Subgroups — C₁₃×#10146 #15696485

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3206	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3151	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
9	C₂²×C₁₃×SD₁₆	—	2	2	no	—
10	C₁₃×#532	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
13	C₁₃×#496	—	2	2	no	—
14	C₁₃×#493	—	2	1	no	—
15	C₁₃×#492	—	2	1	no	—
16	C₁₃×#492	—	2	2	no	—
17	C₁₃×#492	—	2	1	no	—
18	C₁₃×#492	—	2	2	no	—
19	C₁₃×#487	—	2	1	no	—
20	C₁₃×#461	—	2	2	no	—
21	C₁₃×#448	—	2	2	no	—
22	C₁₃×#431	C₂²	1	1	yes	characteristic
23	C₁₃×#423	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#343	—	2	2	no	—
27	C₁₃×#331	—	2	2	no	—
28	C₁₃×D₄⋊C₈	—	2	2	no	—
29	C₂×C₁₃×#41	—	2	1	no	—
30	C₂²×Q₈×C₁₃	D₄	1	2	yes	—
31	C₂²×D₄×C₁₃	—	2	1	no	—
32	C₂²×D₄×C₁₃	—	2	1	no	—
33	C₂²×D₄×C₁₃	—	2	1	no	—
34	C₁₃×AGL(1,ℤ/8)	—	4	4	no	—
35	C₁₃×AGL(1,ℤ/8)	—	4	4	no	—
36	C₂×C₁₃×SD₁₆	—	2	8	no	—
37	C₂×C₁₃×SD₁₆	—	2	4	no	—
38	C₂×C₁₃×SD₁₆	—	2	4	no	—
39	C₂×C₁₃×SD₁₆	—	4	8	no	—
40	C₂×C₁₃×D₈	—	2	4	no	—
41	C₂×C₁₃×M₁₆	—	2	1	no	—
42	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
43	C₂²×C₈×C₁₃	—	2	1	no	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
49	C₁₃×#127	—	2	1	no	—
50	C₁₃×#127	—	2	1	no	—
51	C₁₃×#127	—	2	1	no	—
52	C₁₃×#127	D₄	1	1	yes	characteristic
53	C₁₃×#127	D₄	1	2	yes	—
54	C₁₃×#124	—	2	1	no	—
55	C₁₃×#121	—	2	1	no	—
56	C₁₃×#121	—	2	1	no	—
57	C₄×D₄×C₁₃	—	2	1	no	—
58	C₄×D₄×C₁₃	—	2	1	no	—
59	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
61	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₁₃×C₄⋊C₈	—	2	1	no	—
63	C₁₃×C₄⋊C₈	—	2	1	no	—
64	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
65	C₁₃×#103	—	2	4	no	—
66	C₁₃×C₂²⋊C₈	—	2	1	no	—
67	C₁₃×#97	—	2	1	no	—
68	C₄×C₈×C₁₃	—	2	1	no	—
69	C₄×C₈×C₁₃	—	2	1	no	—
70	C₄×C₈×C₁₃	—	2	1	no	—
71	C₄×C₈×C₁₃	D₄	1	2	yes	—
72	#10146	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂⁴×C₁₃	—	4	1	no	—
74	C₁₃×#41	—	4	2	no	—
75	C₁₃×#41	—	4	2	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	1	no	—
78	C₂×Q₈×C₁₃	—	2	2	no	—
79	C₂×Q₈×C₁₃	—	4	4	no	—
80	C₂×D₄×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	—	2	1	no	—
82	C₂×D₄×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	2	2	no	—
84	C₂×D₄×C₁₃	—	2	1	no	—
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	4	4	no	—
87	C₂×D₄×C₁₃	—	4	1	no	—
88	C₂×D₄×C₁₃	—	4	1	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂×D₄×C₁₃	—	4	1	no	—
91	C₂×D₄×C₁₃	—	4	2	no	—
92	C₂×D₄×C₁₃	—	4	2	no	—
93	C₂×D₄×C₁₃	—	8	2	no	—
94	C₂²×C₄×C₁₃	—	2	2	no	—
95	C₂²×C₄×C₁₃	—	4	1	no	—
96	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
99	C₁₃×SD₁₆	—	4	16	no	—
100	C₁₃×SD₁₆	—	4	8	no	—
101	C₁₃×SD₁₆	—	8	4	no	—
102	C₁₃×D₈	—	4	8	no	—
103	C₁₃×M₁₆	—	2	2	no	—
104	C₁₃×M₁₆	—	4	2	no	—
105	C₁₃×M₁₆	SD₁₆	1	4	yes	complemented
106	C₂×C₈×C₁₃	—	2	4	no	—
107	C₂×C₈×C₁₃	—	2	2	no	—
108	C₂×C₈×C₁₃	—	2	2	no	—
109	C₂×C₈×C₁₃	—	2	2	no	—
110	C₂×C₈×C₁₃	—	4	1	no	—
111	C₁₃×C₄⋊C₄	—	2	1	no	—
112	C₁₃×C₄⋊C₄	—	2	1	no	—
113	C₁₃×C₄⋊C₄	—	4	1	no	—
114	C₁₃×C₄⋊C₄	—	4	1	no	—
115	C₁₃×C₄⋊C₄	—	4	1	no	—
116	C₁₃×C₄⋊C₄	—	4	1	no	—
117	C₁₃×C₂²⋊C₄	—	4	1	no	—
118	C₁₃×C₂²⋊C₄	—	4	1	no	—
119	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
123	#3206	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#3151	C₂×C₁₃	1	1	yes	characteristic
125	C₂×#492	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#1997	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#1545	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#1492	C₂×C₁₃	1	1	yes	characteristic, complemented
129	#1278	C₂×C₁₃	1	1	yes	characteristic, complemented
130	C₂³×C₁₃	—	4	1	no	—
131	C₂³×C₁₃	—	4	2	no	—
132	C₂³×C₁₃	—	8	2	no	—
133	C₂³×C₁₃	—	8	2	no	—
134	C₂³×C₁₃	—	8	2	no	—
135	C₂³×C₁₃	#121	1	1	yes	characteristic
136	Q₈×C₁₃	—	4	4	no	—
137	Q₈×C₁₃	—	4	2	no	—
138	Q₈×C₁₃	—	4	4	no	—
139	D₄×C₁₃	—	4	4	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	4	4	no	—
142	D₄×C₁₃	—	4	2	no	—
143	D₄×C₁₃	—	4	2	no	—
144	D₄×C₁₃	—	8	2	no	—
145	D₄×C₁₃	—	8	2	no	—
146	D₄×C₁₃	—	8	2	no	—
147	D₄×C₁₃	—	8	2	no	—
148	D₄×C₁₃	—	8	1	no	—
149	D₄×C₁₃	—	8	1	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	8	1	no	—
155	C₂×C₄×C₁₃	—	8	1	no	—
156	C₂×C₄×C₁₃	—	8	2	no	—
157	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
158	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
159	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
160	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
161	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
162	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
164	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
165	C₈×C₁₃	—	2	4	no	—
166	C₈×C₁₃	—	2	4	no	—
167	C₈×C₁₃	—	4	2	no	—
168	C₈×C₁₃	—	8	1	no	—
169	C₂×AGL(1,ℤ/8)	—	2	2	no	—
170	C₂²×SD₁₆	—	2	2	no	—
171	#532	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
173	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
174	#496	—	2	2	no	—
175	#493	—	2	1	no	—
176	#492	—	2	1	no	—
177	#492	—	2	2	no	—
178	#492	—	2	1	no	—
179	#492	—	2	2	no	—
180	#487	—	2	1	no	—
181	#461	—	2	2	no	—
182	#448	—	2	2	no	—
183	#431	C₂²×C₁₃	1	1	yes	characteristic
184	#423	C₂²×C₁₃	1	1	yes	characteristic
185	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
187	#343	—	2	2	no	—
188	#331	—	2	2	no	—
189	D₄⋊C₈	—	2	2	no	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	8	2	no	—
193	C₂²×C₁₃	—	8	1	no	—
194	C₂²×C₁₃	—	8	1	no	—
195	C₂²×C₁₃	—	8	2	no	—
196	C₂²×C₁₃	—	8	1	no	—
197	C₂²×C₁₃	—	8	2	no	—
198	C₂²×C₁₃	—	16	4	no	—
199	C₂²×C₁₃	#392	1	1	yes	characteristic
200	C₂²×C₁₃	#460	1	2	yes	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	—	2	1	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	—	8	1	no	—
205	C₄×C₁₃	#450	1	2	yes	—
206	C₄×C₁₃	#453	1	2	yes	—
207	C₄×C₁₃	#471	1	2	yes	—
208	C₄×C₁₃	#492	1	1	yes	characteristic
209	C₄×C₁₃	#496	1	1	yes	characteristic
210	C₂×#41	—	2	1	no	—
211	C₂²×Q₈	D₄×C₁₃	1	2	yes	—
212	C₂²×D₄	—	2	1	no	—
213	C₂²×D₄	—	2	1	no	—
214	C₂²×D₄	—	2	1	no	—
215	AGL(1,ℤ/8)	—	4	4	no	—
216	AGL(1,ℤ/8)	—	4	4	no	—
217	C₂×SD₁₆	—	2	8	no	—
218	C₂×SD₁₆	—	2	4	no	—
219	C₂×SD₁₆	—	2	4	no	—
220	C₂×SD₁₆	—	4	8	no	—
221	C₂×D₈	—	2	4	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
224	C₂²×C₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	D₄×C₁₃	1	1	yes	characteristic
230	#127	—	2	1	no	—
231	#127	—	2	1	no	—
232	#127	—	2	1	no	—
233	#127	D₄×C₁₃	1	2	yes	—
234	#127	D₄×C₁₃	1	1	yes	characteristic
235	#124	—	2	1	no	—
236	#121	—	2	1	no	—
237	#121	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
246	#103	—	2	4	no	—
247	C₂²⋊C₈	—	2	1	no	—
248	#97	—	2	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	D₄×C₁₃	1	2	yes	—
253	C₂×C₁₃	—	2	2	no	—
254	C₂×C₁₃	—	16	1	no	—
255	C₂×C₁₃	—	16	2	no	—
256	C₂×C₁₃	#2002	1	1	yes	characteristic
257	C₂×C₁₃	#2014	1	1	yes	characteristic
258	C₂×C₁₃	#2019	1	1	yes	characteristic
259	C₂⁴	—	4	1	no	—
260	#41	—	4	2	no	—
261	#41	—	4	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	4	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	4	4	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	8	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
284	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
285	SD₁₆	—	4	16	no	—
286	SD₁₆	—	4	8	no	—
287	SD₁₆	—	8	4	no	—
288	D₈	—	4	8	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	4	2	no	—
291	M₁₆	C₁₃×SD₁₆	1	4	yes	complemented
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	4	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	4	1	no	—
304	C₂²⋊C₄	—	4	1	no	—
305	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
306	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
307	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
309	C₁₃	#10146	1	1	yes	characteristic, direct product factor, Sylow
310	C₂³	—	4	1	no	—
311	C₂³	—	4	2	no	—
312	C₂³	—	8	2	no	—
313	C₂³	—	8	2	no	—
314	C₂³	—	8	2	no	—
315	C₂³	C₁₃×#121	1	1	yes	characteristic
316	Q₈	—	4	4	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	4	4	no	—
319	D₄	—	4	4	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	2	no	—
327	D₄	—	8	2	no	—
328	D₄	—	8	1	no	—
329	D₄	—	8	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	1	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
338	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
339	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
340	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
341	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
342	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
343	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
344	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
345	C₈	—	2	4	no	—
346	C₈	—	2	4	no	—
347	C₈	—	4	2	no	—
348	C₈	—	8	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	16	4	no	—
358	C₂²	C₁₃×#392	1	1	yes	characteristic
359	C₂²	C₁₃×#460	1	2	yes	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	C₁₃×#450	1	2	yes	—
365	C₄	C₁₃×#453	1	2	yes	—
366	C₄	C₁₃×#471	1	2	yes	—
367	C₄	C₁₃×#492	1	1	yes	characteristic
368	C₄	C₁₃×#496	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	16	2	no	—
371	C₂	—	16	1	no	—
372	C₂	C₁₃×#2002	1	1	yes	characteristic
373	C₂	C₁₃×#2014	1	1	yes	characteristic
374	C₂	C₁₃×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₁₃×#3151	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
1. C₁₃×#3206	C₂	semidirect
3. C₂×C₁₃×#492	C₂	semidirect
4. C₁₃×#1997	C₂	semidirect
5. C₁₃×#1545	C₂	semidirect
6. C₁₃×#1492	C₂	semidirect
7. C₁₃×#1278	C₂	semidirect
72. #10146	C₁₃	direct (N × H)
105. C₁₃×M₁₆	SD₁₆	semidirect
123. #3206	C₂×C₁₃	semidirect
125. C₂×#492	C₂×C₁₃	semidirect
126. #1997	C₂×C₁₃	semidirect
127. #1545	C₂×C₁₃	semidirect
128. #1492	C₂×C₁₃	semidirect
129. #1278	C₂×C₁₃	semidirect
291. M₁₆	C₁₃×SD₁₆	semidirect
309. C₁₃	#10146	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×#3206	2
C₁₃×#3151	2
C₂×C₁₃×#492	2
C₁₃×#1997	2
C₁₃×#1545	2
C₁₃×#1492	2
C₁₃×#1278	2
#10146	13

Subgroups — C13×#10146 #15696485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10146 #15696485