Subgroups — #28377943

Order 13952.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295912	C₂	1	1	yes	maximal, characteristic
2	C₈×#11724786	C₂	1	1	yes	maximal, characteristic
3	C₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182032	C₄	1	2	yes	—
5	C₄×#11724786	C₄	1	2	yes	—
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	C₈	1	4	yes	—
10	#13182032	C₈	1	4	yes	complemented
11	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
12	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
13	#13182004	C₈	1	2	yes	complemented
14	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₀₉	C₈	1	2	yes	complemented
16	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
17	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
18	#11724786	—	2	4	no	—
19	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
20	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
21	#11724784	C₂×C₈	1	2	yes	—
22	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
23	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
24	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
25	Dic₁₀₉	M₃₂	1	4	yes	—
26	C₂×C₁₀₉	C₈²	1	1	yes	characteristic
27	C₂×C₁₀₉	#346	1	2	yes	—
28	#1311	—	109	1	no	maximal, Sylow
29	C₁₀₉	#1311	1	1	yes	characteristic, complemented, Sylow
30	C₄×C₁₆	—	109	1	no	—
31	C₄×C₁₆	—	109	1	no	—
32	C₈²	—	109	1	no	—
33	C₂×C₁₆	—	109	2	no	—
34	C₂×C₁₆	—	109	1	no	—
35	C₂×C₁₆	#89137	1	1	yes	characteristic
36	C₄×C₈	—	109	2	no	—
37	C₄×C₈	—	109	1	no	—
38	C₂×C₈	—	109	1	no	—
39	C₂×C₈	—	109	4	no	—
40	C₂×C₈	C₂×#89137	1	1	yes	characteristic
41	C₄²	—	109	1	no	—
42	C₁₆	—	109	4	no	—
43	C₁₆	—	109	2	no	—
44	C₁₆	#11724786	1	2	yes	complemented
45	C₂×C₄	—	109	2	no	—
46	C₂×C₄	C₄×#89137	1	1	yes	characteristic
47	C₈	—	109	2	no	—
48	C₈	—	218	4	no	—
49	C₈	C₂×#11724786	1	1	yes	characteristic
50	C₈	D₁₀₉⋊C₈	1	1	yes	characteristic
51	C₂²	C₈×#89137	1	1	yes	characteristic
52	C₄	—	109	4	no	—
53	C₄	C₈×#89137	1	1	yes	characteristic
54	C₄	C₄×#11724786	1	1	yes	characteristic
55	C₂	C₈×#11724786	1	1	yes	characteristic
56	C₂	#20295913	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#89137
commutator	C₂×C₁₀₉	C₈²
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₄×C₈
cosocle	C₈×Dic₁₀₉	C₂²
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	#1311
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
10. #13182032	C₈	semidirect
13. #13182004	C₈	semidirect
15. C₁₆×C₁₀₉	C₈	semidirect
29. C₁₀₉	#1311	semidirect
44. C₁₆	#11724786	semidirect

Maximal subgroups

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

subgroup	index
#20295912	2
C₈×#11724786	2
C₁₆×Dic₁₀₉	2
#1311	109