Subgroups — C₁₃×#8604 #15694943

Order 3328.

462 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1477	C₂	1	2	yes	maximal
5	C₁₃×#1283	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
8	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
9	C₁₃×#431	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₁₃×#405	—	2	1	no	—
12	C₁₃×#405	—	2	1	no	—
13	C₁₃×#405	C₄	1	1	yes	characteristic
14	C₁₃×#405	C₄	1	1	yes	characteristic
15	C₄×C₁₃×M₁₆	—	2	1	no	—
16	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#335	—	2	1	no	—
20	C₁₃×#334	—	2	1	no	—
21	C₁₃×#334	—	2	1	no	—
22	C₁₃×#334	—	2	1	no	—
23	C₁₃×Q₈⋊C₈	—	2	2	no	—
24	C₁₃×D₄⋊C₈	—	2	2	no	—
25	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
26	C₁₃×#131	—	2	2	no	—
27	C₁₃×#131	—	2	2	no	—
28	C₁₃×#131	—	2	2	no	—
29	C₁₃×#131	C₈	1	4	yes	complemented
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	—	2	1	no	—
35	C₂²×C₈×C₁₃	—	2	1	no	—
36	C₂²×C₈×C₁₃	—	2	2	no	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₁₃	Q₈	1	1	yes	characteristic
41	C₄×D₄×C₁₃	—	2	1	no	—
42	C₄×D₄×C₁₃	—	2	1	no	—
43	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
44	C₄×D₄×C₁₃	Q₈	1	1	yes	characteristic
45	C₁₃×#117	—	2	1	no	—
46	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	—	2	1	no	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊C₈	—	2	2	no	—
50	C₁₃×C₄⋊C₈	—	2	2	no	—
51	C₁₃×C₄⋊C₈	—	2	2	no	—
52	C₁₃×C₂²⋊C₈	—	2	2	no	—
53	C₁₃×#97	—	2	1	no	—
54	C₁₃×#97	—	2	1	no	—
55	C₁₃×#97	—	2	1	no	—
56	C₄×C₈×C₁₃	—	2	1	no	—
57	C₄×C₈×C₁₃	—	2	1	no	—
58	C₄×C₈×C₁₃	—	2	1	no	—
59	C₄×C₈×C₁₃	—	2	2	no	—
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	#8604	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₁₃×#41	—	2	1	no	—
66	C₁₃×#41	—	2	1	no	—
67	C₁₃×#41	C₂×C₈	1	2	yes	—
68	C₁₃×#41	M₁₆	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
78	C₁₃×M₁₆	—	2	4	no	—
79	C₁₃×M₁₆	—	2	1	no	—
80	C₁₃×M₁₆	—	2	1	no	—
81	C₁₃×M₁₆	—	2	4	no	—
82	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
83	C₁₃×M₁₆	M₁₆	1	2	yes	—
84	C₂×C₈×C₁₃	—	2	4	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	4	no	—
90	C₂×C₈×C₁₃	—	4	2	no	—
91	C₂×C₈×C₁₃	—	4	2	no	—
92	C₂×C₈×C₁₃	—	8	2	no	—
93	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
98	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
99	C₁₃×C₄⋊C₄	—	2	1	no	—
100	C₁₃×C₄⋊C₄	—	2	1	no	—
101	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
102	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
103	C₁₃×C₂²⋊C₄	—	2	1	no	—
104	C₁₃×C₂²⋊C₄	—	2	1	no	—
105	C₁₃×C₂²⋊C₄	—	2	1	no	—
106	C₄²×C₁₃	—	2	1	no	—
107	C₄²×C₁₃	—	2	1	no	—
108	C₄²×C₁₃	—	2	1	no	—
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
113	C₄×#131	C₂×C₁₃	1	1	yes	characteristic
114	#1564	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂×#334	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#1477	C₂×C₁₃	1	2	yes	—
117	#1283	C₂×C₁₃	1	2	yes	complemented
118	C₂³×C₁₃	—	2	1	no	—
119	C₂³×C₁₃	#117	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	1	no	—
121	Q₈×C₁₃	C₄⋊C₈	1	2	yes	—
122	D₄×C₁₃	—	2	2	no	—
123	D₄×C₁₃	—	2	2	no	—
124	D₄×C₁₃	—	2	1	no	—
125	D₄×C₁₃	C₄⋊C₈	1	2	yes	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
145	C₈×C₁₃	—	2	2	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	2	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	8	2	no	—
150	C₈×C₁₃	—	8	2	no	—
151	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
152	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
153	C₂×#131	C₄×C₁₃	1	1	yes	characteristic
154	C₂×#131	C₄×C₁₃	1	1	yes	characteristic
155	C₄×#41	C₂²×C₁₃	1	1	yes	characteristic
156	#431	C₂²×C₁₃	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
158	#405	—	2	1	no	—
159	#405	—	2	1	no	—
160	#405	C₄×C₁₃	1	1	yes	characteristic
161	#405	C₄×C₁₃	1	1	yes	characteristic
162	C₄×M₁₆	—	2	1	no	—
163	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×C₄×C₈	—	2	1	no	—
165	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
166	#335	—	2	1	no	—
167	#334	—	2	1	no	—
168	#334	—	2	1	no	—
169	#334	—	2	1	no	—
170	Q₈⋊C₈	—	2	2	no	—
171	D₄⋊C₈	—	2	2	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	4	1	no	—
176	C₂²×C₁₃	—	4	1	no	—
177	C₂²×C₁₃	#380	1	1	yes	characteristic
178	C₂²×C₁₃	#431	1	1	yes	characteristic
179	C₂²×C₁₃	#432	1	1	yes	characteristic
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	4	1	no	—
185	C₄×C₁₃	—	4	1	no	—
186	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
187	C₄×C₁₃	#407	1	1	yes	characteristic
188	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
189	C₄×C₁₃	#423	1	1	yes	characteristic
190	C₄×C₁₃	#471	1	2	yes	—
191	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
192	#131	—	2	2	no	—
193	#131	—	2	2	no	—
194	#131	—	2	2	no	—
195	#131	C₈×C₁₃	1	4	yes	complemented
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
199	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
204	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
205	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
206	C₄×Q₈	Q₈×C₁₃	1	1	yes	characteristic
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
210	C₄×D₄	Q₈×C₁₃	1	1	yes	characteristic
211	#117	—	2	1	no	—
212	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₂²⋊C₈	—	2	2	no	—
219	#97	—	2	1	no	—
220	#97	—	2	1	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	2	no	—
226	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
227	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
228	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
229	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
230	C₂×C₁₃	—	2	1	no	—
231	C₂×C₁₃	—	2	1	no	—
232	C₂×C₁₃	—	4	2	no	—
233	C₂×C₁₃	#1797	1	1	yes	characteristic
234	C₂×C₁₃	#1814	1	1	yes	characteristic
235	C₂×C₁₃	#1815	1	1	yes	characteristic
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	#41	C₂×C₈×C₁₃	1	2	yes	—
239	#41	C₁₃×M₁₆	1	2	yes	—
240	C₂×Q₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
247	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
248	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
249	M₁₆	—	2	4	no	—
250	M₁₆	—	2	1	no	—
251	M₁₆	—	2	1	no	—
252	M₁₆	—	2	4	no	—
253	M₁₆	C₂×C₈×C₁₃	1	2	yes	—
254	M₁₆	C₁₃×M₁₆	1	2	yes	—
255	C₂×C₈	—	2	4	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	—	2	4	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	8	2	no	—
264	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
269	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
273	C₄⋊C₄	C₁₃×C₄⋊C₄	1	1	yes	characteristic
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
283	C₄²	C₂×Q₈×C₁₃	1	1	yes	characteristic
284	C₁₃	#8604	1	1	yes	characteristic, direct product factor, Sylow
285	C₂³	—	2	1	no	—
286	C₂³	C₁₃×#117	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	C₁₃×C₄⋊C₈	1	2	yes	—
289	D₄	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	2	1	no	—
292	D₄	C₁₃×C₄⋊C₈	1	2	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
304	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
305	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
306	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
307	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
310	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
311	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	2	1	no	—
316	C₈	—	8	2	no	—
317	C₈	—	8	2	no	—
318	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
319	C₈	C₁₃×C₂²⋊C₈	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	C₁₃×#380	1	1	yes	characteristic
326	C₂²	C₁₃×#431	1	1	yes	characteristic
327	C₂²	C₁₃×#432	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
335	C₄	C₁₃×#407	1	1	yes	characteristic
336	C₄	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
337	C₄	C₁₃×#423	1	1	yes	characteristic
338	C₄	C₁₃×#471	1	2	yes	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	4	2	no	—
342	C₂	C₁₃×#1797	1	1	yes	characteristic
343	C₂	C₁₃×#1814	1	1	yes	characteristic
344	C₂	C₁₃×#1815	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#121
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#380
prime radical	C₂×C₁₃×#41	C₂×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
2. C₁₃×#1564	C₂	semidirect
3. C₂×C₁₃×#334	C₂	semidirect
5. C₁₃×#1283	C₂	semidirect
29. C₁₃×#131	C₈	semidirect
64. #8604	C₁₃	direct (N × H)
114. #1564	C₂×C₁₃	semidirect
115. C₂×#334	C₂×C₁₃	semidirect
117. #1283	C₂×C₁₃	semidirect
195. #131	C₈×C₁₃	semidirect
284. C₁₃	#8604	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#131	2
C₁₃×#1564	2
C₂×C₁₃×#334	2
C₁₃×#1477	2
C₁₃×#1283	2
#8604	13

Subgroups — C13×#8604 #15694943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#8604 #15694943