Subgroups — #28411597

Order 14656.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20301010	C₂	1	1	yes	maximal, characteristic
2	#20300996	C₂	1	2	yes	maximal, complemented
3	#20300914	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₂₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×C₂₂₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16814666	C₂²	1	2	yes	—
7	Dic₉₁₆	C₄	1	1	yes	characteristic, complemented
8	#16814655	C₄	1	2	yes	complemented
9	D₁₈₃₂	C₄	1	1	yes	characteristic, complemented
10	#16814649	—	2	2	no	—
11	C₂×#14270080	C₂²	1	1	yes	characteristic
12	C₈×D₂₂₉	—	2	2	no	—
13	C₄×Dic₂₂₉	C₂²	1	1	yes	characteristic
14	M₁₆×C₂₂₉	C₂²	1	2	yes	—
15	C₂×C₈×C₂₂₉	C₂²	1	1	yes	characteristic
16	Dic₄₅₈	C₂×C₄	1	2	yes	—
17	D₉₁₆	C₂×C₄	1	2	yes	—
18	C₂²⋊D₂₂₉	—	2	2	no	—
19	C₂×Dic₂₂₉	—	2	1	no	—
20	C₄×D₂₂₉	—	2	2	no	—
21	#14270080	D₄	1	1	yes	characteristic
22	#14270080	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₉	C₂³	1	1	yes	characteristic
24	C₈×C₂₂₉	—	2	2	no	—
25	C₈×C₂₂₉	C₂×C₄	1	1	yes	characteristic
26	C₈×C₂₂₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₂₂₉	#41	1	1	yes	characteristic
28	C₂×D₂₂₉	—	2	2	no	—
29	C₄×C₂₂₉	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₂₉	C₂×D₄	1	1	yes	characteristic
31	Dic₂₂₉	—	2	2	no	—
32	Dic₂₂₉	—	2	2	no	—
33	C₂×C₂₂₉	—	2	1	no	—
34	C₂×C₂₂₉	C₄×D₄	1	1	yes	characteristic
35	D₂₂₉	—	4	2	no	—
36	C₂₂₉	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	229	1	no	maximal, Sylow
38	#135	—	229	1	no	—
39	#131	—	229	2	no	—
40	#108	D₂₂₉	1	1	yes	characteristic, complemented
41	#104	—	229	2	no	—
42	C₄×C₈	—	229	1	no	—
43	#41	—	229	2	no	—
44	Q₁₆	—	229	1	no	—
45	SD₁₆	—	229	2	no	—
46	D₈	—	229	1	no	—
47	M₁₆	—	458	2	no	—
48	M₁₆	C₂×D₂₂₉	1	2	yes	—
49	C₂×C₈	—	458	2	no	—
50	C₂×C₈	—	229	1	no	—
51	C₂×C₈	C₂×D₂₂₉	1	1	yes	characteristic
52	C₄²	—	229	1	no	—
53	Q₈	—	229	2	no	—
54	D₄	—	458	2	no	—
55	D₄	—	229	2	no	—
56	C₂×C₄	—	458	1	no	—
57	C₂×C₄	—	458	2	no	—
58	C₂×C₄	C₂²×D₂₂₉	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	229	1	no	—
61	C₈	—	229	1	no	—
62	C₈	C₄×D₂₂₉	1	1	yes	characteristic
63	C₈	C₄×D₂₂₉	1	1	yes	characteristic
64	C₂²	—	458	2	no	—
65	C₂²	#16814667	1	1	yes	characteristic
66	C₄	—	458	2	no	—
67	C₄	—	458	2	no	—
68	C₄	C₂×C₄×D₂₂₉	1	1	yes	characteristic
69	C₄	D₄×D₂₂₉	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	916	2	no	—
72	C₂	#20300950	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₂₉
commutator	C₄×C₂₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₉
Fitting	#108×C₂₂₉	C₂
socle	C₂×C₂₂₉	C₄×D₄
prime radical	#20301010	C₂
cosocle	C₂×C₄×C₂₂₉	C₂³
hypercenter	#108	D₂₂₉
hypocenter	C₂₂₉	#443
LCS	G ▷ C₄×C₂₂₉ ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #20300996	C₂	semidirect
3. #20300914	C₂	semidirect
4. C₈×Dic₂₂₉	C₂	semidirect
5. #108×C₂₂₉	C₂	semidirect
7. Dic₉₁₆	C₄	semidirect
8. #16814655	C₄	semidirect
9. D₁₈₃₂	C₄	semidirect
36. C₂₂₉	#443	semidirect
40. #108	D₂₂₉	semidirect

Maximal subgroups

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

subgroup	index
#20301010	2
#20300996	2
#20300914	2
C₈×Dic₂₂₉	2
#108×C₂₂₉	2
#443	229