Subgroups — C₁₀₉×#12611 #42130605

Order 27904.

486 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#346×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	C₈×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1374	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₀₉	—	2	4	no	—
8	C₂²×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
9	C₂²×C₁₆×C₁₀₉	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₄×C₂²⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
15	C₂²⋊C₁₆×C₁₀₉	C₄	1	4	yes	complemented
16	#346×C₁₀₉	—	2	4	no	—
17	#336×C₁₀₉	C₄	1	2	yes	—
18	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	—	2	2	no	—
20	C₂²×C₈×C₁₀₉	—	2	2	no	—
21	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₀₉	C₈	1	2	yes	—
27	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₁₀₉	—	2	2	no	—
29	C₂×C₁₆×C₁₀₉	—	2	2	no	—
30	C₂×C₁₆×C₁₀₉	—	2	2	no	—
31	C₂×C₁₆×C₁₀₉	—	2	8	no	—
32	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
34	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
36	C₂²⋊C₈×C₁₀₉	C₈	1	4	yes	—
37	C₄×C₈×C₁₀₉	—	2	2	no	—
38	C₄×C₈×C₁₀₉	—	2	2	no	—
39	#95×C₁₀₉	C₈	1	2	yes	—
40	C₂⁴×C₁₀₉	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	—	2	2	no	—
42	C₂²×C₄×C₁₀₉	—	2	2	no	—
43	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
44	C₂²×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	—	2	2	no	—
49	C₂×C₈×C₁₀₉	—	2	2	no	—
50	C₂×C₈×C₁₀₉	—	2	2	no	—
51	C₂×C₈×C₁₀₉	—	2	2	no	—
52	C₂×C₈×C₁₀₉	—	2	2	no	—
53	C₂×C₈×C₁₀₉	—	2	4	no	—
54	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
55	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
56	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₀₉	#41	1	2	yes	—
60	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
61	C₄²×C₁₀₉	—	2	2	no	—
62	C₄²×C₁₀₉	—	2	2	no	—
63	C₁₆×C₁₀₉	—	2	2	no	—
64	C₁₆×C₁₀₉	—	2	4	no	—
65	C₁₆×C₁₀₉	C₂²⋊C₄	1	4	yes	complemented
66	C₂³×C₁₀₉	—	2	2	no	—
67	C₂³×C₁₀₉	—	2	2	no	—
68	C₂³×C₁₀₉	C₄×C₈	1	2	yes	—
69	C₂³×C₁₀₉	M₃₂	1	4	yes	—
70	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	—	2	4	no	—
75	C₂×C₄×C₁₀₉	—	2	4	no	—
76	C₂×C₄×C₁₀₉	—	2	4	no	—
77	C₂×C₄×C₁₀₉	—	2	4	no	—
78	C₂×C₄×C₁₀₉	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	#131	1	2	yes	—
83	C₈×C₁₀₉	—	2	2	no	—
84	C₈×C₁₀₉	—	2	4	no	—
85	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₁₀₉	#117	1	1	yes	characteristic
87	C₈×C₁₀₉	C₄×D₄	1	2	yes	—
88	C₂²×C₁₀₉	—	2	2	no	—
89	C₂²×C₁₀₉	—	2	8	no	—
90	C₂²×C₁₀₉	—	2	2	no	—
91	C₂²×C₁₀₉	#346	1	4	yes	complemented
92	C₂²×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
93	C₂²×C₁₀₉	C₈×D₄	1	2	yes	—
94	C₂²×C₁₀₉	C₂×M₃₂	1	2	yes	—
95	C₂²×C₁₀₉	#504	1	2	yes	—
96	C₄×C₁₀₉	—	2	2	no	—
97	C₄×C₁₀₉	—	4	4	no	—
98	C₄×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₁₀₉	#405	1	1	yes	characteristic
100	C₄×C₁₀₉	C₈×D₄	1	2	yes	—
101	#12611	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₁₀₉	—	2	4	no	—
103	C₂×C₁₀₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₁₀₉	C₂×#346	1	1	yes	characteristic
105	C₂×C₁₀₉	#2108	1	1	yes	characteristic
106	C₂×C₁₀₉	#2168	1	4	yes	—
107	C₂³×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
108	C₂×C₂²⋊C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
109	C₂×#346	C₂×C₁₀₉	1	2	yes	complemented
110	C₈×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
111	#1374	C₂×C₁₀₉	1	2	yes	complemented
112	C₁₀₉	#12611	1	1	yes	characteristic, direct product factor, Sylow
113	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
114	C₂²×C₁₆	—	2	4	no	—
115	C₂²×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
116	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
117	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
119	C₂×C₂²⋊C₈	C₄×C₁₀₉	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
121	C₄×C₂²⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
122	C₂²⋊C₁₆	C₄×C₁₀₉	1	4	yes	complemented
123	#346	—	2	4	no	—
124	#336	C₄×C₁₀₉	1	2	yes	—
125	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
129	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
130	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
131	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
132	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	C₈×C₁₀₉	1	2	yes	—
134	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	2	8	no	—
139	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
140	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
141	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
142	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
143	C₂²⋊C₈	C₈×C₁₀₉	1	4	yes	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	#95	C₈×C₁₀₉	1	2	yes	—
147	C₂⁴	C₂×C₈×C₁₀₉	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
151	C₂²×C₄	C₂×C₈×C₁₀₉	1	4	yes	—
152	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
154	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
162	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
163	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
164	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
165	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
166	C₂×C₈	#41×C₁₀₉	1	2	yes	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₁₆	—	2	2	no	—
171	C₁₆	—	2	4	no	—
172	C₁₆	C₂²⋊C₄×C₁₀₉	1	4	yes	complemented
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	C₄×C₈×C₁₀₉	1	2	yes	—
176	C₂³	M₃₂×C₁₀₉	1	4	yes	—
177	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	C₄×C₈×C₁₀₉	1	2	yes	—
186	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
187	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
188	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
189	C₂×C₄	#131×C₁₀₉	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	—	2	4	no	—
192	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
193	C₈	#117×C₁₀₉	1	1	yes	characteristic
194	C₈	C₄×D₄×C₁₀₉	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	#346×C₁₀₉	1	4	yes	complemented
199	C₂²	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
200	C₂²	C₈×D₄×C₁₀₉	1	2	yes	—
201	C₂²	C₂×M₃₂×C₁₀₉	1	2	yes	—
202	C₂²	#504×C₁₀₉	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	4	4	no	—
205	C₄	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
206	C₄	#405×C₁₀₉	1	1	yes	characteristic
207	C₄	C₈×D₄×C₁₀₉	1	2	yes	—
208	C₂	—	2	4	no	—
209	C₂	C₈×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
210	C₂	C₂×#346×C₁₀₉	1	1	yes	characteristic
211	C₂	C₁₀₉×#2108	1	1	yes	characteristic
212	C₂	C₁₀₉×#2168	1	4	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₀₉	C₂³
commutator	C₂²	C₂×C₄×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂³×C₁₀₉	C₂²×C₈
prime radical	C₂⁴×C₁₀₉	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#346×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1374	C₂	semidirect
8. C₂²×C₁₆×C₁₀₉	C₄	semidirect
15. C₂²⋊C₁₆×C₁₀₉	C₄	semidirect
65. C₁₆×C₁₀₉	C₂²⋊C₄	semidirect
91. C₂²×C₁₀₉	#346	semidirect
101. #12611	C₁₀₉	direct (N × H)
109. C₂×#346	C₂×C₁₀₉	semidirect
111. #1374	C₂×C₁₀₉	semidirect
112. C₁₀₉	#12611	direct (N × H)
115. C₂²×C₁₆	C₄×C₁₀₉	semidirect
122. C₂²⋊C₁₆	C₄×C₁₀₉	semidirect
172. C₁₆	C₂²⋊C₄×C₁₀₉	semidirect
198. C₂²	#346×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₁₆×C₁₀₉	2
C₂×C₂²⋊C₁₆×C₁₀₉	2
C₂×#346×C₁₀₉	2
C₈×C₂²⋊C₄×C₁₀₉	2
C₁₀₉×#1374	2
#12611	109

Subgroups — C109×#12611 #42130605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12611 #42130605