Subgroups — C₁₀₉×#11243 #42129237

Order 27904.

670 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2911	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1742	C₂	1	2	yes	maximal
4	C₂×#339×C₁₀₉	C₂	1	2	yes	maximal
5	C₂×#338×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	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
11	#428×C₁₀₉	—	2	4	no	—
12	#427×C₁₀₉	—	2	2	no	—
13	#423×C₁₀₉	—	2	2	no	—
14	#423×C₁₀₉	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
16	#343×C₁₀₉	—	2	4	no	—
17	#339×C₁₀₉	—	2	2	no	—
18	#339×C₁₀₉	—	2	2	no	—
19	#338×C₁₀₉	—	2	2	no	—
20	C₂³×C₄×C₁₀₉	—	2	1	no	—
21	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₀₉	—	2	2	no	—
23	C₂×M₁₆×C₁₀₉	—	2	2	no	—
24	C₂×M₁₆×C₁₀₉	—	2	2	no	—
25	C₂×M₁₆×C₁₀₉	—	2	1	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	—	2	2	no	—
32	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂×M₁₆×C₁₀₉	D₄	1	2	yes	—
35	C₂×M₁₆×C₁₀₉	Q₈	1	4	yes	—
36	C₂²×C₈×C₁₀₉	—	2	1	no	—
37	C₂²×C₈×C₁₀₉	—	2	2	no	—
38	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
39	#117×C₁₀₉	—	2	2	no	—
40	#117×C₁₀₉	—	2	2	no	—
41	#117×C₁₀₉	—	2	2	no	—
42	#117×C₁₀₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₀₉	D₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
45	C₂×C₄²×C₁₀₉	—	2	1	no	—
46	C₂×C₄²×C₁₀₉	—	2	1	no	—
47	C₂×C₄²×C₁₀₉	—	2	1	no	—
48	C₂×C₄²×C₁₀₉	—	2	2	no	—
49	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₁₀₉	Q₈	1	2	yes	—
53	#107×C₁₀₉	—	2	4	no	—
54	#106×C₁₀₉	—	2	4	no	—
55	C₄⋊C₈×C₁₀₉	—	2	8	no	—
56	C₂⁴×C₁₀₉	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	—	2	1	no	—
58	C₂²×C₄×C₁₀₉	—	2	2	no	—
59	C₂²×C₄×C₁₀₉	—	2	1	no	—
60	C₂²×C₄×C₁₀₉	—	2	2	no	—
61	C₂²×C₄×C₁₀₉	—	2	4	no	—
62	C₂²×C₄×C₁₀₉	—	4	4	no	—
63	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
71	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
72	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
73	M₁₆×C₁₀₉	—	2	4	no	—
74	M₁₆×C₁₀₉	—	2	4	no	—
75	M₁₆×C₁₀₉	—	2	4	no	—
76	M₁₆×C₁₀₉	—	2	4	no	—
77	M₁₆×C₁₀₉	—	2	4	no	—
78	M₁₆×C₁₀₉	—	4	2	no	—
79	M₁₆×C₁₀₉	—	4	2	no	—
80	M₁₆×C₁₀₉	—	4	2	no	—
81	C₂×C₈×C₁₀₉	—	2	2	no	—
82	C₂×C₈×C₁₀₉	—	2	2	no	—
83	C₂×C₈×C₁₀₉	—	2	2	no	—
84	C₂×C₈×C₁₀₉	—	2	4	no	—
85	C₂×C₈×C₁₀₉	—	4	1	no	—
86	C₂×C₈×C₁₀₉	—	4	2	no	—
87	C₂×C₈×C₁₀₉	—	4	1	no	—
88	C₂×C₈×C₁₀₉	—	4	2	no	—
89	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
90	C₄⋊C₄×C₁₀₉	—	2	4	no	—
91	C₄⋊C₄×C₁₀₉	—	2	4	no	—
92	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
93	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
94	C₄²×C₁₀₉	—	2	2	no	—
95	C₄²×C₁₀₉	—	2	1	no	—
96	C₄²×C₁₀₉	—	2	1	no	—
97	C₄²×C₁₀₉	—	2	2	no	—
98	C₄²×C₁₀₉	—	4	2	no	—
99	C₄²×C₁₀₉	—	4	2	no	—
100	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
101	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
102	C₂³×C₁₀₉	—	4	2	no	—
103	C₂³×C₁₀₉	#117	1	1	yes	characteristic
104	C₂³×C₁₀₉	#121	1	1	yes	characteristic
105	C₂³×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₁₀₉	#123	1	2	yes	—
107	C₂³×C₁₀₉	#125	1	2	yes	—
108	C₂×C₄×C₁₀₉	—	2	4	no	—
109	C₂×C₄×C₁₀₉	—	2	2	no	—
110	C₂×C₄×C₁₀₉	—	2	2	no	—
111	C₂×C₄×C₁₀₉	—	4	4	no	—
112	C₂×C₄×C₁₀₉	—	4	4	no	—
113	C₂×C₄×C₁₀₉	—	4	2	no	—
114	C₂×C₄×C₁₀₉	—	4	4	no	—
115	C₂×C₄×C₁₀₉	—	8	4	no	—
116	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	#121	1	2	yes	—
122	C₂×C₄×C₁₀₉	#121	1	2	yes	—
123	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
129	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
130	C₂×C₄×C₁₀₉	#125	1	2	yes	—
131	C₂×C₄×C₁₀₉	#126	1	2	yes	—
132	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
134	C₈×C₁₀₉	—	2	4	no	—
135	C₈×C₁₀₉	—	4	2	no	—
136	C₈×C₁₀₉	—	4	4	no	—
137	C₂²×C₁₀₉	—	2	2	no	—
138	C₂²×C₁₀₉	—	2	4	no	—
139	C₂²×C₁₀₉	—	4	1	no	—
140	C₂²×C₁₀₉	—	4	1	no	—
141	C₂²×C₁₀₉	—	4	2	no	—
142	C₂²×C₁₀₉	#380	1	1	yes	characteristic
143	C₂²×C₁₀₉	#387	1	2	yes	—
144	C₂²×C₁₀₉	#394	1	1	yes	characteristic
145	C₂²×C₁₀₉	#398	1	2	yes	—
146	C₂²×C₁₀₉	#398	1	1	yes	characteristic
147	C₄×C₁₀₉	—	2	4	no	—
148	C₄×C₁₀₉	—	4	2	no	—
149	C₄×C₁₀₉	—	8	4	no	—
150	C₄×C₁₀₉	#386	1	1	yes	characteristic
151	C₄×C₁₀₉	#389	1	2	yes	—
152	C₄×C₁₀₉	#391	1	1	yes	characteristic
153	C₄×C₁₀₉	#395	1	1	yes	characteristic
154	C₄×C₁₀₉	#396	1	2	yes	—
155	C₄×C₁₀₉	#400	1	1	yes	characteristic
156	#11243	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₁₀₉	—	2	1	no	—
158	C₂×C₁₀₉	—	2	2	no	—
159	C₂×C₁₀₉	—	2	1	no	—
160	C₂×C₁₀₉	—	2	2	no	—
161	C₂×C₁₀₉	#1441	1	1	yes	characteristic
162	C₂×C₁₀₉	#2050	1	2	yes	—
163	#2911	C₂×C₁₀₉	1	1	yes	characteristic
164	C₂×#423	C₂×C₁₀₉	1	1	yes	characteristic
165	#1742	C₂×C₁₀₉	1	2	yes	—
166	C₂×#339	C₂×C₁₀₉	1	2	yes	—
167	C₂×#338	C₂×C₁₀₉	1	1	yes	characteristic
168	C₁₀₉	#11243	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
170	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
171	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
172	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
173	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
174	#428	—	2	4	no	—
175	#427	—	2	2	no	—
176	#423	—	2	2	no	—
177	#423	C₄×C₁₀₉	1	4	yes	complemented
178	C₂×C₄⋊C₈	—	2	2	no	—
179	#343	—	2	4	no	—
180	#339	—	2	2	no	—
181	#339	—	2	2	no	—
182	#338	—	2	2	no	—
183	C₂³×C₄	—	2	1	no	—
184	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
196	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
197	C₂×M₁₆	D₄×C₁₀₉	1	2	yes	—
198	C₂×M₁₆	Q₈×C₁₀₉	1	4	yes	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	2	2	no	—
205	#117	—	2	2	no	—
206	C₂×C₄⋊C₄	D₄×C₁₀₉	1	4	yes	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
213	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
214	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
215	C₂×C₄²	Q₈×C₁₀₉	1	2	yes	—
216	#107	—	2	4	no	—
217	#106	—	2	4	no	—
218	C₄⋊C₈	—	2	8	no	—
219	C₂⁴	#41×C₁₀₉	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
231	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
233	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
234	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
235	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	M₁₆	—	4	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
264	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
265	C₂³	—	4	2	no	—
266	C₂³	#117×C₁₀₉	1	1	yes	characteristic
267	C₂³	#121×C₁₀₉	1	1	yes	characteristic
268	C₂³	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
269	C₂³	#123×C₁₀₉	1	2	yes	—
270	C₂³	#125×C₁₀₉	1	2	yes	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	8	4	no	—
279	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
281	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
282	C₂×C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
283	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
284	C₂×C₄	#121×C₁₀₉	1	2	yes	—
285	C₂×C₄	#121×C₁₀₉	1	2	yes	—
286	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
287	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
288	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
289	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
290	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
291	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
292	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
293	C₂×C₄	#125×C₁₀₉	1	2	yes	—
294	C₂×C₄	#126×C₁₀₉	1	2	yes	—
295	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
296	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	#380×C₁₀₉	1	1	yes	characteristic
306	C₂²	#387×C₁₀₉	1	2	yes	—
307	C₂²	#394×C₁₀₉	1	1	yes	characteristic
308	C₂²	#398×C₁₀₉	1	2	yes	—
309	C₂²	#398×C₁₀₉	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	#386×C₁₀₉	1	1	yes	characteristic
314	C₄	#389×C₁₀₉	1	2	yes	—
315	C₄	#391×C₁₀₉	1	1	yes	characteristic
316	C₄	#395×C₁₀₉	1	1	yes	characteristic
317	C₄	#396×C₁₀₉	1	2	yes	—
318	C₄	#400×C₁₀₉	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	C₁₀₉×#1441	1	1	yes	characteristic
324	C₂	C₁₀₉×#2050	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#124
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#394
prime radical	C₂⁴×C₁₀₉	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
14. #423×C₁₀₉	C₄	semidirect
156. #11243	C₁₀₉	direct (N × H)
168. C₁₀₉	#11243	direct (N × H)
177. #423	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×#2911	2
C₂×#423×C₁₀₉	2
C₁₀₉×#1742	2
C₂×#339×C₁₀₉	2
C₂×#338×C₁₀₉	2
#11243	109

Subgroups — C109×#11243 #42129237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11243 #42129237