Subgroups — C₁₀₉×#14692 #42132686

Order 27904.

2286 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#384×C₁₀₉	C₂	1	14	yes	maximal
2	C₂×C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	28	yes	—
4	C₂²×C₄²×C₁₀₉	C₂²	1	7	yes	—
5	#384×C₁₀₉	—	2	56	no	—
6	C₄³×C₁₀₉	C₄	1	8	yes	complemented
7	C₂³×C₄×C₁₀₉	C₂³	1	8	yes	—
8	C₂³×C₄×C₁₀₉	C₂³	1	7	yes	—
9	C₂×C₄⋊C₄×C₁₀₉	—	2	168	no	—
10	C₂×C₄²×C₁₀₉	C₂×C₄	1	28	yes	—
11	C₂×C₄²×C₁₀₉	D₄	1	28	yes	—
12	C₂×C₄²×C₁₀₉	Q₈	1	28	yes	—
13	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₉	—	2	56	no	—
15	C₂²×C₄×C₁₀₉	C₂²×C₄	1	14	yes	—
16	C₂²×C₄×C₁₀₉	C₂×D₄	1	42	yes	—
17	C₂²×C₄×C₁₀₉	C₂×Q₈	1	42	yes	—
18	C₄⋊C₄×C₁₀₉	—	4	112	no	—
19	C₄²×C₁₀₉	C₄⋊C₄	1	112	yes	complemented
20	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
21	C₂³×C₁₀₉	C₂²×D₄	1	7	yes	—
22	C₂³×C₁₀₉	C₂²×Q₈	1	7	yes	—
23	C₂×C₄×C₁₀₉	—	4	56	no	—
24	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	84	yes	—
25	C₂×C₄×C₁₀₉	#127	1	28	yes	—
26	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	84	yes	—
27	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	7	yes	—
28	C₂²×C₁₀₉	C₂×#127	1	7	yes	—
29	C₂²×C₁₀₉	C₂×C₄⋊Q₈	1	21	yes	—
30	C₄×C₁₀₉	—	8	8	no	—
31	C₄×C₁₀₉	#384	1	56	yes	complemented
32	#14692	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₁₀₉	C₂×#384	1	7	yes	—
34	C₂×C₁₀₉	#2867	1	1	yes	characteristic
35	C₂×C₁₀₉	#2868	1	7	yes	—
36	C₂×#384	C₂×C₁₀₉	1	14	yes	—
37	C₂×C₄³	C₂×C₁₀₉	1	1	yes	characteristic
38	C₁₀₉	#14692	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	28	yes	—
40	C₂²×C₄²	C₂²×C₁₀₉	1	7	yes	—
41	#384	—	2	56	no	—
42	C₄³	C₄×C₁₀₉	1	8	yes	complemented
43	C₂³×C₄	C₂³×C₁₀₉	1	8	yes	—
44	C₂³×C₄	C₂³×C₁₀₉	1	7	yes	—
45	C₂×C₄⋊C₄	—	2	168	no	—
46	C₂×C₄²	C₂×C₄×C₁₀₉	1	28	yes	—
47	C₂×C₄²	D₄×C₁₀₉	1	28	yes	—
48	C₂×C₄²	Q₈×C₁₀₉	1	28	yes	—
49	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
50	C₂²×C₄	—	2	56	no	—
51	C₂²×C₄	C₂²×C₄×C₁₀₉	1	14	yes	—
52	C₂²×C₄	C₂×D₄×C₁₀₉	1	42	yes	—
53	C₂²×C₄	C₂×Q₈×C₁₀₉	1	42	yes	—
54	C₄⋊C₄	—	4	112	no	—
55	C₄²	C₄⋊C₄×C₁₀₉	1	112	yes	complemented
56	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
57	C₂³	C₂²×D₄×C₁₀₉	1	7	yes	—
58	C₂³	C₂²×Q₈×C₁₀₉	1	7	yes	—
59	C₂×C₄	—	4	56	no	—
60	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	84	yes	—
61	C₂×C₄	#127×C₁₀₉	1	28	yes	—
62	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	84	yes	—
63	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	7	yes	—
64	C₂²	C₂×#127×C₁₀₉	1	7	yes	—
65	C₂²	C₂×C₄⋊Q₈×C₁₀₉	1	21	yes	—
66	C₄	—	8	8	no	—
67	C₄	#384×C₁₀₉	1	56	yes	complemented
68	C₂	C₂×#384×C₁₀₉	1	7	yes	—
69	C₂	C₁₀₉×#2867	1	1	yes	characteristic
70	C₂	C₁₀₉×#2868	1	7	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₀₉	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₀₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₄³×C₁₀₉	C₄	semidirect
19. C₄²×C₁₀₉	C₄⋊C₄	semidirect
31. C₄×C₁₀₉	#384	semidirect
32. #14692	C₁₀₉	direct (N × H)
38. C₁₀₉	#14692	direct (N × H)
42. C₄³	C₄×C₁₀₉	semidirect
55. C₄²	C₄⋊C₄×C₁₀₉	semidirect
67. C₄	#384×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#384×C₁₀₉	2
C₂×C₄³×C₁₀₉	2
#14692	109

Subgroups — C109×#14692 #42132686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#14692 #42132686