Subgroups — #28378058

Order 13952.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295887	C₂	1	1	yes	maximal, characteristic
2	#20295887	C₂	1	1	yes	maximal, characteristic
3	#20294801	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11724786	—	2	1	no	—
5	C₄⋊#11724786	—	2	1	no	—
6	C₂×#13182004	C₄	1	1	yes	characteristic
7	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic
9	C₂×#11724786	—	4	1	no	—
10	C₂×#11724786	—	4	1	no	—
11	C₂×#11724784	Q₈	1	1	yes	characteristic
12	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
13	#13182004	C₈	1	2	yes	complemented
14	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₀₉	—	2	1	no	—
16	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
17	C₂×Dic₁₀₉	SD₁₆	1	1	yes	characteristic
18	C₂×Dic₁₀₉	SD₁₆	1	1	yes	characteristic
19	#11724786	—	8	1	no	—
20	#11724786	—	8	1	no	—
21	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
22	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
23	#11724784	—	2	1	no	—
24	C₂²×C₁₀₉	#106	1	1	yes	characteristic
25	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
26	C₄×C₁₀₉	#108	1	1	yes	characteristic
27	Dic₁₀₉	—	2	1	no	—
28	Dic₁₀₉	—	2	1	no	—
29	C₂×C₁₀₉	#334	1	1	yes	characteristic
30	C₂×C₁₀₉	#365	1	1	yes	characteristic
31	C₂×C₁₀₉	#365	1	1	yes	characteristic
32	#1368	—	109	1	no	maximal, Sylow
33	C₁₀₉	#1368	1	1	yes	characteristic, complemented, Sylow
34	#346	—	109	1	no	—
35	#334	—	109	1	no	—
36	#334	—	109	1	no	—
37	C₂×C₁₆	—	109	1	no	—
38	C₂×C₁₆	#89137	1	1	yes	characteristic
39	C₄⋊C₈	—	218	1	no	—
40	C₄⋊C₈	—	218	1	no	—
41	C₄×C₈	—	109	1	no	—
42	C₂×C₈	—	109	1	no	—
43	C₂×C₈	—	436	1	no	—
44	C₂×C₈	—	436	1	no	—
45	C₂×C₈	C₂×#89137	1	1	yes	characteristic
46	C₄²	—	109	1	no	—
47	C₁₆	—	2	1	no	—
48	C₁₆	—	109	2	no	—
49	C₂×C₄	—	109	1	no	—
50	C₂×C₄	—	109	1	no	—
51	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
52	C₈	—	872	1	no	—
53	C₈	—	872	1	no	—
54	C₈	—	218	1	no	—
55	C₈	D₁₀₉⋊C₈	1	1	yes	characteristic
56	C₈	#13182038	1	1	yes	characteristic
57	C₂²	#16812767	1	1	yes	characteristic
58	C₄	—	218	1	no	—
59	C₄	—	218	1	no	—
60	C₄	#16812766	1	1	yes	characteristic
61	C₄	#16812770	1	1	yes	characteristic
62	C₂	#20295888	1	1	yes	characteristic
63	C₂	#20295950	1	1	yes	characteristic
64	C₂	#20295951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812767
commutator	C₈×C₁₀₉	C₂×C₈
Frattini	C₂×C₈	C₂×#89137
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	#106
cosocle	C₈×Dic₁₀₉	C₂²
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	#1368
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
13. #13182004	C₈	semidirect
33. C₁₀₉	#1368	semidirect

Maximal subgroups

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

subgroup	index
#20295887	2
#20294801	2
#1368	109