Subgroups — C₁₀₉×#12541 #42130535

Order 27904.

402 conjugacy classes of subgroups.

The subgroup lattice has 134 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#347×C₁₀₉	C₂	1	4	yes	maximal
2	C₄×M₃₂×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
5	C₂×#97×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₀₉	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	#376×C₁₀₉	C₄	1	1	yes	characteristic
10	#347×C₁₀₉	C₄	1	16	yes	complemented
11	#346×C₁₀₉	C₄	1	4	yes	complemented
12	C₄×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
13	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	M₃₂×C₁₀₉	C₂×C₄	1	16	yes	—
19	C₂×C₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
20	#97×C₁₀₉	—	2	2	no	—
21	#97×C₁₀₉	C₂×C₄	1	8	yes	—
22	C₄×C₈×C₁₀₉	—	2	4	no	—
23	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	#95×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	—	2	2	no	—
27	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
28	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	—	2	2	no	—
31	C₂×C₈×C₁₀₉	C₄²	1	4	yes	—
32	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	M₁₆	1	4	yes	—
35	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
36	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
37	C₄²×C₁₀₉	—	2	2	no	—
38	C₄²×C₁₀₉	C₄²	1	4	yes	—
39	C₄²×C₁₀₉	C₄²	1	2	yes	—
40	C₄²×C₁₀₉	M₁₆	1	2	yes	—
41	C₁₆×C₁₀₉	C₄²	1	16	yes	—
42	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	—	2	4	no	—
44	C₂×C₄×C₁₀₉	—	4	2	no	—
45	C₂×C₄×C₁₀₉	—	4	4	no	—
46	C₂×C₄×C₁₀₉	#97	1	4	yes	—
47	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
51	C₈×C₁₀₉	—	2	4	no	—
52	C₈×C₁₀₉	#97	1	2	yes	—
53	C₈×C₁₀₉	#97	1	2	yes	—
54	C₈×C₁₀₉	C₂×C₄²	1	4	yes	—
55	C₂²×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	C₂×#97	1	1	yes	characteristic
57	C₂²×C₁₀₉	C₄×M₁₆	1	2	yes	—
58	C₄×C₁₀₉	—	2	4	no	—
59	C₄×C₁₀₉	—	4	4	no	—
60	C₄×C₁₀₉	C₄³	1	1	yes	characteristic
61	C₄×C₁₀₉	C₂×#97	1	1	yes	characteristic
62	C₄×C₁₀₉	C₄×M₁₆	1	2	yes	—
63	#12541	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₁₀₉	—	2	2	no	—
65	C₂×C₁₀₉	C₄×#97	1	1	yes	characteristic
66	C₂×C₁₀₉	#2110	1	2	yes	—
67	C₂×#347	C₂×C₁₀₉	1	4	yes	—
68	C₄×M₃₂	C₂×C₁₀₉	1	2	yes	—
69	#1726	C₂×C₁₀₉	1	1	yes	characteristic
70	C₁₀₉	#12541	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
72	C₂×#97	C₄×C₁₀₉	1	1	yes	characteristic
73	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
74	C₂×C₄×C₈	C₄×C₁₀₉	1	2	yes	—
75	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
76	#376	C₄×C₁₀₉	1	1	yes	characteristic
77	#347	C₄×C₁₀₉	1	16	yes	complemented
78	#346	C₄×C₁₀₉	1	4	yes	complemented
79	C₄×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
80	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
81	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
82	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
83	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
84	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
85	M₃₂	C₂×C₄×C₁₀₉	1	16	yes	—
86	C₂×C₁₆	C₂×C₄×C₁₀₉	1	8	yes	—
87	#97	—	2	2	no	—
88	#97	C₂×C₄×C₁₀₉	1	8	yes	—
89	C₄×C₈	—	2	4	no	—
90	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
91	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
92	#95	—	2	2	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
95	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	C₄²×C₁₀₉	1	4	yes	—
99	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
100	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
101	C₂×C₈	M₁₆×C₁₀₉	1	4	yes	—
102	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
103	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
104	C₄²	—	2	2	no	—
105	C₄²	C₄²×C₁₀₉	1	4	yes	—
106	C₄²	C₄²×C₁₀₉	1	2	yes	—
107	C₄²	M₁₆×C₁₀₉	1	2	yes	—
108	C₁₆	C₄²×C₁₀₉	1	16	yes	—
109	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	#97×C₁₀₉	1	4	yes	—
114	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
116	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
117	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
118	C₈	—	2	4	no	—
119	C₈	#97×C₁₀₉	1	2	yes	—
120	C₈	#97×C₁₀₉	1	2	yes	—
121	C₈	C₂×C₄²×C₁₀₉	1	4	yes	—
122	C₂²	—	2	2	no	—
123	C₂²	C₂×#97×C₁₀₉	1	1	yes	characteristic
124	C₂²	C₄×M₁₆×C₁₀₉	1	2	yes	—
125	C₄	—	2	4	no	—
126	C₄	—	4	4	no	—
127	C₄	C₄³×C₁₀₉	1	1	yes	characteristic
128	C₄	C₂×#97×C₁₀₉	1	1	yes	characteristic
129	C₄	C₄×M₁₆×C₁₀₉	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₄×#97×C₁₀₉	1	1	yes	characteristic
132	C₂	C₁₀₉×#2110	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₀₉	C₄²
commutator	C₄	C₄³×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×#97
prime radical	C₂³×C₁₀₉	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
10. #347×C₁₀₉	C₄	semidirect
11. #346×C₁₀₉	C₄	semidirect
12. C₄×C₁₆×C₁₀₉	C₄	semidirect
63. #12541	C₁₀₉	direct (N × H)
70. C₁₀₉	#12541	direct (N × H)
77. #347	C₄×C₁₀₉	semidirect
78. #346	C₄×C₁₀₉	semidirect
79. C₄×C₁₆	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×#347×C₁₀₉	2
C₄×M₃₂×C₁₀₉	2
C₁₀₉×#1726	2
#12541	109

Subgroups — C109×#12541 #42130535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12541 #42130535