Subgroups — #28383538

Order 14272.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#20297263	C₂	1	1	yes	maximal, characteristic
3	#20297214	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₂₃	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₂₃⋊C₈	C₄	1	4	yes	complemented
7	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₂₃	C₄	1	2	yes	—
10	C₂²⋊Dic₂₂₃	C₄	1	2	yes	—
11	C₄×Dic₂₂₃	C₂²	1	2	yes	—
12	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	M₁₆×C₂₂₃	C₄	1	4	yes	complemented
14	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
15	C₂×Dic₂₂₃	C₂×C₄	1	4	yes	—
16	C₂₂₃⋊C₈	C₂×C₄	1	2	yes	—
17	C₂₂₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
23	C₈×C₂₂₃	C₂×C₄	1	4	yes	—
24	C₂²×C₂₂₃	—	2	2	no	—
25	C₂²×C₂₂₃	C₄²	1	2	yes	—
26	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₂₂₃	C₄²	1	2	yes	—
28	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
30	Dic₂₂₃	—	2	4	no	—
31	C₂×C₂₂₃	—	2	2	no	—
32	C₂×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
33	C₂×C₂₂₃	#131	1	2	yes	—
34	C₂₂₃	#405	1	1	yes	characteristic, complemented, Sylow
35	#405	—	223	1	no	maximal, Sylow
36	C₂×M₁₆	D₂₂₃	1	1	yes	characteristic
37	C₂²×C₈	—	223	1	no	—
38	#117	—	223	1	no	—
39	#97	—	223	2	no	—
40	C₄×C₈	—	223	2	no	—
41	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
42	M₁₆	Dic₂₂₃	1	4	yes	complemented
43	C₂×C₈	—	223	4	no	—
44	C₂×C₈	—	223	1	no	—
45	C₂×C₈	—	223	1	no	—
46	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
47	C₄⋊C₄	—	223	2	no	—
48	C₂²⋊C₄	—	223	2	no	—
49	C₄²	—	223	2	no	—
50	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
51	C₂×C₄	—	223	4	no	—
52	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
53	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
54	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
55	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
56	C₈	—	223	2	no	—
57	C₈	—	223	2	no	—
58	C₈	C₂×Dic₂₂₃	1	4	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	C₄×Dic₂₂₃	1	2	yes	—
61	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
62	C₄	—	446	4	no	—
63	C₄	C₄×Dic₂₂₃	1	2	yes	—
64	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
65	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	C₂×C₄×Dic₂₂₃	1	1	yes	characteristic
68	C₂	#20297312	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator	C₂×C₂₂₃	C₂×C₄²
Fitting	C₂×M₁₆×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²×C₄
prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×M₁₆	D₂₂₃
hypocenter	C₂₂₃	#405
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20297214	C₂	semidirect
4. C₈×Dic₂₂₃	C₂	semidirect
6. C₂×C₂₂₃⋊C₈	C₄	semidirect
13. M₁₆×C₂₂₃	C₄	semidirect
34. C₂₂₃	#405	semidirect
42. M₁₆	Dic₂₂₃	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
#20297263	2
#20297214	2
C₈×Dic₂₂₃	2
C₂×M₁₆×C₂₂₃	2
#405	223