Subgroups — C₁₃×#8898 #15695237

Order 3328.

390 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2990	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1848	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	C₁₃×#446	—	2	1	no	—
10	C₁₃×#435	—	2	2	no	—
11	C₈×D₄×C₁₃	—	2	1	no	—
12	C₁₃×#432	—	2	1	no	—
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#108	—	2	2	no	—
16	C₁₃×#423	C₂²	1	1	yes	characteristic
17	C₁₃×#423	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
19	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
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	1	no	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	—	2	1	no	—
34	C₂²×C₈×C₁₃	—	2	2	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×#108	—	4	4	no	—
46	C₁₃×C₄⋊C₈	—	2	2	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	—	2	1	no	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
55	C₁₃×C₂²⋊C₈	—	2	2	no	—
56	C₁₃×C₂²⋊C₈	—	2	1	no	—
57	C₁₃×C₂²⋊C₈	—	2	1	no	—
58	C₁₃×#97	—	2	2	no	—
59	C₁₃×#97	—	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	—	2	1	no	—
64	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
66	#8898	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₃×#41	—	2	2	no	—
68	C₁₃×#41	—	4	1	no	—
69	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂×D₄×C₁₃	—	2	1	no	—
71	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	—	2	1	no	—
74	C₂²×C₄×C₁₃	—	2	1	no	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
78	C₁₃×M₁₆	—	2	2	no	—
79	C₁₃×M₁₆	—	4	1	no	—
80	C₁₃×M₁₆	—	4	2	no	—
81	C₁₃×M₁₆	—	4	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	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	1	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₂×C₈×C₁₃	—	8	2	no	—
94	C₁₃×C₄⋊C₄	—	2	1	no	—
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
98	C₁₃×C₂²⋊C₄	—	2	1	no	—
99	C₁₃×C₂²⋊C₄	—	2	1	no	—
100	C₁₃×C₂²⋊C₄	—	2	1	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃	#41	1	1	yes	characteristic
108	#2990	C₂×C₁₃	1	1	yes	characteristic
109	#1848	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#1574	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#1489	C₂×C₁₃	1	1	yes	characteristic
112	#1476	C₂×C₁₃	1	1	yes	characteristic
113	#1299	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#1278	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	—	2	1	no	—
116	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
117	Q₈×C₁₃	—	2	1	no	—
118	Q₈×C₁₃	—	2	1	no	—
119	D₄×C₁₃	—	2	1	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	#108	1	2	yes	complemented
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	—	2	1	no	—
131	C₂×C₄×C₁₃	—	4	2	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	—	4	1	no	—
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
142	C₈×C₁₃	—	2	2	no	—
143	C₈×C₁₃	—	2	2	no	—
144	C₈×C₁₃	—	4	2	no	—
145	C₈×C₁₃	—	4	1	no	—
146	C₈×C₁₃	—	4	1	no	—
147	C₈×C₁₃	—	4	1	no	—
148	C₈×C₁₃	—	4	2	no	—
149	C₈×C₁₃	—	4	1	no	—
150	C₈×C₁₃	—	8	1	no	—
151	C₄×#41	C₂²×C₁₃	1	1	yes	characteristic
152	#446	—	2	1	no	—
153	#435	—	2	2	no	—
154	C₈×D₄	—	2	1	no	—
155	#432	—	2	1	no	—
156	#431	C₂²×C₁₃	1	1	yes	characteristic
157	#431	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#108	—	2	2	no	—
159	#423	C₂²×C₁₃	1	1	yes	characteristic
160	#423	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄⋊C₈	—	2	1	no	—
162	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
164	#335	—	2	1	no	—
165	#334	—	2	1	no	—
166	Q₈⋊C₈	—	2	1	no	—
167	Q₈⋊C₈	C₄×C₁₃	1	1	yes	characteristic
168	Q₈⋊C₈	C₄×C₁₃	1	1	yes	characteristic
169	D₄⋊C₈	—	2	1	no	—
170	D₄⋊C₈	C₄×C₁₃	1	1	yes	characteristic
171	D₄⋊C₈	C₄×C₁₃	1	1	yes	characteristic
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	4	1	no	—
177	C₂²×C₁₃	—	4	1	no	—
178	C₂²×C₁₃	#385	1	1	yes	characteristic
179	C₄×C₁₃	—	2	1	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	—	4	1	no	—
185	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
186	C₄×C₁₃	#430	1	1	yes	characteristic
187	C₄×C₁₃	#443	1	1	yes	characteristic
188	C₄×C₁₃	#444	1	1	yes	characteristic
189	C₄×C₁₃	#453	1	1	yes	characteristic
190	C₄×C₁₃	#456	1	1	yes	characteristic
191	C₂×#41	Q₈×C₁₃	1	1	yes	characteristic
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	C₂²×C₈	—	2	2	no	—
197	C₄×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
198	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
202	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
203	#117	—	2	1	no	—
204	#117	Q₈×C₁₃	1	1	yes	characteristic
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
207	#108	—	4	4	no	—
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
214	C₄⋊C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
215	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
216	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
217	C₂²⋊C₈	—	2	2	no	—
218	C₂²⋊C₈	—	2	1	no	—
219	C₂²⋊C₈	—	2	1	no	—
220	#97	—	2	2	no	—
221	#97	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
227	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
228	C₂×C₁₃	—	4	1	no	—
229	C₂×C₁₃	—	4	2	no	—
230	C₂×C₁₃	#1855	1	1	yes	characteristic
231	C₂×C₁₃	#1859	1	1	yes	characteristic
232	C₂×C₁₃	#1866	1	1	yes	characteristic
233	#41	—	2	2	no	—
234	#41	—	4	1	no	—
235	C₂×Q₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
243	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
244	M₁₆	—	2	2	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	2	no	—
247	M₁₆	—	4	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	8	2	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
263	C₄⋊C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₄²	C₁₃×#41	1	1	yes	characteristic
274	C₁₃	#8898	1	1	yes	characteristic, direct product factor, Sylow
275	C₂³	—	2	1	no	—
276	C₂³	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	4	2	no	—
281	D₄	C₁₃×#108	1	2	yes	complemented
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	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
298	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
299	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
300	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
301	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	2	no	—
304	C₈	—	4	2	no	—
305	C₈	—	4	1	no	—
306	C₈	—	4	1	no	—
307	C₈	—	4	1	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	1	no	—
310	C₈	—	8	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₁₃×#385	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	C₂×C₁₃×#108	1	1	yes	characteristic
325	C₄	C₁₃×#430	1	1	yes	characteristic
326	C₄	C₁₃×#443	1	1	yes	characteristic
327	C₄	C₁₃×#444	1	1	yes	characteristic
328	C₄	C₁₃×#453	1	1	yes	characteristic
329	C₄	C₁₃×#456	1	1	yes	characteristic
330	C₂	—	4	1	no	—
331	C₂	—	4	2	no	—
332	C₂	C₁₃×#1855	1	1	yes	characteristic
333	C₂	C₁₃×#1859	1	1	yes	characteristic
334	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×#1848	C₂	semidirect
3. C₁₃×#1574	C₂	semidirect
6. C₁₃×#1299	C₂	semidirect
7. C₁₃×#1278	C₂	semidirect
66. #8898	C₁₃	direct (N × H)
109. #1848	C₂×C₁₃	semidirect
110. #1574	C₂×C₁₃	semidirect
113. #1299	C₂×C₁₃	semidirect
114. #1278	C₂×C₁₃	semidirect
121. D₄×C₁₃	#108	semidirect
274. C₁₃	#8898	direct (N × H)
281. D₄	C₁₃×#108	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#2990	2
C₁₃×#1848	2
C₁₃×#1574	2
C₁₃×#1489	2
C₁₃×#1476	2
C₁₃×#1299	2
C₁₃×#1278	2
#8898	13

Subgroups — C13×#8898 #15695237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#8898 #15695237