Subgroups — C₂²×#401×C₁₀₉ #42152542

Order 27904.

2942 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#401×C₁₀₉	C₂	1	24	yes	maximal, direct product factor
2	C₂²×#95×C₁₀₉	C₂	1	7	yes	maximal
3	C₂⁴×C₄×C₁₀₉	C₂²	1	7	yes	—
4	#401×C₁₀₉	C₂²	1	64	yes	direct product factor
5	C₂×#95×C₁₀₉	C₂²	1	84	yes	—
6	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
7	C₂³×C₄×C₁₀₉	—	2	84	no	—
8	C₂³×C₄×C₁₀₉	C₂³	1	42	yes	—
9	#95×C₁₀₉	C₂³	1	112	yes	—
10	C₂⁴×C₁₀₉	#41	1	28	yes	—
11	C₂⁴×C₁₀₉	C₂⁴	1	3	yes	—
12	C₂²×C₄×C₁₀₉	—	2	252	no	—
13	C₂²×C₄×C₁₀₉	—	4	112	no	—
14	C₂²×C₄×C₁₀₉	C₂⁴	1	28	yes	—
15	C₂³×C₁₀₉	#126	1	112	yes	—
16	C₂³×C₁₀₉	C₂×#41	1	42	yes	—
17	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
18	C₂×C₄×C₁₀₉	—	2	84	no	—
19	C₂×C₄×C₁₀₉	—	4	168	no	—
20	C₂²×C₁₀₉	#401	1	64	yes	direct product factor
21	C₂²×C₁₀₉	C₂×#126	1	84	yes	—
22	C₂²×C₁₀₉	C₂²×#41	1	7	yes	—
23	C₄×C₁₀₉	—	4	28	no	—
24	C₂²×#401	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₁₀₉	C₂×#401	1	24	yes	direct product factor
26	C₂×C₁₀₉	C₂²×#126	1	7	yes	—
27	C₂×#401	C₂×C₁₀₉	1	24	yes	direct product factor
28	C₂²×#95	C₂×C₁₀₉	1	7	yes	—
29	C₁₀₉	C₂²×#401	1	1	yes	characteristic, direct product factor, Sylow
30	C₂⁴×C₄	C₂²×C₁₀₉	1	7	yes	—
31	#401	C₂²×C₁₀₉	1	64	yes	direct product factor
32	C₂×#95	C₂²×C₁₀₉	1	84	yes	—
33	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
34	C₂³×C₄	—	2	84	no	—
35	C₂³×C₄	C₂³×C₁₀₉	1	42	yes	—
36	#95	C₂³×C₁₀₉	1	112	yes	—
37	C₂⁴	#41×C₁₀₉	1	28	yes	—
38	C₂⁴	C₂⁴×C₁₀₉	1	3	yes	—
39	C₂²×C₄	—	2	252	no	—
40	C₂²×C₄	—	4	112	no	—
41	C₂²×C₄	C₂⁴×C₁₀₉	1	28	yes	—
42	C₂³	#126×C₁₀₉	1	112	yes	—
43	C₂³	C₂×#41×C₁₀₉	1	42	yes	—
44	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
45	C₂×C₄	—	2	84	no	—
46	C₂×C₄	—	4	168	no	—
47	C₂²	#401×C₁₀₉	1	64	yes	direct product factor
48	C₂²	C₂×#126×C₁₀₉	1	84	yes	—
49	C₂²	C₂²×#41×C₁₀₉	1	7	yes	—
50	C₄	—	4	28	no	—
51	C₂	C₂×#401×C₁₀₉	1	24	yes	direct product factor
52	C₂	C₂²×#126×C₁₀₉	1	7	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₁₀₉	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#401×C₁₀₉	C₂	direct (N × H)
4. #401×C₁₀₉	C₂²	direct (N × H)
20. C₂²×C₁₀₉	#401	direct (N × H)
24. C₂²×#401	C₁₀₉	direct (N × H)
25. C₂×C₁₀₉	C₂×#401	direct (N × H)
27. C₂×#401	C₂×C₁₀₉	direct (N × H)
29. C₁₀₉	C₂²×#401	direct (N × H)
31. #401	C₂²×C₁₀₉	direct (N × H)
47. C₂²	#401×C₁₀₉	direct (N × H)
51. C₂	C₂×#401×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#401×C₁₀₉	2
C₂²×#95×C₁₀₉	2
C₂²×#401	109

Subgroups — C22×#401×C109 #42152542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#401×C₁₀₉ #42152542