Subgroups — #28365313

Order 13952.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295430	C₂	1	2	yes	maximal
2	#20294854	C₂	1	1	yes	maximal, characteristic
3	#20294853	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#367×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812698	—	2	2	no	—
7	C₂×#13182004	C₂²	1	1	yes	characteristic
8	#16812633	C₂²	1	2	yes	—
9	#16812632	—	2	2	no	—
10	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	#107×C₁₀₉	C₂²	1	2	yes	—
13	C₂×Dic₂₁₈	—	4	2	no	—
14	C₂×#11724784	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₀₉	—	2	2	no	—
16	#13182008	—	4	2	no	—
17	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
18	#13182004	Q₈	1	2	yes	complemented
19	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₉	—	2	2	no	—
21	C₁₆×C₁₀₉	Q₈	1	2	yes	complemented
22	C₂×C₄×C₁₀₉	—	4	2	no	—
23	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₁₀₉	—	4	2	no	—
25	C₂×Dic₁₀₉	D₈	1	2	yes	—
26	Dic₂₁₈	—	8	4	no	—
27	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
28	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
29	#11724784	Q₁₆	1	2	yes	—
30	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
31	C₄×C₁₀₉	—	8	2	no	—
32	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
33	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
34	Dic₁₀₉	—	8	2	no	—
35	Dic₁₀₉	SD₃₂	1	4	yes	—
36	C₂×C₁₀₉	#500	1	1	yes	characteristic
37	C₂×C₁₀₉	C₂×SD₃₂	1	2	yes	—
38	#2254	—	109	1	no	maximal, Sylow
39	C₁₀₉	#2254	1	1	yes	characteristic, complemented, Sylow
40	#500	—	109	2	no	—
41	#367	—	109	1	no	—
42	#367	—	109	2	no	—
43	#367	D₁₀₉	1	1	yes	characteristic
44	C₄×C₁₆	—	109	1	no	—
45	C₄⋊Q₈	—	218	2	no	—
46	C₂×C₁₆	—	109	1	no	—
47	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
48	#107	—	109	2	no	—
49	#107	—	218	2	no	—
50	#107	C₂×D₁₀₉	1	2	yes	—
51	C₄×C₈	—	109	1	no	—
52	C₂×Q₈	—	436	2	no	—
53	C₂×C₈	—	109	1	no	—
54	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	436	2	no	—
57	C₄⋊C₄	—	218	2	no	—
58	C₄²	—	109	1	no	—
59	C₁₆	—	109	2	no	—
60	C₁₆	Dic₂₁₈	1	2	yes	complemented
61	Q₈	—	872	4	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	109	2	no	—
64	C₂×C₄	—	436	2	no	—
65	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
66	C₈	—	109	2	no	—
67	C₈	Q₈×D₁₀₉	1	1	yes	characteristic
68	C₈	C₂×Dic₂₁₈	1	1	yes	characteristic
69	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
70	C₄	—	8	2	no	—
71	C₄	—	872	2	no	—
72	C₄	—	109	4	no	—
73	C₄	#16812698	1	1	yes	characteristic
74	C₄	Q₁₆×D₁₀₉	1	1	yes	characteristic
75	C₂	#20295430	1	1	yes	characteristic
76	C₂	SD₃₂×D₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#367×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂×D₈
hypercenter	#367	D₁₀₉
hypocenter	C₁₀₉	#2254
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
18. #13182004	Q₈	semidirect
21. C₁₆×C₁₀₉	Q₈	semidirect
39. C₁₀₉	#2254	semidirect
60. C₁₆	Dic₂₁₈	semidirect

Maximal subgroups

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

subgroup	index
#20295430	2
#20294854	2
#20294853	2
C₁₆×Dic₁₀₉	2
#367×C₁₀₉	2
#2254	109