Subgroups — #28365341

Order 13952.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295458	C₂	1	2	yes	maximal, complemented
2	Dic₁₇₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294865	C₂	1	2	yes	maximal, complemented
4	D₃₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812721	—	2	2	no	—
7	Dic₈₇₂	C₂²	1	2	yes	—
8	#16812644	—	2	2	no	—
9	D₁₇₄₄	C₂²	1	2	yes	—
10	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
13	#13182029	—	4	2	no	—
14	Dic₄₃₆	—	2	2	no	—
15	#13182018	—	4	2	no	—
16	D₈₇₂	—	2	2	no	—
17	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
18	C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
19	C₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
21	C₂²⋊D₁₀₉	—	8	2	no	—
22	D₄₃₆	—	4	2	no	—
23	C₄×D₁₀₉	—	8	2	no	—
24	Dic₂₁₈	—	4	2	no	—
25	C₈×C₁₀₉	D₈	1	1	yes	characteristic
26	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₁₀₉	D₁₆	1	1	yes	characteristic
28	C₂×D₁₀₉	—	8	2	no	—
29	C₄×C₁₀₉	D₁₆	1	1	yes	characteristic
30	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
31	Dic₁₀₉	—	8	2	no	—
32	C₂×C₁₀₉	—	2	1	no	—
33	C₂×C₁₀₉	C₂×D₁₆	1	1	yes	characteristic
34	D₁₀₉	—	16	2	no	—
35	#2262	—	109	1	no	maximal, Sylow
36	C₁₀₉	#2262	1	1	yes	characteristic, complemented, Sylow
37	#508	—	109	2	no	—
38	Q₆₄	—	109	1	no	—
39	SD₆₄	—	109	2	no	—
40	D₃₂	—	109	1	no	—
41	C₂×C₃₂	D₁₀₉	1	1	yes	characteristic, complemented
42	#135	—	218	2	no	—
43	Q₃₂	—	109	2	no	—
44	SD₃₂	—	218	2	no	—
45	D₁₆	—	109	2	no	—
46	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
47	C₃₂	C₂×D₁₀₉	1	1	yes	characteristic
48	C₃₂	C₂×D₁₀₉	1	1	yes	characteristic
49	#41	—	436	2	no	—
50	Q₁₆	—	218	2	no	—
51	SD₁₆	—	436	2	no	—
52	D₈	—	218	2	no	—
53	C₂×C₈	D₄₃₆	1	1	yes	characteristic
54	C₁₆	D₄₃₆	1	1	yes	characteristic
55	C₁₆	C₂²×D₁₀₉	1	1	yes	characteristic
56	Q₈	—	436	2	no	—
57	D₄	—	872	2	no	—
58	D₄	—	436	2	no	—
59	C₂×C₄	—	872	2	no	—
60	C₂×C₄	D₈₇₂	1	1	yes	characteristic
61	C₈	D₈₇₂	1	1	yes	characteristic
62	C₈	C₂×D₄₃₆	1	1	yes	characteristic
63	C₂²	—	872	2	no	—
64	C₂²	D₁₇₄₄	1	1	yes	characteristic
65	C₄	—	872	2	no	—
66	C₄	D₁₇₄₄	1	1	yes	characteristic
67	C₄	C₂×D₈₇₂	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1744	2	no	—
70	C₂	C₂×D₁₇₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₇₄₄
commutator, cosocle	C₁₆×C₁₀₉	C₂³
Frattini	C₁₆	C₂²×D₁₀₉
Fitting	C₂×C₃₂×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×D₁₆
hypercenter	C₂×C₃₂	D₁₀₉
hypocenter	C₁₀₉	#2262
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #20295458	C₂	semidirect
2. Dic₁₇₄₄	C₂	semidirect
3. #20294865	C₂	semidirect
4. D₃₄₈₈	C₂	semidirect
5. C₂×C₃₂×C₁₀₉	C₂	semidirect
36. C₁₀₉	#2262	semidirect
41. C₂×C₃₂	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295458	2
Dic₁₇₄₄	2
#20294865	2
D₃₄₈₈	2
C₂×C₃₂×C₁₀₉	2
#2262	109