Subgroups — #28378959

Order 13952.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295892	C₂	1	6	yes	maximal
2	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11724786	C₂²	1	4	yes	—
4	C₂×C₄×Dic₁₀₉	C₄	1	3	yes	—
5	C₂×C₄×Dic₁₀₉	C₂²	1	3	yes	—
6	C₂×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#11724786	—	2	12	no	—
8	C₂²×Dic₁₀₉	C₂×C₄	1	3	yes	—
9	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
10	C₄×Dic₁₀₉	—	2	6	no	—
11	C₄×Dic₁₀₉	—	2	6	no	—
12	C₂²×C₄×C₁₀₉	C₂×C₄	1	3	yes	—
13	C₄²×C₁₀₉	C₈	1	4	yes	complemented
14	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	—	2	3	no	—
16	C₂×C₄×C₁₀₉	—	2	3	no	—
17	C₂×C₄×C₁₀₉	C₂×C₈	1	6	yes	—
18	C₂×Dic₁₀₉	—	2	3	no	—
19	C₂×Dic₁₀₉	—	2	3	no	—
20	C₂×Dic₁₀₉	M₁₆	1	6	yes	—
21	C₂×Dic₁₀₉	#41	1	6	yes	—
22	#11724786	—	4	4	no	—
23	C₂²×C₁₀₉	#117	1	3	yes	—
24	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
25	C₂²×C₁₀₉	C₂×M₁₆	1	3	yes	—
26	C₄×C₁₀₉	—	2	6	no	—
27	Dic₁₀₉	—	2	6	no	—
28	Dic₁₀₉	#126	1	4	yes	—
29	C₂×C₁₀₉	#383	1	1	yes	characteristic
30	C₂×C₁₀₉	#431	1	3	yes	—
31	C₂×C₁₀₉	#432	1	3	yes	—
32	#1839	—	109	1	no	maximal, Sylow
33	C₁₀₉	#1839	1	1	yes	characteristic, complemented, Sylow
34	C₄³	—	109	1	no	—
35	#336	—	109	6	no	—
36	C₂²×C₈	—	109	4	no	—
37	C₂×C₄²	—	109	3	no	—
38	C₂×C₄²	—	109	3	no	—
39	C₂×C₄²	#89137	1	1	yes	characteristic
40	C₂²×C₄	—	109	1	no	—
41	C₂²×C₄	—	109	3	no	—
42	C₂²×C₄	C₂×#89137	1	3	yes	—
43	C₂×C₈	—	218	12	no	—
44	C₄²	—	218	6	no	—
45	C₄²	—	218	6	no	—
46	C₄²	#11724786	1	4	yes	complemented
47	C₂³	C₂²×#89137	1	1	yes	characteristic
48	C₂×C₄	—	2	3	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	—	109	6	no	—
51	C₂×C₄	—	109	6	no	—
52	C₂×C₄	—	218	3	no	—
53	C₂×C₄	—	218	3	no	—
54	C₂×C₄	C₂×#11724786	1	6	yes	—
55	C₈	—	436	4	no	—
56	C₂²	#16812778	1	3	yes	—
57	C₂²	C₂²×#11724786	1	1	yes	characteristic
58	C₂²	D₁₀₉⋊M₁₆	1	3	yes	—
59	C₄	—	2	6	no	—
60	C₄	—	109	4	no	—
61	C₄	—	218	6	no	—
62	C₂	#20295971	1	1	yes	characteristic
63	C₂	#20296042	1	3	yes	—
64	C₂	#20296049	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#89137
commutator	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₂×C₄²×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×Dic₁₀₉	C₂³
hypercenter	C₂×C₄²	#89137
hypocenter	C₁₀₉	#1839
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
13. C₄²×C₁₀₉	C₈	semidirect
33. C₁₀₉	#1839	semidirect
46. C₄²	#11724786	semidirect

Maximal subgroups

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

subgroup	index
#20295892	2
C₄²×Dic₁₀₉	2
#1839	109