Subgroups — #28361064

Order 13952.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295391	C₂	1	1	yes	maximal, characteristic
2	#20294804	C₂	1	1	yes	maximal, characteristic, complemented
3	#347×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812696	C₄	1	1	yes	characteristic
5	#16812691	C₄	1	1	yes	characteristic
6	#16812639	—	2	1	no	—
7	#16812621	—	2	1	no	—
8	#16812621	—	2	1	no	—
9	M₃₂×C₁₀₉	—	2	1	no	—
10	#97×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×D₄₃₆	—	2	1	no	—
12	C₂×D₄₃₆	—	4	1	no	—
13	C₄⋊Dic₁₀₉	—	2	1	no	—
14	#13182008	—	4	1	no	—
15	#13182004	—	2	2	no	—
16	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
18	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₀₉	—	2	2	no	—
20	C₂²×D₁₀₉	—	4	1	no	—
21	C₂×C₄×C₁₀₉	—	2	1	no	—
22	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₁₀₉	—	4	1	no	—
24	D₄₃₆	—	4	1	no	—
25	D₄₃₆	—	4	1	no	—
26	D₄₃₆	—	8	2	no	—
27	C₈×C₁₀₉	—	2	1	no	—
28	C₈×C₁₀₉	—	2	1	no	—
29	C₂²×C₁₀₉	#100	1	1	yes	characteristic
30	C₂×D₁₀₉	—	8	2	no	—
31	C₂×D₁₀₉	—	8	1	no	—
32	C₄×C₁₀₉	—	4	1	no	—
33	C₄×C₁₀₉	#104	1	1	yes	characteristic
34	C₄×C₁₀₉	#104	1	1	yes	characteristic
35	Dic₁₀₉	—	8	1	no	—
36	C₂×C₁₀₉	—	2	1	no	—
37	C₂×C₁₀₉	#329	1	1	yes	characteristic
38	D₁₀₉	—	16	1	no	—
39	#1355	—	109	1	no	maximal, Sylow
40	C₁₀₉	#1355	1	1	yes	characteristic, complemented, Sylow
41	#490	—	109	1	no	—
42	#347	—	109	1	no	—
43	#347	D₁₀₉	1	1	yes	characteristic, complemented
44	#127	—	109	1	no	—
45	#125	—	109	1	no	—
46	M₃₂	—	2	1	no	—
47	M₃₂	—	218	1	no	—
48	#102	—	218	1	no	—
49	#102	—	218	1	no	—
50	#97	C₂×D₁₀₉	1	1	yes	characteristic
51	C₂×D₄	—	436	1	no	—
52	C₂×D₄	—	218	1	no	—
53	C₂×C₈	D₄₃₆	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
55	C₄⋊C₄	—	436	1	no	—
56	C₄⋊C₄	—	218	1	no	—
57	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
58	C₁₆	—	2	2	no	—
59	C₁₆	—	218	2	no	—
60	C₂³	—	436	1	no	—
61	D₄	—	872	2	no	—
62	D₄	—	436	1	no	—
63	D₄	—	436	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	436	1	no	—
66	C₂×C₄	#13182006	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₂²	—	872	2	no	—
70	C₂²	—	872	1	no	—
71	C₂²	#16812615	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	872	1	no	—
74	C₄	#16812625	1	1	yes	characteristic
75	C₄	#16812627	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	1744	1	no	—
78	C₂	#20294765	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20294765
commutator	C₄²×C₁₀₉	C₂×C₄
Frattini	#97	C₂×D₁₀₉
Fitting	#347×C₁₀₉	C₂
socle	C₂×C₁₀₉	#329
prime radical	#16812696	C₄
cosocle	#97×C₁₀₉	C₂²
hypercenter	#347	D₁₀₉
hypocenter	C₁₀₉	#1355
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #20294804	C₂	semidirect
3. #347×C₁₀₉	C₂	semidirect
40. C₁₀₉	#1355	semidirect
43. #347	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295391	2
#20294804	2
#347×C₁₀₉	2
#1355	109