yes	characteristic
56	C₂³	C₄×Dic₂₂₃	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
61	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
62	C₂×C₄	Dic₂₂₃⋊C₄	1	2	yes	—
63	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
64	C₈	—	4	2	no	—
65	C₈	—	892	4	no	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	2	no	—
72	C₄	#20297221	1	1	yes	characteristic
73	C₄	#20297223	1	1	yes	characteristic
74	C₂	—	4	1	no	—
75	C₂	#28383349	1	1	yes	characteristic
76	C₂	#28383352	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂₂₃⋊#95
commutator, prime radical	C₂³×C₂₂₃	C₄²
Frattini	C₂×C₄²	C₂×D₂₂₃
Fitting	#423×C₂₂₃	C₂
socle	C₂²×C₂₂₃	#95
cosocle	C₂×C₄²×C₂₂₃	C₂²
hypercenter	#423	D₂₂₃
hypocenter	C₂₂₃	#1307
LCS	G ▷ C₂³×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
38. C₂₂₃	#1307	semidirect

Maximal subgroups

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

subgroup	index
#28383637	2
#423×C₂₂₃	2
#1307	223