Subgroups — C₁₃×#8959 #15695298

Order 3328.

722 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×#2972	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1478	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1460	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1276	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#127	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	—	2	2	no	—
8	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₈×D₄×C₁₃	—	2	4	no	—
11	C₁₃×#431	C₂²	1	2	yes	—
12	C₁₃×#431	C₂²	1	2	yes	—
13	C₂×C₁₃×C₂²⋊C₈	—	2	2	no	—
14	C₁₃×#390	—	2	2	no	—
15	C₁₃×#390	C₄	1	1	yes	characteristic
16	C₁₃×#390	C₄	1	1	yes	characteristic
17	C₁₃×#384	C₄	1	1	yes	characteristic
18	C₄³×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×D₄⋊C₈	—	2	4	no	—
20	C₁₃×#323	—	2	2	no	—
21	C₂²×D₄×C₁₃	—	2	2	no	—
22	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	—	2	1	no	—
25	C₂³×C₄×C₁₃	D₄	1	2	yes	—
26	C₂²×C₈×C₁₃	—	2	4	no	—
27	C₁₃×#127	—	2	2	no	—
28	C₁₃×#127	C₈	1	4	yes	complemented
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₄×D₄×C₁₃	—	2	2	no	—
33	C₄×D₄×C₁₃	—	4	8	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
40	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃	—	2	1	no	—
42	C₂×C₄²×C₁₃	—	2	1	no	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊C₈	—	2	2	no	—
47	C₁₃×C₄⋊C₈	D₄	1	4	yes	complemented
48	C₁₃×C₂²⋊C₈	—	2	4	no	—
49	C₁₃×C₂²⋊C₈	—	2	2	no	—
50	C₁₃×C₂²⋊C₈	—	2	2	no	—
51	C₄×C₈×C₁₃	—	2	2	no	—
52	C₄×C₈×C₁₃	—	2	2	no	—
53	#8959	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂⁴×C₁₃	—	2	1	no	—
55	C₂⁴×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₂×D₄×C₁₃	—	2	8	no	—
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	2	2	no	—
59	C₂×D₄×C₁₃	—	2	1	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	4	4	no	—
63	C₂×D₄×C₁₃	—	4	4	no	—
64	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
65	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	complemented
66	C₂×D₄×C₁₃	M₁₆	1	2	yes	—
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	4	4	no	—
73	C₂²×C₄×C₁₃	—	4	4	no	—
74	C₂²×C₄×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	8	4	no	—
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	4	4	no	—
86	C₁₃×C₂²⋊C₄	—	2	4	no	—
87	C₁₃×C₂²⋊C₄	—	4	4	no	—
88	C₁₃×C₂²⋊C₄	—	4	4	no	—
89	C₁₃×C₂²⋊C₄	—	4	2	no	—
90	C₄²×C₁₃	—	2	4	no	—
91	C₄²×C₁₃	—	2	1	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	—	4	2	no	—
94	C₄²×C₁₃	C₂×C₈	1	2	yes	complemented
95	C₄²×C₁₃	C₂×D₄	1	2	yes	—
96	C₄²×C₁₃	#41	1	2	yes	—
97	#2972	C₂×C₁₃	1	1	yes	characteristic, complemented
98	C₄×#127	C₂×C₁₃	1	1	yes	characteristic
99	#1478	C₂×C₁₃	1	2	yes	complemented
100	#1460	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#1276	C₂×C₁₃	1	2	yes	complemented
102	C₂³×C₁₃	—	2	2	no	—
103	C₂³×C₁₃	—	2	2	no	—
104	C₂³×C₁₃	—	2	2	no	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	—	4	2	no	—
107	C₂³×C₁₃	—	4	2	no	—
108	C₂³×C₁₃	—	4	2	no	—
109	C₂³×C₁₃	—	4	1	no	—
110	C₂³×C₁₃	—	4	1	no	—
111	C₂³×C₁₃	C₂²⋊C₈	1	4	yes	—
112	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	D₄×C₁₃	—	2	4	no	—
114	D₄×C₁₃	—	2	4	no	—
115	D₄×C₁₃	—	2	4	no	—
116	D₄×C₁₃	—	4	16	no	—
117	D₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	8	2	no	—
132	C₂×C₄×C₁₃	—	8	2	no	—
133	C₂×C₄×C₁₃	—	8	2	no	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#121	1	2	yes	—
137	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#131	1	2	yes	—
141	C₈×C₁₃	—	4	4	no	—
142	C₈×C₁₃	—	8	2	no	—
143	C₂×#127	C₄×C₁₃	1	1	yes	characteristic
144	C₂×C₄×D₄	—	2	2	no	—
145	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
146	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
147	C₈×D₄	—	2	4	no	—
148	#431	C₂²×C₁₃	1	2	yes	—
149	#431	C₂²×C₁₃	1	2	yes	—
150	C₂×C₂²⋊C₈	—	2	2	no	—
151	#390	—	2	2	no	—
152	#390	C₄×C₁₃	1	1	yes	characteristic
153	#390	C₄×C₁₃	1	1	yes	characteristic
154	#384	C₄×C₁₃	1	1	yes	characteristic
155	C₄³	C₂²×C₁₃	1	1	yes	characteristic
156	D₄⋊C₈	—	2	4	no	—
157	#323	—	2	2	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	4	no	—
161	C₂²×C₁₃	—	4	4	no	—
162	C₂²×C₁₃	—	4	4	no	—
163	C₂²×C₁₃	—	4	2	no	—
164	C₂²×C₁₃	—	4	2	no	—
165	C₂²×C₁₃	—	4	2	no	—
166	C₂²×C₁₃	—	8	2	no	—
167	C₂²×C₁₃	—	8	2	no	—
168	C₂²×C₁₃	—	8	1	no	—
169	C₂²×C₁₃	—	8	1	no	—
170	C₂²×C₁₃	#386	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
172	C₂²×C₁₃	#408	1	1	yes	characteristic
173	C₄×C₁₃	—	2	4	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	4	2	no	—
177	C₄×C₁₃	—	4	2	no	—
178	C₄×C₁₃	—	8	1	no	—
179	C₄×C₁₃	C₈×D₄	1	2	yes	—
180	C₄×C₁₃	#435	1	2	yes	—
181	C₄×C₁₃	#453	1	2	yes	—
182	C₂²×D₄	—	2	2	no	—
183	C₂²×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
184	C₂²×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
185	C₂³×C₄	—	2	1	no	—
186	C₂³×C₄	D₄×C₁₃	1	2	yes	—
187	C₂²×C₈	—	2	4	no	—
188	#127	—	2	2	no	—
189	#127	C₈×C₁₃	1	4	yes	complemented
190	C₄×D₄	—	2	2	no	—
191	C₄×D₄	—	2	2	no	—
192	C₄×D₄	—	2	2	no	—
193	C₄×D₄	—	2	2	no	—
194	C₄×D₄	—	4	8	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	2	1	no	—
201	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	2	1	no	—
204	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
205	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	D₄×C₁₃	1	4	yes	complemented
209	C₂²⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	2	2	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	2	1	no	—
216	C₂×C₁₃	—	4	4	no	—
217	C₂×C₁₃	—	8	2	no	—
218	C₂×C₁₃	#1869	1	1	yes	characteristic
219	C₂×C₁₃	#1884	1	1	yes	characteristic
220	C₂×C₁₃	#1887	1	1	yes	characteristic
221	C₂⁴	—	2	1	no	—
222	C₂⁴	C₁₃×C₂²⋊C₄	1	2	yes	—
223	C₂×D₄	—	2	8	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
232	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	complemented
233	C₂×D₄	C₁₃×M₁₆	1	2	yes	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	8	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	C₂×C₈×C₁₃	1	2	yes	complemented
262	C₄²	C₂×D₄×C₁₃	1	2	yes	—
263	C₄²	C₁₃×#41	1	2	yes	—
264	C₁₃	#8959	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	1	no	—
274	C₂³	C₁₃×C₂²⋊C₈	1	4	yes	—
275	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
276	D₄	—	2	4	no	—
277	D₄	—	2	4	no	—
278	D₄	—	2	4	no	—
279	D₄	—	4	16	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
298	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
299	C₂×C₄	C₁₃×#121	1	2	yes	—
300	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
302	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
303	C₂×C₄	C₁₃×#131	1	2	yes	—
304	C₈	—	4	4	no	—
305	C₈	—	8	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	C₁₃×#386	1	1	yes	characteristic
319	C₂²	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
320	C₂²	C₁₃×#408	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	1	no	—
327	C₄	C₈×D₄×C₁₃	1	2	yes	—
328	C₄	C₁₃×#435	1	2	yes	—
329	C₄	C₁₃×#453	1	2	yes	—
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	4	4	no	—
333	C₂	—	8	2	no	—
334	C₂	C₁₃×#1869	1	1	yes	characteristic
335	C₂	C₁₃×#1884	1	1	yes	characteristic
336	C₂	C₁₃×#1887	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#120
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#386
prime radical	C₂×C₁₃×#127	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₁₃×#2972	C₂	semidirect
3. C₁₃×#1478	C₂	semidirect
4. C₁₃×#1460	C₂	semidirect
5. C₁₃×#1276	C₂	semidirect
28. C₁₃×#127	C₈	semidirect
47. C₁₃×C₄⋊C₈	D₄	semidirect
53. #8959	C₁₃	direct (N × H)
65. C₂×D₄×C₁₃	C₂×C₈	semidirect
94. C₄²×C₁₃	C₂×C₈	semidirect
97. #2972	C₂×C₁₃	semidirect
99. #1478	C₂×C₁₃	semidirect
100. #1460	C₂×C₁₃	semidirect
101. #1276	C₂×C₁₃	semidirect
189. #127	C₈×C₁₃	semidirect
208. C₄⋊C₈	D₄×C₁₃	semidirect
232. C₂×D₄	C₂×C₈×C₁₃	semidirect
261. C₄²	C₂×C₈×C₁₃	semidirect
264. C₁₃	#8959	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×#2972	2
C₄×C₁₃×#127	2
C₁₃×#1478	2
C₁₃×#1460	2
C₁₃×#1276	2
#8959	13

Subgroups — C13×#8959 #15695298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#8959 #15695298