Subgroups — C₁₃×#8901 #15695240

Order 3328.

372 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2991	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1574	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1489	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1476	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₁₃	—	2	1	no	—
10	C₁₃×#436	—	2	2	no	—
11	C₁₃×#436	—	2	1	no	—
12	C₁₃×#431	—	2	1	no	—
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₁₃×#423	—	2	1	no	—
16	C₁₃×#423	C₂²	1	1	yes	characteristic
17	C₁₃×#423	C₂²	1	1	yes	characteristic
18	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#341	—	2	2	no	—
21	C₁₃×#335	—	2	1	no	—
22	C₁₃×#334	—	2	1	no	—
23	C₁₃×Q₈⋊C₈	—	2	1	no	—
24	C₁₃×Q₈⋊C₈	C₄	1	1	yes	characteristic
25	C₁₃×Q₈⋊C₈	C₄	1	1	yes	characteristic
26	C₁₃×D₄⋊C₈	—	2	1	no	—
27	C₁₃×D₄⋊C₈	C₄	1	1	yes	characteristic
28	C₁₃×D₄⋊C₈	C₄	1	1	yes	characteristic
29	C₂×C₁₃×#41	Q₈	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	—	2	2	no	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	—	2	1	no	—
33	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
34	C₂²×C₈×C₁₃	—	2	1	no	—
35	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
37	C₄×D₄×C₁₃	—	2	1	no	—
38	C₄×D₄×C₁₃	—	2	1	no	—
39	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
41	C₁₃×#117	—	2	1	no	—
42	C₁₃×#117	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	—	2	1	no	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊C₈	—	2	2	no	—
46	C₁₃×C₄⋊C₈	—	2	1	no	—
47	C₁₃×C₄⋊C₈	—	2	1	no	—
48	C₁₃×C₄⋊C₈	—	2	1	no	—
49	C₁₃×C₄⋊C₈	—	2	1	no	—
50	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
53	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₁₃×C₂²⋊C₈	—	2	2	no	—
55	C₁₃×C₂²⋊C₈	—	2	1	no	—
56	C₁₃×C₂²⋊C₈	—	2	1	no	—
57	C₁₃×#97	—	2	1	no	—
58	C₄×C₈×C₁₃	—	2	2	no	—
59	C₄×C₈×C₁₃	—	2	1	no	—
60	C₄×C₈×C₁₃	—	2	1	no	—
61	C₄×C₈×C₁₃	—	2	1	no	—
62	C₄×C₈×C₁₃	—	2	1	no	—
63	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
65	#8901	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₁₃×#41	—	2	2	no	—
67	C₁₃×#41	—	4	1	no	—
68	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₁₃	—	2	1	no	—
70	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	—	2	1	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
77	C₁₃×M₁₆	—	2	2	no	—
78	C₁₃×M₁₆	—	4	1	no	—
79	C₁₃×M₁₆	—	4	2	no	—
80	C₁₃×M₁₆	—	4	2	no	—
81	C₁₃×M₁₆	—	8	2	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	2	1	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	4	1	no	—
90	C₂×C₈×C₁₃	—	4	1	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₁₃×C₄⋊C₄	—	2	1	no	—
94	C₁₃×C₄⋊C₄	—	2	1	no	—
95	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
97	C₁₃×C₂²⋊C₄	—	2	1	no	—
98	C₁₃×C₂²⋊C₄	—	2	1	no	—
99	C₁₃×C₂²⋊C₄	—	2	1	no	—
100	C₄²×C₁₃	—	2	1	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃	#41	1	1	yes	characteristic
107	#2991	C₂×C₁₃	1	1	yes	characteristic
108	#1837	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#1574	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#1489	C₂×C₁₃	1	1	yes	characteristic
111	#1476	C₂×C₁₃	1	1	yes	characteristic
112	#1299	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#1278	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂³×C₁₃	—	2	1	no	—
115	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
116	Q₈×C₁₃	—	2	1	no	—
117	Q₈×C₁₃	#108	1	2	yes	—
118	D₄×C₁₃	—	2	1	no	—
119	D₄×C₁₃	—	2	1	no	—
120	D₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
141	C₈×C₁₃	—	4	1	no	—
142	C₈×C₁₃	—	4	1	no	—
143	C₈×C₁₃	—	4	2	no	—
144	C₈×C₁₃	—	4	1	no	—
145	C₈×C₁₃	—	4	1	no	—
146	C₈×C₁₃	—	4	1	no	—
147	C₈×C₁₃	—	4	2	no	—
148	C₈×C₁₃	—	4	1	no	—
149	C₈×C₁₃	—	8	1	no	—
150	C₄×#41	C₂²×C₁₃	1	1	yes	characteristic
151	C₈×Q₈	—	2	1	no	—
152	#436	—	2	2	no	—
153	#436	—	2	1	no	—
154	#431	—	2	1	no	—
155	#431	C₂²×C₁₃	1	1	yes	characteristic
156	#431	C₂²×C₁₃	1	1	yes	characteristic
157	#423	—	2	1	no	—
158	#423	C₂²×C₁₃	1	1	yes	characteristic
159	#423	C₂²×C₁₃	1	1	yes	characteristic
160	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
162	#341	—	2	2	no	—
163	#335	—	2	1	no	—
164	#334	—	2	1	no	—
165	Q₈⋊C₈	—	2	1	no	—
166	Q₈⋊C₈	C₄×C₁₃	1	1	yes	characteristic
167	Q₈⋊C₈	C₄×C₁₃	1	1	yes	characteristic
168	D₄⋊C₈	—	2	1	no	—
169	D₄⋊C₈	C₄×C₁₃	1	1	yes	characteristic
170	D₄⋊C₈	C₄×C₁₃	1	1	yes	characteristic
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	4	1	no	—
175	C₂²×C₁₃	—	4	1	no	—
176	C₂²×C₁₃	—	4	1	no	—
177	C₂²×C₁₃	#385	1	1	yes	characteristic
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	—	2	2	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	4	1	no	—
183	C₄×C₁₃	—	4	1	no	—
184	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
185	C₄×C₁₃	#430	1	1	yes	characteristic
186	C₄×C₁₃	#443	1	1	yes	characteristic
187	C₄×C₁₃	#444	1	1	yes	characteristic
188	C₄×C₁₃	#454	1	1	yes	characteristic
189	C₄×C₁₃	#455	1	1	yes	characteristic
190	C₂×#41	Q₈×C₁₃	1	1	yes	characteristic
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
195	C₂²×C₈	—	2	1	no	—
196	C₄×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
197	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
201	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
202	#117	—	2	1	no	—
203	#117	Q₈×C₁₃	1	1	yes	characteristic
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
212	C₄⋊C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
213	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
214	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
215	C₂²⋊C₈	—	2	2	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	#97	—	2	1	no	—
219	C₄×C₈	—	2	2	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
226	C₂×C₁₃	—	4	1	no	—
227	C₂×C₁₃	—	8	1	no	—
228	C₂×C₁₃	#1855	1	1	yes	characteristic
229	C₂×C₁₃	#1860	1	1	yes	characteristic
230	C₂×C₁₃	#1866	1	1	yes	characteristic
231	#41	—	2	2	no	—
232	#41	—	4	1	no	—
233	C₂×Q₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
242	M₁₆	—	2	2	no	—
243	M₁₆	—	4	1	no	—
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	8	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
261	C₄⋊C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
270	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₄²	C₁₃×#41	1	1	yes	characteristic
272	C₁₃	#8901	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³	—	2	1	no	—
274	C₂³	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
275	Q₈	—	2	1	no	—
276	Q₈	C₁₃×#108	1	2	yes	—
277	D₄	—	2	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	4	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
296	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
297	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
298	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
299	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
300	C₈	—	4	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	2	no	—
303	C₈	—	4	1	no	—
304	C₈	—	4	1	no	—
305	C₈	—	4	1	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	1	no	—
308	C₈	—	8	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₁₃×#385	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	C₂×C₁₃×#108	1	1	yes	characteristic
323	C₄	C₁₃×#430	1	1	yes	characteristic
324	C₄	C₁₃×#443	1	1	yes	characteristic
325	C₄	C₁₃×#444	1	1	yes	characteristic
326	C₄	C₁₃×#454	1	1	yes	characteristic
327	C₄	C₁₃×#455	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	1	no	—
330	C₂	C₁₃×#1855	1	1	yes	characteristic
331	C₂	C₁₃×#1860	1	1	yes	characteristic
332	C₂	C₁₃×#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#120
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#385
prime radical	C₂×C₁₃×#41	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1837	C₂	semidirect
3. C₁₃×#1574	C₂	semidirect
6. C₁₃×#1299	C₂	semidirect
7. C₁₃×#1278	C₂	semidirect
65. #8901	C₁₃	direct (N × H)
108. #1837	C₂×C₁₃	semidirect
109. #1574	C₂×C₁₃	semidirect
112. #1299	C₂×C₁₃	semidirect
113. #1278	C₂×C₁₃	semidirect
272. C₁₃	#8901	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#2991	2
C₁₃×#1837	2
C₁₃×#1574	2
C₁₃×#1489	2
C₁₃×#1476	2
C₁₃×#1299	2
C₁₃×#1278	2
#8901	13

Subgroups — C13×#8901 #15695240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#8901 #15695240