Subgroups — #28417347

Order 14912.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21418608	C₂	1	1	yes	maximal, characteristic, complemented
2	#21418604	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₂₃₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11745400	—	2	1	no	—
5	#16836597	—	2	1	no	—
6	D₄×D₂₃₃	—	2	1	no	—
7	D₁₈₆₄	C₄	1	1	yes	characteristic, complemented
8	#16836579	—	2	1	no	—
9	C₈×D₂₃₃	C₂²	1	1	yes	characteristic
10	D₈×C₂₃₃	C₄	1	1	yes	characteristic, complemented
11	C₂×#11745400	—	4	1	no	—
12	C₂²×D₂₃₃	—	4	1	no	—
13	D₉₃₂	—	2	1	no	—
14	C₂²⋊D₂₃₃	—	4	1	no	—
15	C₄×D₂₃₃	D₄	1	1	yes	characteristic
16	#14272338	D₄	1	1	yes	characteristic
17	D₄×C₂₃₃	—	2	1	no	—
18	C₈×C₂₃₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₂₃₃	—	4	1	no	—
20	C₂×D₂₃₃	—	4	1	no	—
21	C₂×D₂₃₃	—	8	1	no	—
22	C₂×D₂₃₃	—	8	1	no	—
23	C₂×D₂₃₃	D₈	1	1	yes	characteristic
24	#11745400	—	8	1	no	—
25	C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
26	Dic₂₃₃	SD₁₆	1	1	yes	characteristic
27	C₂×C₂₃₃	—	8	1	no	—
28	C₂×C₂₃₃	#102	1	1	yes	characteristic
29	D₂₃₃	—	8	1	no	—
30	D₂₃₃	D₁₆	1	1	yes	characteristic
31	D₂₃₃	SD₃₂	1	1	yes	characteristic
32	C₂₃₃	#357	1	1	yes	characteristic, complemented, Sylow
33	#357	—	233	1	no	maximal, Sylow
34	C₂×D₈	—	233	1	no	—
35	C₂×C₁₆	—	233	1	no	—
36	#107	—	233	1	no	—
37	C₂×D₄	—	466	1	no	—
38	D₈	—	466	1	no	—
39	D₈	—	233	1	no	—
40	D₈	#11745400	1	1	yes	characteristic, complemented
41	C₂×C₈	—	233	1	no	—
42	C₄⋊C₄	—	466	1	no	—
43	C₁₆	—	466	1	no	—
44	C₂³	—	932	1	no	—
45	D₄	—	2	1	no	—
46	D₄	—	932	1	no	—
47	D₄	—	466	1	no	—
48	C₂×C₄	—	932	1	no	—
49	C₂×C₄	—	233	1	no	—
50	C₈	—	233	1	no	—
51	C₈	C₂×#11745400	1	1	yes	characteristic
52	C₂²	—	4	1	no	—
53	C₂²	—	1864	1	no	—
54	C₂²	—	1864	1	no	—
55	C₂²	—	932	1	no	—
56	C₂²	—	233	1	no	—
57	C₄	—	1864	1	no	—
58	C₄	—	233	1	no	—
59	C₄	C₂²⋊#11745400	1	1	yes	characteristic
60	C₂	—	8	1	no	—
61	C₂	—	1864	1	no	—
62	C₂	—	233	1	no	—
63	C₂	—	233	1	no	—
64	C₂	#21418597	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21418597
commutator	C₈×C₂₃₃	C₂×C₄
Frattini	C₈	C₂×#11745400
Fitting	D₈×C₂₃₃	C₄
socle	C₂×C₂₃₃	#102
prime radical	D₈×D₂₃₃	C₂
cosocle	C₈×D₂₃₃	C₂²
hypercenter	D₈	#11745400
hypocenter	C₂₃₃	#357
LCS	G ▷ C₈×C₂₃₃ ▷ C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #21418608	C₂	semidirect
2. #21418604	C₂	semidirect
7. D₁₈₆₄	C₄	semidirect
10. D₈×C₂₃₃	C₄	semidirect
32. C₂₃₃	#357	semidirect
40. D₈	#11745400	semidirect

Maximal subgroups

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

subgroup	index
#21418608	2
#21418604	2
D₈×D₂₃₃	2
#357	233