Subgroups — C₁₃×#9815 #15696154

Order 3328.

548 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#466	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1810	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1507	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1471	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×D₈	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#121	C₂²	1	2	yes	—
9	C₁₃×#466	—	2	1	no	—
10	C₁₃×#466	—	2	1	no	—
11	C₁₃×#466	—	2	1	no	—
12	C₁₃×#466	—	2	1	no	—
13	C₂×C₁₃×#108	—	2	1	no	—
14	C₂×C₁₃×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₃×#102	C₄	1	2	yes	complemented
16	C₁₃×#407	C₂²	1	2	yes	—
17	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	—	2	1	no	—
19	C₄×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₁₃×#341	—	2	1	no	—
21	C₁₃×#336	—	2	1	no	—
22	C₁₃×#327	—	2	4	no	—
23	C₁₃×#323	—	2	4	no	—
24	C₂²×D₄×C₁₃	—	2	2	no	—
25	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	1	no	—
29	C₂×C₁₃×D₈	—	4	2	no	—
30	C₂×C₁₃×M₁₆	—	2	2	no	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	—	4	2	no	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₁₃×#121	—	2	2	no	—
40	C₁₃×#121	—	2	2	no	—
41	C₁₃×#121	—	2	2	no	—
42	C₁₃×#121	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
45	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₄²×C₁₃	—	2	1	no	—
47	C₁₃×#108	—	4	2	no	—
48	C₁₃×#107	—	2	1	no	—
49	C₁₃×#107	—	2	1	no	—
50	C₁₃×#102	—	2	2	no	—
51	C₁₃×#102	—	2	2	no	—
52	C₁₃×C₂²⋊C₈	—	2	1	no	—
53	C₁₃×C₂²⋊C₈	—	2	2	no	—
54	C₁₃×C₂²⋊C₈	—	2	2	no	—
55	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
56	C₄×C₈×C₁₃	—	4	2	no	—
57	C₁₃×#95	D₄	1	2	yes	—
58	#9815	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂⁴×C₁₃	—	2	2	no	—
60	C₂⁴×C₁₃	#41	1	1	yes	characteristic
61	C₂×D₄×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	4	2	no	—
65	C₂×D₄×C₁₃	—	4	2	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	—	8	4	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	4	1	no	—
71	C₂²×C₄×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
78	C₁₃×D₈	—	4	2	no	—
79	C₁₃×D₈	—	4	2	no	—
80	C₁₃×M₁₆	—	4	2	no	—
81	C₁₃×M₁₆	—	4	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	—	2	2	no	—
86	C₂×C₈×C₁₃	—	4	1	no	—
87	C₂×C₈×C₁₃	—	4	1	no	—
88	C₂×C₈×C₁₃	—	4	1	no	—
89	C₂×C₈×C₁₃	—	4	1	no	—
90	C₂×C₈×C₁₃	—	4	1	no	—
91	C₂×C₈×C₁₃	—	4	2	no	—
92	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₂²⋊C₄	—	2	4	no	—
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₁₃×C₂²⋊C₄	—	4	2	no	—
101	C₄²×C₁₃	—	4	1	no	—
102	C₄²×C₁₃	—	4	1	no	—
103	C₂×#466	C₂×C₁₃	1	1	yes	characteristic
104	#1810	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₈×C₂²⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#1507	C₂×C₁₃	1	2	yes	complemented
107	#1471	C₂×C₁₃	1	2	yes	complemented
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	—	4	1	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	—	4	4	no	—
113	C₂³×C₁₃	—	8	2	no	—
114	C₂³×C₁₃	—	8	2	no	—
115	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂³×C₁₃	#121	1	1	yes	characteristic
117	C₂³×C₁₃	#124	1	1	yes	characteristic
118	D₄×C₁₃	—	4	2	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	8	2	no	—
121	D₄×C₁₃	—	8	4	no	—
122	D₄×C₁₃	—	8	4	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	8	2	no	—
137	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	—	4	2	no	—
145	C₈×C₁₃	—	8	2	no	—
146	C₈×C₁₃	#99	1	2	yes	—
147	C₂²×D₈	C₄×C₁₃	1	1	yes	characteristic, complemented
148	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×#121	C₂²×C₁₃	1	2	yes	—
150	#466	—	2	1	no	—
151	#466	—	2	1	no	—
152	#466	—	2	1	no	—
153	#466	—	2	1	no	—
154	C₂×#108	—	2	1	no	—
155	C₂×#107	C₄×C₁₃	1	1	yes	characteristic, complemented
156	C₂×#102	C₄×C₁₃	1	2	yes	complemented
157	#407	C₂²×C₁₃	1	2	yes	—
158	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×C₄×C₈	—	2	1	no	—
160	C₄×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
161	#341	—	2	1	no	—
162	#336	—	2	1	no	—
163	#327	—	2	4	no	—
164	#323	—	2	4	no	—
165	C₂²×C₁₃	—	2	1	no	—
166	C₂²×C₁₃	—	2	1	no	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	—	4	4	no	—
173	C₂²×C₁₃	—	8	2	no	—
174	C₂²×C₁₃	—	8	2	no	—
175	C₂²×C₁₃	—	8	2	no	—
176	C₂²×C₁₃	—	16	4	no	—
177	C₂²×C₁₃	#390	1	1	yes	characteristic
178	C₂²×C₁₃	#443	1	2	yes	—
179	C₂²×C₁₃	#472	1	2	yes	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	4	4	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	C₂×#99	1	1	yes	characteristic
186	C₄×C₁₃	#410	1	1	yes	characteristic
187	C₂²×D₄	—	2	2	no	—
188	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
189	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	4	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂²×C₈	—	2	1	no	—
195	C₂²×C₈	—	4	2	no	—
196	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
198	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
202	#121	—	2	2	no	—
203	#121	—	2	2	no	—
204	#121	—	2	2	no	—
205	#121	—	2	2	no	—
206	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
207	C₂×C₂²⋊C₄	—	2	1	no	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₄²	—	2	1	no	—
210	#108	—	4	2	no	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#102	—	2	2	no	—
214	#102	—	2	2	no	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	—	2	2	no	—
217	C₂²⋊C₈	—	2	2	no	—
218	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
219	C₄×C₈	—	4	2	no	—
220	#95	D₄×C₁₃	1	2	yes	—
221	C₂×C₁₃	—	2	2	no	—
222	C₂×C₁₃	—	2	2	no	—
223	C₂×C₁₃	—	4	1	no	—
224	C₂×C₁₃	—	16	2	no	—
225	C₂×C₁₃	#1964	1	1	yes	characteristic
226	C₂×C₁₃	#1974	1	1	yes	characteristic
227	C₂×C₁₃	#1978	1	1	yes	characteristic
228	C₂⁴	—	2	2	no	—
229	C₂⁴	C₁₃×#41	1	1	yes	characteristic
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	8	4	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
247	D₈	—	4	2	no	—
248	D₈	—	4	2	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₁₃	#9815	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	4	1	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	8	2	no	—
279	C₂³	—	8	2	no	—
280	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
281	C₂³	C₁₃×#121	1	1	yes	characteristic
282	C₂³	C₁₃×#124	1	1	yes	characteristic
283	D₄	—	4	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	8	2	no	—
286	D₄	—	8	4	no	—
287	D₄	—	8	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
304	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
305	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	—	8	2	no	—
311	C₈	C₁₃×#99	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	4	no	—
324	C₂²	C₁₃×#390	1	1	yes	characteristic
325	C₂²	C₁₃×#443	1	2	yes	—
326	C₂²	C₁₃×#472	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	4	no	—
331	C₄	—	8	2	no	—
332	C₄	C₂×C₁₃×#99	1	1	yes	characteristic
333	C₄	C₁₃×#410	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	2	2	no	—
336	C₂	—	4	1	no	—
337	C₂	—	16	2	no	—
338	C₂	C₁₃×#1964	1	1	yes	characteristic
339	C₂	C₁₃×#1974	1	1	yes	characteristic
340	C₂	C₁₃×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#390
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
prime radical	C₂×C₁₃×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1810	C₂	semidirect
3. C₈×C₁₃×C₂²⋊C₄	C₂	semidirect
4. C₁₃×#1507	C₂	semidirect
5. C₁₃×#1471	C₂	semidirect
6. C₂²×C₁₃×D₈	C₄	semidirect
14. C₂×C₁₃×#107	C₄	semidirect
15. C₂×C₁₃×#102	C₄	semidirect
58. #9815	C₁₃	direct (N × H)
104. #1810	C₂×C₁₃	semidirect
105. C₈×C₂²⋊C₄	C₂×C₁₃	semidirect
106. #1507	C₂×C₁₃	semidirect
107. #1471	C₂×C₁₃	semidirect
147. C₂²×D₈	C₄×C₁₃	semidirect
155. C₂×#107	C₄×C₁₃	semidirect
156. C₂×#102	C₄×C₁₃	semidirect
272. C₁₃	#9815	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×#466	2
C₁₃×#1810	2
C₈×C₁₃×C₂²⋊C₄	2
C₁₃×#1507	2
C₁₃×#1471	2
#9815	13

Subgroups — C13×#9815 #15696154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9815 #15696154