Subgroups — C₁₃×#9879 #15696218

Order 3328.

616 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#389	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1280	C₂	1	4	yes	maximal
5	C₂²×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
6	C₂²×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
7	C₂²×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
10	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
11	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	—	2	1	no	—
13	C₁₃×#389	—	2	1	no	—
14	C₁₃×#389	—	2	1	no	—
15	C₁₃×#389	—	2	1	no	—
16	C₁₃×#389	—	2	1	no	—
17	C₂×C₁₃×#95	C₂²	1	2	yes	—
18	C₁₃×#336	—	2	1	no	—
19	C₁₃×#336	—	2	1	no	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	4	4	no	—
36	C₂×C₁₃×C₄⋊C₄	C₈	1	4	yes	complemented
37	C₂×C₄²×C₁₃	—	4	2	no	—
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₁₃×C₄⋊C₈	—	4	2	no	—
43	C₁₃×C₂²⋊C₈	—	2	2	no	—
44	C₁₃×C₂²⋊C₈	—	2	4	no	—
45	C₁₃×C₂²⋊C₈	Q₈	1	4	yes	—
46	C₄×C₈×C₁₃	—	4	2	no	—
47	C₁₃×#95	—	2	2	no	—
48	C₁₃×#95	—	2	2	no	—
49	#9879	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
72	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₃	#41	1	2	yes	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	4	2	no	—
86	C₁₃×C₄⋊C₄	—	4	4	no	—
87	C₁₃×C₄⋊C₄	—	4	2	no	—
88	C₁₃×C₄⋊C₄	—	4	4	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	4	1	no	—
92	C₄²×C₁₃	—	4	1	no	—
93	C₂×#389	C₂×C₁₃	1	1	yes	characteristic
94	#1797	C₂×C₁₃	1	1	yes	characteristic
95	C₄×C₂²⋊C₈	C₂×C₁₃	1	1	yes	characteristic
96	#1280	C₂×C₁₃	1	4	yes	—
97	C₂³×C₁₃	—	2	1	no	—
98	C₂³×C₁₃	—	2	1	no	—
99	C₂³×C₁₃	—	2	1	no	—
100	C₂³×C₁₃	—	2	1	no	—
101	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
103	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
104	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂³×C₁₃	#131	1	2	yes	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	8	4	no	—
131	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
133	C₂×C₄×C₁₃	#99	1	2	yes	—
134	C₂×C₄×C₁₃	#100	1	2	yes	—
135	C₂×C₄×C₁₃	#101	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
139	C₈×C₁₃	—	4	4	no	—
140	C₈×C₁₃	—	8	2	no	—
141	C₂²×C₄⋊C₄	C₄×C₁₃	1	1	yes	characteristic
142	C₂²×C₄⋊C₄	C₄×C₁₃	1	2	yes	—
143	C₂²×C₄⋊C₄	C₄×C₁₃	1	1	yes	characteristic
144	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×C₄⋊C₈	—	2	1	no	—
146	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
147	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
148	C₂×C₄×C₈	—	2	1	no	—
149	#389	—	2	1	no	—
150	#389	—	2	1	no	—
151	#389	—	2	1	no	—
152	#389	—	2	1	no	—
153	C₂×#95	C₂²×C₁₃	1	2	yes	—
154	#336	—	2	1	no	—
155	#336	—	2	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	#391	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
166	C₂²×C₁₃	#408	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×#99	1	1	yes	characteristic
168	C₂²×C₁₃	#410	1	1	yes	characteristic
169	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
170	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
171	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
172	C₂²×C₁₃	#446	1	2	yes	—
173	C₄×C₁₃	—	2	2	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	—	4	1	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	#324	1	4	yes	—
182	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
183	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
185	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
186	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	4	2	no	—
197	C₂×C₄⋊C₄	—	4	4	no	—
198	C₂×C₄⋊C₄	C₈×C₁₃	1	4	yes	complemented
199	C₂×C₄²	—	4	2	no	—
200	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
202	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
203	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
204	C₄⋊C₈	—	4	2	no	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₂²⋊C₈	—	2	4	no	—
207	C₂²⋊C₈	Q₈×C₁₃	1	4	yes	—
208	C₄×C₈	—	4	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	C₂×C₁₃	—	2	2	no	—
212	C₂×C₁₃	—	2	2	no	—
213	C₂×C₁₃	C₂×#324	1	1	yes	characteristic
214	C₂×C₁₃	#1460	1	1	yes	characteristic
215	C₂×C₁₃	#1470	1	1	yes	characteristic
216	C₂×C₁₃	#1471	1	1	yes	characteristic
217	C₂×C₁₃	#1984	1	1	yes	characteristic
218	C₂×C₁₃	#1996	1	1	yes	characteristic
219	C₂×C₁₃	#1997	1	1	yes	characteristic
220	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
238	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
239	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
241	C₂²×C₄	C₂×C₈×C₁₃	1	4	yes	—
242	C₂²×C₄	C₁₃×M₁₆	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
246	C₂²×C₄	C₁₃×#41	1	2	yes	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	1	no	—
263	C₁₃	#9879	1	1	yes	characteristic, direct product factor, Sylow
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
269	C₂³	C₄×Q₈×C₁₃	1	2	yes	—
270	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
271	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
272	C₂³	C₁₃×#131	1	2	yes	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
299	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
300	C₂×C₄	C₁₃×#99	1	2	yes	—
301	C₂×C₄	C₁₃×#100	1	2	yes	—
302	C₂×C₄	C₁₃×#101	1	2	yes	—
303	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
304	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
305	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
306	C₈	—	4	4	no	—
307	C₈	—	8	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	C₁₃×#391	1	1	yes	characteristic
317	C₂²	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
318	C₂²	C₁₃×#408	1	1	yes	characteristic
319	C₂²	C₂×C₁₃×#99	1	1	yes	characteristic
320	C₂²	C₁₃×#410	1	1	yes	characteristic
321	C₂²	C₂×C₁₃×#100	1	1	yes	characteristic
322	C₂²	C₂×C₁₃×#101	1	1	yes	characteristic
323	C₂²	C₈×Q₈×C₁₃	1	2	yes	—
324	C₂²	C₁₃×#446	1	2	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	4	no	—
332	C₄	—	8	2	no	—
333	C₄	C₁₃×#324	1	4	yes	—
334	C₂	—	2	2	no	—
335	C₂	—	2	2	no	—
336	C₂	C₂×C₁₃×#324	1	1	yes	characteristic
337	C₂	C₁₃×#1460	1	1	yes	characteristic
338	C₂	C₁₃×#1470	1	1	yes	characteristic
339	C₂	C₁₃×#1471	1	1	yes	characteristic
340	C₂	C₁₃×#1984	1	1	yes	characteristic
341	C₂	C₁₃×#1996	1	1	yes	characteristic
342	C₂	C₁₃×#1997	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
prime radical	C₂⁴×C₁₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
36. C₂×C₁₃×C₄⋊C₄	C₈	semidirect
49. #9879	C₁₃	direct (N × H)
198. C₂×C₄⋊C₄	C₈×C₁₃	semidirect
263. C₁₃	#9879	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×#389	2
C₁₃×#1797	2
C₄×C₁₃×C₂²⋊C₈	2
C₁₃×#1280	2
#9879	13

Subgroups — C13×#9879 #15696218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9879 #15696218