Subgroups — C₁₀₉×#12597 #42130591

Order 27904.

258 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2110	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2108	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1747	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1379	C₂	1	2	yes	maximal
5	C₁₀₉×#1378	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
7	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	#405×C₁₀₉	C₂²	1	2	yes	—
10	#405×C₁₀₉	C₂²	1	1	yes	characteristic
11	#347×C₁₀₉	C₄	1	4	yes	complemented
12	#346×C₁₀₉	C₄	1	2	yes	complemented
13	C₄×C₁₆×C₁₀₉	C₄	1	2	yes	complemented
14	#338×C₁₀₉	C₄	1	1	yes	characteristic
15	H(3,ℤ/4)×C₁₀₉	C₄	1	1	yes	characteristic
16	H(3,ℤ/4)×C₁₀₉	C₄	1	2	yes	—
17	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂×M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
20	#117×C₁₀₉	C₂×C₄	1	2	yes	—
21	#117×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₁₀₉	—	2	2	no	—
23	M₃₂×C₁₀₉	—	2	2	no	—
24	M₃₂×C₁₀₉	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₁₀₉	—	2	2	no	—
26	C₂×C₁₆×C₁₀₉	—	2	2	no	—
27	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
30	#97×C₁₀₉	—	2	2	no	—
31	#97×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₀₉	—	2	2	no	—
33	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₁₀₉	—	2	2	no	—
36	M₁₆×C₁₀₉	—	2	2	no	—
37	M₁₆×C₁₀₉	—	2	2	no	—
38	C₂×C₈×C₁₀₉	—	2	2	no	—
39	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
40	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	#41	1	2	yes	—
43	C₂×C₈×C₁₀₉	#41	1	2	yes	—
44	C₄⋊C₄×C₁₀₉	—	2	1	no	—
45	C₄⋊C₄×C₁₀₉	—	2	2	no	—
46	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
47	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
48	C₄²×C₁₀₉	—	2	2	no	—
49	C₄²×C₁₀₉	C₄²	1	2	yes	—
50	C₁₆×C₁₀₉	—	2	1	no	—
51	C₁₆×C₁₀₉	—	2	1	no	—
52	C₁₆×C₁₀₉	—	2	4	no	—
53	C₁₆×C₁₀₉	—	2	2	no	—
54	C₂³×C₁₀₉	#131	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	—	2	2	no	—
56	C₂×C₄×C₁₀₉	—	4	2	no	—
57	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	#117	1	2	yes	—
59	C₂×C₄×C₁₀₉	#131	1	2	yes	—
60	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
61	C₈×C₁₀₉	—	2	2	no	—
62	C₈×C₁₀₉	—	4	2	no	—
63	C₈×C₁₀₉	#117	1	1	yes	characteristic
64	C₈×C₁₀₉	#117	1	2	yes	—
65	C₈×C₁₀₉	#117	1	1	yes	characteristic
66	C₂²×C₁₀₉	—	4	1	no	—
67	C₂²×C₁₀₉	#405	1	1	yes	characteristic
68	C₂²×C₁₀₉	#433	1	2	yes	—
69	C₄×C₁₀₉	—	4	2	no	—
70	C₄×C₁₀₉	—	8	2	no	—
71	C₄×C₁₀₉	#376	1	1	yes	characteristic
72	C₄×C₁₀₉	#405	1	1	yes	characteristic
73	C₄×C₁₀₉	#433	1	2	yes	—
74	#12597	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₁₀₉	—	2	1	no	—
76	C₂×C₁₀₉	—	2	2	no	—
77	C₂×C₁₀₉	#1746	1	1	yes	characteristic
78	#2110	C₂×C₁₀₉	1	1	yes	characteristic
79	#2108	C₂×C₁₀₉	1	1	yes	characteristic
80	#1747	C₂×C₁₀₉	1	1	yes	characteristic
81	#1379	C₂×C₁₀₉	1	2	yes	—
82	#1378	C₂×C₁₀₉	1	2	yes	—
83	C₁₀₉	#12597	1	1	yes	characteristic, direct product factor, Sylow
84	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
85	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
86	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
87	#405	C₂²×C₁₀₉	1	2	yes	—
88	#405	C₂²×C₁₀₉	1	1	yes	characteristic
89	#347	C₄×C₁₀₉	1	4	yes	complemented
90	#346	C₄×C₁₀₉	1	2	yes	complemented
91	C₄×C₁₆	C₄×C₁₀₉	1	2	yes	complemented
92	#338	C₄×C₁₀₉	1	1	yes	characteristic
93	H(3,ℤ/4)	C₄×C₁₀₉	1	1	yes	characteristic
94	H(3,ℤ/4)	C₄×C₁₀₉	1	2	yes	—
95	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
96	C₂×M₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
97	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
98	#117	C₂×C₄×C₁₀₉	1	2	yes	—
99	#117	C₂×C₄×C₁₀₉	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	2	2	no	—
102	M₃₂	C₂×C₄×C₁₀₉	1	4	yes	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
106	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
107	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
108	#97	—	2	2	no	—
109	#97	C₂×C₄×C₁₀₉	1	2	yes	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
112	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
118	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
119	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
120	C₂×C₈	#41×C₁₀₉	1	2	yes	—
121	C₂×C₈	#41×C₁₀₉	1	2	yes	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	2	1	no	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	C₄²×C₁₀₉	1	2	yes	—
128	C₁₆	—	2	1	no	—
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	4	no	—
131	C₁₆	—	2	2	no	—
132	C₂³	#131×C₁₀₉	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
136	C₂×C₄	#117×C₁₀₉	1	2	yes	—
137	C₂×C₄	#131×C₁₀₉	1	2	yes	—
138	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	2	no	—
141	C₈	#117×C₁₀₉	1	1	yes	characteristic
142	C₈	#117×C₁₀₉	1	2	yes	—
143	C₈	#117×C₁₀₉	1	1	yes	characteristic
144	C₂²	—	4	1	no	—
145	C₂²	#405×C₁₀₉	1	1	yes	characteristic
146	C₂²	#433×C₁₀₉	1	2	yes	—
147	C₄	—	4	2	no	—
148	C₄	—	8	2	no	—
149	C₄	#376×C₁₀₉	1	1	yes	characteristic
150	C₄	#405×C₁₀₉	1	1	yes	characteristic
151	C₄	#433×C₁₀₉	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	2	no	—
154	C₂	C₁₀₉×#1746	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₀₉	#117
commutator	C₂×C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#1746
prime radical	C₂³×C₁₀₉	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₉ ◁ #405×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
11. #347×C₁₀₉	C₄	semidirect
12. #346×C₁₀₉	C₄	semidirect
13. C₄×C₁₆×C₁₀₉	C₄	semidirect
74. #12597	C₁₀₉	direct (N × H)
83. C₁₀₉	#12597	direct (N × H)
89. #347	C₄×C₁₀₉	semidirect
90. #346	C₄×C₁₀₉	semidirect
91. 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
C₁₀₉×#2110	2
C₁₀₉×#2108	2
C₁₀₉×#1747	2
C₁₀₉×#1379	2
C₁₀₉×#1378	2
#12597	109

Subgroups — C109×#12597 #42130591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12597 #42130591