Subgroups — C₁₃×#20820 #15707159

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2314	C₂	1	2	yes	maximal
3	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1836	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1809	C₂	1	2	yes	maximal
7	C₁₃×#1807	C₂	1	2	yes	maximal
8	C₁₃×#1776	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×#106	C₂	1	2	yes	maximal, complemented
10	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×C₁₃×#106	C₂²	1	2	yes	—
17	C₁₃×#405	C₄	1	8	yes	complemented
18	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×#388	—	2	4	no	—
23	C₁₃×#387	—	2	4	no	—
24	C₁₃×#384	C₂²	1	2	yes	—
25	C₁₃×#382	C₂²	1	2	yes	—
26	C₁₃×#380	C₂²	1	4	yes	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
28	C₁₃×#340	C₂²	1	4	yes	—
29	C₁₃×#340	C₂²	1	4	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	—	2	2	no	—
32	C₂×C₁₃×M₁₆	—	2	2	no	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₃	—	2	2	no	—
35	C₂²×C₈×C₁₃	—	2	2	no	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×#117	C₂×C₄	1	4	yes	—
42	C₁₃×#117	Q₈	1	4	yes	—
43	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
48	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	4	no	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×#107	—	2	4	no	—
53	C₁₃×#106	—	2	4	no	—
54	C₁₃×#106	—	2	4	no	—
55	C₁₃×#106	—	2	4	no	—
56	C₁₃×#97	C₂×C₄	1	4	yes	—
57	C₁₃×#97	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
60	C₁₃×#95	—	2	4	no	—
61	C₁₃×#95	—	2	4	no	—
62	C₁₃×#95	—	2	4	no	—
63	#20820	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₁₃×M₁₆	—	2	8	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	8	no	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₂×C₈×C₁₃	#41	1	2	yes	—
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂×C₈×C₁₃	#41	1	2	yes	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	—	2	4	no	—
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	2	4	no	—
95	C₁₃×C₄⋊C₄	—	8	4	no	—
96	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
97	C₁₃×C₂²⋊C₄	—	8	4	no	—
98	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
99	C₄²×C₁₃	—	8	4	no	—
100	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
103	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
104	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
105	#2314	C₂×C₁₃	1	2	yes	—
106	#1846	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#1844	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#1836	C₂×C₁₃	1	2	yes	complemented
109	#1809	C₂×C₁₃	1	2	yes	—
110	#1807	C₂×C₁₃	1	2	yes	—
111	#1776	C₂×C₁₃	1	2	yes	complemented
112	C₄×#106	C₂×C₁₃	1	2	yes	complemented
113	C₂³×C₁₃	—	2	1	no	—
114	C₂³×C₁₃	—	2	1	no	—
115	C₂³×C₁₃	—	2	4	no	—
116	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	—	4	4	no	—
131	C₂×C₄×C₁₃	—	4	4	no	—
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	—	2	2	no	—
143	C₈×C₁₃	#117	1	4	yes	—
144	C₈×C₁₃	#117	1	4	yes	—
145	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
146	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×#107	C₂²×C₁₃	1	2	yes	—
149	C₂×#106	C₂²×C₁₃	1	2	yes	—
150	C₂×#106	C₂²×C₁₃	1	2	yes	—
151	C₂×#106	C₂²×C₁₃	1	2	yes	—
152	#405	C₄×C₁₃	1	8	yes	complemented
153	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
157	#388	—	2	4	no	—
158	#387	—	2	4	no	—
159	#384	C₂²×C₁₃	1	2	yes	—
160	#382	C₂²×C₁₃	1	2	yes	—
161	#380	C₂²×C₁₃	1	4	yes	—
162	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
163	#340	C₂²×C₁₃	1	4	yes	—
164	#340	C₂²×C₁₃	1	4	yes	—
165	C₂²×C₁₃	—	2	4	no	—
166	C₂²×C₁₃	—	2	2	no	—
167	C₂²×C₁₃	—	4	2	no	—
168	C₂²×C₁₃	—	4	2	no	—
169	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₁₃	#532	1	1	yes	characteristic
171	C₂²×C₁₃	#532	1	1	yes	characteristic
172	C₂²×C₁₃	#577	1	4	yes	—
173	C₄×C₁₃	—	2	4	no	—
174	C₄×C₁₃	—	2	4	no	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	8	4	no	—
177	C₄×C₁₃	—	8	4	no	—
178	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
179	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
180	C₄×C₁₃	#533	1	1	yes	characteristic
181	C₄×C₁₃	#533	1	1	yes	characteristic
182	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
189	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
193	#117	C₂×C₄×C₁₃	1	4	yes	—
194	#117	Q₈×C₁₃	1	4	yes	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
199	C₂×C₂²⋊C₄	—	2	4	no	—
200	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
201	C₂×C₄²	—	2	4	no	—
202	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
204	#107	—	2	4	no	—
205	#106	—	2	4	no	—
206	#106	—	2	4	no	—
207	#106	—	2	4	no	—
208	#97	C₂×C₄×C₁₃	1	4	yes	—
209	#97	C₂×C₄×C₁₃	1	4	yes	—
210	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
211	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
212	#95	—	2	4	no	—
213	#95	—	2	4	no	—
214	#95	—	2	4	no	—
215	C₂×C₁₃	—	4	2	no	—
216	C₂×C₁₃	#2285	1	1	yes	characteristic
217	C₂×C₁₃	#2912	1	2	yes	—
218	C₂×C₁₃	#3173	1	2	yes	—
219	C₂×C₁₃	#3174	1	2	yes	—
220	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
231	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
233	M₁₆	—	2	8	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
243	C₂×C₈	C₁₃×#41	1	2	yes	—
244	C₂×C₈	C₁₃×#41	1	2	yes	—
245	C₂×C₈	C₁₃×#41	1	2	yes	—
246	C₂×C₈	C₁₃×#41	1	2	yes	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	8	4	no	—
252	C₄⋊C₄	C₁₃×C₄⋊C₄	1	8	yes	—
253	C₂²⋊C₄	—	8	4	no	—
254	C₂²⋊C₄	C₁₃×C₄⋊C₄	1	8	yes	—
255	C₄²	—	8	4	no	—
256	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
257	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
258	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
259	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
260	C₁₃	#20820	1	1	yes	characteristic, direct product factor, Sylow
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	4	no	—
264	C₂³	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
265	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
281	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
282	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
283	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
285	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
286	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
287	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
288	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	2	no	—
291	C₈	C₁₃×#117	1	4	yes	—
292	C₈	C₁₃×#117	1	4	yes	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
298	C₂²	C₁₃×#532	1	1	yes	characteristic
299	C₂²	C₁₃×#532	1	1	yes	characteristic
300	C₂²	C₁₃×#577	1	4	yes	—
301	C₄	—	2	2	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	4	no	—
304	C₄	—	8	4	no	—
305	C₄	—	8	4	no	—
306	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
307	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
308	C₄	C₁₃×#533	1	1	yes	characteristic
309	C₄	C₁₃×#533	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	C₁₃×#2285	1	1	yes	characteristic
312	C₂	C₁₃×#2912	1	2	yes	—
313	C₂	C₁₃×#3173	1	2	yes	—
314	C₂	C₁₃×#3174	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1846	C₂	semidirect
4. C₁₃×#1844	C₂	semidirect
5. C₁₃×#1836	C₂	semidirect
8. C₁₃×#1776	C₂	semidirect
9. C₄×C₁₃×#106	C₂	semidirect
17. C₁₃×#405	C₄	semidirect
63. #20820	C₁₃	direct (N × H)
106. #1846	C₂×C₁₃	semidirect
107. #1844	C₂×C₁₃	semidirect
108. #1836	C₂×C₁₃	semidirect
111. #1776	C₂×C₁₃	semidirect
112. C₄×#106	C₂×C₁₃	semidirect
152. #405	C₄×C₁₃	semidirect
260. C₁₃	#20820	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×#405	2
C₁₃×#2314	2
C₁₃×#1846	2
C₁₃×#1844	2
C₁₃×#1836	2
C₁₃×#1809	2
C₁₃×#1807	2
C₁₃×#1776	2
C₄×C₁₃×#106	2
#20820	13

Subgroups — C13×#20820 #15707159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20820 #15707159