Subgroups — #28378184

Order 13952.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295954	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295459	C₂	1	1	yes	maximal, characteristic
4	#16812770	—	2	1	no	—
5	#16812749	—	2	1	no	—
6	#16812722	—	2	1	no	—
7	#16812642	—	2	1	no	—
8	D₁₇₄₄	C₄	1	1	yes	characteristic
9	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
10	Q₃₂×C₁₀₉	C₄	1	1	yes	characteristic
11	#13182039	—	4	1	no	—
12	#13182030	—	4	1	no	—
13	#13182016	—	4	1	no	—
14	D₈₇₂	—	2	1	no	—
15	C₈×D₁₀₉	D₄	1	1	yes	characteristic
16	#13182004	D₄	1	1	yes	characteristic
17	Q₁₆×C₁₀₉	—	2	1	no	—
18	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	Q₈×C₁₀₉	—	4	1	no	—
20	D₄₃₆	—	4	1	no	—
21	D₄₃₆	—	8	1	no	—
22	C₄×D₁₀₉	—	8	1	no	—
23	C₄×D₁₀₉	D₈	1	1	yes	characteristic
24	#11724786	—	8	1	no	—
25	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
26	#11724784	SD₁₆	1	1	yes	characteristic
27	C₂×D₁₀₉	—	8	1	no	—
28	C₂×D₁₀₉	SD₃₂	1	1	yes	characteristic
29	C₄×C₁₀₉	—	8	1	no	—
30	C₄×C₁₀₉	#102	1	1	yes	characteristic
31	Dic₁₀₉	D₁₆	1	1	yes	characteristic
32	C₂×C₁₀₉	#357	1	1	yes	characteristic
33	D₁₀₉	—	2	1	no	—
34	D₁₀₉	—	16	1	no	—
35	#1417	—	109	1	no	maximal, Sylow
36	C₁₀₉	#1417	1	1	yes	characteristic, complemented, Sylow
37	#508	—	109	1	no	—
38	C₂×C₃₂	—	109	1	no	—
39	#368	—	109	1	no	—
40	#135	—	218	1	no	—
41	Q₃₂	#89137	1	1	yes	characteristic
42	SD₃₂	—	218	1	no	—
43	D₁₆	—	109	1	no	—
44	C₂×C₁₆	—	109	1	no	—
45	#108	—	218	1	no	—
46	C₃₂	—	218	1	no	—
47	#41	—	436	1	no	—
48	Q₁₆	—	2	1	no	—
49	SD₁₆	—	436	1	no	—
50	D₈	—	218	1	no	—
51	M₁₆	—	436	1	no	—
52	C₂×C₈	—	109	1	no	—
53	C₁₆	—	109	1	no	—
54	C₁₆	C₂×#89137	1	1	yes	characteristic
55	Q₈	—	4	1	no	—
56	D₄	—	872	1	no	—
57	D₄	—	436	1	no	—
58	C₂×C₄	—	872	1	no	—
59	C₂×C₄	—	109	1	no	—
60	C₈	—	872	1	no	—
61	C₈	—	109	1	no	—
62	C₈	C₂²⋊#89137	1	1	yes	characteristic
63	C₂²	—	872	1	no	—
64	C₂²	—	109	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	109	1	no	—
67	C₄	#16812761	1	1	yes	characteristic
68	C₂	—	1744	1	no	—
69	C₂	—	218	1	no	—
70	C₂	#20295938	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295938
commutator	C₁₆×C₁₀₉	C₂×C₄
Frattini	C₁₆	C₂×#89137
Fitting	Q₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	#357
prime radical	#20295459	C₂
cosocle	C₁₆×D₁₀₉	C₂²
hypercenter	Q₃₂	#89137
hypocenter	C₁₀₉	#1417
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. D₁₀₉⋊C₃₂	C₂	semidirect
2. #20295954	C₂	semidirect
36. C₁₀₉	#1417	semidirect

Maximal subgroups

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

subgroup	index
D₁₀₉⋊C₃₂	2
#20295954	2
#20295459	2
#1417	109