Subgroups — C₁₃×#10283 #15696622

Order 3328.

626 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3227	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3206	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2868	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1962	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#532	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊Q₈	—	2	2	no	—
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₁₃×#498	—	2	1	no	—
13	C₁₃×#492	—	2	1	no	—
14	C₁₃×#487	—	2	1	no	—
15	C₁₃×#486	—	2	1	no	—
16	C₁₃×#481	—	2	2	no	—
17	C₁₃×#476	—	2	2	no	—
18	C₁₃×#461	—	2	2	no	—
19	C₁₃×#448	—	2	2	no	—
20	C₁₃×#431	C₂²	1	1	yes	characteristic
21	C₁₃×#431	C₂²	1	1	yes	characteristic
22	C₁₃×#423	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×#104	—	2	2	no	—
24	C₁₃×#384	—	2	2	no	—
25	C₁₃×#384	—	2	2	no	—
26	C₄³×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#333	—	2	2	no	—
28	C₁₃×#332	—	2	2	no	—
29	C₂×C₁₃×#41	—	2	1	no	—
30	C₂²×Q₈×C₁₃	—	2	1	no	—
31	C₂²×Q₈×C₁₃	D₄	1	2	yes	—
32	C₂×C₁₃×SD₁₆	—	2	4	no	—
33	C₂×C₁₃×M₁₆	—	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	—	2	1	no	—
38	C₁₃×C₄⋊Q₈	—	2	1	no	—
39	C₁₃×C₄⋊Q₈	—	2	1	no	—
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	—	4	8	no	—
43	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₁₃×#127	D₄	1	1	yes	characteristic
45	C₁₃×#124	—	2	1	no	—
46	C₁₃×#121	—	2	1	no	—
47	C₁₃×#121	—	2	1	no	—
48	C₄×D₄×C₁₃	—	2	1	no	—
49	C₄×D₄×C₁₃	—	2	1	no	—
50	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
51	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
55	C₂×C₄²×C₁₃	—	2	1	no	—
56	C₂×C₄²×C₁₃	—	2	1	no	—
57	C₂×C₄²×C₁₃	D₄	1	2	yes	—
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₁₃×#106	—	2	4	no	—
60	C₁₃×C₄⋊C₈	—	2	1	no	—
61	C₁₃×C₄⋊C₈	—	2	1	no	—
62	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
63	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
64	C₁₃×#104	—	4	4	no	—
65	C₁₃×#103	—	2	4	no	—
66	C₁₃×#102	—	2	4	no	—
67	C₁₃×C₂²⋊C₈	—	2	1	no	—
68	C₁₃×C₂²⋊C₈	—	2	1	no	—
69	C₄×C₈×C₁₃	—	2	1	no	—
70	C₄×C₈×C₁₃	—	2	1	no	—
71	#10283	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃×#41	—	4	2	no	—
73	C₁₃×#41	—	4	2	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	2	1	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	4	4	no	—
78	C₂×Q₈×C₁₃	—	4	4	no	—
79	C₂×Q₈×C₁₃	—	4	1	no	—
80	C₂×Q₈×C₁₃	—	4	1	no	—
81	C₂×D₄×C₁₃	—	2	1	no	—
82	C₂×D₄×C₁₃	—	2	1	no	—
83	C₂×D₄×C₁₃	—	2	1	no	—
84	C₂×D₄×C₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	—	2	2	no	—
86	C₂²×C₄×C₁₃	—	2	2	no	—
87	C₂²×C₄×C₁₃	—	2	2	no	—
88	C₂²×C₄×C₁₃	—	4	1	no	—
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₁₃×SD₁₆	—	8	4	no	—
93	C₁₃×M₁₆	—	4	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	4	1	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	1	no	—
99	C₁₃×C₄⋊C₄	—	2	1	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	4	16	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₂²⋊C₄	—	4	1	no	—
109	C₁₃×C₂²⋊C₄	—	4	1	no	—
110	C₄²×C₁₃	—	2	4	no	—
111	C₄²×C₁₃	—	2	4	no	—
112	C₄²×C₁₃	—	2	2	no	—
113	C₄²×C₁₃	SD₁₆	1	4	yes	complemented
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃	#41	1	1	yes	characteristic
118	#3227	C₂×C₁₃	1	1	yes	characteristic
119	#3206	C₂×C₁₃	1	1	yes	characteristic
120	#2868	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#1962	C₂×C₁₃	1	1	yes	characteristic
122	#1549	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#1548	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#1276	C₂×C₁₃	1	1	yes	characteristic, complemented
125	C₂³×C₁₃	—	4	1	no	—
126	C₂³×C₁₃	#120	1	1	yes	characteristic
127	Q₈×C₁₃	—	4	4	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	8	2	no	—
130	Q₈×C₁₃	—	8	2	no	—
131	Q₈×C₁₃	—	8	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	8	2	no	—
136	C₂×C₄×C₁₃	—	2	4	no	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	—	2	4	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	4	4	no	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	8	1	no	—
147	C₂×C₄×C₁₃	—	8	1	no	—
148	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
150	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
155	C₂×C₄×C₁₃	#137	1	2	yes	—
156	C₈×C₁₃	—	4	2	no	—
157	C₈×C₁₃	—	4	2	no	—
158	C₈×C₁₃	—	8	1	no	—
159	#532	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×C₄⋊Q₈	—	2	2	no	—
161	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
163	#498	—	2	1	no	—
164	#492	—	2	1	no	—
165	#487	—	2	1	no	—
166	#486	—	2	1	no	—
167	#481	—	2	2	no	—
168	#476	—	2	2	no	—
169	#461	—	2	2	no	—
170	#448	—	2	2	no	—
171	#431	C₂²×C₁₃	1	1	yes	characteristic
172	#431	C₂²×C₁₃	1	1	yes	characteristic
173	#423	C₂²×C₁₃	1	1	yes	characteristic
174	C₂×#104	—	2	2	no	—
175	#384	—	2	2	no	—
176	#384	—	2	2	no	—
177	C₄³	C₂²×C₁₃	1	1	yes	characteristic
178	#333	—	2	2	no	—
179	#332	—	2	2	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	8	1	no	—
183	C₂²×C₁₃	—	8	1	no	—
184	C₂²×C₁₃	—	8	1	no	—
185	C₂²×C₁₃	#392	1	1	yes	characteristic
186	C₂²×C₁₃	#448	1	2	yes	—
187	C₄×C₁₃	—	2	8	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	—	8	1	no	—
193	C₄×C₁₃	#453	1	2	yes	—
194	C₄×C₁₃	#456	1	2	yes	—
195	C₄×C₁₃	#461	1	2	yes	—
196	C₄×C₁₃	#492	1	1	yes	characteristic
197	C₄×C₁₃	#497	1	1	yes	characteristic
198	C₂×#41	—	2	1	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	D₄×C₁₃	1	2	yes	—
201	C₂×SD₁₆	—	2	4	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	4	8	no	—
212	C₄⋊Q₈	D₄×C₁₃	1	1	yes	characteristic
213	#127	D₄×C₁₃	1	1	yes	characteristic
214	#124	—	2	1	no	—
215	#121	—	2	1	no	—
216	#121	—	2	1	no	—
217	C₄×D₄	—	2	1	no	—
218	C₄×D₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	8	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
224	C₂×C₄²	—	2	1	no	—
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	D₄×C₁₃	1	2	yes	—
227	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
228	#106	—	2	4	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
232	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
233	#104	—	4	4	no	—
234	#103	—	2	4	no	—
235	#102	—	2	4	no	—
236	C₂²⋊C₈	—	2	1	no	—
237	C₂²⋊C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	—	16	1	no	—
242	C₂×C₁₃	#2002	1	1	yes	characteristic
243	C₂×C₁₃	#2005	1	1	yes	characteristic
244	C₂×C₁₃	#2013	1	1	yes	characteristic
245	#41	—	4	2	no	—
246	#41	—	4	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	4	4	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
265	SD₁₆	—	8	4	no	—
266	M₁₆	—	4	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	16	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	2	no	—
286	C₄²	C₁₃×SD₁₆	1	4	yes	complemented
287	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
290	C₄²	C₁₃×#41	1	1	yes	characteristic
291	C₁₃	#10283	1	1	yes	characteristic, direct product factor, Sylow
292	C₂³	—	4	1	no	—
293	C₂³	C₁₃×#120	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	8	2	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
316	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
317	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
318	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
319	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
320	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
321	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
322	C₂×C₄	C₁₃×#137	1	2	yes	—
323	C₈	—	4	2	no	—
324	C₈	—	4	2	no	—
325	C₈	—	8	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	C₁₃×#392	1	1	yes	characteristic
332	C₂²	C₁₃×#448	1	2	yes	—
333	C₄	—	2	8	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	1	no	—
339	C₄	C₁₃×#453	1	2	yes	—
340	C₄	C₁₃×#456	1	2	yes	—
341	C₄	C₁₃×#461	1	2	yes	—
342	C₄	C₁₃×#492	1	1	yes	characteristic
343	C₄	C₁₃×#497	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	16	1	no	—
346	C₂	C₁₃×#2002	1	1	yes	characteristic
347	C₂	C₁₃×#2005	1	1	yes	characteristic
348	C₂	C₁₃×#2013	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₁₃×#532	C₂²
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×#2868	C₂	semidirect
5. C₁₃×#1549	C₂	semidirect
6. C₁₃×#1548	C₂	semidirect
7. C₁₃×#1276	C₂	semidirect
71. #10283	C₁₃	direct (N × H)
113. C₄²×C₁₃	SD₁₆	semidirect
120. #2868	C₂×C₁₃	semidirect
122. #1549	C₂×C₁₃	semidirect
123. #1548	C₂×C₁₃	semidirect
124. #1276	C₂×C₁₃	semidirect
286. C₄²	C₁₃×SD₁₆	semidirect
291. C₁₃	#10283	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×#3227	2
C₁₃×#3206	2
C₁₃×#2868	2
C₁₃×#1962	2
C₁₃×#1549	2
C₁₃×#1548	2
C₁₃×#1276	2
#10283	13

Subgroups — C13×#10283 #15696622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10283 #15696622