Subgroups — C₁₀₉×#12594 #42130588

Order 27904.

354 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₁₀₉	C₂	1	2	yes	maximal
2	C₄×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1375	C₂	1	4	yes	maximal
4	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
5	C₂×#97×C₁₀₉	C₄	1	2	yes	—
6	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₄³×C₁₀₉	C₄	1	1	yes	characteristic
10	#346×C₁₀₉	C₄	1	4	yes	complemented
11	C₄×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
12	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₉	—	2	2	no	—
16	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	M₃₂×C₁₀₉	—	2	8	no	—
19	C₂×C₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
20	#97×C₁₀₉	—	2	4	no	—
21	#97×C₁₀₉	C₈	1	8	yes	—
22	C₄×C₈×C₁₀₉	—	2	2	no	—
23	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
31	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
34	C₂×C₈×C₁₀₉	#41	1	4	yes	—
35	C₄²×C₁₀₉	—	2	2	no	—
36	C₄²×C₁₀₉	—	4	4	no	—
37	C₄²×C₁₀₉	C₄²	1	1	yes	characteristic
38	C₄²×C₁₀₉	C₄²	1	1	yes	characteristic
39	C₄²×C₁₀₉	C₂×C₈	1	4	yes	—
40	C₄²×C₁₀₉	M₁₆	1	2	yes	—
41	C₁₆×C₁₀₉	—	2	8	no	—
42	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	—	2	4	no	—
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	—	4	4	no	—
46	C₂×C₄×C₁₀₉	#97	1	2	yes	—
47	C₂×C₄×C₁₀₉	#97	1	2	yes	—
48	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	#117	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
51	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
52	C₈×C₁₀₉	—	2	4	no	—
53	C₈×C₁₀₉	C₄×C₈	1	4	yes	—
54	C₈×C₁₀₉	#117	1	4	yes	—
55	C₂²×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	C₂×#97	1	1	yes	characteristic
57	C₂²×C₁₀₉	#431	1	2	yes	—
58	C₄×C₁₀₉	—	2	2	no	—
59	C₄×C₁₀₉	—	4	4	no	—
60	C₄×C₁₀₉	#347	1	4	yes	—
61	C₄×C₁₀₉	#376	1	1	yes	characteristic
62	C₄×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₁₀₉	#431	1	2	yes	—
64	#12594	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₁₀₉	—	2	2	no	—
66	C₂×C₁₀₉	#1744	1	1	yes	characteristic
67	C₂×C₁₀₉	C₂×#347	1	2	yes	—
68	C₄×M₃₂	C₂×C₁₀₉	1	2	yes	—
69	C₄×#97	C₂×C₁₀₉	1	1	yes	characteristic
70	#1375	C₂×C₁₀₉	1	4	yes	—
71	C₁₀₉	#12594	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
73	C₂×#97	C₄×C₁₀₉	1	2	yes	—
74	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
75	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
77	C₄³	C₄×C₁₀₉	1	1	yes	characteristic
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	—	2	2	no	—
84	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
85	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
86	M₃₂	—	2	8	no	—
87	C₂×C₁₆	C₂×C₄×C₁₀₉	1	8	yes	—
88	#97	—	2	4	no	—
89	#97	C₈×C₁₀₉	1	8	yes	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
92	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
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	2	yes	—
99	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
100	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
101	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
102	C₂×C₈	#41×C₁₀₉	1	4	yes	—
103	C₄²	—	2	2	no	—
104	C₄²	—	4	4	no	—
105	C₄²	C₄²×C₁₀₉	1	1	yes	characteristic
106	C₄²	C₄²×C₁₀₉	1	1	yes	characteristic
107	C₄²	C₂×C₈×C₁₀₉	1	4	yes	—
108	C₄²	M₁₆×C₁₀₉	1	2	yes	—
109	C₁₆	—	2	8	no	—
110	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	#97×C₁₀₉	1	2	yes	—
115	C₂×C₄	#97×C₁₀₉	1	2	yes	—
116	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
117	C₂×C₄	#117×C₁₀₉	1	2	yes	—
118	C₂×C₄	C₂²×C₈×C₁₀₉	1	2	yes	—
119	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
120	C₈	—	2	4	no	—
121	C₈	C₄×C₈×C₁₀₉	1	4	yes	—
122	C₈	#117×C₁₀₉	1	4	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂×#97×C₁₀₉	1	1	yes	characteristic
125	C₂²	#431×C₁₀₉	1	2	yes	—
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	#347×C₁₀₉	1	4	yes	—
129	C₄	#376×C₁₀₉	1	1	yes	characteristic
130	C₄	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
131	C₄	#431×C₁₀₉	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₁₀₉×#1744	1	1	yes	characteristic
134	C₂	C₂×#347×C₁₀₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
10. #346×C₁₀₉	C₄	semidirect
11. C₄×C₁₆×C₁₀₉	C₄	semidirect
64. #12594	C₁₀₉	direct (N × H)
71. C₁₀₉	#12594	direct (N × H)
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
C₄×M₃₂×C₁₀₉	2
C₄×#97×C₁₀₉	2
C₁₀₉×#1375	2
#12594	109

Subgroups — C109×#12594 #42130588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12594 #42130588