Subgroups — #28383380

Order 14272.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₇₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#107×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₉₂	—	2	1	no	—
5	D₁₇₈₄	—	2	1	no	—
6	D₁₇₈₄	C₄	1	2	yes	complemented
7	C₂₂₃⋊C₁₆	—	2	1	no	—
8	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₂₂₃	—	2	1	no	—
10	C₂²×D₂₂₃	—	4	1	no	—
11	D₈₉₂	—	2	2	no	—
12	D₈₉₂	—	4	1	no	—
13	C₂×C₄×C₂₂₃	—	4	1	no	—
14	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
15	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₂₂₃	D₈	1	1	yes	characteristic
18	C₂×D₂₂₃	—	4	2	no	—
19	C₂×D₂₂₃	—	8	1	no	—
20	C₂×D₂₂₃	—	8	1	no	—
21	C₄×C₂₂₃	—	8	1	no	—
22	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
24	C₂×C₂₂₃	#102	1	1	yes	characteristic
25	C₂×C₂₂₃	D₁₆	1	1	yes	characteristic
26	C₂×C₂₂₃	SD₃₂	1	1	yes	characteristic
27	D₂₂₃	—	8	2	no	—
28	C₂₂₃	#357	1	1	yes	characteristic, complemented, Sylow
29	#357	—	223	1	no	maximal, Sylow
30	C₂×D₈	—	223	1	no	—
31	C₂×C₁₆	—	223	1	no	—
32	#107	D₂₂₃	1	1	yes	characteristic, complemented
33	C₂×D₄	—	446	1	no	—
34	D₈	—	446	1	no	—
35	D₈	—	223	2	no	—
36	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
37	C₄⋊C₄	—	2	1	no	—
38	C₁₆	—	446	1	no	—
39	C₂³	—	892	1	no	—
40	D₄	—	892	1	no	—
41	D₄	—	446	2	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
44	C₈	C₄×D₂₂₃	1	1	yes	characteristic
45	C₈	D₈₉₂	1	1	yes	characteristic
46	C₂²	—	892	2	no	—
47	C₂²	—	1784	1	no	—
48	C₂²	—	1784	1	no	—
49	C₂²	#16813155	1	1	yes	characteristic
50	C₄	—	8	1	no	—
51	C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
52	C₄	#16813157	1	1	yes	characteristic
53	C₂	—	1784	2	no	—
54	C₂	#20297224	1	1	yes	characteristic
55	C₂	#20297245	1	1	yes	characteristic
56	C₂	#20297247	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16813155
commutator	C₈×C₂₂₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₂₃
Fitting	#107×C₂₂₃	C₂
socle	C₂²×C₂₂₃	D₈
prime radical	C₂×D₁₇₈₄	C₂
cosocle	C₂×C₈×C₂₂₃	C₂²
hypercenter	#107	D₂₂₃
hypocenter	C₂₂₃	#357
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₂₃⋊C₁₆	C₂	semidirect
3. #107×C₂₂₃	C₂	semidirect
6. D₁₇₈₄	C₄	semidirect
28. C₂₂₃	#357	semidirect
32. #107	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₇₈₄	2
C₂×C₂₂₃⋊C₁₆	2
#107×C₂₂₃	2
#357	223