Subgroups — C₁₃×#10820 #15697159

Order 3328.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#340	C₂	1	2	yes	maximal
3	C₂×C₁₃×#340	C₂	1	2	yes	maximal
4	C₁₃×C₄⋊#95	C₂	1	2	yes	maximal
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	—	2	4	no	—
10	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	complemented
11	C₂×C₁₃×#95	—	2	2	no	—
12	C₂×C₁₃×#95	—	2	2	no	—
13	C₁₃×#340	—	2	4	no	—
14	C₁₃×#340	C₄	1	8	yes	complemented
15	C₂³×C₄×C₁₃	—	2	4	no	—
16	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	D₄	1	4	yes	—
27	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
33	C₂×C₄²×C₁₃	—	2	2	no	—
34	C₂×C₄²×C₁₃	—	2	2	no	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
37	C₄×C₈×C₁₃	—	2	8	no	—
38	C₁₃×#95	—	4	4	no	—
39	C₁₃×#95	—	4	4	no	—
40	#10820	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	—	2	8	no	—
47	C₂²×C₄×C₁₃	—	4	4	no	—
48	C₂²×C₄×C₁₃	—	4	4	no	—
49	C₂²×C₄×C₁₃	—	4	4	no	—
50	C₂²×C₄×C₁₃	—	4	4	no	—
51	C₂²×C₄×C₁₃	—	4	4	no	—
52	C₂²×C₄×C₁₃	—	4	4	no	—
53	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	D₈	1	2	yes	—
56	C₂²×C₄×C₁₃	SD₁₆	1	4	yes	—
57	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	8	no	—
71	C₂×C₈×C₁₃	—	2	8	no	—
72	C₂×C₈×C₁₃	C₄²	1	4	yes	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
74	C₁₃×C₄⋊C₄	—	2	8	no	—
75	C₁₃×C₄⋊C₄	—	2	8	no	—
76	C₁₃×C₄⋊C₄	—	4	4	no	—
77	C₁₃×C₄⋊C₄	—	4	4	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	2	4	no	—
81	C₂²×C₄×C₈	C₂×C₁₃	1	1	yes	characteristic
82	C₂×#340	C₂×C₁₃	1	2	yes	—
83	C₂×#340	C₂×C₁₃	1	2	yes	—
84	C₄⋊#95	C₂×C₁₃	1	2	yes	—
85	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂³×C₁₃	C₄×D₄	1	2	yes	—
87	C₂³×C₁₃	#121	1	2	yes	—
88	C₂³×C₁₃	#124	1	2	yes	—
89	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
90	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
91	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
92	C₂³×C₁₃	#135	1	2	yes	—
93	C₂³×C₁₃	#135	1	2	yes	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	8	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	4	8	no	—
102	C₂×C₄×C₁₃	—	4	8	no	—
103	C₂×C₄×C₁₃	—	4	8	no	—
104	C₂×C₄×C₁₃	—	8	4	no	—
105	C₂×C₄×C₁₃	—	8	4	no	—
106	C₂×C₄×C₁₃	—	8	4	no	—
107	C₂×C₄×C₁₃	—	8	4	no	—
108	C₂×C₄×C₁₃	#106	1	4	yes	—
109	C₂×C₄×C₁₃	#107	1	4	yes	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#117	1	4	yes	—
115	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
118	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₃	#123	1	4	yes	—
121	C₂×C₄×C₁₃	#125	1	2	yes	—
122	C₂×C₄×C₁₃	#126	1	4	yes	—
123	C₈×C₁₃	—	2	4	no	—
124	C₈×C₁₃	—	2	4	no	—
125	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
126	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
128	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×C₄×C₈	—	2	4	no	—
130	C₂×C₄×C₈	C₄×C₁₃	1	4	yes	complemented
131	C₂×#95	—	2	2	no	—
132	C₂×#95	—	2	2	no	—
133	#340	—	2	4	no	—
134	#340	C₄×C₁₃	1	8	yes	complemented
135	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₁₃	#380	1	1	yes	characteristic
137	C₂²×C₁₃	#382	1	1	yes	characteristic
138	C₂²×C₁₃	#388	1	4	yes	—
139	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
141	C₂²×C₁₃	#427	1	1	yes	characteristic
142	C₂²×C₁₃	#427	1	1	yes	characteristic
143	C₂²×C₁₃	C₄×D₈	1	2	yes	—
144	C₂²×C₁₃	C₄×SD₁₆	1	4	yes	—
145	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
146	C₂²×C₁₃	#465	1	4	yes	—
147	C₂²×C₁₃	#466	1	2	yes	—
148	C₂²×C₁₃	#467	1	2	yes	—
149	C₂²×C₁₃	#486	1	2	yes	—
150	C₂²×C₁₃	#487	1	2	yes	—
151	C₂²×C₁₃	#488	1	4	yes	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	4	no	—
154	C₄×C₁₃	—	8	8	no	—
155	C₄×C₁₃	#376	1	1	yes	characteristic
156	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₁₃	#381	1	1	yes	characteristic
158	C₄×C₁₃	#383	1	1	yes	characteristic
159	C₄×C₁₃	#387	1	4	yes	—
160	C₂³×C₄	—	2	4	no	—
161	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
162	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
163	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
169	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
170	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
171	C₂²×C₈	D₄×C₁₃	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	8	yes	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
181	C₂×C₄²	Q₈×C₁₃	1	2	yes	—
182	C₄×C₈	—	2	8	no	—
183	#95	—	4	4	no	—
184	#95	—	4	4	no	—
185	C₂×C₁₃	#1434	1	1	yes	characteristic
186	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
187	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
188	C₂×C₁₃	#1807	1	1	yes	characteristic
189	C₂×C₁₃	#1808	1	1	yes	characteristic
190	C₂×C₁₃	#1834	1	1	yes	characteristic
191	C₂×C₁₃	#1835	1	1	yes	characteristic
192	C₂×C₁₃	#1933	1	4	yes	—
193	C₂×C₁₃	#1934	1	4	yes	—
194	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
207	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
208	C₂²×C₄	C₁₃×D₈	1	2	yes	—
209	C₂²×C₄	C₁₃×SD₁₆	1	4	yes	—
210	C₂²×C₄	C₁₃×Q₁₆	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
214	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₁₃×#41	1	2	yes	—
216	C₂²×C₄	C₁₃×#41	1	2	yes	—
217	C₂²×C₄	C₁₃×#41	1	2	yes	—
218	C₂²×C₄	C₁₃×#41	1	2	yes	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	C₄²×C₁₃	1	4	yes	—
226	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	4	no	—
234	C₁₃	#10820	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
236	C₂³	C₄×D₄×C₁₃	1	2	yes	—
237	C₂³	C₁₃×#121	1	2	yes	—
238	C₂³	C₁₃×#124	1	2	yes	—
239	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
240	C₂³	C₂×C₁₃×SD₁₆	1	2	yes	—
241	C₂³	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
242	C₂³	C₁₃×#135	1	2	yes	—
243	C₂³	C₁₃×#135	1	2	yes	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	C₁₃×#106	1	4	yes	—
259	C₂×C₄	C₁₃×#107	1	4	yes	—
260	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
263	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
264	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
265	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
266	C₂×C₄	C₁₃×#117	1	4	yes	—
267	C₂×C₄	C₄×D₄×C₁₃	1	4	yes	—
268	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
269	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
270	C₂×C₄	C₁₃×#123	1	4	yes	—
271	C₂×C₄	C₁₃×#125	1	2	yes	—
272	C₂×C₄	C₁₃×#126	1	4	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	2	4	no	—
275	C₂²	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
276	C₂²	C₁₃×#380	1	1	yes	characteristic
277	C₂²	C₁₃×#382	1	1	yes	characteristic
278	C₂²	C₁₃×#388	1	4	yes	—
279	C₂²	C₂×C₁₃×#106	1	1	yes	characteristic
280	C₂²	C₂×C₁₃×#107	1	1	yes	characteristic
281	C₂²	C₁₃×#427	1	1	yes	characteristic
282	C₂²	C₁₃×#427	1	1	yes	characteristic
283	C₂²	C₄×C₁₃×D₈	1	2	yes	—
284	C₂²	C₄×C₁₃×SD₁₆	1	4	yes	—
285	C₂²	C₄×C₁₃×Q₁₆	1	2	yes	—
286	C₂²	C₁₃×#465	1	4	yes	—
287	C₂²	C₁₃×#466	1	2	yes	—
288	C₂²	C₁₃×#467	1	2	yes	—
289	C₂²	C₁₃×#486	1	2	yes	—
290	C₂²	C₁₃×#487	1	2	yes	—
291	C₂²	C₁₃×#488	1	4	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	2	4	no	—
294	C₄	—	8	8	no	—
295	C₄	C₁₃×#376	1	1	yes	characteristic
296	C₄	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
297	C₄	C₁₃×#381	1	1	yes	characteristic
298	C₄	C₁₃×#383	1	1	yes	characteristic
299	C₄	C₁₃×#387	1	4	yes	—
300	C₂	C₁₃×#1434	1	1	yes	characteristic
301	C₂	C₄×C₁₃×#106	1	1	yes	characteristic
302	C₂	C₄×C₁₃×#107	1	1	yes	characteristic
303	C₂	C₁₃×#1807	1	1	yes	characteristic
304	C₂	C₁₃×#1808	1	1	yes	characteristic
305	C₂	C₁₃×#1834	1	1	yes	characteristic
306	C₂	C₁₃×#1835	1	1	yes	characteristic
307	C₂	C₁₃×#1933	1	4	yes	—
308	C₂	C₁₃×#1934	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×C₈×C₁₃	C₄	semidirect
14. C₁₃×#340	C₄	semidirect
40. #10820	C₁₃	direct (N × H)
130. C₂×C₄×C₈	C₄×C₁₃	semidirect
134. #340	C₄×C₁₃	semidirect
234. C₁₃	#10820	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₁₃	2
C₂×C₁₃×#340	2
C₁₃×C₄⋊#95	2
#10820	13

Subgroups — C13×#10820 #15697159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10820 #15697159