Subgroups — #28378150

Order 13952.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295958	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812749	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295444	C₂	1	1	yes	maximal, characteristic
4	#16812749	—	2	1	no	—
5	#16812749	—	2	1	no	—
6	#16812705	C₄	1	1	yes	characteristic
7	#16812638	—	2	1	no	—
8	C₂×#13182004	C₂²	1	1	yes	characteristic
9	C₁₆×D₁₀₉	—	2	1	no	—
10	M₃₂×C₁₀₉	C₄	1	1	yes	characteristic
11	#13182029	C₈	1	1	yes	characteristic
12	#13182013	C₈	1	1	yes	characteristic
13	#13182012	—	2	1	no	—
14	C₈×D₁₀₉	—	2	1	no	—
15	#13182004	D₄	1	1	yes	characteristic
16	#13182004	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₀₉	—	2	1	no	—
19	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
20	C₂²⋊D₁₀₉	—	2	1	no	—
21	D₄₃₆	C₁₆	1	1	yes	characteristic
22	C₄×D₁₀₉	—	2	1	no	—
23	Dic₂₁₈	C₁₆	1	1	yes	characteristic
24	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
26	#11724784	—	2	1	no	—
27	C₂²×C₁₀₉	M₃₂	1	1	yes	characteristic
28	C₂×D₁₀₉	—	2	1	no	—
29	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
31	Dic₁₀₉	—	2	1	no	—
32	C₂×C₁₀₉	—	2	1	no	—
33	C₂×C₁₀₉	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₁₀₉	—	4	1	no	—
35	#1401	—	109	1	no	maximal, Sylow
36	C₁₀₉	#1401	1	1	yes	characteristic, complemented, Sylow
37	#504	—	109	1	no	—
38	M₆₄	—	109	1	no	—
39	C₂×C₃₂	—	109	1	no	—
40	#131	—	109	1	no	—
41	M₃₂	—	218	1	no	—
42	M₃₂	#89137	1	1	yes	characteristic
43	C₂×C₁₆	—	109	1	no	—
44	C₂×C₁₆	—	218	1	no	—
45	C₃₂	—	218	1	no	—
46	C₃₂	—	218	1	no	—
47	#41	—	109	1	no	—
48	M₁₆	—	109	1	no	—
49	M₁₆	—	218	1	no	—
50	C₂×C₈	—	218	1	no	—
51	C₂×C₈	C₂×#89137	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	—	109	1	no	—
54	C₁₆	—	109	1	no	—
55	Q₈	—	109	1	no	—
56	D₄	—	109	1	no	—
57	D₄	—	218	1	no	—
58	C₂×C₄	—	218	1	no	—
59	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
60	C₈	—	218	1	no	—
61	C₈	C₂²⋊#89137	1	1	yes	characteristic
62	C₈	#13182039	1	1	yes	characteristic
63	C₂²	—	218	1	no	—
64	C₂²	#16812774	1	1	yes	characteristic
65	C₄	—	218	1	no	—
66	C₄	#16812756	1	1	yes	characteristic
67	C₄	D₁₀₉⋊C₁₆	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	436	1	no	—
70	C₂	#20295921	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₂×C₁₆
Frattini	C₂×C₈	C₂×#89137
Fitting	M₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂²⋊C₁₆
prime radical	#13182029	C₈
cosocle	C₂×#13182004	C₂²
hypercenter	M₃₂	#89137
hypocenter	C₁₀₉	#1401
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #20295958	C₂	semidirect
2. C₂×#16812749	C₂	semidirect
36. C₁₀₉	#1401	semidirect

Maximal subgroups

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

subgroup	index
#20295958	2
C₂×#16812749	2
#20295444	2
#1401	109