Subgroups — C₁₃×#20481 #15706820

Order 3328.

962 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1771	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1752	C₂	1	2	yes	maximal
6	C₈×C₁₃×C₂²⋊C₄	C₂	1	2	yes	maximal
7	C₁₃×#1746	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1726	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#431	C₄	1	8	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#97	C₄	1	8	yes	complemented
16	C₂×C₁₃×#97	C₂²	1	4	yes	—
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
22	C₁₃×#383	C₄	1	2	yes	—
23	C₁₃×#382	C₄	1	2	yes	—
24	C₁₃×#381	C₄	1	2	yes	—
25	C₁₃×#380	C₄	1	2	yes	—
26	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₁₃×#376	C₂²	1	2	yes	—
29	C₁₃×#336	C₂²	1	4	yes	—
30	C₁₃×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	—	2	8	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
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	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
50	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
51	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
52	C₁₃×#97	—	2	8	no	—
53	C₁₃×#97	C₂×C₄	1	4	yes	—
54	C₁₃×#97	C₂×C₄	1	8	yes	—
55	C₁₃×#97	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₁₃	—	2	8	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
58	C₁₃×#95	C₂×C₄	1	4	yes	—
59	C₁₃×#95	C₂×C₄	1	4	yes	—
60	C₁₃×#95	C₂×C₄	1	4	yes	—
61	C₁₃×#95	C₂×C₄	1	4	yes	—
62	#20481	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
68	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	16	no	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₃	#41	1	8	yes	—
87	C₁₃×C₄⋊C₄	—	4	4	no	—
88	C₁₃×C₂²⋊C₄	—	2	8	no	—
89	C₄²×C₁₃	—	2	4	no	—
90	C₄²×C₁₃	—	2	4	no	—
91	C₄²×C₁₃	C₄²	1	4	yes	—
92	C₄²×C₁₃	C₄²	1	4	yes	—
93	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
98	C₂²×#97	C₂×C₁₃	1	1	yes	characteristic
99	#2283	C₂×C₁₃	1	1	yes	characteristic
100	#1771	C₂×C₁₃	1	4	yes	complemented
101	#1752	C₂×C₁₃	1	2	yes	—
102	C₈×C₂²⋊C₄	C₂×C₁₃	1	2	yes	—
103	#1746	C₂×C₁₃	1	2	yes	complemented
104	#1726	C₂×C₁₃	1	2	yes	complemented
105	C₂³×C₁₃	—	2	2	no	—
106	C₂³×C₁₃	—	2	2	no	—
107	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
108	C₂³×C₁₃	#131	1	4	yes	—
109	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	8	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	8	no	—
119	C₂×C₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	—	2	8	no	—
121	C₂×C₄×C₁₃	—	4	4	no	—
122	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	—	2	4	no	—
132	C₈×C₁₃	—	2	4	no	—
133	C₈×C₁₃	#117	1	8	yes	—
134	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
135	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
136	#431	C₄×C₁₃	1	8	yes	complemented
137	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
138	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×#97	C₄×C₁₃	1	8	yes	complemented
141	C₂×#97	C₂²×C₁₃	1	4	yes	—
142	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×#97	C₂²×C₁₃	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
146	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
147	#383	C₄×C₁₃	1	2	yes	—
148	#382	C₄×C₁₃	1	2	yes	—
149	#381	C₄×C₁₃	1	2	yes	—
150	#380	C₄×C₁₃	1	2	yes	—
151	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
152	C₄×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
153	#376	C₂²×C₁₃	1	2	yes	—
154	#336	C₂²×C₁₃	1	4	yes	—
155	#336	C₂²×C₁₃	1	4	yes	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	—	2	8	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	#405	1	4	yes	—
160	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
161	C₂²×C₁₃	C₄×#41	1	2	yes	—
162	C₂²×C₁₃	C₂×#131	1	2	yes	—
163	C₂²×C₁₃	#568	1	2	yes	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	4	8	no	—
170	C₄×C₁₃	C₂×#117	1	2	yes	—
171	C₄×C₁₃	C₄×#41	1	2	yes	—
172	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
173	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
175	C₂²×C₈	—	2	8	no	—
176	C₂²×C₈	C₂×C₄×C₁₃	1	8	yes	—
177	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
178	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
179	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
180	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
182	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
183	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
185	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
186	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
187	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
188	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
190	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
191	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
192	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
193	#97	—	2	8	no	—
194	#97	C₂×C₄×C₁₃	1	4	yes	—
195	#97	C₂×C₄×C₁₃	1	8	yes	—
196	#97	C₂×C₄×C₁₃	1	4	yes	—
197	C₄×C₈	—	2	8	no	—
198	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
199	#95	C₂×C₄×C₁₃	1	4	yes	—
200	#95	C₂×C₄×C₁₃	1	4	yes	—
201	#95	C₂×C₄×C₁₃	1	4	yes	—
202	#95	C₂×C₄×C₁₃	1	4	yes	—
203	C₂×C₁₃	—	2	4	no	—
204	C₂×C₁₃	C₄×#117	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
206	C₂×C₁₃	#2873	1	1	yes	characteristic
207	C₂×C₁₃	#2968	1	4	yes	—
208	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
213	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
215	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
222	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	16	no	—
227	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
228	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
229	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
230	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
231	C₂×C₈	C₁₃×#41	1	8	yes	—
232	C₄⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	2	8	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	C₄²×C₁₃	1	4	yes	—
237	C₄²	C₄²×C₁₃	1	4	yes	—
238	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
239	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
240	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
241	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
242	C₁₃	#20481	1	1	yes	characteristic, direct product factor, Sylow
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
246	C₂³	C₁₃×#131	1	4	yes	—
247	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
261	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
262	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
263	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
264	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
266	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₁₃×#41	1	4	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	2	4	no	—
270	C₈	—	2	4	no	—
271	C₈	C₁₃×#117	1	8	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	8	no	—
275	C₂²	C₁₃×#405	1	4	yes	—
276	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
277	C₂²	C₄×C₁₃×#41	1	2	yes	—
278	C₂²	C₂×C₁₃×#131	1	2	yes	—
279	C₂²	C₁₃×#568	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	8	no	—
286	C₄	C₂×C₁₃×#117	1	2	yes	—
287	C₄	C₄×C₁₃×#41	1	2	yes	—
288	C₂	—	2	4	no	—
289	C₂	C₄×C₁₃×#117	1	1	yes	characteristic
290	C₂	C₂×C₁₃×#405	1	1	yes	characteristic
291	C₂	C₁₃×#2873	1	1	yes	characteristic
292	C₂	C₁₃×#2968	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×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
4. C₁₃×#1771	C₂	semidirect
7. C₁₃×#1746	C₂	semidirect
8. C₁₃×#1726	C₂	semidirect
11. C₁₃×#431	C₄	semidirect
15. C₂×C₁₃×#97	C₄	semidirect
62. #20481	C₁₃	direct (N × H)
100. #1771	C₂×C₁₃	semidirect
103. #1746	C₂×C₁₃	semidirect
104. #1726	C₂×C₁₃	semidirect
136. #431	C₄×C₁₃	semidirect
140. C₂×#97	C₄×C₁₃	semidirect
242. C₁₃	#20481	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×#431	2
C₂²×C₁₃×#97	2
C₁₃×#2283	2
C₁₃×#1771	2
C₁₃×#1752	2
C₈×C₁₃×C₂²⋊C₄	2
C₁₃×#1746	2
C₁₃×#1726	2
#20481	13

Subgroups — C13×#20481 #15706820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20481 #15706820