Subgroups — C₁₀₉×#19613 #42137607

Order 27904.

734 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₀₉×#3169	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#3167	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#3131	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1720	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1719	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1718	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1717	C₂	1	2	yes	maximal, complemented
9	C₂×#322×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#533×C₁₀₉	C₂²	1	2	yes	—
11	#532×C₁₀₉	C₂²	1	2	yes	—
12	#500×C₁₀₉	C₂²	1	1	yes	characteristic
13	#500×C₁₀₉	C₂²	1	1	yes	characteristic
14	#498×C₁₀₉	C₂²	1	2	yes	—
15	#494×C₁₀₉	C₂²	1	1	yes	characteristic
16	#493×C₁₀₉	C₂²	1	1	yes	characteristic
17	#492×C₁₀₉	C₂²	1	2	yes	—
18	#488×C₁₀₉	—	2	4	no	—
19	#487×C₁₀₉	C₂²	1	2	yes	—
20	#487×C₁₀₉	C₂²	1	2	yes	—
21	#487×C₁₀₉	C₂²	1	2	yes	—
22	#487×C₁₀₉	C₂²	1	2	yes	—
23	#486×C₁₀₉	C₂²	1	2	yes	—
24	#486×C₁₀₉	C₂²	1	2	yes	—
25	#486×C₁₀₉	C₂²	1	2	yes	—
26	#486×C₁₀₉	C₂²	1	2	yes	—
27	#467×C₁₀₉	—	2	2	no	—
28	#466×C₁₀₉	—	2	2	no	—
29	#465×C₁₀₉	—	2	4	no	—
30	C₄×Q₁₆×C₁₀₉	—	2	2	no	—
31	C₄×SD₁₆×C₁₀₉	—	2	4	no	—
32	C₄×D₈×C₁₀₉	—	2	2	no	—
33	#416×C₁₀₉	—	2	4	no	—
34	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
37	#322×C₁₀₉	C₂²	1	2	yes	—
38	#322×C₁₀₉	C₂²	1	2	yes	—
39	C₂×#41×C₁₀₉	—	2	2	no	—
40	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
41	C₂×SD₁₆×C₁₀₉	—	2	4	no	—
42	C₂×D₈×C₁₀₉	—	2	2	no	—
43	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
44	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
45	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
46	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
49	#127×C₁₀₉	C₂³	1	2	yes	—
50	#125×C₁₀₉	—	2	2	no	—
51	#124×C₁₀₉	—	2	2	no	—
52	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
53	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
54	#121×C₁₀₉	—	2	2	no	—
55	#121×C₁₀₉	—	2	2	no	—
56	C₄×Q₈×C₁₀₉	—	2	2	no	—
57	C₄×Q₈×C₁₀₉	—	2	2	no	—
58	C₄×D₄×C₁₀₉	—	2	2	no	—
59	C₄×D₄×C₁₀₉	—	2	2	no	—
60	#117×C₁₀₉	—	2	2	no	—
61	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
62	#107×C₁₀₉	—	2	2	no	—
63	#107×C₁₀₉	—	2	2	no	—
64	#106×C₁₀₉	—	2	4	no	—
65	#103×C₁₀₉	—	2	4	no	—
66	#103×C₁₀₉	—	2	4	no	—
67	#103×C₁₀₉	—	2	4	no	—
68	#103×C₁₀₉	—	2	4	no	—
69	#102×C₁₀₉	—	2	4	no	—
70	#102×C₁₀₉	—	2	4	no	—
71	#102×C₁₀₉	—	2	4	no	—
72	#102×C₁₀₉	—	2	4	no	—
73	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
74	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
75	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
76	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
77	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
80	#41×C₁₀₉	—	4	8	no	—
81	C₂×Q₈×C₁₀₉	—	2	2	no	—
82	C₂×Q₈×C₁₀₉	—	2	2	no	—
83	C₂×Q₈×C₁₀₉	—	2	2	no	—
84	C₂×D₄×C₁₀₉	—	2	2	no	—
85	C₂×D₄×C₁₀₉	—	2	2	no	—
86	C₂×D₄×C₁₀₉	—	2	2	no	—
87	C₂×D₄×C₁₀₉	—	4	2	no	—
88	C₂²×C₄×C₁₀₉	—	4	2	no	—
89	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₁₀₉	—	4	4	no	—
93	SD₁₆×C₁₀₉	—	4	8	no	—
94	D₈×C₁₀₉	—	4	4	no	—
95	C₂×C₈×C₁₀₉	—	2	4	no	—
96	C₂×C₈×C₁₀₉	—	4	2	no	—
97	C₂×C₈×C₁₀₉	—	4	2	no	—
98	C₂×C₈×C₁₀₉	C₂×D₄	1	4	yes	—
99	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₁₀₉	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₁₀₉	—	2	2	no	—
103	C₄⋊C₄×C₁₀₉	—	2	2	no	—
104	C₄⋊C₄×C₁₀₉	—	2	2	no	—
105	C₄⋊C₄×C₁₀₉	—	2	2	no	—
106	C₄⋊C₄×C₁₀₉	—	2	2	no	—
107	C₄⋊C₄×C₁₀₉	—	2	2	no	—
108	C₄⋊C₄×C₁₀₉	—	4	2	no	—
109	C₄⋊C₄×C₁₀₉	—	4	2	no	—
110	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
111	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
112	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
113	C₄²×C₁₀₉	—	4	2	no	—
114	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
118	C₂³×C₁₀₉	—	4	2	no	—
119	C₂³×C₁₀₉	#127	1	1	yes	characteristic
120	Q₈×C₁₀₉	—	4	4	no	—
121	Q₈×C₁₀₉	—	4	4	no	—
122	Q₈×C₁₀₉	—	4	4	no	—
123	D₄×C₁₀₉	—	4	4	no	—
124	D₄×C₁₀₉	—	4	4	no	—
125	D₄×C₁₀₉	—	4	4	no	—
126	D₄×C₁₀₉	—	8	4	no	—
127	C₂×C₄×C₁₀₉	—	2	2	no	—
128	C₂×C₄×C₁₀₉	—	2	2	no	—
129	C₂×C₄×C₁₀₉	—	4	2	no	—
130	C₂×C₄×C₁₀₉	—	4	2	no	—
131	C₂×C₄×C₁₀₉	—	4	2	no	—
132	C₂×C₄×C₁₀₉	—	8	4	no	—
133	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
134	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
135	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
136	C₂×C₄×C₁₀₉	#135	1	4	yes	—
137	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₁₀₉	—	2	4	no	—
141	C₈×C₁₀₉	—	2	2	no	—
142	C₈×C₁₀₉	—	2	4	no	—
143	C₈×C₁₀₉	—	2	2	no	—
144	C₂²×C₁₀₉	—	2	1	no	—
145	C₂²×C₁₀₉	—	2	1	no	—
146	C₂²×C₁₀₉	—	8	2	no	—
147	C₂²×C₁₀₉	—	8	2	no	—
148	C₂²×C₁₀₉	—	8	2	no	—
149	C₂²×C₁₀₉	#495	1	2	yes	—
150	C₂²×C₁₀₉	C₂×#127	1	1	yes	characteristic
151	C₄×C₁₀₉	—	2	1	no	—
152	C₄×C₁₀₉	—	2	1	no	—
153	C₄×C₁₀₉	—	8	2	no	—
154	C₄×C₁₀₉	—	8	2	no	—
155	C₄×C₁₀₉	—	8	2	no	—
156	C₄×C₁₀₉	#495	1	2	yes	—
157	C₄×C₁₀₉	C₂×#135	1	2	yes	—
158	C₄×C₁₀₉	#575	1	2	yes	—
159	C₄×C₁₀₉	#575	1	2	yes	—
160	#19613	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₁₀₉	—	2	2	no	—
162	C₂×C₁₀₉	—	16	2	no	—
163	C₂×C₁₀₉	C₂×#495	1	1	yes	characteristic
164	C₂×C₁₀₉	#3150	1	1	yes	characteristic
165	C₂×C₁₀₉	#3150	1	1	yes	characteristic
166	#3170	C₂×C₁₀₉	1	1	yes	characteristic, complemented
167	#3169	C₂×C₁₀₉	1	1	yes	characteristic
168	#3167	C₂×C₁₀₉	1	2	yes	complemented
169	#3131	C₂×C₁₀₉	1	2	yes	complemented
170	#1720	C₂×C₁₀₉	1	2	yes	complemented
171	#1719	C₂×C₁₀₉	1	2	yes	complemented
172	#1718	C₂×C₁₀₉	1	2	yes	complemented
173	#1717	C₂×C₁₀₉	1	2	yes	complemented
174	C₂×#322	C₂×C₁₀₉	1	1	yes	characteristic, complemented
175	C₁₀₉	#19613	1	1	yes	characteristic, direct product factor, Sylow
176	#533	C₂²×C₁₀₉	1	2	yes	—
177	#532	C₂²×C₁₀₉	1	2	yes	—
178	#500	C₂²×C₁₀₉	1	1	yes	characteristic
179	#500	C₂²×C₁₀₉	1	1	yes	characteristic
180	#498	C₂²×C₁₀₉	1	2	yes	—
181	#494	C₂²×C₁₀₉	1	1	yes	characteristic
182	#493	C₂²×C₁₀₉	1	1	yes	characteristic
183	#492	C₂²×C₁₀₉	1	2	yes	—
184	#488	—	2	4	no	—
185	#487	C₂²×C₁₀₉	1	2	yes	—
186	#487	C₂²×C₁₀₉	1	2	yes	—
187	#487	C₂²×C₁₀₉	1	2	yes	—
188	#487	C₂²×C₁₀₉	1	2	yes	—
189	#486	C₂²×C₁₀₉	1	2	yes	—
190	#486	C₂²×C₁₀₉	1	2	yes	—
191	#486	C₂²×C₁₀₉	1	2	yes	—
192	#486	C₂²×C₁₀₉	1	2	yes	—
193	#467	—	2	2	no	—
194	#466	—	2	2	no	—
195	#465	—	2	4	no	—
196	C₄×Q₁₆	—	2	2	no	—
197	C₄×SD₁₆	—	2	4	no	—
198	C₄×D₈	—	2	2	no	—
199	#416	—	2	4	no	—
200	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
202	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
203	#322	C₂²×C₁₀₉	1	2	yes	—
204	#322	C₂²×C₁₀₉	1	2	yes	—
205	C₂×#41	—	2	2	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
210	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
211	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
214	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
215	#127	C₂³×C₁₀₉	1	2	yes	—
216	#125	—	2	2	no	—
217	#124	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	#121	—	2	2	no	—
221	#121	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	2	2	no	—
226	#117	—	2	2	no	—
227	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
228	#107	—	2	2	no	—
229	#107	—	2	2	no	—
230	#106	—	2	4	no	—
231	#103	—	2	4	no	—
232	#103	—	2	4	no	—
233	#103	—	2	4	no	—
234	#103	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	4	no	—
239	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
240	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
241	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
242	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
243	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
245	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
246	#41	—	4	8	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
258	Q₁₆	—	4	4	no	—
259	SD₁₆	—	4	8	no	—
260	D₈	—	4	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	C₂×D₄×C₁₀₉	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₁₀₉	1	4	yes	—
267	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
283	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	#127×C₁₀₉	1	1	yes	characteristic
286	Q₈	—	4	4	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	8	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
300	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
301	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
302	C₂×C₄	#135×C₁₀₉	1	4	yes	—
303	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
306	C₈	—	2	4	no	—
307	C₈	—	2	2	no	—
308	C₈	—	2	4	no	—
309	C₈	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	#495×C₁₀₉	1	2	yes	—
316	C₂²	C₂×#127×C₁₀₉	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	#495×C₁₀₉	1	2	yes	—
323	C₄	C₂×#135×C₁₀₉	1	2	yes	—
324	C₄	#575×C₁₀₉	1	2	yes	—
325	C₄	#575×C₁₀₉	1	2	yes	—
326	C₂	—	2	2	no	—
327	C₂	—	16	2	no	—
328	C₂	C₂×#495×C₁₀₉	1	1	yes	characteristic
329	C₂	C₁₀₉×#3150	1	1	yes	characteristic
330	C₂	C₁₀₉×#3150	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂×#127
prime radical	C₁₀₉×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₉×#3170	C₂	semidirect
3. C₁₀₉×#3167	C₂	semidirect
4. C₁₀₉×#3131	C₂	semidirect
5. C₁₀₉×#1720	C₂	semidirect
6. C₁₀₉×#1719	C₂	semidirect
7. C₁₀₉×#1718	C₂	semidirect
8. C₁₀₉×#1717	C₂	semidirect
9. C₂×#322×C₁₀₉	C₂	semidirect
160. #19613	C₁₀₉	direct (N × H)
166. #3170	C₂×C₁₀₉	semidirect
168. #3167	C₂×C₁₀₉	semidirect
169. #3131	C₂×C₁₀₉	semidirect
170. #1720	C₂×C₁₀₉	semidirect
171. #1719	C₂×C₁₀₉	semidirect
172. #1718	C₂×C₁₀₉	semidirect
173. #1717	C₂×C₁₀₉	semidirect
174. C₂×#322	C₂×C₁₀₉	semidirect
175. C₁₀₉	#19613	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#3170	2
C₁₀₉×#3169	2
C₁₀₉×#3167	2
C₁₀₉×#3131	2
C₁₀₉×#1720	2
C₁₀₉×#1719	2
C₁₀₉×#1718	2
C₁₀₉×#1717	2
C₂×#322×C₁₀₉	2
#19613	109

Subgroups — C109×#19613 #42137607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#19613 #42137607