Subgroups — #28361334

Order 13952.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13182012	C₂	1	3	yes	maximal
2	#20294747	C₂	1	3	yes	maximal, complemented
3	C₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13182012	C₄	1	6	yes	—
5	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
6	#16812609	C₄	1	6	yes	—
7	C₄×#11724784	C₂²	1	3	yes	—
8	C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
9	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
10	#13182012	—	2	12	no	—
11	C₂×#11724784	C₂×C₄	1	6	yes	—
12	C₄×Dic₁₀₉	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
14	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
15	C₂²×D₁₀₉	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
17	C₂×Dic₁₀₉	C₄²	1	3	yes	—
18	C₄×D₁₀₉	—	2	6	no	—
19	C₈×C₁₀₉	M₁₆	1	12	yes	complemented
20	#11724784	—	2	6	no	—
21	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
22	C₂×D₁₀₉	—	2	3	no	—
23	C₄×C₁₀₉	C₂×M₁₆	1	6	yes	—
24	Dic₁₀₉	—	2	3	no	—
25	C₂×C₁₀₉	C₄×M₁₆	1	3	yes	—
26	D₁₀₉	—	4	1	no	—
27	#1455	—	109	1	no	maximal, Sylow
28	C₁₀₉	#1455	1	1	yes	characteristic, complemented, Sylow
29	C₄×M₁₆	—	109	3	no	—
30	#322	—	109	3	no	—
31	C₈²	D₁₀₉	1	1	yes	characteristic, complemented
32	C₂×M₁₆	—	109	6	no	—
33	C₂×C₄²	—	109	1	no	—
34	#97	—	109	6	no	—
35	C₄×C₈	—	109	3	no	—
36	C₄×C₈	C₂×D₁₀₉	1	3	yes	—
37	C₂²×C₄	—	109	3	no	—
38	M₁₆	—	218	12	no	—
39	C₂×C₈	—	109	6	no	—
40	C₂×C₈	C₄×D₁₀₉	1	6	yes	—
41	C₄²	—	109	3	no	—
42	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
43	C₂³	—	109	1	no	—
44	C₂×C₄	—	109	3	no	—
45	C₂×C₄	—	218	6	no	—
46	C₂×C₄	C₂×C₄×D₁₀₉	1	3	yes	—
47	C₈	—	218	6	no	—
48	C₈	#13182012	1	12	yes	complemented
49	C₂²	—	218	3	no	—
50	C₂²	C₄²×D₁₀₉	1	1	yes	characteristic
51	C₄	—	218	3	no	—
52	C₄	C₂×#13182012	1	6	yes	—
53	C₂	—	436	1	no	—
54	C₂	C₄×#13182012	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂×C₄²
Fitting	C₈²×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₄²
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	C₈²	D₁₀₉
hypocenter	C₁₀₉	#1455
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #20294747	C₂	semidirect
3. C₈²×C₁₀₉	C₂	semidirect
19. C₈×C₁₀₉	M₁₆	semidirect
28. C₁₀₉	#1455	semidirect
31. C₈²	D₁₀₉	semidirect
48. C₈	#13182012	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13182012	2
#20294747	2
C₈²×C₁₀₉	2
#1455	109