Subgroups — C₁₃×#25919 #15712258

Order 3328.

954 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3502	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3139	C₂	1	2	yes	maximal
3	C₁₃×#2546	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2028	C₂	1	4	yes	maximal
5	C₁₃×#1986	C₂	1	2	yes	maximal
6	C₁₃×#1886	C₂	1	4	yes	maximal
7	C₄×C₁₃×#101	C₂	1	1	yes	maximal, characteristic
8	Q₈²×C₁₃	—	2	2	no	—
9	C₁₃×#557	—	2	2	no	—
10	C₁₃×#557	—	2	4	no	—
11	C₁₃×#544	—	2	2	no	—
12	C₁₃×#544	C₂²	1	2	yes	—
13	C₁₃×#533	—	2	2	no	—
14	C₁₃×#533	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#519	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#491	—	2	4	no	—
20	C₁₃×#487	—	2	2	no	—
21	C₁₃×#487	—	2	2	no	—
22	C₁₃×#419	C₂²	1	8	yes	—
23	C₂×C₁₃×#101	C₂²	1	2	yes	—
24	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
25	C₁₃×#398	—	2	2	no	—
26	C₁₃×#391	C₂²	1	2	yes	—
27	C₁₃×#389	—	2	2	no	—
28	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₁₃×#343	C₂²	1	2	yes	—
30	C₁₃×#339	C₂²	1	8	yes	—
31	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
32	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
34	C₁₃×C₄⋊Q₈	—	2	2	no	—
35	C₁₃×C₄⋊Q₈	—	2	2	no	—
36	C₁₃×C₄⋊Q₈	—	2	8	no	—
37	C₁₃×C₄⋊Q₈	—	2	4	no	—
38	C₁₃×C₄⋊Q₈	—	2	8	no	—
39	C₁₃×C₄⋊Q₈	—	2	2	no	—
40	C₁₃×C₄⋊Q₈	—	2	2	no	—
41	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
42	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
43	C₁₃×#125	—	2	4	no	—
44	C₁₃×#125	—	2	8	no	—
45	C₁₃×#125	—	2	4	no	—
46	C₁₃×#125	D₄	1	4	yes	—
47	C₁₃×C₂²⋊Q₈	—	2	2	no	—
48	C₁₃×C₂²⋊Q₈	—	2	4	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₄×Q₈×C₁₃	—	2	4	no	—
51	C₄×Q₈×C₁₃	—	2	4	no	—
52	C₄×Q₈×C₁₃	—	2	4	no	—
53	C₄×Q₈×C₁₃	—	2	4	no	—
54	C₄×Q₈×C₁₃	—	2	2	no	—
55	C₄×Q₈×C₁₃	—	2	2	no	—
56	C₄×Q₈×C₁₃	—	2	2	no	—
57	C₄×Q₈×C₁₃	—	2	2	no	—
58	C₁₃×#117	—	2	4	no	—
59	C₁₃×#117	C₂³	1	4	yes	—
60	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
61	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
64	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
65	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₁₃	D₄	1	2	yes	—
67	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
68	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
69	C₁₃×#103	—	2	8	no	—
70	C₁₃×#103	—	2	8	no	—
71	C₁₃×#101	—	2	4	no	—
72	C₁₃×#97	—	2	2	no	—
73	C₄×C₈×C₁₃	—	2	2	no	—
74	C₁₃×#95	—	2	1	no	—
75	C₁₃×#95	—	2	4	no	—
76	#25919	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×Q₈×C₁₃	—	2	4	no	—
78	C₂×Q₈×C₁₃	—	2	2	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×Q₈×C₁₃	—	2	4	no	—
83	C₂×Q₈×C₁₃	—	2	4	no	—
84	C₂×Q₈×C₁₃	—	2	4	no	—
85	C₂×Q₈×C₁₃	#41	1	4	yes	—
86	C₂²×C₄×C₁₃	—	2	2	no	—
87	C₂²×C₄×C₁₃	—	2	2	no	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×M₁₆	—	2	4	no	—
94	C₁₃×M₁₆	—	2	4	no	—
95	C₂×C₈×C₁₃	—	2	4	no	—
96	C₁₃×C₄⋊C₄	—	2	4	no	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	—	2	4	no	—
99	C₁₃×C₄⋊C₄	—	2	4	no	—
100	C₁₃×C₄⋊C₄	—	2	8	no	—
101	C₁₃×C₄⋊C₄	—	2	4	no	—
102	C₁₃×C₄⋊C₄	—	2	4	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	2	4	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₄⋊C₄	—	4	2	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	8	no	—
111	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
112	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
113	C₁₃×C₂²⋊C₄	—	2	4	no	—
114	C₄²×C₁₃	—	2	8	no	—
115	C₄²×C₁₃	—	2	2	no	—
116	C₄²×C₁₃	—	2	2	no	—
117	C₄²×C₁₃	—	2	4	no	—
118	C₄²×C₁₃	—	2	4	no	—
119	C₄²×C₁₃	C₂×D₄	1	2	yes	—
120	C₄²×C₁₃	C₂×D₄	1	2	yes	—
121	#3502	C₂×C₁₃	1	1	yes	characteristic
122	#3139	C₂×C₁₃	1	2	yes	—
123	#2546	C₂×C₁₃	1	1	yes	characteristic
124	#2028	C₂×C₁₃	1	4	yes	—
125	#1986	C₂×C₁₃	1	2	yes	—
126	#1886	C₂×C₁₃	1	4	yes	—
127	C₄×#101	C₂×C₁₃	1	1	yes	characteristic
128	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₁₃	—	2	8	no	—
130	Q₈×C₁₃	—	4	4	no	—
131	Q₈×C₁₃	—	4	8	no	—
132	Q₈×C₁₃	—	8	2	no	—
133	Q₈×C₁₃	—	8	4	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	—	2	4	no	—
138	C₂×C₄×C₁₃	—	2	4	no	—
139	C₂×C₄×C₁₃	—	2	4	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	4	1	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	#120	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#120	1	2	yes	—
150	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#124	1	4	yes	—
152	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
153	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
154	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
155	C₈×C₁₃	—	4	4	no	—
156	Q₈²	—	2	2	no	—
157	#557	—	2	2	no	—
158	#557	—	2	4	no	—
159	#544	—	2	2	no	—
160	#544	C₂²×C₁₃	1	2	yes	—
161	#533	—	2	2	no	—
162	#533	C₂²×C₁₃	1	2	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
165	#519	C₂²×C₁₃	1	2	yes	—
166	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
167	#491	—	2	4	no	—
168	#487	—	2	2	no	—
169	#487	—	2	2	no	—
170	#419	C₂²×C₁₃	1	8	yes	—
171	C₂×#101	C₂²×C₁₃	1	2	yes	—
172	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
173	#398	—	2	2	no	—
174	#391	C₂²×C₁₃	1	2	yes	—
175	#389	—	2	2	no	—
176	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
177	#343	C₂²×C₁₃	1	2	yes	—
178	#339	C₂²×C₁₃	1	8	yes	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
182	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
183	C₂²×C₁₃	#536	1	1	yes	characteristic
184	C₄×C₁₃	—	2	4	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	—	4	4	no	—
188	C₄×C₁₃	—	4	4	no	—
189	C₄×C₁₃	—	4	2	no	—
190	C₄×C₁₃	—	4	4	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	#456	1	2	yes	—
193	C₄×C₁₃	#548	1	4	yes	—
194	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
195	C₂²×Q₈	C₂³×C₁₃	1	1	yes	characteristic
196	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	8	no	—
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	—	2	8	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
205	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
206	#125	—	2	4	no	—
207	#125	—	2	8	no	—
208	#125	—	2	4	no	—
209	#125	D₄×C₁₃	1	4	yes	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	2	4	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	4	no	—
214	C₄×Q₈	—	2	4	no	—
215	C₄×Q₈	—	2	4	no	—
216	C₄×Q₈	—	2	4	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	#117	—	2	4	no	—
222	#117	C₂³×C₁₃	1	4	yes	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	4	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₁₃	1	2	yes	—
230	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
231	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
232	#103	—	2	8	no	—
233	#103	—	2	8	no	—
234	#101	—	2	4	no	—
235	#97	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	#95	—	2	1	no	—
238	#95	—	2	4	no	—
239	C₂×C₁₃	—	2	1	no	—
240	C₂×C₁₃	—	2	1	no	—
241	C₂×C₁₃	#2436	1	1	yes	characteristic
242	C₂×C₁₃	C₂×#456	1	1	yes	characteristic
243	C₂×C₁₃	#3022	1	1	yes	characteristic
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	C₁₃×#41	1	4	yes	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
258	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
260	M₁₆	—	2	4	no	—
261	M₁₆	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	8	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	8	no	—
278	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
279	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₄²	—	2	8	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	4	no	—
286	C₄²	C₂×D₄×C₁₃	1	2	yes	—
287	C₄²	C₂×D₄×C₁₃	1	2	yes	—
288	C₁₃	#25919	1	1	yes	characteristic, direct product factor, Sylow
289	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
290	Q₈	—	2	8	no	—
291	Q₈	—	4	4	no	—
292	Q₈	—	4	8	no	—
293	Q₈	—	8	2	no	—
294	Q₈	—	8	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
310	C₂×C₄	C₁₃×#120	1	2	yes	—
311	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
312	C₂×C₄	C₁₃×#124	1	4	yes	—
313	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
314	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
315	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
316	C₈	—	4	4	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
320	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
321	C₂²	C₁₃×#536	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	4	no	—
325	C₄	—	4	4	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	4	no	—
329	C₄	—	8	2	no	—
330	C₄	C₁₃×#456	1	2	yes	—
331	C₄	C₁₃×#548	1	4	yes	—
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	C₁₃×#2436	1	1	yes	characteristic
335	C₂	C₂×C₁₃×#456	1	1	yes	characteristic
336	C₂	C₁₃×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂³×C₁₃	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
76. #25919	C₁₃	direct (N × H)
288. C₁₃	#25919	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×#3502	2
C₁₃×#3139	2
C₁₃×#2546	2
C₁₃×#2028	2
C₁₃×#1986	2
C₁₃×#1886	2
C₄×C₁₃×#101	2
#25919	13

Subgroups — C13×#25919 #15712258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#25919 #15712258