Subgroups — C₁₀₉×#13551 #42131545

Order 27904.

214 conjugacy classes of subgroups.

The subgroup lattice has 118 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄×C₁₀₉	C₂	1	2	yes	maximal
2	C₁₀₉×#2108	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1398	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₀₉	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
6	#405×C₁₀₉	C₄	1	1	yes	characteristic
7	M₆₄×C₁₀₉	C₄	1	4	yes	complemented
8	M₆₄×C₁₀₉	C₂²	1	4	yes	—
9	C₂×C₃₂×C₁₀₉	C₄	1	4	yes	complemented
10	#346×C₁₀₉	C₂²	1	2	yes	—
11	C₄×C₁₆×C₁₀₉	C₄	1	2	yes	—
12	C₂×M₁₆×C₁₀₉	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	#117×C₁₀₉	C₈	1	1	yes	characteristic
15	M₃₂×C₁₀₉	C₈	1	4	yes	—
16	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
21	#97×C₁₀₉	C₈	1	2	yes	—
22	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₃₂×C₁₀₉	C₂×C₄	1	8	yes	—
24	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
25	M₁₆×C₁₀₉	—	2	2	no	—
26	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
27	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₉	—	2	1	no	—
32	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
33	C₄²×C₁₀₉	C₂×C₈	1	2	yes	—
34	C₁₆×C₁₀₉	C₄²	1	1	yes	characteristic
35	C₁₆×C₁₀₉	C₄²	1	1	yes	characteristic
36	C₁₆×C₁₀₉	C₂×C₈	1	4	yes	—
37	C₁₆×C₁₀₉	C₂²×C₄	1	2	yes	—
38	C₂³×C₁₀₉	M₃₂	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	C₄×C₈	1	2	yes	—
41	C₂×C₄×C₁₀₉	M₃₂	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	M₃₂	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
44	C₈×C₁₀₉	—	2	2	no	—
45	C₈×C₁₀₉	C₄×C₈	1	2	yes	—
46	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
47	C₈×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
48	C₂²×C₁₀₉	—	4	1	no	—
49	C₂²×C₁₀₉	#346	1	2	yes	—
50	C₂²×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
51	C₄×C₁₀₉	—	4	2	no	—
52	C₄×C₁₀₉	#346	1	2	yes	—
53	C₄×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
54	C₄×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
55	#13551	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₁₀₉	—	2	1	no	—
57	C₂×C₁₀₉	—	2	2	no	—
58	C₂×C₁₀₉	C₂×#346	1	1	yes	characteristic
59	C₂×M₆₄	C₂×C₁₀₉	1	2	yes	—
60	#2108	C₂×C₁₀₉	1	1	yes	characteristic
61	#1398	C₂×C₁₀₉	1	4	yes	complemented
62	C₁₀₉	#13551	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×M₃₂	C₄×C₁₀₉	1	1	yes	characteristic
64	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
65	#405	C₄×C₁₀₉	1	1	yes	characteristic
66	M₆₄	C₄×C₁₀₉	1	4	yes	complemented
67	M₆₄	C₂²×C₁₀₉	1	4	yes	—
68	C₂×C₃₂	C₄×C₁₀₉	1	4	yes	complemented
69	#346	C₂²×C₁₀₉	1	2	yes	—
70	C₄×C₁₆	C₄×C₁₀₉	1	2	yes	—
71	C₂×M₁₆	C₈×C₁₀₉	1	1	yes	characteristic
72	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
73	#117	C₈×C₁₀₉	1	1	yes	characteristic
74	M₃₂	C₈×C₁₀₉	1	4	yes	—
75	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
76	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
77	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
78	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
79	C₂×C₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
80	#97	C₈×C₁₀₉	1	2	yes	—
81	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
82	C₃₂	C₂×C₄×C₁₀₉	1	8	yes	—
83	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
86	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
87	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
88	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
89	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	C₂×C₈×C₁₀₉	1	2	yes	—
93	C₁₆	C₄²×C₁₀₉	1	1	yes	characteristic
94	C₁₆	C₄²×C₁₀₉	1	1	yes	characteristic
95	C₁₆	C₂×C₈×C₁₀₉	1	4	yes	—
96	C₁₆	C₂²×C₄×C₁₀₉	1	2	yes	—
97	C₂³	M₃₂×C₁₀₉	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₄×C₈×C₁₀₉	1	2	yes	—
100	C₂×C₄	M₃₂×C₁₀₉	1	2	yes	—
101	C₂×C₄	M₃₂×C₁₀₉	1	1	yes	characteristic
102	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	C₄×C₈×C₁₀₉	1	2	yes	—
105	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
106	C₈	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	#346×C₁₀₉	1	2	yes	—
109	C₂²	C₂×M₃₂×C₁₀₉	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	#346×C₁₀₉	1	2	yes	—
112	C₄	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
113	C₄	C₂×M₃₂×C₁₀₉	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×#346×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₁₀₉	C₄²
commutator	C₄	C₂×C₄×C₈×C₁₀₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₀₉
socle	C₂×C₁₀₉	C₂×#346
prime radical	C₂³×C₁₀₉	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1398	C₂	semidirect
7. M₆₄×C₁₀₉	C₄	semidirect
9. C₂×C₃₂×C₁₀₉	C₄	semidirect
55. #13551	C₁₀₉	direct (N × H)
61. #1398	C₂×C₁₀₉	semidirect
62. C₁₀₉	#13551	direct (N × H)
66. M₆₄	C₄×C₁₀₉	semidirect
68. C₂×C₃₂	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×M₆₄×C₁₀₉	2
C₁₀₉×#2108	2
C₁₀₉×#1398	2
#13551	109

Subgroups — C109×#13551 #42131545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#13551 #42131545