Subgroups — C₁₃×#10440 #15696779

Order 3328.

492 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#459	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1800	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1478	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#459	—	2	2	no	—
12	C₁₃×#459	—	2	2	no	—
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₁₃×#423	C₂²	1	1	yes	characteristic
16	C₁₃×#423	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#102	—	2	1	no	—
19	C₂×C₁₃×#100	—	2	1	no	—
20	C₁₃×#343	—	2	1	no	—
21	C₁₃×#335	—	2	2	no	—
22	C₁₃×#331	—	2	2	no	—
23	C₁₃×#331	—	2	2	no	—
24	C₁₃×D₄⋊C₈	—	2	2	no	—
25	C₂²×D₄×C₁₃	—	2	1	no	—
26	C₂²×D₄×C₁₃	—	2	1	no	—
27	C₂²×D₄×C₁₃	—	2	1	no	—
28	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	—	2	1	no	—
30	C₂×C₁₃×D₈	—	2	4	no	—
31	C₂×C₁₃×D₈	—	4	2	no	—
32	C₂×C₁₃×D₈	—	4	2	no	—
33	C₂×C₁₃×M₁₆	—	2	1	no	—
34	C₂×C₁₃×M₁₆	—	2	1	no	—
35	C₂²×C₈×C₁₃	—	2	1	no	—
36	C₁₃×#127	—	2	1	no	—
37	C₁₃×#127	—	2	1	no	—
38	C₁₃×#127	—	2	1	no	—
39	C₁₃×#127	D₄	1	2	yes	—
40	C₄×D₄×C₁₃	—	2	1	no	—
41	C₄×D₄×C₁₃	—	2	1	no	—
42	C₄×D₄×C₁₃	—	2	1	no	—
43	C₄×D₄×C₁₃	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊C₈	—	2	1	no	—
48	C₁₃×C₄⋊C₈	—	2	1	no	—
49	C₁₃×C₄⋊C₈	—	2	1	no	—
50	C₁₃×C₄⋊C₈	—	2	1	no	—
51	C₁₃×C₄⋊C₈	—	2	1	no	—
52	C₁₃×C₄⋊C₈	—	2	1	no	—
53	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
54	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
55	C₁₃×#102	—	2	4	no	—
56	C₁₃×#100	—	4	2	no	—
57	C₁₃×C₂²⋊C₈	—	2	1	no	—
58	C₁₃×C₂²⋊C₈	—	2	1	no	—
59	C₄×C₈×C₁₃	—	2	1	no	—
60	C₄×C₈×C₁₃	—	2	1	no	—
61	#10440	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	—	2	1	no	—
63	C₂⁴×C₁₃	—	4	1	no	—
64	C₂×D₄×C₁₃	—	2	1	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	1	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	—	2	1	no	—
70	C₂×D₄×C₁₃	—	2	1	no	—
71	C₂×D₄×C₁₃	—	2	1	no	—
72	C₂×D₄×C₁₃	—	4	1	no	—
73	C₂×D₄×C₁₃	—	4	1	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂×D₄×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	4	1	no	—
78	C₂×D₄×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	8	2	no	—
80	C₂×D₄×C₁₃	—	8	2	no	—
81	C₂×D₄×C₁₃	D₈	1	2	yes	—
82	C₂²×C₄×C₁₃	—	2	1	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	—	8	2	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
89	C₁₃×D₈	—	4	8	no	—
90	C₁₃×M₁₆	—	4	1	no	—
91	C₁₃×M₁₆	—	4	1	no	—
92	C₁₃×M₁₆	—	4	1	no	—
93	C₁₃×M₁₆	—	4	1	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	4	1	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₂×C₈×C₁₃	—	4	1	no	—
99	C₂×C₈×C₁₃	—	4	1	no	—
100	C₂×C₈×C₁₃	—	4	1	no	—
101	C₁₃×C₄⋊C₄	—	2	1	no	—
102	C₁₃×C₄⋊C₄	—	2	1	no	—
103	C₁₃×C₂²⋊C₄	—	4	1	no	—
104	C₁₃×C₂²⋊C₄	—	4	1	no	—
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	C₂×#459	C₂×C₁₃	1	1	yes	characteristic
110	#1800	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#1571	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#1545	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#1545	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#1478	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#1301	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂³×C₁₃	—	2	2	no	—
117	C₂³×C₁₃	—	4	1	no	—
118	C₂³×C₁₃	—	4	1	no	—
119	C₂³×C₁₃	—	4	2	no	—
120	C₂³×C₁₃	—	8	1	no	—
121	C₂³×C₁₃	—	8	1	no	—
122	C₂³×C₁₃	—	8	2	no	—
123	C₂³×C₁₃	—	8	2	no	—
124	C₂³×C₁₃	—	8	1	no	—
125	C₂³×C₁₃	—	8	1	no	—
126	C₂³×C₁₃	#123	1	1	yes	characteristic
127	D₄×C₁₃	—	2	2	no	—
128	D₄×C₁₃	—	2	2	no	—
129	D₄×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	4	1	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	4	1	no	—
133	D₄×C₁₃	—	8	2	no	—
134	D₄×C₁₃	—	8	2	no	—
135	D₄×C₁₃	—	8	2	no	—
136	D₄×C₁₃	—	8	1	no	—
137	D₄×C₁₃	—	8	1	no	—
138	D₄×C₁₃	—	8	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	4	1	no	—
142	C₂×C₄×C₁₃	—	8	2	no	—
143	C₂×C₄×C₁₃	—	8	2	no	—
144	C₂×C₄×C₁₃	—	8	1	no	—
145	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
150	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
153	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
155	C₈×C₁₃	—	4	2	no	—
156	C₈×C₁₃	—	4	2	no	—
157	C₈×C₁₃	—	8	1	no	—
158	C₈×C₁₃	—	8	1	no	—
159	C₈×C₁₃	—	8	1	no	—
160	C₂²×D₈	—	2	1	no	—
161	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
163	#459	—	2	2	no	—
164	#459	—	2	2	no	—
165	#431	C₂²×C₁₃	1	1	yes	characteristic
166	#431	C₂²×C₁₃	1	1	yes	characteristic
167	#423	C₂²×C₁₃	1	1	yes	characteristic
168	#423	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×#102	—	2	1	no	—
171	C₂×#100	—	2	1	no	—
172	#343	—	2	1	no	—
173	#335	—	2	2	no	—
174	#331	—	2	2	no	—
175	#331	—	2	2	no	—
176	D₄⋊C₈	—	2	2	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	4	2	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	—	8	1	no	—
183	C₂²×C₁₃	—	8	1	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	8	2	no	—
186	C₂²×C₁₃	—	8	2	no	—
187	C₂²×C₁₃	—	16	2	no	—
188	C₂²×C₁₃	—	16	2	no	—
189	C₂²×C₁₃	#394	1	1	yes	characteristic
190	C₂²×C₁₃	#480	1	1	yes	characteristic
191	C₂²×C₁₃	#482	1	1	yes	characteristic
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	8	1	no	—
196	C₄×C₁₃	#447	1	1	yes	characteristic
197	C₄×C₁₃	#452	1	1	yes	characteristic
198	C₄×C₁₃	#466	1	1	yes	characteristic
199	C₄×C₁₃	#468	1	1	yes	characteristic
200	C₄×C₁₃	#472	1	2	yes	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	—	2	1	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂²×D₄	D₄×C₁₃	1	1	yes	characteristic
205	C₂³×C₄	—	2	1	no	—
206	C₂×D₈	—	2	4	no	—
207	C₂×D₈	—	4	2	no	—
208	C₂×D₈	—	4	2	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	#127	—	2	1	no	—
213	#127	—	2	1	no	—
214	#127	—	2	1	no	—
215	#127	D₄×C₁₃	1	2	yes	—
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	—	2	1	no	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	—	2	1	no	—
220	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
230	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
231	#102	—	2	4	no	—
232	#100	—	4	2	no	—
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₂×C₁₃	—	2	1	no	—
238	C₂×C₁₃	—	2	1	no	—
239	C₂×C₁₃	—	8	2	no	—
240	C₂×C₁₃	—	16	2	no	—
241	C₂×C₁₃	#2033	1	1	yes	characteristic
242	C₂×C₁₃	#2045	1	1	yes	characteristic
243	C₂×C₁₃	#2048	1	1	yes	characteristic
244	C₂⁴	—	2	1	no	—
245	C₂⁴	—	4	1	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	8	2	no	—
262	C₂×D₄	—	8	2	no	—
263	C₂×D₄	C₁₃×D₈	1	2	yes	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	—	4	1	no	—
267	C₂²×C₄	—	8	2	no	—
268	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
269	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
270	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
271	D₈	—	4	8	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	4	1	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
290	C₄²	C₁₃×#41	1	1	yes	characteristic
291	C₁₃	#10440	1	1	yes	characteristic, direct product factor, Sylow
292	C₂³	—	2	2	no	—
293	C₂³	—	4	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	8	1	no	—
297	C₂³	—	8	1	no	—
298	C₂³	—	8	2	no	—
299	C₂³	—	8	2	no	—
300	C₂³	—	8	1	no	—
301	C₂³	—	8	1	no	—
302	C₂³	C₁₃×#123	1	1	yes	characteristic
303	D₄	—	2	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	1	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	1	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	2	no	—
311	D₄	—	8	2	no	—
312	D₄	—	8	1	no	—
313	D₄	—	8	1	no	—
314	D₄	—	8	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	8	1	no	—
321	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
322	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
323	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
324	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
325	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
326	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
327	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
328	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
329	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
330	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	4	2	no	—
333	C₈	—	8	1	no	—
334	C₈	—	8	1	no	—
335	C₈	—	8	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	1	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	16	2	no	—
347	C₂²	—	16	2	no	—
348	C₂²	C₁₃×#394	1	1	yes	characteristic
349	C₂²	C₁₃×#480	1	1	yes	characteristic
350	C₂²	C₁₃×#482	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	8	1	no	—
355	C₄	C₁₃×#447	1	1	yes	characteristic
356	C₄	C₁₃×#452	1	1	yes	characteristic
357	C₄	C₁₃×#466	1	1	yes	characteristic
358	C₄	C₁₃×#468	1	1	yes	characteristic
359	C₄	C₁₃×#472	1	2	yes	—
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	8	2	no	—
363	C₂	—	16	2	no	—
364	C₂	C₁₃×#2033	1	1	yes	characteristic
365	C₂	C₁₃×#2045	1	1	yes	characteristic
366	C₂	C₁₃×#2048	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂×C₁₃×#459	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1800	C₂	semidirect
3. C₁₃×#1571	C₂	semidirect
4. C₁₃×#1545	C₂	semidirect
5. C₁₃×#1545	C₂	semidirect
6. C₁₃×#1478	C₂	semidirect
7. C₁₃×#1301	C₂	semidirect
61. #10440	C₁₃	direct (N × H)
110. #1800	C₂×C₁₃	semidirect
111. #1571	C₂×C₁₃	semidirect
112. #1545	C₂×C₁₃	semidirect
113. #1545	C₂×C₁₃	semidirect
114. #1478	C₂×C₁₃	semidirect
115. #1301	C₂×C₁₃	semidirect
291. C₁₃	#10440	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×#459	2
C₁₃×#1800	2
C₁₃×#1571	2
C₁₃×#1545	2
C₁₃×#1478	2
C₁₃×#1301	2
#10440	13

Subgroups — C13×#10440 #15696779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10440 #15696779