Subgroups — #28378053

Order 13952.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295949	C₂	1	1	yes	maximal, characteristic
2	#20295949	C₂	1	1	yes	maximal, characteristic
3	#20295425	C₂	1	1	yes	maximal, characteristic
4	#16812773	—	2	1	no	—
5	#16812773	—	2	1	no	—
6	#16812691	C₄	1	1	yes	characteristic
7	#16812632	—	2	1	no	—
8	#16812630	—	2	1	no	—
9	C₄×#11724784	C₂²	1	1	yes	characteristic
10	#125×C₁₀₉	C₄	1	1	yes	characteristic
11	#13182032	—	4	1	no	—
12	#13182032	—	4	1	no	—
13	C₂×#11724784	D₄	1	1	yes	characteristic
14	C₂×#11724784	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₀₉	—	2	1	no	—
16	#13182008	—	4	1	no	—
17	C₄⋊C₄×C₁₀₉	—	2	1	no	—
18	C₄⋊C₄×C₁₀₉	—	4	1	no	—
19	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₉	—	2	1	no	—
21	C₂×C₄×C₁₀₉	—	4	1	no	—
22	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₁₀₉	—	4	1	no	—
24	#11724784	SD₁₆	1	1	yes	characteristic
25	#11724784	SD₁₆	1	1	yes	characteristic
26	#11724784	Q₁₆	1	1	yes	characteristic
27	#11724784	Q₁₆	1	1	yes	characteristic
28	C₂²×C₁₀₉	#100	1	1	yes	characteristic
29	C₄×C₁₀₉	—	4	1	no	—
30	C₄×C₁₀₉	—	8	1	no	—
31	C₄×C₁₀₉	#103	1	1	yes	characteristic
32	C₄×C₁₀₉	#103	1	1	yes	characteristic
33	Dic₁₀₉	—	8	1	no	—
34	C₂×C₁₀₉	—	2	1	no	—
35	C₂×C₁₀₉	#333	1	1	yes	characteristic
36	#1366	—	109	1	no	maximal, Sylow
37	C₁₀₉	#1366	1	1	yes	characteristic, complemented, Sylow
38	#499	—	109	1	no	—
39	#364	—	109	1	no	—
40	#364	—	109	1	no	—
41	#125	—	109	1	no	—
42	#125	#89137	1	1	yes	characteristic
43	M₃₂	—	218	1	no	—
44	M₃₂	—	218	1	no	—
45	#107	—	218	1	no	—
46	#106	—	218	1	no	—
47	C₄×C₈	—	109	1	no	—
48	C₂×C₈	—	109	1	no	—
49	C₂×C₈	—	109	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	4	1	no	—
52	C₄⋊C₄	—	436	1	no	—
53	C₄⋊C₄	—	218	1	no	—
54	C₄²	C₂×#89137	1	1	yes	characteristic
55	C₁₆	—	436	1	no	—
56	C₁₆	—	436	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	436	1	no	—
60	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
61	C₈	—	109	1	no	—
62	C₈	—	109	1	no	—
63	C₈	—	109	1	no	—
64	C₈	—	109	1	no	—
65	C₂²	#16812759	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	8	1	no	—
68	C₄	—	872	1	no	—
69	C₄	#16812762	1	1	yes	characteristic
70	C₄	#16812762	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#20295886	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295886
commutator	C₄²×C₁₀₉	C₂×C₄
Frattini	C₄²	C₂×#89137
Fitting	#125×C₁₀₉	C₄
socle	C₂×C₁₀₉	#333
prime radical	C₂²×C₁₀₉	#100
cosocle	C₄×#11724784	C₂²
hypercenter	#125	#89137
hypocenter	C₁₀₉	#1366
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
37. C₁₀₉	#1366	semidirect

Maximal subgroups

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

subgroup	index
#20295949	2
#20295425	2
#1366	109