Subgroups — C₁₃×#20519 #15706858

Order 3328.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#446	C₂	1	2	yes	maximal
2	C₁₃×#2295	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1771	C₂	1	2	yes	maximal
4	C₁₃×#1771	C₂	1	2	yes	maximal
5	C₈×C₁₃×C₄⋊C₄	C₂	1	2	yes	maximal
6	C₁₃×#1768	C₂	1	2	yes	maximal
7	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1726	C₂	1	2	yes	maximal
9	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#446	C₄	1	16	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×#97	C₂²	1	2	yes	—
16	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#97	C₂²	1	2	yes	—
18	C₂×C₁₃×#97	C₂²	1	2	yes	—
19	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
24	C₁₃×#391	C₄	1	2	yes	—
25	C₁₃×#389	C₄	1	2	yes	—
26	C₁₃×#387	C₄	1	4	yes	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₁₃×#376	C₂²	1	2	yes	—
30	C₄³×C₁₃	C₂²	1	1	yes	characteristic
31	C₁₃×#336	C₂²	1	1	yes	characteristic
32	C₁₃×#336	C₂²	1	1	yes	characteristic
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
53	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
55	C₁₃×#97	—	2	4	no	—
56	C₁₃×#97	—	2	2	no	—
57	C₁₃×#97	C₂×C₄	1	8	yes	—
58	C₁₃×#97	C₂×C₄	1	8	yes	—
59	C₁₃×#97	Q₈	1	4	yes	complemented
60	C₄×C₈×C₁₃	—	2	4	no	—
61	C₄×C₈×C₁₃	—	2	4	no	—
62	C₄×C₈×C₁₃	—	2	4	no	—
63	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
64	C₁₃×#95	C₂×C₄	1	2	yes	—
65	C₁₃×#95	C₂×C₄	1	4	yes	—
66	C₁₃×#95	C₂×C₄	1	4	yes	—
67	C₁₃×#95	C₂×C₄	1	4	yes	—
68	#20519	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×Q₈×C₁₃	—	4	2	no	—
70	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
90	C₂×C₈×C₁₃	#41	1	4	yes	—
91	C₁₃×C₄⋊C₄	—	2	8	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₄⋊C₄	C₄²	1	8	yes	—
94	C₁₃×C₄⋊C₄	C₄²	1	4	yes	—
95	C₄²×C₁₃	—	2	4	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	4	2	no	—
100	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
101	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
102	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
103	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
104	C₄²×C₁₃	#41	1	2	yes	—
105	C₂×#446	C₂×C₁₃	1	2	yes	—
106	#2295	C₂×C₁₃	1	1	yes	characteristic
107	#1771	C₂×C₁₃	1	2	yes	—
108	#1771	C₂×C₁₃	1	2	yes	—
109	C₈×C₄⋊C₄	C₂×C₁₃	1	2	yes	—
110	#1768	C₂×C₁₃	1	2	yes	—
111	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
112	#1726	C₂×C₁₃	1	2	yes	—
113	C₄×#97	C₂×C₁₃	1	1	yes	characteristic
114	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	Q₈×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
128	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
132	C₂×C₄×C₁₃	#131	1	4	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₈×C₁₃	—	2	4	no	—
138	C₈×C₁₃	—	2	4	no	—
139	C₈×C₁₃	—	2	4	no	—
140	C₈×C₁₃	C₄×Q₈	1	8	yes	—
141	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
142	#446	C₄×C₁₃	1	16	yes	complemented
143	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
144	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
146	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×#97	C₂²×C₁₃	1	2	yes	—
148	C₂×#97	C₂²×C₁₃	1	2	yes	—
149	C₂×#97	C₂²×C₁₃	1	2	yes	—
150	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
153	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
155	#391	C₄×C₁₃	1	2	yes	—
156	#389	C₄×C₁₃	1	2	yes	—
157	#387	C₄×C₁₃	1	4	yes	—
158	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
159	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
160	#376	C₂²×C₁₃	1	2	yes	—
161	C₄³	C₂²×C₁₃	1	1	yes	characteristic
162	#336	C₂²×C₁₃	1	1	yes	characteristic
163	#336	C₂²×C₁₃	1	1	yes	characteristic
164	#336	C₂²×C₁₃	1	2	yes	—
165	#336	C₂²×C₁₃	1	2	yes	—
166	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
169	C₂²×C₁₃	C₂×#131	1	2	yes	—
170	C₂²×C₁₃	#568	1	2	yes	—
171	C₄×C₁₃	—	2	2	no	—
172	C₄×C₁₃	—	2	4	no	—
173	C₄×C₁₃	—	2	2	no	—
174	C₄×C₁₃	—	2	4	no	—
175	C₄×C₁₃	—	2	4	no	—
176	C₄×C₁₃	—	2	4	no	—
177	C₄×C₁₃	—	4	4	no	—
178	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
179	C₄×C₁₃	C₄×#41	1	2	yes	—
180	C₂²×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
182	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
183	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
184	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
185	C₄×Q₈	C₂×C₄×C₁₃	1	8	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
190	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
191	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
192	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
194	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
196	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
197	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
198	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
199	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
200	#97	—	2	4	no	—
201	#97	—	2	2	no	—
202	#97	C₂×C₄×C₁₃	1	8	yes	—
203	#97	C₂×C₄×C₁₃	1	8	yes	—
204	#97	Q₈×C₁₃	1	4	yes	complemented
205	C₄×C₈	—	2	4	no	—
206	C₄×C₈	—	2	4	no	—
207	C₄×C₈	—	2	4	no	—
208	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
209	#95	C₂×C₄×C₁₃	1	4	yes	—
210	#95	C₂×C₄×C₁₃	1	4	yes	—
211	#95	C₂×C₄×C₁₃	1	2	yes	—
212	#95	C₂×C₄×C₁₃	1	4	yes	—
213	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
214	C₂×C₁₃	#2873	1	1	yes	characteristic
215	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
216	C₂×C₁₃	#2962	1	2	yes	—
217	C₂×C₁₃	#2968	1	2	yes	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	C₄²×C₁₃	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
227	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
235	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
236	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
237	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
238	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
239	C₂×C₈	C₁₃×#41	1	4	yes	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	C₄²×C₁₃	1	8	yes	—
243	C₄⋊C₄	C₄²×C₁₃	1	4	yes	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
250	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
251	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
252	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
253	C₄²	C₁₃×#41	1	2	yes	—
254	C₁₃	#20519	1	1	yes	characteristic, direct product factor, Sylow
255	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
256	Q₈	—	4	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
269	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
270	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
271	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
272	C₂×C₄	C₄×Q₈×C₁₃	1	4	yes	—
273	C₂×C₄	C₁₃×#131	1	4	yes	—
274	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
275	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
277	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	—	2	4	no	—
280	C₈	—	2	4	no	—
281	C₈	C₄×Q₈×C₁₃	1	8	yes	—
282	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
283	C₂²	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
284	C₂²	C₄×C₁₃×#41	1	1	yes	characteristic
285	C₂²	C₂×C₁₃×#131	1	2	yes	—
286	C₂²	C₁₃×#568	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	2	4	no	—
293	C₄	—	4	4	no	—
294	C₄	C₂×C₄×Q₈×C₁₃	1	2	yes	—
295	C₄	C₄×C₁₃×#41	1	2	yes	—
296	C₂	C₄²×Q₈×C₁₃	1	1	yes	characteristic
297	C₂	C₁₃×#2873	1	1	yes	characteristic
298	C₂	C₄×C₁₃×#131	1	1	yes	characteristic
299	C₂	C₁₃×#2962	1	2	yes	—
300	C₂	C₁₃×#2968	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₁₃×#446	C₄	semidirect
59. C₁₃×#97	Q₈	semidirect
68. #20519	C₁₃	direct (N × H)
142. #446	C₄×C₁₃	semidirect
204. #97	Q₈×C₁₃	semidirect
254. C₁₃	#20519	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×#446	2
C₁₃×#2295	2
C₁₃×#1771	2
C₈×C₁₃×C₄⋊C₄	2
C₁₃×#1768	2
C₄×C₁₃×C₄⋊C₈	2
C₁₃×#1726	2
C₄×C₁₃×#97	2
#20519	13

Subgroups — C13×#20519 #15706858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20519 #15706858