Subgroups — #28383351

Order 14272.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813154	C₂	1	1	yes	maximal, characteristic
2	#20297263	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#16813154	—	2	1	no	—
5	#16813154	C₄	1	2	yes	complemented
6	C₂×C₂₂₃⋊C₈	—	2	1	no	—
7	C₄⋊Dic₂₂₃	—	2	1	no	—
8	C₂²⋊Dic₂₂₃	C₄	1	2	yes	—
9	C₄×Dic₂₂₃	—	2	1	no	—
10	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
11	M₁₆×C₂₂₃	—	2	1	no	—
12	M₁₆×C₂₂₃	—	2	1	no	—
13	C₂×C₈×C₂₂₃	C₄	1	2	yes	complemented
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂₂₃⋊C₈	—	2	2	no	—
16	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
23	C₈×C₂₂₃	—	2	2	no	—
24	C₂²×C₂₂₃	—	4	1	no	—
25	C₂²×C₂₂₃	C₄²	1	1	yes	characteristic
26	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
32	Dic₂₂₃	—	4	2	no	—
33	C₂×C₂₂₃	—	2	1	no	—
34	C₂×C₂₂₃	—	2	1	no	—
35	C₂×C₂₂₃	—	2	1	no	—
36	C₂×C₂₂₃	#95	1	1	yes	characteristic
37	C₂₂₃	#344	1	1	yes	characteristic, complemented, Sylow
38	#344	—	223	1	no	maximal, Sylow
39	C₂×M₁₆	—	223	1	no	—
40	C₂×M₁₆	D₂₂₃	1	1	yes	characteristic
41	#117	—	223	1	no	—
42	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
43	M₁₆	—	2	1	no	—
44	M₁₆	—	2	1	no	—
45	M₁₆	—	446	1	no	—
46	M₁₆	—	223	2	no	—
47	C₂×C₈	—	446	1	no	—
48	C₂×C₈	Dic₂₂₃	1	2	yes	complemented
49	C₄⋊C₄	—	446	1	no	—
50	C₂²⋊C₄	—	223	2	no	—
51	C₄²	—	446	1	no	—
52	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
53	C₂×C₄	—	446	2	no	—
54	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
55	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
58	C₂×C₄	D₈₉₂	1	1	yes	characteristic
59	C₂×C₄	Dic₄₄₆	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	446	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₄×Dic₂₂₃	1	1	yes	characteristic
64	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
65	C₂²	Dic₂₂₃⋊C₄	1	1	yes	characteristic
66	C₄	—	892	2	no	—
67	C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
68	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
69	C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
70	C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂₂₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊Dic₂₂₃
commutator	C₂²×C₂₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	C₂×M₁₆×C₂₂₃	C₂
socle	C₂×C₂₂₃	#95
prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂×M₁₆	D₂₂₃
hypocenter	C₂₂₃	#344
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
5. #16813154	C₄	semidirect
13. C₂×C₈×C₂₂₃	C₄	semidirect
37. C₂₂₃	#344	semidirect
48. C₂×C₈	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813154	2
#20297263	2
C₂×M₁₆×C₂₂₃	2
#344	223