Subgroups — C₁₀₉×#14691 #42132685

Order 27904.

1278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1434	C₂	1	14	yes	maximal
3	C₂²×C₄²×C₁₀₉	C₂²	1	7	yes	—
4	C₂×#95×C₁₀₉	C₂²	1	28	yes	—
5	C₄³×C₁₀₉	C₄	1	8	yes	complemented
6	C₂³×C₄×C₁₀₉	C₂³	1	8	yes	—
7	C₂³×C₄×C₁₀₉	C₂³	1	7	yes	—
8	C₂×C₄²×C₁₀₉	—	2	28	no	—
9	C₂×C₄²×C₁₀₉	C₂×C₄	1	28	yes	—
10	#95×C₁₀₉	—	2	56	no	—
11	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
12	C₂²×C₄×C₁₀₉	—	2	28	no	—
13	C₂²×C₄×C₁₀₉	—	2	56	no	—
14	C₂²×C₄×C₁₀₉	C₂²×C₄	1	14	yes	—
15	C₂²×C₄×C₁₀₉	#41	1	28	yes	—
16	C₄²×C₁₀₉	—	2	56	no	—
17	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
18	C₂³×C₁₀₉	C₂×#41	1	14	yes	—
19	C₂×C₄×C₁₀₉	—	2	28	no	—
20	C₂×C₄×C₁₀₉	—	2	56	no	—
21	C₂×C₄×C₁₀₉	—	4	56	no	—
22	C₂×C₄×C₁₀₉	#117	1	28	yes	—
23	C₂²×C₁₀₉	C₂×#117	1	7	yes	—
24	C₂²×C₁₀₉	#529	1	28	yes	—
25	C₄×C₁₀₉	—	2	28	no	—
26	C₄×C₁₀₉	—	8	8	no	—
27	#14691	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₁₀₉	#2283	1	7	yes	—
29	C₂×C₁₀₉	#2866	1	8	yes	—
30	C₂×C₄³	C₂×C₁₀₉	1	1	yes	characteristic
31	#1434	C₂×C₁₀₉	1	14	yes	—
32	C₁₀₉	#14691	1	1	yes	characteristic, direct product factor, Sylow
33	C₂²×C₄²	C₂²×C₁₀₉	1	7	yes	—
34	C₂×#95	C₂²×C₁₀₉	1	28	yes	—
35	C₄³	C₄×C₁₀₉	1	8	yes	complemented
36	C₂³×C₄	C₂³×C₁₀₉	1	8	yes	—
37	C₂³×C₄	C₂³×C₁₀₉	1	7	yes	—
38	C₂×C₄²	—	2	28	no	—
39	C₂×C₄²	C₂×C₄×C₁₀₉	1	28	yes	—
40	#95	—	2	56	no	—
41	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
42	C₂²×C₄	—	2	28	no	—
43	C₂²×C₄	—	2	56	no	—
44	C₂²×C₄	C₂²×C₄×C₁₀₉	1	14	yes	—
45	C₂²×C₄	#41×C₁₀₉	1	28	yes	—
46	C₄²	—	2	56	no	—
47	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
48	C₂³	C₂×#41×C₁₀₉	1	14	yes	—
49	C₂×C₄	—	2	28	no	—
50	C₂×C₄	—	2	56	no	—
51	C₂×C₄	—	4	56	no	—
52	C₂×C₄	#117×C₁₀₉	1	28	yes	—
53	C₂²	C₂×#117×C₁₀₉	1	7	yes	—
54	C₂²	#529×C₁₀₉	1	28	yes	—
55	C₄	—	2	28	no	—
56	C₄	—	8	8	no	—
57	C₂	C₁₀₉×#2283	1	7	yes	—
58	C₂	C₁₀₉×#2866	1	8	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
5. C₄³×C₁₀₉	C₄	semidirect
27. #14691	C₁₀₉	direct (N × H)
32. C₁₀₉	#14691	direct (N × H)
35. C₄³	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄³×C₁₀₉	2
C₁₀₉×#1434	2
#14691	109

Subgroups — C109×#14691 #42132685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#14691 #42132685