Subgroups — C₁₀₉×#12894 #42130888

Order 27904.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1374	C₂	1	6	yes	maximal
3	C₂²×C₁₆×C₁₀₉	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₁₀₉	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
6	C₄³×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
8	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
9	C₂×C₄²×C₁₀₉	C₈	1	3	yes	—
10	C₂×C₄²×C₁₀₉	C₈	1	1	yes	characteristic
11	C₂×C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
12	C₂×C₁₆×C₁₀₉	—	2	12	no	—
13	C₄×C₈×C₁₀₉	—	2	6	no	—
14	C₄×C₈×C₁₀₉	—	2	6	no	—
15	C₂²×C₄×C₁₀₉	C₂×C₈	1	3	yes	—
16	C₂²×C₄×C₁₀₉	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	—	2	3	no	—
19	C₂×C₈×C₁₀₉	—	2	3	no	—
20	C₂×C₈×C₁₀₉	M₁₆	1	6	yes	—
21	C₂×C₈×C₁₀₉	#41	1	6	yes	—
22	C₄²×C₁₀₉	—	2	6	no	—
23	C₄²×C₁₀₉	—	2	3	no	—
24	C₄²×C₁₀₉	—	2	3	no	—
25	C₄²×C₁₀₉	C₁₆	1	4	yes	complemented
26	C₁₆×C₁₀₉	—	4	4	no	—
27	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	—	2	3	no	—
29	C₂×C₄×C₁₀₉	—	2	3	no	—
30	C₂×C₄×C₁₀₉	—	2	3	no	—
31	C₂×C₄×C₁₀₉	—	2	3	no	—
32	C₂×C₄×C₁₀₉	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₁₀₉	M₃₂	1	6	yes	—
34	C₂×C₄×C₁₀₉	#117	1	3	yes	—
35	C₂×C₄×C₁₀₉	C₂×M₁₆	1	3	yes	—
36	C₈×C₁₀₉	—	2	6	no	—
37	C₈×C₁₀₉	#126	1	4	yes	—
38	C₂²×C₁₀₉	#431	1	3	yes	—
39	C₂²×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
40	C₂²×C₁₀₉	C₂×M₃₂	1	3	yes	—
41	C₄×C₁₀₉	—	2	6	no	—
42	C₄×C₁₀₉	—	2	6	no	—
43	C₄×C₁₀₉	#383	1	1	yes	characteristic
44	C₄×C₁₀₉	#432	1	3	yes	—
45	#12894	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₁₀₉	#1839	1	1	yes	characteristic
47	C₂×C₁₀₉	#2162	1	3	yes	—
48	C₂×C₁₀₉	#2163	1	3	yes	—
49	C₄²×C₈	C₂×C₁₀₉	1	1	yes	characteristic
50	#1374	C₂×C₁₀₉	1	6	yes	—
51	C₁₀₉	#12894	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×C₁₆	C₂²×C₁₀₉	1	4	yes	—
53	C₂×C₄×C₈	C₄×C₁₀₉	1	3	yes	—
54	C₂×C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
55	C₄³	C₄×C₁₀₉	1	1	yes	characteristic
56	C₂²×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
57	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
58	C₂×C₄²	C₈×C₁₀₉	1	3	yes	—
59	C₂×C₄²	C₈×C₁₀₉	1	1	yes	characteristic
60	C₂×C₄²	C₂×C₄×C₁₀₉	1	3	yes	—
61	C₂×C₁₆	—	2	12	no	—
62	C₄×C₈	—	2	6	no	—
63	C₄×C₈	—	2	6	no	—
64	C₂²×C₄	C₂×C₈×C₁₀₉	1	3	yes	—
65	C₂²×C₄	C₂×C₈×C₁₀₉	1	3	yes	—
66	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	M₁₆×C₁₀₉	1	6	yes	—
70	C₂×C₈	#41×C₁₀₉	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	C₁₆×C₁₀₉	1	4	yes	complemented
75	C₁₆	—	4	4	no	—
76	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×C₁₆×C₁₀₉	1	6	yes	—
82	C₂×C₄	M₃₂×C₁₀₉	1	6	yes	—
83	C₂×C₄	#117×C₁₀₉	1	3	yes	—
84	C₂×C₄	C₂×M₁₆×C₁₀₉	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#126×C₁₀₉	1	4	yes	—
87	C₂²	#431×C₁₀₉	1	3	yes	—
88	C₂²	C₂²×C₁₆×C₁₀₉	1	1	yes	characteristic
89	C₂²	C₂×M₃₂×C₁₀₉	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#383×C₁₀₉	1	1	yes	characteristic
93	C₄	#432×C₁₀₉	1	3	yes	—
94	C₂	C₁₀₉×#1839	1	1	yes	characteristic
95	C₂	C₁₀₉×#2162	1	3	yes	—
96	C₂	C₁₀₉×#2163	1	3	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₀₉	C₂³
commutator	C₂²	C₂²×C₁₆×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₄²×C₁₀₉	C₁₆	semidirect
45. #12894	C₁₀₉	direct (N × H)
51. C₁₀₉	#12894	direct (N × H)
74. C₄²	C₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₈×C₁₀₉	2
C₁₀₉×#1374	2
#12894	109

Subgroups — C109×#12894 #42130888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12894 #42130888