Subgroups — #28383386

Order 14272.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297326	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813170	—	2	1	no	—
5	Dic₈₉₂	C₄	1	1	yes	characteristic
6	#16813159	—	2	1	no	—
7	D₁₇₈₄	C₄	1	1	yes	characteristic
8	C₂₂₃⋊C₁₆	—	2	1	no	—
9	M₁₆×C₂₂₃	—	2	1	no	—
10	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	Dic₄₄₆	—	2	1	no	—
12	D₈₉₂	—	2	1	no	—
13	C₂²⋊D₂₂₃	—	4	1	no	—
14	C₄×D₂₂₃	—	4	1	no	—
15	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
16	C₈×C₂₂₃	—	4	1	no	—
17	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₈×C₂₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₂₂₃	SD₁₆	1	1	yes	characteristic
20	C₂×D₂₂₃	—	4	1	no	—
21	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₂₂₃	D₈	1	1	yes	characteristic
23	Dic₂₂₃	—	4	1	no	—
24	C₂×C₂₂₃	—	2	1	no	—
25	C₂×C₂₂₃	#102	1	1	yes	characteristic
26	D₂₂₃	—	8	1	no	—
27	C₂₂₃	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	223	1	no	maximal, Sylow
29	#135	—	223	1	no	—
30	C₂×C₁₆	—	223	1	no	—
31	#108	D₂₂₃	1	1	yes	characteristic, complemented
32	#41	—	446	1	no	—
33	Q₁₆	—	223	1	no	—
34	SD₁₆	—	446	1	no	—
35	D₈	—	223	1	no	—
36	M₁₆	—	2	1	no	—
37	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
38	C₁₆	—	446	1	no	—
39	Q₈	—	446	1	no	—
40	D₄	—	892	1	no	—
41	D₄	—	446	1	no	—
42	C₂×C₄	—	892	1	no	—
43	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
44	C₈	—	4	1	no	—
45	C₈	C₄×D₂₂₃	1	1	yes	characteristic
46	C₈	D₈₉₂	1	1	yes	characteristic
47	C₂²	—	892	1	no	—
48	C₂²	#16813157	1	1	yes	characteristic
49	C₄	—	892	1	no	—
50	C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
51	C₄	#16813155	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	1784	1	no	—
54	C₂	#20297224	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16813155
commutator	C₈×C₂₂₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₂₃
Fitting	#108×C₂₂₃	C₂
socle	C₂×C₂₂₃	#102
prime radical	#20297326	C₂
cosocle	C₂×C₈×C₂₂₃	C₂²
hypercenter	#108	D₂₂₃
hypocenter	C₂₂₃	#359
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₂₃⋊C₁₆	C₂	semidirect
3. #108×C₂₂₃	C₂	semidirect
27. C₂₂₃	#359	semidirect
31. #108	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297326	2
C₂×C₂₂₃⋊C₁₆	2
#108×C₂₂₃	2
#359	223