Subgroups — C₁₀₉×#23847 #42141841

Order 27904.

604 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2104	C₂	1	8	yes	maximal
2	C₁₀₉×#1740	C₂	1	7	yes	maximal
3	C₂×#117×C₁₀₉	C₂²	1	7	yes	—
4	#342×C₁₀₉	C₂²	1	28	yes	—
5	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
6	#117×C₁₀₉	—	2	14	no	—
7	#117×C₁₀₉	—	2	14	no	—
8	C₂×C₄⋊C₄×C₁₀₉	—	2	7	no	—
9	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	14	yes	—
10	C₂×C₄²×C₁₀₉	—	2	7	no	—
11	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
12	C₂²×C₄×C₁₀₉	—	2	14	no	—
13	C₂²×C₄×C₁₀₉	#41	1	7	yes	—
14	C₂²×C₄×C₁₀₉	#41	1	7	yes	—
15	C₄⋊C₄×C₁₀₉	—	4	14	no	—
16	C₂²⋊C₄×C₁₀₉	—	2	28	no	—
17	C₄²×C₁₀₉	—	4	7	no	—
18	C₄²×C₁₀₉	—	4	7	no	—
19	C₂³×C₁₀₉	—	8	1	no	—
20	C₂³×C₁₀₉	C₂×#41	1	7	yes	—
21	C₂×C₄×C₁₀₉	—	4	14	no	—
22	C₂×C₄×C₁₀₉	—	8	14	no	—
23	C₂×C₄×C₁₀₉	#126	1	7	yes	—
24	C₂×C₄×C₁₀₉	#126	1	14	yes	—
25	C₂×C₄×C₁₀₉	#126	1	7	yes	—
26	C₂²×C₁₀₉	—	4	7	no	—
27	C₂²×C₁₀₉	C₂×#126	1	7	yes	—
28	C₄×C₁₀₉	—	8	14	no	—
29	C₄×C₁₀₉	#401	1	7	yes	—
30	C₄×C₁₀₉	#401	1	1	yes	characteristic
31	#23847	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₁₀₉	—	2	7	no	—
33	C₂×C₁₀₉	C₂×#401	1	1	yes	characteristic
34	#2104	C₂×C₁₀₉	1	8	yes	—
35	#1740	C₂×C₁₀₉	1	7	yes	—
36	C₁₀₉	#23847	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×#117	C₂²×C₁₀₉	1	7	yes	—
38	#342	C₂²×C₁₀₉	1	28	yes	—
39	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
40	#117	—	2	14	no	—
41	#117	—	2	14	no	—
42	C₂×C₄⋊C₄	—	2	7	no	—
43	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	14	yes	—
44	C₂×C₄²	—	2	7	no	—
45	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
46	C₂²×C₄	—	2	14	no	—
47	C₂²×C₄	#41×C₁₀₉	1	7	yes	—
48	C₂²×C₄	#41×C₁₀₉	1	7	yes	—
49	C₄⋊C₄	—	4	14	no	—
50	C₂²⋊C₄	—	2	28	no	—
51	C₄²	—	4	7	no	—
52	C₄²	—	4	7	no	—
53	C₂³	—	8	1	no	—
54	C₂³	C₂×#41×C₁₀₉	1	7	yes	—
55	C₂×C₄	—	4	14	no	—
56	C₂×C₄	—	8	14	no	—
57	C₂×C₄	#126×C₁₀₉	1	7	yes	—
58	C₂×C₄	#126×C₁₀₉	1	7	yes	—
59	C₂×C₄	#126×C₁₀₉	1	14	yes	—
60	C₂²	—	4	7	no	—
61	C₂²	C₂×#126×C₁₀₉	1	7	yes	—
62	C₄	—	8	14	no	—
63	C₄	#401×C₁₀₉	1	7	yes	—
64	C₄	#401×C₁₀₉	1	1	yes	characteristic
65	C₂	—	2	7	no	—
66	C₂	C₂×#401×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₀₉	#401
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₀₉
socle	C₂×C₁₀₉	C₂×#401
prime radical	C₂⁴×C₁₀₉	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
31. #23847	C₁₀₉	direct (N × H)
36. C₁₀₉	#23847	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2104	2
C₁₀₉×#1740	2
#23847	109

Subgroups — C109×#23847 #42141841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#23847 #42141841