Subgroups — #28361105

Order 13952.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13182004	C₂	1	1	yes	maximal, characteristic
2	#20294779	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182004	C₄	1	1	yes	characteristic
5	C₂×#13182004	C₄	1	1	yes	characteristic
6	C₄⋊#11724784	—	2	1	no	—
7	C₄⋊C₈×C₁₀₉	—	2	1	no	—
8	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#11724784	—	4	1	no	—
10	#13182004	—	2	1	no	—
11	#13182004	C₈	1	2	yes	complemented
12	C₂×C₈×C₁₀₉	—	4	1	no	—
13	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
15	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
18	C₂×C₄×C₁₀₉	Q₁₆	1	1	yes	characteristic
19	C₈×C₁₀₉	—	2	1	no	—
20	C₈×C₁₀₉	—	8	1	no	—
21	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
22	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
23	#11724784	—	8	1	no	—
24	C₂²×C₁₀₉	#107	1	1	yes	characteristic
25	C₄×C₁₀₉	—	2	1	no	—
26	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₁₀₉	#108	1	1	yes	characteristic
28	C₄×C₁₀₉	SD₃₂	1	1	yes	characteristic
29	C₄×C₁₀₉	SD₃₂	1	1	yes	characteristic
30	C₂×C₁₀₉	#335	1	1	yes	characteristic
31	C₂×C₁₀₉	#367	1	1	yes	characteristic
32	C₂×C₁₀₉	#368	1	1	yes	characteristic
33	#1372	—	109	1	no	maximal, Sylow
34	C₁₀₉	#1372	1	1	yes	characteristic, complemented, Sylow
35	C₄×C₁₆	—	109	1	no	—
36	#335	—	109	1	no	—
37	#335	D₁₀₉	1	1	yes	characteristic
38	C₂×C₁₆	—	109	1	no	—
39	C₂×C₁₆	—	109	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	218	1	no	—
42	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	—	436	1	no	—
45	C₂×C₈	Dic₂₁₈	1	1	yes	characteristic
46	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
47	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
48	C₁₆	—	109	2	no	—
49	C₁₆	—	218	1	no	—
50	C₂×C₄	#13182008	1	1	yes	characteristic
51	C₂×C₄	#13182014	1	1	yes	characteristic
52	C₂×C₄	#13182019	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	8	1	no	—
55	C₈	—	872	1	no	—
56	C₈	C₈×D₁₀₉	1	1	yes	characteristic
57	C₈	#13182012	1	1	yes	characteristic
58	C₂²	#16812632	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	#16812628	1	1	yes	characteristic
61	C₄	#16812634	1	1	yes	characteristic
62	C₄	#16812642	1	1	yes	characteristic
63	C₄	#16812643	1	1	yes	characteristic
64	C₂	#20294778	1	1	yes	characteristic
65	C₂	#20294853	1	1	yes	characteristic
66	C₂	#20294855	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13182014
commutator	C₈×C₁₀₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₉
Fitting	#335×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#107
cosocle	C₄×C₈×C₁₀₉	C₂²
hypercenter	#335	D₁₀₉
hypocenter	C₁₀₉	#1372
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
11. #13182004	C₈	semidirect
34. C₁₀₉	#1372	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13182004	2
#20294779	2
#335×C₁₀₉	2
#1372	109