Subgroups — C₁₃×#32539 #15718878

Order 3328.

590 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×#2514	C₂	1	2	yes	maximal
2	C₁₃×#2097	C₂	1	2	yes	maximal
3	C₁₃×#2068	C₂	1	2	yes	maximal
4	C₁₃×#2056	C₂	1	2	yes	maximal
5	C₁₃×#1971	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1949	C₂	1	2	yes	maximal
7	C₁₃×#1944	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1940	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1928	C₂	1	2	yes	maximal
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#103	C₂²	1	2	yes	—
18	C₂×C₁₃×#103	C₂²	1	2	yes	—
19	C₂×C₁₃×#103	C₂²	1	2	yes	—
20	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
21	C₁₃×#401	—	2	2	no	—
22	C₁₃×#400	—	2	2	no	—
23	C₁₃×#391	C₂²	1	2	yes	—
24	C₁₃×#391	C₂²	1	2	yes	—
25	C₁₃×#391	C₂²	1	2	yes	—
26	C₁₃×#389	C₂²	1	2	yes	—
27	C₁₃×#387	—	2	2	no	—
28	C₁₃×#383	—	2	2	no	—
29	C₁₃×#382	C₂²	1	2	yes	—
30	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
31	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
36	C₂×C₁₃×Q₁₆	—	2	2	no	—
37	C₂×C₁₃×Q₁₆	—	4	1	no	—
38	C₂×C₁₃×Q₁₆	—	4	1	no	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₃	—	2	2	no	—
48	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×#107	—	4	1	no	—
51	C₁₃×#107	—	4	2	no	—
52	C₁₃×#106	—	2	2	no	—
53	C₁₃×#106	—	4	1	no	—
54	C₁₃×C₄⋊C₈	—	2	4	no	—
55	C₁₃×C₄⋊C₈	—	4	1	no	—
56	C₁₃×#103	—	2	4	no	—
57	C₁₃×#103	—	4	2	no	—
58	C₁₃×#103	—	4	2	no	—
59	C₁₃×#97	—	4	1	no	—
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	—	2	2	no	—
63	C₁₃×#95	—	2	2	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	—	2	2	no	—
68	C₁₃×#95	—	2	2	no	—
69	C₁₃×#95	—	2	2	no	—
70	C₁₃×#95	D₄	1	1	yes	characteristic
71	C₁₃×#95	D₄	1	1	yes	characteristic
72	#32539	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	2	2	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×Q₁₆	—	4	4	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	#41	1	2	yes	—
97	C₂×C₈×C₁₃	#41	1	2	yes	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	4	2	no	—
105	C₁₃×C₄⋊C₄	—	4	4	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₄²×C₁₃	—	2	1	no	—
108	C₄²×C₁₃	—	2	1	no	—
109	C₄²×C₁₃	—	2	2	no	—
110	C₄²×C₁₃	—	4	4	no	—
111	#2514	C₂×C₁₃	1	2	yes	—
112	#2097	C₂×C₁₃	1	2	yes	—
113	#2068	C₂×C₁₃	1	2	yes	—
114	#2056	C₂×C₁₃	1	2	yes	—
115	#1971	C₂×C₁₃	1	1	yes	characteristic
116	#1949	C₂×C₁₃	1	2	yes	—
117	#1944	C₂×C₁₃	1	1	yes	characteristic
118	#1940	C₂×C₁₃	1	1	yes	characteristic
119	#1928	C₂×C₁₃	1	2	yes	—
120	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	4	2	no	—
123	Q₈×C₁₃	—	4	2	no	—
124	Q₈×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
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	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	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	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	—	2	2	no	—
155	C₈×C₁₃	—	4	2	no	—
156	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#107	C₂²×C₁₃	1	2	yes	—
158	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×#103	C₂²×C₁₃	1	2	yes	—
164	C₂×#103	C₂²×C₁₃	1	2	yes	—
165	C₂×#103	C₂²×C₁₃	1	2	yes	—
166	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
167	#401	—	2	2	no	—
168	#400	—	2	2	no	—
169	#391	C₂²×C₁₃	1	2	yes	—
170	#391	C₂²×C₁₃	1	2	yes	—
171	#391	C₂²×C₁₃	1	2	yes	—
172	#389	C₂²×C₁₃	1	2	yes	—
173	#387	—	2	2	no	—
174	#383	—	2	2	no	—
175	#382	C₂²×C₁₃	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
177	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
178	#340	C₂²×C₁₃	1	2	yes	—
179	#340	C₂²×C₁₃	1	2	yes	—
180	#336	C₂²×C₁₃	1	2	yes	—
181	C₂²×C₁₃	#537	1	1	yes	characteristic
182	C₂²×C₁₃	#547	1	1	yes	characteristic
183	C₂²×C₁₃	#550	1	1	yes	characteristic
184	C₂²×C₁₃	#575	1	2	yes	—
185	C₂²×C₁₃	#578	1	2	yes	—
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	—	4	2	no	—
188	C₄×C₁₃	—	4	1	no	—
189	C₄×C₁₃	—	4	1	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	—	8	2	no	—
194	C₄×C₁₃	#538	1	1	yes	characteristic
195	C₄×C₁₃	#539	1	1	yes	characteristic
196	C₄×C₁₃	#556	1	1	yes	characteristic
197	C₄×C₁₃	#563	1	1	yes	characteristic
198	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
199	C₂×Q₁₆	—	2	2	no	—
200	C₂×Q₁₆	—	4	1	no	—
201	C₂×Q₁₆	—	4	1	no	—
202	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
203	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
212	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
213	#107	—	4	1	no	—
214	#107	—	4	2	no	—
215	#106	—	2	2	no	—
216	#106	—	4	1	no	—
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	4	1	no	—
219	#103	—	2	4	no	—
220	#103	—	4	2	no	—
221	#103	—	4	2	no	—
222	#97	—	4	1	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	D₄×C₁₃	1	1	yes	characteristic
234	#95	D₄×C₁₃	1	1	yes	characteristic
235	C₂×C₁₃	#2727	1	1	yes	characteristic
236	C₂×C₁₃	#3256	1	1	yes	characteristic
237	C₂×C₁₃	#3261	1	1	yes	characteristic
238	C₂×C₁₃	#3349	1	2	yes	—
239	C₂×C₁₃	#3402	1	2	yes	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
254	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
255	Q₁₆	—	4	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	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	C₁₃×#41	1	2	yes	—
264	C₂×C₈	C₁₃×#41	1	2	yes	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	4	4	no	—
278	C₁₃	#32539	1	1	yes	characteristic, direct product factor, Sylow
279	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
280	Q₈	—	4	2	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
307	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
308	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
309	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
310	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
311	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	2	no	—
314	C₈	—	4	2	no	—
315	C₂²	C₁₃×#537	1	1	yes	characteristic
316	C₂²	C₁₃×#547	1	1	yes	characteristic
317	C₂²	C₁₃×#550	1	1	yes	characteristic
318	C₂²	C₁₃×#575	1	2	yes	—
319	C₂²	C₁₃×#578	1	2	yes	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	C₁₃×#538	1	1	yes	characteristic
329	C₄	C₁₃×#539	1	1	yes	characteristic
330	C₄	C₁₃×#556	1	1	yes	characteristic
331	C₄	C₁₃×#563	1	1	yes	characteristic
332	C₂	C₁₃×#2727	1	1	yes	characteristic
333	C₂	C₁₃×#3256	1	1	yes	characteristic
334	C₂	C₁₃×#3261	1	1	yes	characteristic
335	C₂	C₁₃×#3349	1	2	yes	—
336	C₂	C₁₃×#3402	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
72. #32539	C₁₃	direct (N × H)
278. C₁₃	#32539	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×#2514	2
C₁₃×#2097	2
C₁₃×#2068	2
C₁₃×#2056	2
C₁₃×#1971	2
C₁₃×#1949	2
C₁₃×#1944	2
C₁₃×#1940	2
C₁₃×#1928	2
#32539	13

Subgroups — C13×#32539 #15718878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32539 #15718878