Subgroups — #28383335

Order 14272.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₂₃⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₂₃⋊C₈	—	2	2	no	—
4	C₂×C₂₂₃⋊C₈	—	2	2	no	—
5	C₂×C₂₂₃⋊C₈	C₄	1	4	yes	complemented
6	C₂²×C₄×C₂₂₃	C₄	1	2	yes	—
7	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₄²×C₂₂₃	—	2	2	no	—
9	C₂₂₃⋊C₈	—	2	4	no	—
10	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₂₂₃	—	2	2	no	—
12	C₂×C₄×C₂₂₃	—	2	2	no	—
13	C₂×C₄×C₂₂₃	C₈	1	4	yes	complemented
14	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
16	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
18	C₂²×C₂₂₃	C₄²	1	1	yes	characteristic
19	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₂₂₃	C₂×C₈	1	2	yes	—
22	C₂²×C₂₂₃	M₁₆	1	2	yes	—
23	C₄×C₂₂₃	—	2	4	no	—
24	C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
25	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₂₂₃	#95	1	1	yes	characteristic
27	C₂×C₂₂₃	C₄×C₈	1	1	yes	characteristic
28	C₂×C₂₂₃	#97	1	1	yes	characteristic
29	C₂×C₂₂₃	C₂²⋊C₈	1	2	yes	—
30	C₂×C₂₂₃	C₄⋊C₈	1	2	yes	—
31	C₂₂₃	#336	1	1	yes	characteristic, complemented, Sylow
32	#336	—	223	1	no	maximal, Sylow
33	C₂²×C₈	—	223	2	no	—
34	C₂×C₄²	D₂₂₃	1	1	yes	characteristic
35	C₂²×C₄	Dic₂₂₃	1	2	yes	—
36	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
37	C₂×C₈	—	446	2	no	—
38	C₂×C₈	—	446	2	no	—
39	C₂×C₈	—	223	4	no	—
40	C₄²	—	2	2	no	—
41	C₂³	C₂×Dic₂₂₃	1	1	yes	characteristic
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	C₂₂₃⋊C₈	1	4	yes	complemented
45	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
46	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
47	C₂×C₄	D₈₉₂	1	1	yes	characteristic
48	C₂×C₄	Dic₄₄₆	1	1	yes	characteristic
49	C₈	—	446	4	no	—
50	C₂²	C₄×Dic₂₂₃	1	1	yes	characteristic
51	C₂²	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
52	C₂²	C₄⋊Dic₂₂₃	1	1	yes	characteristic
53	C₂²	C₂×C₂₂₃⋊C₈	1	2	yes	—
54	C₂²	#16813154	1	2	yes	—
55	C₄	—	2	4	no	—
56	C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
57	C₄	Dic₂₂₃⋊C₄	1	2	yes	—
58	C₂	C₂₂₃⋊#95	1	1	yes	characteristic
59	C₂	C₄×C₂₂₃⋊C₈	1	1	yes	characteristic
60	C₂	#20297215	1	1	yes	characteristic
61	C₂	C₂²⋊(C₂₂₃⋊C₈)	1	2	yes	—
62	C₂	C₄⋊(C₂₂₃⋊C₈)	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₄×C₈
Fitting	C₂×C₄²×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂×C₄²	D₂₂₃
hypocenter	C₂₂₃	#336
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. C₂×C₂₂₃⋊C₈	C₄	semidirect
13. C₂×C₄×C₂₂₃	C₈	semidirect
31. C₂₂₃	#336	semidirect
44. C₂×C₄	C₂₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₂₃⋊C₈	2
C₂×C₄²×C₂₂₃	2
#336	223