Subgroups — C₂×C₁₀₉×#1313 #42129397

Order 27904.

302 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#322×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1313	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₁₆×C₁₀₉	C₄	1	2	yes	—
6	C₂²×C₁₆×C₁₀₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₀₉	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	#346×C₁₀₉	C₄	1	2	yes	—
10	#346×C₁₀₉	C₄	1	2	yes	—
11	#346×C₁₀₉	C₂²	1	2	yes	—
12	#346×C₁₀₉	C₂²	1	2	yes	—
13	#322×C₁₀₉	C₄	1	2	yes	—
14	#322×C₁₀₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₀₉	—	2	2	no	—
16	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₁₀₉	—	2	4	no	—
20	C₂×C₁₆×C₁₀₉	C₈	1	8	yes	complemented
21	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
22	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₀₉	—	2	2	no	—
24	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
31	C₂×C₈×C₁₀₉	—	2	4	no	—
32	C₂×C₈×C₁₀₉	—	4	4	no	—
33	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₁₀₉	M₁₆	1	4	yes	—
37	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₁₀₉	C₄²	1	1	yes	characteristic
40	C₄²×C₁₀₉	M₁₆	1	2	yes	—
41	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₁₀₉	—	2	4	no	—
43	C₁₆×C₁₀₉	C₂×C₈	1	8	yes	complemented
44	C₂³×C₁₀₉	#97	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	—	2	4	no	—
46	C₂×C₄×C₁₀₉	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₁₀₉	#97	1	2	yes	—
48	C₂×C₄×C₁₀₉	#97	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	#97	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
52	C₈×C₁₀₉	—	4	4	no	—
53	C₈×C₁₀₉	C₄×C₈	1	2	yes	—
54	C₈×C₁₀₉	#97	1	2	yes	—
55	C₈×C₁₀₉	C₂²×C₈	1	2	yes	—
56	C₈×C₁₀₉	C₂×M₁₆	1	2	yes	—
57	C₂²×C₁₀₉	#322	1	2	yes	—
58	C₂²×C₁₀₉	#347	1	4	yes	—
59	C₂²×C₁₀₉	C₂×#97	1	1	yes	characteristic
60	C₄×C₁₀₉	—	2	2	no	—
61	C₄×C₁₀₉	—	2	2	no	—
62	C₄×C₁₀₉	#322	1	2	yes	—
63	C₄×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₁₀₉	C₂×#97	1	1	yes	characteristic
65	C₂×#1313	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₁₀₉	#1313	1	4	yes	direct product factor
67	C₂×C₁₀₉	C₂×#322	1	1	yes	characteristic
68	C₂×C₁₀₉	C₂×#347	1	2	yes	—
69	C₂×#346	C₂×C₁₀₉	1	1	yes	characteristic
70	C₂×#346	C₂×C₁₀₉	1	1	yes	characteristic
71	C₂×#322	C₂×C₁₀₉	1	1	yes	characteristic
72	#1313	C₂×C₁₀₉	1	4	yes	direct product factor
73	C₁₀₉	C₂×#1313	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₁₆	C₄×C₁₀₉	1	2	yes	—
75	C₂²×C₁₆	C₄×C₁₀₉	1	2	yes	—
76	C₂×C₄×C₈	C₄×C₁₀₉	1	2	yes	—
77	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
78	#346	C₄×C₁₀₉	1	2	yes	—
79	#346	C₄×C₁₀₉	1	2	yes	—
80	#346	C₂²×C₁₀₉	1	2	yes	—
81	#346	C₂²×C₁₀₉	1	2	yes	—
82	#322	C₄×C₁₀₉	1	2	yes	—
83	#322	C₂²×C₁₀₉	1	2	yes	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
86	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
87	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
88	C₂×C₁₆	—	2	4	no	—
89	C₂×C₁₆	C₈×C₁₀₉	1	8	yes	complemented
90	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
91	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
94	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
95	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
96	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
97	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
98	C₂²×C₄	C₄²×C₁₀₉	1	1	yes	characteristic
99	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
103	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
104	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
105	C₂×C₈	M₁₆×C₁₀₉	1	4	yes	—
106	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
107	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
108	C₄²	C₄²×C₁₀₉	1	1	yes	characteristic
109	C₄²	M₁₆×C₁₀₉	1	2	yes	—
110	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
111	C₁₆	—	2	4	no	—
112	C₁₆	C₂×C₈×C₁₀₉	1	8	yes	complemented
113	C₂³	#97×C₁₀₉	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₄×C₈×C₁₀₉	1	2	yes	—
116	C₂×C₄	#97×C₁₀₉	1	2	yes	—
117	C₂×C₄	#97×C₁₀₉	1	1	yes	characteristic
118	C₂×C₄	#97×C₁₀₉	1	2	yes	—
119	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
120	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
121	C₈	—	4	4	no	—
122	C₈	C₄×C₈×C₁₀₉	1	2	yes	—
123	C₈	#97×C₁₀₉	1	2	yes	—
124	C₈	C₂²×C₈×C₁₀₉	1	2	yes	—
125	C₈	C₂×M₁₆×C₁₀₉	1	2	yes	—
126	C₂²	#322×C₁₀₉	1	2	yes	—
127	C₂²	#347×C₁₀₉	1	4	yes	—
128	C₂²	C₂×#97×C₁₀₉	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	2	no	—
131	C₄	#322×C₁₀₉	1	2	yes	—
132	C₄	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
133	C₄	C₂×#97×C₁₀₉	1	1	yes	characteristic
134	C₂	C₁₀₉×#1313	1	4	yes	direct product factor
135	C₂	C₂×#322×C₁₀₉	1	1	yes	characteristic
136	C₂	C₂×#347×C₁₀₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#1313	C₂	direct (N × H)
20. C₂×C₁₆×C₁₀₉	C₈	semidirect
43. C₁₆×C₁₀₉	C₂×C₈	semidirect
65. C₂×#1313	C₁₀₉	direct (N × H)
66. C₂×C₁₀₉	#1313	direct (N × H)
72. #1313	C₂×C₁₀₉	direct (N × H)
73. C₁₀₉	C₂×#1313	direct (N × H)
89. C₂×C₁₆	C₈×C₁₀₉	semidirect
112. C₁₆	C₂×C₈×C₁₀₉	semidirect
134. C₂	C₁₀₉×#1313	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×#346×C₁₀₉	2
C₂×#322×C₁₀₉	2
C₁₀₉×#1313	2
C₂×#1313	109

Subgroups — C2×C109×#1313 #42129397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#1313 #42129397