Subgroups — C₁₀₉×#11269 #42129263

Order 27904.

642 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#341×C₁₀₉	C₂	1	2	yes	maximal
3	C₂×#341×C₁₀₉	C₂	1	2	yes	maximal
4	C₁₀₉×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂×#336×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₀₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
11	#423×C₁₀₉	—	2	4	no	—
12	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
13	#341×C₁₀₉	—	2	4	no	—
14	#341×C₁₀₉	—	2	4	no	—
15	#336×C₁₀₉	—	2	2	no	—
16	#336×C₁₀₉	C₄	1	2	yes	—
17	#336×C₁₀₉	C₄	1	2	yes	—
18	C₂³×C₄×C₁₀₉	Q₈	1	2	yes	—
19	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₀₉	—	2	2	no	—
21	C₂×M₁₆×C₁₀₉	—	2	1	no	—
22	C₂×M₁₆×C₁₀₉	—	2	1	no	—
23	C₂×M₁₆×C₁₀₉	—	2	2	no	—
24	C₂×M₁₆×C₁₀₉	—	2	2	no	—
25	C₂×M₁₆×C₁₀₉	—	2	2	no	—
26	C₂×M₁₆×C₁₀₉	—	2	2	no	—
27	C₂×M₁₆×C₁₀₉	—	2	2	no	—
28	C₂×M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×M₁₆×C₁₀₉	—	2	2	no	—
30	C₂×M₁₆×C₁₀₉	—	2	2	no	—
31	C₂×M₁₆×C₁₀₉	D₄	1	2	yes	—
32	C₂×M₁₆×C₁₀₉	D₄	1	4	yes	—
33	C₂×M₁₆×C₁₀₉	Q₈	1	2	yes	—
34	C₂²×C₈×C₁₀₉	—	2	2	no	—
35	C₂²×C₈×C₁₀₉	—	2	2	no	—
36	C₂²×C₈×C₁₀₉	—	2	1	no	—
37	C₂²×C₈×C₁₀₉	—	2	1	no	—
38	C₂²×C₈×C₁₀₉	—	2	2	no	—
39	C₂²×C₈×C₁₀₉	—	2	2	no	—
40	C₂²×C₈×C₁₀₉	—	2	2	no	—
41	C₂²×C₈×C₁₀₉	—	2	2	no	—
42	C₂²×C₈×C₁₀₉	—	2	2	no	—
43	C₂²×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₁₀₉	—	2	2	no	—
45	C₂×C₄²×C₁₀₉	—	2	2	no	—
46	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
49	C₄⋊C₈×C₁₀₉	—	2	8	no	—
50	C₂⁴×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	—	2	4	no	—
52	C₂²×C₄×C₁₀₉	—	2	4	no	—
53	C₂²×C₄×C₁₀₉	—	4	2	no	—
54	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
62	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
63	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
66	M₁₆×C₁₀₉	—	2	4	no	—
67	M₁₆×C₁₀₉	—	2	4	no	—
68	M₁₆×C₁₀₉	—	2	4	no	—
69	M₁₆×C₁₀₉	—	2	4	no	—
70	M₁₆×C₁₀₉	—	4	2	no	—
71	M₁₆×C₁₀₉	—	4	2	no	—
72	M₁₆×C₁₀₉	—	4	2	no	—
73	M₁₆×C₁₀₉	—	4	2	no	—
74	C₂×C₈×C₁₀₉	—	2	4	no	—
75	C₂×C₈×C₁₀₉	—	2	4	no	—
76	C₂×C₈×C₁₀₉	—	2	4	no	—
77	C₂×C₈×C₁₀₉	—	2	4	no	—
78	C₂×C₈×C₁₀₉	—	2	2	no	—
79	C₂×C₈×C₁₀₉	—	2	2	no	—
80	C₂×C₈×C₁₀₉	—	4	1	no	—
81	C₂×C₈×C₁₀₉	—	4	2	no	—
82	C₂×C₈×C₁₀₉	—	4	2	no	—
83	C₂×C₈×C₁₀₉	—	4	2	no	—
84	C₂×C₈×C₁₀₉	—	4	1	no	—
85	C₂×C₈×C₁₀₉	—	4	2	no	—
86	C₂×C₈×C₁₀₉	—	4	1	no	—
87	C₂×C₈×C₁₀₉	—	4	2	no	—
88	C₂×C₈×C₁₀₉	—	4	1	no	—
89	C₂×C₈×C₁₀₉	—	4	2	no	—
90	C₄²×C₁₀₉	—	2	2	no	—
91	C₄²×C₁₀₉	—	2	2	no	—
92	C₄²×C₁₀₉	—	2	2	no	—
93	C₄²×C₁₀₉	—	2	2	no	—
94	C₂³×C₁₀₉	—	2	2	no	—
95	C₂³×C₁₀₉	—	2	2	no	—
96	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₁₀₉	#125	1	2	yes	—
99	C₂³×C₁₀₉	#125	1	1	yes	characteristic
100	C₂³×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	—	2	2	no	—
102	C₂×C₄×C₁₀₉	—	2	2	no	—
103	C₂×C₄×C₁₀₉	—	2	2	no	—
104	C₂×C₄×C₁₀₉	—	2	2	no	—
105	C₂×C₄×C₁₀₉	—	2	2	no	—
106	C₂×C₄×C₁₀₉	—	4	8	no	—
107	C₂×C₄×C₁₀₉	#108	1	4	yes	—
108	C₂×C₄×C₁₀₉	#117	1	2	yes	—
109	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	#121	1	2	yes	—
114	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	#123	1	2	yes	—
118	C₂×C₄×C₁₀₉	#123	1	2	yes	—
119	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	#124	1	2	yes	—
121	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	#125	1	2	yes	—
124	C₂×C₄×C₁₀₉	#126	1	2	yes	—
125	C₂×C₄×C₁₀₉	#126	1	2	yes	—
126	C₈×C₁₀₉	—	4	4	no	—
127	C₈×C₁₀₉	—	4	2	no	—
128	C₈×C₁₀₉	—	4	4	no	—
129	C₈×C₁₀₉	—	4	2	no	—
130	C₂²×C₁₀₉	—	2	2	no	—
131	C₂²×C₁₀₉	—	2	2	no	—
132	C₂²×C₁₀₉	—	2	4	no	—
133	C₂²×C₁₀₉	—	2	2	no	—
134	C₂²×C₁₀₉	—	2	2	no	—
135	C₂²×C₁₀₉	#382	1	1	yes	characteristic
136	C₂²×C₁₀₉	#385	1	1	yes	characteristic
137	C₂²×C₁₀₉	#389	1	1	yes	characteristic
138	C₂²×C₁₀₉	#398	1	2	yes	—
139	C₂²×C₁₀₉	#400	1	2	yes	—
140	C₂²×C₁₀₉	C₂×#108	1	2	yes	—
141	C₂²×C₁₀₉	#430	1	2	yes	—
142	C₄×C₁₀₉	—	2	4	no	—
143	C₄×C₁₀₉	—	4	2	no	—
144	C₄×C₁₀₉	#387	1	2	yes	—
145	C₄×C₁₀₉	#388	1	2	yes	—
146	C₄×C₁₀₉	#393	1	1	yes	characteristic
147	C₄×C₁₀₉	#394	1	1	yes	characteristic
148	C₄×C₁₀₉	#397	1	1	yes	characteristic
149	C₄×C₁₀₉	#400	1	1	yes	characteristic
150	#11269	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₁₀₉	—	2	2	no	—
152	C₂×C₁₀₉	—	2	2	no	—
153	C₂×C₁₀₉	#1442	1	1	yes	characteristic
154	C₂×C₁₀₉	#1837	1	1	yes	characteristic
155	C₂×C₁₀₉	#1838	1	1	yes	characteristic
156	C₂×C₁₀₉	#1855	1	2	yes	—
157	C₂×C₁₀₉	#1950	1	2	yes	—
158	C₂×#423	C₂×C₁₀₉	1	1	yes	characteristic
159	C₂×#341	C₂×C₁₀₉	1	2	yes	—
160	C₂×#341	C₂×C₁₀₉	1	2	yes	—
161	#1728	C₂×C₁₀₉	1	1	yes	characteristic
162	C₂×#336	C₂×C₁₀₉	1	1	yes	characteristic
163	C₁₀₉	#11269	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
165	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
166	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
167	C₂³×C₈	C₂²×C₁₀₉	1	2	yes	—
168	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
169	#423	—	2	4	no	—
170	C₂×C₄⋊C₈	—	2	2	no	—
171	#341	—	2	4	no	—
172	#341	—	2	4	no	—
173	#336	—	2	2	no	—
174	#336	C₄×C₁₀₉	1	2	yes	—
175	#336	C₄×C₁₀₉	1	2	yes	—
176	C₂³×C₄	Q₈×C₁₀₉	1	2	yes	—
177	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	D₄×C₁₀₉	1	2	yes	—
190	C₂×M₁₆	D₄×C₁₀₉	1	4	yes	—
191	C₂×M₁₆	Q₈×C₁₀₉	1	2	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	1	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
205	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
206	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
207	C₄⋊C₈	—	2	8	no	—
208	C₂⁴	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
213	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
219	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
220	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
221	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
222	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
223	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
255	C₂³	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
256	C₂³	#125×C₁₀₉	1	2	yes	—
257	C₂³	#125×C₁₀₉	1	1	yes	characteristic
258	C₂³	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	#108×C₁₀₉	1	4	yes	—
266	C₂×C₄	#117×C₁₀₉	1	2	yes	—
267	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
268	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
269	C₂×C₄	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
270	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
271	C₂×C₄	#121×C₁₀₉	1	2	yes	—
272	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
273	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
274	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
275	C₂×C₄	#123×C₁₀₉	1	2	yes	—
276	C₂×C₄	#123×C₁₀₉	1	2	yes	—
277	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
278	C₂×C₄	#124×C₁₀₉	1	2	yes	—
279	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
280	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
281	C₂×C₄	#125×C₁₀₉	1	2	yes	—
282	C₂×C₄	#126×C₁₀₉	1	2	yes	—
283	C₂×C₄	#126×C₁₀₉	1	2	yes	—
284	C₈	—	4	4	no	—
285	C₈	—	4	2	no	—
286	C₈	—	4	4	no	—
287	C₈	—	4	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	4	no	—
293	C₂²	#382×C₁₀₉	1	1	yes	characteristic
294	C₂²	#385×C₁₀₉	1	1	yes	characteristic
295	C₂²	#389×C₁₀₉	1	1	yes	characteristic
296	C₂²	#398×C₁₀₉	1	2	yes	—
297	C₂²	#400×C₁₀₉	1	2	yes	—
298	C₂²	C₂×#108×C₁₀₉	1	2	yes	—
299	C₂²	#430×C₁₀₉	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	#387×C₁₀₉	1	2	yes	—
303	C₄	#388×C₁₀₉	1	2	yes	—
304	C₄	#393×C₁₀₉	1	1	yes	characteristic
305	C₄	#394×C₁₀₉	1	1	yes	characteristic
306	C₄	#397×C₁₀₉	1	1	yes	characteristic
307	C₄	#400×C₁₀₉	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	C₁₀₉×#1442	1	1	yes	characteristic
311	C₂	C₁₀₉×#1837	1	1	yes	characteristic
312	C₂	C₁₀₉×#1838	1	1	yes	characteristic
313	C₂	C₁₀₉×#1855	1	2	yes	—
314	C₂	C₁₀₉×#1950	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
socle	C₂³×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₀₉	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
150. #11269	C₁₀₉	direct (N × H)
163. C₁₀₉	#11269	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#423×C₁₀₉	2
C₂×#341×C₁₀₉	2
C₁₀₉×#1728	2
C₂×#336×C₁₀₉	2
#11269	109

Subgroups — C109×#11269 #42129263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11269 #42129263