Subgroups — C₁₃×#29185 #15715524

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2647	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2330	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2088	C₂	1	2	yes	maximal
4	C₁₃×#2088	C₂	1	2	yes	maximal
5	C₁₃×#1992	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1991	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1920	C₂	1	2	yes	maximal
8	C₁₃×#1919	C₂	1	2	yes	maximal
9	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#1836	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1768	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#400	—	2	2	no	—
20	C₁₃×#398	—	2	2	no	—
21	C₁₃×#389	—	2	2	no	—
22	C₁₃×#389	C₂²	1	2	yes	—
23	C₁₃×#389	C₂²	1	2	yes	—
24	C₁₃×#389	C₂²	1	2	yes	—
25	C₁₃×#387	—	2	2	no	—
26	C₁₃×#384	C₂²	1	1	yes	characteristic
27	C₁₃×#384	C₂²	1	1	yes	characteristic
28	C₁₃×#382	C₂²	1	1	yes	characteristic
29	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
30	C₁₃×#376	C₂²	1	1	yes	characteristic
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	C₁₃×#340	C₂²	1	2	yes	—
35	C₁₃×#340	C₂²	1	2	yes	—
36	C₁₃×#340	C₂²	1	2	yes	—
37	C₁₃×#336	C₂²	1	1	yes	characteristic
38	C₁₃×#336	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#125	—	2	2	no	—
44	C₁₃×#125	Q₈	1	4	yes	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₃	—	2	2	no	—
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₁₃×#106	—	2	4	no	—
60	C₁₃×#106	—	2	4	no	—
61	C₁₃×C₄⋊C₈	—	2	2	no	—
62	C₁₃×C₄⋊C₈	—	2	2	no	—
63	C₁₃×#97	—	2	2	no	—
64	C₄×C₈×C₁₃	—	2	2	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	—	2	2	no	—
68	C₁₃×#95	—	2	2	no	—
69	C₁₃×#95	—	2	2	no	—
70	C₁₃×#95	D₄	1	2	yes	—
71	C₁₃×#95	D₄	1	2	yes	—
72	#29185	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
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	1	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	#41	1	2	yes	—
93	C₂×C₈×C₁₃	#41	1	2	yes	—
94	C₁₃×C₄⋊C₄	—	2	8	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	8	2	no	—
103	C₁₃×C₄⋊C₄	—	8	2	no	—
104	C₁₃×C₄⋊C₄	—	8	2	no	—
105	C₁₃×C₄⋊C₄	—	8	2	no	—
106	C₁₃×C₄⋊C₄	—	8	2	no	—
107	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
108	C₄²×C₁₃	—	2	1	no	—
109	C₄²×C₁₃	—	2	1	no	—
110	C₄²×C₁₃	—	2	1	no	—
111	C₄²×C₁₃	—	2	1	no	—
112	C₄²×C₁₃	—	8	2	no	—
113	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
114	C₄²×C₁₃	#41	1	2	yes	—
115	#2647	C₂×C₁₃	1	1	yes	characteristic
116	#2330	C₂×C₁₃	1	1	yes	characteristic
117	#2088	C₂×C₁₃	1	2	yes	—
118	#2088	C₂×C₁₃	1	2	yes	—
119	#1992	C₂×C₁₃	1	1	yes	characteristic
120	#1991	C₂×C₁₃	1	1	yes	characteristic
121	#1920	C₂×C₁₃	1	2	yes	—
122	#1919	C₂×C₁₃	1	2	yes	—
123	#1844	C₂×C₁₃	1	1	yes	characteristic
124	#1836	C₂×C₁₃	1	1	yes	characteristic
125	#1768	C₂×C₁₃	1	1	yes	characteristic
126	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
149	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
155	C₈×C₁₃	—	2	2	no	—
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	—	4	1	no	—
158	C₈×C₁₃	—	4	1	no	—
159	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×#106	C₂²×C₁₃	1	2	yes	—
161	C₂×#106	C₂²×C₁₃	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
166	#400	—	2	2	no	—
167	#398	—	2	2	no	—
168	#389	—	2	2	no	—
169	#389	C₂²×C₁₃	1	2	yes	—
170	#389	C₂²×C₁₃	1	2	yes	—
171	#389	C₂²×C₁₃	1	2	yes	—
172	#387	—	2	2	no	—
173	#384	C₂²×C₁₃	1	1	yes	characteristic
174	#384	C₂²×C₁₃	1	1	yes	characteristic
175	#382	C₂²×C₁₃	1	1	yes	characteristic
176	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
177	#376	C₂²×C₁₃	1	1	yes	characteristic
178	#340	C₂²×C₁₃	1	2	yes	—
179	#340	C₂²×C₁₃	1	2	yes	—
180	#340	C₂²×C₁₃	1	2	yes	—
181	#340	C₂²×C₁₃	1	2	yes	—
182	#340	C₂²×C₁₃	1	2	yes	—
183	#340	C₂²×C₁₃	1	2	yes	—
184	#336	C₂²×C₁₃	1	1	yes	characteristic
185	#336	C₂²×C₁₃	1	1	yes	characteristic
186	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
187	C₂²×C₁₃	#532	1	1	yes	characteristic
188	C₂²×C₁₃	#536	1	1	yes	characteristic
189	C₂²×C₁₃	#577	1	2	yes	—
190	C₂²×C₁₃	#577	1	2	yes	—
191	C₄×C₁₃	—	2	4	no	—
192	C₄×C₁₃	—	4	4	no	—
193	C₄×C₁₃	—	4	1	no	—
194	C₄×C₁₃	—	4	1	no	—
195	C₄×C₁₃	—	4	2	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	#552	1	1	yes	characteristic
200	C₄×C₁₃	#552	1	1	yes	characteristic
201	C₄×C₁₃	#557	1	1	yes	characteristic
202	C₄×C₁₃	#557	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
207	#125	—	2	2	no	—
208	#125	Q₈×C₁₃	1	4	yes	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
223	#106	—	2	4	no	—
224	#106	—	2	4	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	#97	—	2	2	no	—
228	C₄×C₈	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	D₄×C₁₃	1	2	yes	—
235	#95	D₄×C₁₃	1	2	yes	—
236	C₂×C₁₃	#2565	1	1	yes	characteristic
237	C₂×C₁₃	#3084	1	2	yes	—
238	C₂×C₁₃	#3174	1	1	yes	characteristic
239	C₂×C₁₃	#3174	1	1	yes	characteristic
240	C₂×C₁₃	#3244	1	2	yes	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
249	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
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	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	C₁₃×#41	1	2	yes	—
261	C₂×C₈	C₁₃×#41	1	2	yes	—
262	C₄⋊C₄	—	2	8	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	8	2	no	—
271	C₄⋊C₄	—	8	2	no	—
272	C₄⋊C₄	—	8	2	no	—
273	C₄⋊C₄	—	8	2	no	—
274	C₄⋊C₄	—	8	2	no	—
275	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
276	C₄²	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	8	2	no	—
281	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
282	C₄²	C₁₃×#41	1	2	yes	—
283	C₁₃	#29185	1	1	yes	characteristic, direct product factor, Sylow
284	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	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	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
307	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
309	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
310	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
311	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
312	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	2	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
318	C₂²	C₁₃×#532	1	1	yes	characteristic
319	C₂²	C₁₃×#536	1	1	yes	characteristic
320	C₂²	C₁₃×#577	1	2	yes	—
321	C₂²	C₁₃×#577	1	2	yes	—
322	C₄	—	2	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	C₁₃×#552	1	1	yes	characteristic
331	C₄	C₁₃×#552	1	1	yes	characteristic
332	C₄	C₁₃×#557	1	1	yes	characteristic
333	C₄	C₁₃×#557	1	1	yes	characteristic
334	C₂	C₁₃×#2565	1	1	yes	characteristic
335	C₂	C₁₃×#3084	1	2	yes	—
336	C₂	C₁₃×#3174	1	1	yes	characteristic
337	C₂	C₁₃×#3174	1	1	yes	characteristic
338	C₂	C₁₃×#3244	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
72. #29185	C₁₃	direct (N × H)
283. C₁₃	#29185	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×#2647	2
C₁₃×#2330	2
C₁₃×#2088	2
C₁₃×#1992	2
C₁₃×#1991	2
C₁₃×#1920	2
C₁₃×#1919	2
C₁₃×#1844	2
C₁₃×#1836	2
C₁₃×#1768	2
#29185	13

Subgroups — C13×#29185 #15715524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29185 #15715524