Subgroups — C₁₀₉×#20477 #42138471

Order 27904.

922 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2283	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1771	C₂	1	4	yes	maximal, complemented
5	C₁₀₉×#1753	C₂	1	4	yes	maximal
6	C₁₀₉×#1746	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1726	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
10	#432×C₁₀₉	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
13	C₄×M₁₆×C₁₀₉	C₄	1	8	yes	complemented
14	C₂×#97×C₁₀₉	C₂²	1	4	yes	—
15	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
16	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₀₉	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	#383×C₁₀₉	C₄	1	2	yes	—
21	#382×C₁₀₉	C₄	1	2	yes	—
22	#381×C₁₀₉	C₄	1	2	yes	—
23	#380×C₁₀₉	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
25	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
26	#376×C₁₀₉	C₂²	1	2	yes	—
27	#336×C₁₀₉	C₂²	1	8	yes	—
28	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₁₀₉	—	2	8	no	—
32	C₂×M₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₀₉	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
43	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
46	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
47	#97×C₁₀₉	—	2	8	no	—
48	#97×C₁₀₉	C₂×C₄	1	8	yes	—
49	#97×C₁₀₉	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₁₀₉	—	2	8	no	—
51	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
53	#95×C₁₀₉	C₂×C₄	1	4	yes	—
54	#95×C₁₀₉	C₂×C₄	1	4	yes	—
55	#95×C₁₀₉	C₂×C₄	1	4	yes	—
56	#95×C₁₀₉	C₂×C₄	1	4	yes	—
57	C₂⁴×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	—	2	4	no	—
59	C₂²×C₄×C₁₀₉	—	4	2	no	—
60	C₂²×C₄×C₁₀₉	—	4	2	no	—
61	C₂²×C₄×C₁₀₉	C₄²	1	4	yes	—
62	C₂²×C₄×C₁₀₉	C₄²	1	4	yes	—
63	C₂²×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₁₀₉	—	2	16	no	—
73	C₂×C₈×C₁₀₉	—	2	4	no	—
74	C₂×C₈×C₁₀₉	—	2	4	no	—
75	C₂×C₈×C₁₀₉	—	2	4	no	—
76	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₁₀₉	C₂²×C₄	1	8	yes	—
78	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₀₉	#41	1	8	yes	—
80	C₄⋊C₄×C₁₀₉	—	4	4	no	—
81	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
82	C₄²×C₁₀₉	—	2	4	no	—
83	C₄²×C₁₀₉	—	2	4	no	—
84	C₄²×C₁₀₉	C₄²	1	4	yes	—
85	C₄²×C₁₀₉	C₄²	1	4	yes	—
86	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
87	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂³×C₁₀₉	—	2	2	no	—
91	C₂³×C₁₀₉	—	2	2	no	—
92	C₂³×C₁₀₉	—	2	2	no	—
93	C₂³×C₁₀₉	C₂×C₄²	1	2	yes	—
94	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	2	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	2	no	—
100	C₂×C₄×C₁₀₉	—	2	4	no	—
101	C₂×C₄×C₁₀₉	—	2	4	no	—
102	C₂×C₄×C₁₀₉	—	2	8	no	—
103	C₂×C₄×C₁₀₉	—	2	8	no	—
104	C₂×C₄×C₁₀₉	—	2	8	no	—
105	C₂×C₄×C₁₀₉	—	4	4	no	—
106	C₂×C₄×C₁₀₉	—	4	4	no	—
107	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂×#41	1	4	yes	—
115	C₈×C₁₀₉	—	2	4	no	—
116	C₈×C₁₀₉	—	2	8	no	—
117	C₈×C₁₀₉	#117	1	8	yes	—
118	C₂²×C₁₀₉	—	2	2	no	—
119	C₂²×C₁₀₉	—	2	2	no	—
120	C₂²×C₁₀₉	—	2	2	no	—
121	C₂²×C₁₀₉	—	4	4	no	—
122	C₂²×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
123	C₂²×C₁₀₉	C₄×#41	1	2	yes	—
124	C₂²×C₁₀₉	#568	1	2	yes	—
125	C₂²×C₁₀₉	#568	1	2	yes	—
126	C₄×C₁₀₉	—	2	2	no	—
127	C₄×C₁₀₉	—	2	2	no	—
128	C₄×C₁₀₉	—	2	2	no	—
129	C₄×C₁₀₉	—	2	2	no	—
130	C₄×C₁₀₉	—	2	2	no	—
131	C₄×C₁₀₉	—	4	8	no	—
132	C₄×C₁₀₉	C₂×#117	1	2	yes	—
133	C₄×C₁₀₉	C₄×#41	1	2	yes	—
134	#20477	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂×C₁₀₉	—	4	2	no	—
136	C₂×C₁₀₉	C₄×#117	1	1	yes	characteristic
137	C₂×C₁₀₉	#2873	1	1	yes	characteristic
138	C₂×C₁₀₉	#2873	1	1	yes	characteristic
139	C₂×C₁₀₉	#2967	1	4	yes	—
140	C₂×#432	C₂×C₁₀₉	1	1	yes	characteristic
141	C₂×C₄×M₁₆	C₂×C₁₀₉	1	1	yes	characteristic
142	#2283	C₂×C₁₀₉	1	1	yes	characteristic
143	#1771	C₂×C₁₀₉	1	4	yes	complemented
144	#1753	C₂×C₁₀₉	1	4	yes	—
145	#1746	C₂×C₁₀₉	1	2	yes	complemented
146	#1726	C₂×C₁₀₉	1	2	yes	complemented
147	C₁₀₉	#20477	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
149	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
150	#432	C₄×C₁₀₉	1	8	yes	complemented
151	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
152	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
153	C₄×M₁₆	C₄×C₁₀₉	1	8	yes	complemented
154	C₂×#97	C₂²×C₁₀₉	1	4	yes	—
155	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
156	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₁₀₉	1	4	yes	—
158	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
160	#383	C₄×C₁₀₉	1	2	yes	—
161	#382	C₄×C₁₀₉	1	2	yes	—
162	#381	C₄×C₁₀₉	1	2	yes	—
163	#380	C₄×C₁₀₉	1	2	yes	—
164	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
165	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
166	#376	C₂²×C₁₀₉	1	2	yes	—
167	#336	C₂²×C₁₀₉	1	8	yes	—
168	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
169	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
170	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
171	C₂×M₁₆	—	2	8	no	—
172	C₂×M₁₆	C₂×C₄×C₁₀₉	1	8	yes	—
173	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
174	C₂²×C₈	C₂³×C₁₀₉	1	4	yes	—
175	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
176	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
179	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
181	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
182	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
183	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
185	C₄⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
186	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
187	#97	—	2	8	no	—
188	#97	C₂×C₄×C₁₀₉	1	8	yes	—
189	#97	C₂×C₄×C₁₀₉	1	8	yes	—
190	C₄×C₈	—	2	8	no	—
191	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
192	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
193	#95	C₂×C₄×C₁₀₉	1	4	yes	—
194	#95	C₂×C₄×C₁₀₉	1	4	yes	—
195	#95	C₂×C₄×C₁₀₉	1	4	yes	—
196	#95	C₂×C₄×C₁₀₉	1	4	yes	—
197	C₂⁴	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₄²×C₁₀₉	1	4	yes	—
202	C₂²×C₄	C₄²×C₁₀₉	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
204	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
211	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
212	M₁₆	—	2	16	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
217	C₂×C₈	C₂²×C₄×C₁₀₉	1	8	yes	—
218	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
219	C₂×C₈	#41×C₁₀₉	1	8	yes	—
220	C₄⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	4	no	—
224	C₄²	C₄²×C₁₀₉	1	4	yes	—
225	C₄²	C₄²×C₁₀₉	1	4	yes	—
226	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
227	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
228	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
229	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	C₂×C₄²×C₁₀₉	1	2	yes	—
234	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
248	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
249	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
250	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
251	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
252	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
253	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₁₀₉	1	4	yes	—
255	C₈	—	2	4	no	—
256	C₈	—	2	8	no	—
257	C₈	#117×C₁₀₉	1	8	yes	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
263	C₂²	C₄×#41×C₁₀₉	1	2	yes	—
264	C₂²	#568×C₁₀₉	1	2	yes	—
265	C₂²	#568×C₁₀₉	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	8	no	—
272	C₄	C₂×#117×C₁₀₉	1	2	yes	—
273	C₄	C₄×#41×C₁₀₉	1	2	yes	—
274	C₂	—	4	2	no	—
275	C₂	C₄×#117×C₁₀₉	1	1	yes	characteristic
276	C₂	C₁₀₉×#2873	1	1	yes	characteristic
277	C₂	C₁₀₉×#2873	1	1	yes	characteristic
278	C₂	C₁₀₉×#2967	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂³×C₁₀₉	C₂³×C₄
prime radical	C₂⁴×C₁₀₉	C₂²×C₄
LCS	G ▷ C₂² ▷ 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×#1771	C₂	semidirect
6. C₁₀₉×#1746	C₂	semidirect
7. C₁₀₉×#1726	C₂	semidirect
10. #432×C₁₀₉	C₄	semidirect
13. C₄×M₁₆×C₁₀₉	C₄	semidirect
134. #20477	C₁₀₉	direct (N × H)
143. #1771	C₂×C₁₀₉	semidirect
145. #1746	C₂×C₁₀₉	semidirect
146. #1726	C₂×C₁₀₉	semidirect
147. C₁₀₉	#20477	direct (N × H)
150. #432	C₄×C₁₀₉	semidirect
153. C₄×M₁₆	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×#432×C₁₀₉	2
C₂×C₄×M₁₆×C₁₀₉	2
C₁₀₉×#2283	2
C₁₀₉×#1771	2
C₁₀₉×#1753	2
C₁₀₉×#1746	2
C₁₀₉×#1726	2
#20477	109

Subgroups — C109×#20477 #42138471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20477 #42138471