Subgroups — C₁₃×#9269 #15695608

Order 3328.

480 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	C₁₃×#3227	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2935	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1997	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1547	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#131	—	2	1	no	—
9	C₁₃×#532	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₁₃×#498	—	2	1	no	—
12	C₁₃×#486	—	2	1	no	—
13	C₁₃×#481	C₄	1	4	yes	—
14	C₁₃×#476	—	2	2	no	—
15	C₁₃×#436	—	2	1	no	—
16	C₁₃×#435	—	2	2	no	—
17	C₈×D₄×C₁₃	—	2	1	no	—
18	C₁₃×#431	C₂²	1	1	yes	characteristic
19	C₁₃×#423	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₁₃×#408	—	2	1	no	—
23	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
24	C₁₃×#343	—	2	1	no	—
25	C₁₃×#343	—	2	1	no	—
26	C₁₃×#332	—	2	1	no	—
27	C₁₃×#332	—	2	1	no	—
28	C₁₃×#332	—	2	1	no	—
29	C₁₃×#331	—	2	1	no	—
30	C₂×C₁₃×#41	—	2	1	no	—
31	C₂²×Q₈×C₁₃	—	2	1	no	—
32	C₁₃×#131	—	4	2	no	—
33	C₁₃×#131	—	4	2	no	—
34	C₂×C₁₃×M₁₆	—	2	2	no	—
35	C₂×C₁₃×M₁₆	—	2	1	no	—
36	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
37	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	—	2	2	no	—
39	C₂²×C₈×C₁₃	—	2	1	no	—
40	C₂²×C₈×C₁₃	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₁₃×#127	D₄	1	1	yes	characteristic
47	C₁₃×#124	—	2	1	no	—
48	C₁₃×#121	—	2	1	no	—
49	C₁₃×#121	C₂×C₄	1	2	yes	—
50	C₄×D₄×C₁₃	—	2	1	no	—
51	C₄×D₄×C₁₃	D₄	1	2	yes	—
52	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×#106	—	2	4	no	—
56	C₁₃×C₄⋊C₈	—	2	1	no	—
57	C₁₃×C₄⋊C₈	—	2	1	no	—
58	C₁₃×C₄⋊C₈	—	2	1	no	—
59	C₁₃×C₄⋊C₈	—	2	1	no	—
60	C₁₃×C₄⋊C₈	—	2	1	no	—
61	C₁₃×C₄⋊C₈	—	2	1	no	—
62	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₁₃×#102	—	2	4	no	—
65	C₁₃×C₂²⋊C₈	—	2	2	no	—
66	C₁₃×C₂²⋊C₈	—	2	2	no	—
67	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
68	C₁₃×#97	—	2	1	no	—
69	C₄×C₈×C₁₃	—	2	1	no	—
70	C₄×C₈×C₁₃	—	2	1	no	—
71	#9269	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃×#41	—	4	1	no	—
73	C₁₃×#41	—	4	1	no	—
74	C₁₃×#41	—	4	1	no	—
75	C₁₃×#41	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	2	1	no	—
77	C₂×Q₈×C₁₃	—	4	1	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×Q₈×C₁₃	—	8	1	no	—
80	C₂×Q₈×C₁₃	—	8	1	no	—
81	C₂×D₄×C₁₃	—	2	2	no	—
82	C₂×D₄×C₁₃	—	2	1	no	—
83	C₂×D₄×C₁₃	—	2	1	no	—
84	C₂×D₄×C₁₃	C₂²⋊C₄	1	2	yes	—
85	C₂²×C₄×C₁₃	—	2	2	no	—
86	C₂²×C₄×C₁₃	—	2	2	no	—
87	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
90	C₁₃×M₁₆	—	2	1	no	—
91	C₁₃×M₁₆	—	2	1	no	—
92	C₁₃×M₁₆	—	2	1	no	—
93	C₁₃×M₁₆	—	4	4	no	—
94	C₁₃×M₁₆	—	4	1	no	—
95	C₁₃×M₁₆	—	4	1	no	—
96	C₁₃×M₁₆	SD₁₆	1	2	yes	—
97	C₂×C₈×C₁₃	—	2	1	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	1	no	—
100	C₂×C₈×C₁₃	—	2	1	no	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	—	4	4	no	—
103	C₂×C₈×C₁₃	—	4	1	no	—
104	C₂×C₈×C₁₃	—	4	1	no	—
105	C₂×C₈×C₁₃	—	4	1	no	—
106	C₂×C₈×C₁₃	—	4	1	no	—
107	C₂×C₈×C₁₃	—	8	1	no	—
108	C₁₃×C₄⋊C₄	—	2	1	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
114	C₁₃×C₂²⋊C₄	—	2	2	no	—
115	C₁₃×C₂²⋊C₄	—	2	2	no	—
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₃	#41	1	1	yes	characteristic
120	#3227	C₂×C₁₃	1	1	yes	characteristic
121	#2935	C₂×C₁₃	1	1	yes	characteristic
122	#1997	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#1547	C₂×C₁₃	1	1	yes	characteristic
124	C₂×#332	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#1301	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#1283	C₂×C₁₃	1	1	yes	characteristic, complemented
127	C₂³×C₁₃	—	2	2	no	—
128	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
129	Q₈×C₁₃	—	4	1	no	—
130	Q₈×C₁₃	—	4	1	no	—
131	Q₈×C₁₃	—	8	2	no	—
132	Q₈×C₁₃	—	8	1	no	—
133	Q₈×C₁₃	—	8	1	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	1	no	—
138	D₄×C₁₃	—	4	1	no	—
139	D₄×C₁₃	—	4	1	no	—
140	D₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	1	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	8	2	no	—
149	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
152	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
157	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	—	4	2	no	—
161	C₈×C₁₃	—	4	1	no	—
162	C₈×C₁₃	—	4	1	no	—
163	C₈×C₁₃	—	4	1	no	—
164	C₈×C₁₃	—	8	1	no	—
165	C₈×C₁₃	—	8	1	no	—
166	C₂×#131	—	2	1	no	—
167	#532	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
169	#498	—	2	1	no	—
170	#486	—	2	1	no	—
171	#481	C₄×C₁₃	1	4	yes	—
172	#476	—	2	2	no	—
173	#436	—	2	1	no	—
174	#435	—	2	2	no	—
175	C₈×D₄	—	2	1	no	—
176	#431	C₂²×C₁₃	1	1	yes	characteristic
177	#423	C₂²×C₁₃	1	1	yes	characteristic
178	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
180	#408	—	2	1	no	—
181	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
182	#343	—	2	1	no	—
183	#343	—	2	1	no	—
184	#332	—	2	1	no	—
185	#332	—	2	1	no	—
186	#332	—	2	1	no	—
187	#331	—	2	1	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	—	2	1	no	—
190	C₂²×C₁₃	—	4	2	no	—
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	—	4	2	no	—
193	C₂²×C₁₃	#386	1	1	yes	characteristic
194	C₂²×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
195	C₂²×C₁₃	#440	1	1	yes	characteristic
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	—	4	2	no	—
200	C₄×C₁₃	—	8	2	no	—
201	C₄×C₁₃	#450	1	1	yes	characteristic
202	C₄×C₁₃	#452	1	1	yes	characteristic
203	C₄×C₁₃	#461	1	1	yes	characteristic
204	C₄×C₁₃	#463	1	1	yes	characteristic
205	C₄×C₁₃	#471	1	1	yes	characteristic
206	C₄×C₁₃	#472	1	1	yes	characteristic
207	C₂×#41	—	2	1	no	—
208	C₂²×Q₈	—	2	1	no	—
209	#131	—	4	2	no	—
210	#131	—	4	2	no	—
211	C₂×M₁₆	—	2	2	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
214	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	D₄×C₁₃	1	1	yes	characteristic
223	#127	D₄×C₁₃	1	1	yes	characteristic
224	#124	—	2	1	no	—
225	#121	—	2	1	no	—
226	#121	C₂×C₄×C₁₃	1	2	yes	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	D₄×C₁₃	1	2	yes	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
231	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
232	#106	—	2	4	no	—
233	C₄⋊C₈	—	2	1	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
240	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
241	#102	—	2	4	no	—
242	C₂²⋊C₈	—	2	2	no	—
243	C₂²⋊C₈	—	2	2	no	—
244	C₂²⋊C₈	C₂×C₄×C₁₃	1	2	yes	—
245	#97	—	2	1	no	—
246	C₄×C₈	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	C₂×C₁₃	—	2	1	no	—
249	C₂×C₁₃	—	2	1	no	—
250	C₂×C₁₃	—	8	2	no	—
251	C₂×C₁₃	#1877	1	1	yes	characteristic
252	C₂×C₁₃	#1907	1	1	yes	characteristic
253	C₂×C₁₃	#1915	1	1	yes	characteristic
254	#41	—	4	1	no	—
255	#41	—	4	1	no	—
256	#41	—	4	1	no	—
257	#41	—	4	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	—	8	1	no	—
262	C₂×Q₈	—	8	1	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	C₁₃×C₂²⋊C₄	1	2	yes	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
272	M₁₆	—	2	1	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	4	4	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	C₁₃×SD₁₆	1	2	yes	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	4	4	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	8	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
301	C₄²	C₁₃×#41	1	1	yes	characteristic
302	C₁₃	#9269	1	1	yes	characteristic, direct product factor, Sylow
303	C₂³	—	2	2	no	—
304	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
305	Q₈	—	4	1	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	8	2	no	—
308	Q₈	—	8	1	no	—
309	Q₈	—	8	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	1	no	—
314	D₄	—	4	1	no	—
315	D₄	—	4	1	no	—
316	D₄	—	4	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
326	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
327	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
328	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
329	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
330	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
331	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
332	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
333	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	1	no	—
340	C₈	—	8	1	no	—
341	C₈	—	8	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	2	no	—
347	C₂²	C₁₃×#386	1	1	yes	characteristic
348	C₂²	C₄×C₁₃×SD₁₆	1	1	yes	characteristic
349	C₂²	C₁₃×#440	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	8	2	no	—
355	C₄	C₁₃×#450	1	1	yes	characteristic
356	C₄	C₁₃×#452	1	1	yes	characteristic
357	C₄	C₁₃×#461	1	1	yes	characteristic
358	C₄	C₁₃×#463	1	1	yes	characteristic
359	C₄	C₁₃×#471	1	1	yes	characteristic
360	C₄	C₁₃×#472	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	8	2	no	—
364	C₂	C₁₃×#1877	1	1	yes	characteristic
365	C₂	C₁₃×#1907	1	1	yes	characteristic
366	C₂	C₁₃×#1915	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₁₃×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ 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
3. C₁₃×#1997	C₂	semidirect
5. C₂×C₁₃×#332	C₂	semidirect
6. C₁₃×#1301	C₂	semidirect
7. C₁₃×#1283	C₂	semidirect
71. #9269	C₁₃	direct (N × H)
122. #1997	C₂×C₁₃	semidirect
124. C₂×#332	C₂×C₁₃	semidirect
125. #1301	C₂×C₁₃	semidirect
126. #1283	C₂×C₁₃	semidirect
302. C₁₃	#9269	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#3227	2
C₁₃×#2935	2
C₁₃×#1997	2
C₁₃×#1547	2
C₂×C₁₃×#332	2
C₁₃×#1301	2
C₁₃×#1283	2
#9269	13

Subgroups — C13×#9269 #15695608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9269 #15695608