Subgroups — #28365342

Order 13952.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295459	C₂	1	2	yes	maximal, complemented
2	#20294863	C₂	1	2	yes	maximal, complemented
3	D₃₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812722	—	2	2	no	—
7	#16812642	—	2	2	no	—
8	D₁₇₄₄	C₂²	1	2	yes	—
9	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
10	#16812605	C₂²	1	1	yes	characteristic
11	Q₃₂×C₁₀₉	C₂²	1	2	yes	—
12	C₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
13	#13182030	—	4	2	no	—
14	#13182016	—	4	2	no	—
15	D₈₇₂	—	2	2	no	—
16	C₈×D₁₀₉	D₄	1	1	yes	characteristic
17	#13182004	D₄	1	1	yes	characteristic
18	Q₁₆×C₁₀₉	—	2	2	no	—
19	C₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
20	Q₈×C₁₀₉	—	4	2	no	—
21	D₄₃₆	—	4	2	no	—
22	D₄₃₆	—	8	2	no	—
23	C₄×D₁₀₉	—	8	2	no	—
24	C₄×D₁₀₉	D₈	1	1	yes	characteristic
25	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
26	#11724784	D₈	1	1	yes	characteristic
27	C₂×D₁₀₉	—	8	2	no	—
28	C₂×D₁₀₉	D₁₆	1	1	yes	characteristic
29	C₄×C₁₀₉	—	8	2	no	—
30	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
31	Dic₁₀₉	D₁₆	1	1	yes	characteristic
32	C₂×C₁₀₉	C₂×D₁₆	1	1	yes	characteristic
33	D₁₀₉	—	2	1	no	—
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	D₁₀₉	1	1	yes	characteristic, complemented
39	SD₆₄	—	109	2	no	—
40	D₃₂	—	109	1	no	—
41	C₂×C₃₂	—	109	1	no	—
42	#135	—	218	2	no	—
43	Q₃₂	C₂×D₁₀₉	1	2	yes	—
44	SD₃₂	—	218	2	no	—
45	D₁₆	—	109	2	no	—
46	C₂×C₁₆	—	109	1	no	—
47	C₃₂	—	109	1	no	—
48	C₃₂	C₂×D₁₀₉	1	1	yes	characteristic
49	#41	—	436	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	436	2	no	—
52	D₈	—	218	2	no	—
53	C₂×C₈	—	109	1	no	—
54	C₁₆	—	109	1	no	—
55	C₁₆	C₂²×D₁₀₉	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	872	2	no	—
58	D₄	—	436	2	no	—
59	C₂×C₄	—	872	2	no	—
60	C₂×C₄	—	109	1	no	—
61	C₈	—	109	1	no	—
62	C₈	D₄×D₁₀₉	1	1	yes	characteristic
63	C₂²	—	872	2	no	—
64	C₂²	—	109	1	no	—
65	C₄	—	8	2	no	—
66	C₄	—	109	1	no	—
67	C₄	D₈×D₁₀₉	1	1	yes	characteristic
68	C₂	—	1744	2	no	—
69	C₂	—	218	1	no	—
70	C₂	D₁₆×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₁₀₉
commutator, cosocle	C₁₆×C₁₀₉	C₂³
Frattini	C₁₆	C₂²×D₁₀₉
Fitting	Q₆₄×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×D₁₆
hypercenter	Q₆₄	D₁₀₉
hypocenter	C₁₀₉	#2262
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #20295459	C₂	semidirect
2. #20294863	C₂	semidirect
3. D₃₄₈₈	C₂	semidirect
4. C₃₂×D₁₀₉	C₂	semidirect
5. Q₆₄×C₁₀₉	C₂	semidirect
36. C₁₀₉	#2262	semidirect
38. Q₆₄	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295459	2
#20294863	2
D₃₄₈₈	2
C₃₂×D₁₀₉	2
Q₆₄×C₁₀₉	2
#2262	109