Subgroups — #28365307

Order 13952.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295431	C₂	1	2	yes	maximal
2	#20294852	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812699	—	2	2	no	—
5	#16812633	—	2	2	no	—
6	#16812633	C₂²	1	4	yes	—
7	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₁₈	—	4	2	no	—
10	C₄⋊Dic₁₀₉	—	2	4	no	—
11	C₄⋊Dic₁₀₉	—	4	2	no	—
12	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
14	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
15	C₁₆×C₁₀₉	Q₈	1	4	yes	complemented
16	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
18	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
19	C₂×Dic₁₀₉	—	4	4	no	—
20	Dic₂₁₈	—	8	4	no	—
21	C₈×C₁₀₉	Q₁₆	1	2	yes	—
22	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
23	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
24	C₄×C₁₀₉	D₁₆	1	2	yes	—
25	C₄×C₁₀₉	Q₃₂	1	2	yes	—
26	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₁₀₉	—	8	4	no	—
29	C₂×C₁₀₉	#500	1	1	yes	characteristic
30	C₂×C₁₀₉	C₂×D₁₆	1	1	yes	characteristic
31	C₂×C₁₀₉	C₂×Q₃₂	1	1	yes	characteristic
32	#2252	—	109	1	no	maximal, Sylow
33	C₁₀₉	#2252	1	1	yes	characteristic, complemented, Sylow
34	#500	—	109	2	no	—
35	#366	—	109	4	no	—
36	C₄×C₁₆	D₁₀₉	1	1	yes	characteristic
37	C₄⋊Q₈	—	218	2	no	—
38	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
39	#107	—	109	4	no	—
40	#107	—	218	2	no	—
41	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
42	C₂×Q₈	—	436	2	no	—
43	C₂×C₈	D₄₃₆	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
45	C₄⋊C₄	—	436	2	no	—
46	C₄⋊C₄	—	218	4	no	—
47	C₄²	D₄₃₆	1	1	yes	characteristic
48	C₁₆	Dic₂₁₈	1	4	yes	complemented
49	Q₈	—	872	4	no	—
50	C₂×C₄	—	436	4	no	—
51	C₂×C₄	D₈₇₂	1	1	yes	characteristic
52	C₂×C₄	D₈₇₂	1	1	yes	characteristic
53	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
54	C₈	Dic₄₃₆	1	2	yes	—
55	C₈	C₂×Dic₂₁₈	1	2	yes	—
56	C₂²	C₂×D₈₇₂	1	1	yes	characteristic
57	C₄	—	872	4	no	—
58	C₄	D₁₇₄₄	1	2	yes	—
59	C₄	Dic₈₇₂	1	2	yes	—
60	C₄	#16812699	1	1	yes	characteristic
61	C₄	C₂×Dic₄₃₆	1	1	yes	characteristic
62	C₂	#20295431	1	1	yes	characteristic
63	C₂	C₂×D₁₇₄₄	1	1	yes	characteristic
64	C₂	C₂×Dic₈₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₇₂
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	C₄×C₁₆×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂×D₈
hypercenter	C₄×C₁₆	D₁₀₉
hypocenter	C₁₀₉	#2252
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₁₀₉	Q₈	semidirect
33. C₁₀₉	#2252	semidirect
48. C₁₆	Dic₂₁₈	semidirect

Maximal subgroups

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

subgroup	index
#20295431	2
#20294852	2
C₄×C₁₆×C₁₀₉	2
#2252	109