Subgroups — C₁₃×#10445 #15696784

Order 3328.

440 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#461	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1800	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1478	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×SD₁₆	—	2	1	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#461	—	2	2	no	—
12	C₁₃×#461	—	2	2	no	—
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₁₃×#423	C₂²	1	1	yes	characteristic
16	C₁₃×#423	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#103	—	2	1	no	—
19	C₂×C₁₃×#100	—	2	1	no	—
20	C₁₃×#343	—	2	1	no	—
21	C₁₃×#334	—	2	2	no	—
22	C₁₃×#332	—	2	2	no	—
23	C₁₃×#332	—	2	2	no	—
24	C₁₃×D₄⋊C₈	—	2	2	no	—
25	C₂²×Q₈×C₁₃	—	2	1	no	—
26	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	—	2	1	no	—
28	C₂×C₁₃×SD₁₆	—	2	4	no	—
29	C₂×C₁₃×SD₁₆	—	4	2	no	—
30	C₂×C₁₃×SD₁₆	—	4	2	no	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂×C₁₃×M₁₆	—	2	1	no	—
33	C₂²×C₈×C₁₃	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×C₄⋊Q₈	—	2	1	no	—
36	C₁₃×C₄⋊Q₈	—	2	1	no	—
37	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
38	C₄×D₄×C₁₃	—	2	1	no	—
39	C₄×D₄×C₁₃	—	2	1	no	—
40	C₄×D₄×C₁₃	—	2	1	no	—
41	C₄×D₄×C₁₃	—	2	1	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
52	C₁₃×C₄⋊C₈	—	2	1	no	—
53	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
54	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
55	C₁₃×#103	—	2	4	no	—
56	C₁₃×#100	—	4	2	no	—
57	C₁₃×C₂²⋊C₈	—	2	1	no	—
58	C₁₃×C₂²⋊C₈	—	2	1	no	—
59	C₄×C₈×C₁₃	—	2	1	no	—
60	C₄×C₈×C₁₃	—	2	1	no	—
61	#10445	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	—	2	1	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	4	1	no	—
65	C₂×Q₈×C₁₃	—	4	1	no	—
66	C₂×Q₈×C₁₃	—	4	1	no	—
67	C₂×Q₈×C₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	2	1	no	—
70	C₂×D₄×C₁₃	—	2	1	no	—
71	C₂×D₄×C₁₃	—	2	1	no	—
72	C₂×D₄×C₁₃	—	2	1	no	—
73	C₂×D₄×C₁₃	—	2	1	no	—
74	C₂×D₄×C₁₃	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₁₃	—	2	1	no	—
76	C₂²×C₄×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	—	4	1	no	—
79	C₂²×C₄×C₁₃	—	8	2	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
83	C₁₃×SD₁₆	—	4	8	no	—
84	C₁₃×M₁₆	—	4	1	no	—
85	C₁₃×M₁₆	—	4	1	no	—
86	C₁₃×M₁₆	—	4	1	no	—
87	C₁₃×M₁₆	—	4	1	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	4	1	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
94	C₂×C₈×C₁₃	—	4	1	no	—
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₂²⋊C₄	—	4	1	no	—
102	C₁₃×C₂²⋊C₄	—	4	1	no	—
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃	#41	1	1	yes	characteristic
107	C₂×#461	C₂×C₁₃	1	1	yes	characteristic
108	#1800	C₂×C₁₃	1	1	yes	characteristic
109	#1570	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#1546	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#1546	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#1478	C₂×C₁₃	1	1	yes	characteristic
113	#1301	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	—	4	1	no	—
116	C₂³×C₁₃	—	4	1	no	—
117	C₂³×C₁₃	—	8	1	no	—
118	C₂³×C₁₃	—	8	1	no	—
119	C₂³×C₁₃	#123	1	1	yes	characteristic
120	Q₈×C₁₃	—	4	2	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	8	1	no	—
123	Q₈×C₁₃	—	8	1	no	—
124	D₄×C₁₃	—	2	2	no	—
125	D₄×C₁₃	—	2	2	no	—
126	D₄×C₁₃	—	4	1	no	—
127	D₄×C₁₃	—	4	1	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	4	1	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	8	2	no	—
133	C₂×C₄×C₁₃	—	8	1	no	—
134	C₂×C₄×C₁₃	—	8	1	no	—
135	C₂×C₄×C₁₃	—	8	2	no	—
136	C₂×C₄×C₁₃	—	8	1	no	—
137	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
145	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₈×C₁₃	—	4	2	no	—
148	C₈×C₁₃	—	4	2	no	—
149	C₈×C₁₃	—	8	1	no	—
150	C₈×C₁₃	—	8	1	no	—
151	C₈×C₁₃	—	8	1	no	—
152	C₂²×SD₁₆	—	2	1	no	—
153	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
155	#461	—	2	2	no	—
156	#461	—	2	2	no	—
157	#431	C₂²×C₁₃	1	1	yes	characteristic
158	#431	C₂²×C₁₃	1	1	yes	characteristic
159	#423	C₂²×C₁₃	1	1	yes	characteristic
160	#423	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×#103	—	2	1	no	—
163	C₂×#100	—	2	1	no	—
164	#343	—	2	1	no	—
165	#334	—	2	2	no	—
166	#332	—	2	2	no	—
167	#332	—	2	2	no	—
168	D₄⋊C₈	—	2	2	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	4	2	no	—
172	C₂²×C₁₃	—	8	2	no	—
173	C₂²×C₁₃	—	8	2	no	—
174	C₂²×C₁₃	—	8	1	no	—
175	C₂²×C₁₃	—	8	1	no	—
176	C₂²×C₁₃	#394	1	1	yes	characteristic
177	C₂²×C₁₃	#481	1	1	yes	characteristic
178	C₂²×C₁₃	#482	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	—	8	2	no	—
183	C₄×C₁₃	—	8	1	no	—
184	C₄×C₁₃	#449	1	1	yes	characteristic
185	C₄×C₁₃	#450	1	1	yes	characteristic
186	C₄×C₁₃	#465	1	1	yes	characteristic
187	C₄×C₁₃	#469	1	1	yes	characteristic
188	C₄×C₁₃	#471	1	2	yes	—
189	C₂²×Q₈	—	2	1	no	—
190	C₂²×D₄	D₄×C₁₃	1	1	yes	characteristic
191	C₂³×C₄	—	2	1	no	—
192	C₂×SD₁₆	—	2	4	no	—
193	C₂×SD₁₆	—	4	2	no	—
194	C₂×SD₁₆	—	4	2	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
218	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
219	#103	—	2	4	no	—
220	#100	—	4	2	no	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₂×C₁₃	—	2	1	no	—
226	C₂×C₁₃	—	2	1	no	—
227	C₂×C₁₃	—	8	2	no	—
228	C₂×C₁₃	#2034	1	1	yes	characteristic
229	C₂×C₁₃	#2046	1	1	yes	characteristic
230	C₂×C₁₃	#2049	1	1	yes	characteristic
231	C₂⁴	—	2	1	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	C₁₃×SD₁₆	1	2	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	8	2	no	—
249	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
251	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
252	SD₁₆	—	4	8	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	4	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	4	1	no	—
272	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₄²	C₁₃×#41	1	1	yes	characteristic
276	C₁₃	#10445	1	1	yes	characteristic, direct product factor, Sylow
277	C₂³	—	2	2	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	8	1	no	—
282	C₂³	C₁₃×#123	1	1	yes	characteristic
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	8	1	no	—
286	Q₈	—	8	1	no	—
287	D₄	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
301	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
302	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
303	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
304	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
305	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
306	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
307	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
308	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	8	1	no	—
313	C₈	—	8	1	no	—
314	C₈	—	8	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	C₁₃×#394	1	1	yes	characteristic
323	C₂²	C₁₃×#481	1	1	yes	characteristic
324	C₂²	C₁₃×#482	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	1	no	—
330	C₄	C₁₃×#449	1	1	yes	characteristic
331	C₄	C₁₃×#450	1	1	yes	characteristic
332	C₄	C₁₃×#465	1	1	yes	characteristic
333	C₄	C₁₃×#469	1	1	yes	characteristic
334	C₄	C₁₃×#471	1	2	yes	—
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	8	2	no	—
338	C₂	C₁₃×#2034	1	1	yes	characteristic
339	C₂	C₁₃×#2046	1	1	yes	characteristic
340	C₂	C₁₃×#2049	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂²×D₄×C₁₃	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ 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
3. C₁₃×#1570	C₂	semidirect
4. C₁₃×#1546	C₂	semidirect
5. C₁₃×#1546	C₂	semidirect
7. C₁₃×#1301	C₂	semidirect
61. #10445	C₁₃	direct (N × H)
109. #1570	C₂×C₁₃	semidirect
110. #1546	C₂×C₁₃	semidirect
111. #1546	C₂×C₁₃	semidirect
113. #1301	C₂×C₁₃	semidirect
276. C₁₃	#10445	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#461	2
C₁₃×#1800	2
C₁₃×#1570	2
C₁₃×#1546	2
C₁₃×#1478	2
C₁₃×#1301	2
#10445	13

Subgroups — C13×#10445 #15696784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10445 #15696784