Subgroups — C₁₃×#10281 #15696620

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 380 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×#3205	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2858	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1962	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#532	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#124	—	2	2	no	—
12	C₁₃×#493	—	2	1	no	—
13	C₁₃×#492	—	2	1	no	—
14	C₁₃×#487	—	2	1	no	—
15	C₁₃×#486	—	2	1	no	—
16	C₁₃×#461	—	2	2	no	—
17	C₁₃×#459	—	2	2	no	—
18	C₁₃×#448	—	2	2	no	—
19	C₁₃×#447	—	2	2	no	—
20	C₁₃×#431	C₂²	1	1	yes	characteristic
21	C₁₃×#431	C₂²	1	1	yes	characteristic
22	C₁₃×#423	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×#104	—	2	2	no	—
24	C₁₃×#390	—	2	2	no	—
25	C₁₃×#390	—	2	2	no	—
26	C₄³×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#333	—	2	2	no	—
28	C₁₃×#331	—	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	D₄	1	2	yes	—
34	C₂×C₁₃×SD₁₆	—	2	4	no	—
35	C₂×C₁₃×D₈	—	2	4	no	—
36	C₂×C₁₃×M₁₆	—	2	1	no	—
37	C₁₃×C₄⋊Q₈	—	2	1	no	—
38	C₁₃×C₄⋊Q₈	—	2	1	no	—
39	C₁₃×C₄⋊Q₈	—	2	1	no	—
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₁₃×#127	—	2	1	no	—
43	C₁₃×#127	—	2	1	no	—
44	C₁₃×#127	—	2	1	no	—
45	C₁₃×#127	—	2	1	no	—
46	C₁₃×#127	D₄	1	1	yes	characteristic
47	C₁₃×#124	—	2	1	no	—
48	C₁₃×#124	—	4	8	no	—
49	C₁₃×#121	—	2	1	no	—
50	C₁₃×#121	—	2	1	no	—
51	C₄×D₄×C₁₃	—	2	1	no	—
52	C₄×D₄×C₁₃	—	2	1	no	—
53	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₂×C₁₃×C₂²⋊C₄	—	2	8	no	—
56	C₂×C₄²×C₁₃	—	2	1	no	—
57	C₂×C₄²×C₁₃	—	2	1	no	—
58	C₂×C₄²×C₁₃	D₄	1	2	yes	—
59	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×C₄⋊C₈	—	2	1	no	—
61	C₁₃×C₄⋊C₈	—	2	1	no	—
62	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
63	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
64	C₁₃×#104	—	4	4	no	—
65	C₁₃×#103	—	2	4	no	—
66	C₁₃×#102	—	2	4	no	—
67	C₁₃×C₂²⋊C₈	—	2	1	no	—
68	C₁₃×C₂²⋊C₈	—	2	1	no	—
69	C₄×C₈×C₁₃	—	2	1	no	—
70	C₄×C₈×C₁₃	—	2	1	no	—
71	#10281	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂⁴×C₁₃	—	2	2	no	—
73	C₁₃×#41	—	4	2	no	—
74	C₁₃×#41	—	4	2	no	—
75	C₂×Q₈×C₁₃	—	2	2	no	—
76	C₂×Q₈×C₁₃	—	2	1	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	4	4	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	—	2	1	no	—
82	C₂×D₄×C₁₃	—	2	1	no	—
83	C₂×D₄×C₁₃	—	2	2	no	—
84	C₂×D₄×C₁₃	—	4	4	no	—
85	C₂×D₄×C₁₃	—	4	4	no	—
86	C₂×D₄×C₁₃	—	4	1	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	C₂²×C₄×C₁₃	—	2	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	C₂×D₄	1	1	yes	characteristic
99	C₁₃×SD₁₆	—	8	4	no	—
100	C₁₃×D₈	—	8	4	no	—
101	C₁₃×M₁₆	—	4	2	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	2	1	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	1	no	—
111	C₁₃×C₂²⋊C₄	—	4	16	no	—
112	C₁₃×C₂²⋊C₄	—	4	1	no	—
113	C₁₃×C₂²⋊C₄	—	4	1	no	—
114	C₄²×C₁₃	—	2	4	no	—
115	C₄²×C₁₃	—	2	2	no	—
116	C₄²×C₁₃	—	2	2	no	—
117	C₄²×C₁₃	—	4	2	no	—
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₃	#41	1	1	yes	characteristic
122	#3206	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#3205	C₂×C₁₃	1	1	yes	characteristic
124	#2858	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#1962	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#1548	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#1545	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#1276	C₂×C₁₃	1	1	yes	characteristic, complemented
129	C₂³×C₁₃	—	4	4	no	—
130	C₂³×C₁₃	—	4	4	no	—
131	C₂³×C₁₃	—	4	1	no	—
132	C₂³×C₁₃	—	4	2	no	—
133	C₂³×C₁₃	—	4	4	no	—
134	C₂³×C₁₃	#120	1	1	yes	characteristic
135	Q₈×C₁₃	—	4	4	no	—
136	Q₈×C₁₃	—	4	2	no	—
137	Q₈×C₁₃	—	8	2	no	—
138	D₄×C₁₃	—	4	2	no	—
139	D₄×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	4	4	no	—
141	D₄×C₁₃	—	4	2	no	—
142	D₄×C₁₃	—	8	2	no	—
143	D₄×C₁₃	—	8	2	no	—
144	D₄×C₁₃	—	8	2	no	—
145	D₄×C₁₃	—	8	2	no	—
146	D₄×C₁₃	—	8	2	no	—
147	D₄×C₁₃	—	8	1	no	—
148	D₄×C₁₃	—	8	1	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	4	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	4	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	8	1	no	—
158	C₂×C₄×C₁₃	—	8	1	no	—
159	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
160	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
161	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
162	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
163	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
164	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
165	C₂×C₄×C₁₃	#135	1	2	yes	—
166	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
167	C₈×C₁₃	—	4	2	no	—
168	C₈×C₁₃	—	4	2	no	—
169	C₈×C₁₃	—	8	1	no	—
170	#532	C₂²×C₁₃	1	1	yes	characteristic
171	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
173	C₂×#124	—	2	2	no	—
174	#493	—	2	1	no	—
175	#492	—	2	1	no	—
176	#487	—	2	1	no	—
177	#486	—	2	1	no	—
178	#461	—	2	2	no	—
179	#459	—	2	2	no	—
180	#448	—	2	2	no	—
181	#447	—	2	2	no	—
182	#431	C₂²×C₁₃	1	1	yes	characteristic
183	#431	C₂²×C₁₃	1	1	yes	characteristic
184	#423	C₂²×C₁₃	1	1	yes	characteristic
185	C₂×#104	—	2	2	no	—
186	#390	—	2	2	no	—
187	#390	—	2	2	no	—
188	C₄³	C₂²×C₁₃	1	1	yes	characteristic
189	#333	—	2	2	no	—
190	#331	—	2	2	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	8	8	no	—
194	C₂²×C₁₃	—	8	1	no	—
195	C₂²×C₁₃	—	8	2	no	—
196	C₂²×C₁₃	—	8	2	no	—
197	C₂²×C₁₃	—	8	1	no	—
198	C₂²×C₁₃	—	8	1	no	—
199	C₂²×C₁₃	—	8	2	no	—
200	C₂²×C₁₃	#392	1	1	yes	characteristic
201	C₂²×C₁₃	#449	1	2	yes	—
202	C₄×C₁₃	—	2	1	no	—
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	—	4	4	no	—
205	C₄×C₁₃	—	8	2	no	—
206	C₄×C₁₃	—	8	1	no	—
207	C₄×C₁₃	#453	1	2	yes	—
208	C₄×C₁₃	#455	1	2	yes	—
209	C₄×C₁₃	#463	1	2	yes	—
210	C₄×C₁₃	#495	1	1	yes	characteristic
211	C₄×C₁₃	#496	1	1	yes	characteristic
212	C₂×#41	—	2	1	no	—
213	C₂²×Q₈	D₄×C₁₃	1	2	yes	—
214	C₂²×D₄	—	2	1	no	—
215	C₂²×D₄	—	2	1	no	—
216	C₂²×D₄	D₄×C₁₃	1	2	yes	—
217	C₂×SD₁₆	—	2	4	no	—
218	C₂×D₈	—	2	4	no	—
219	C₂×M₁₆	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	D₄×C₁₃	1	1	yes	characteristic
225	#127	—	2	1	no	—
226	#127	—	2	1	no	—
227	#127	—	2	1	no	—
228	#127	—	2	1	no	—
229	#127	D₄×C₁₃	1	1	yes	characteristic
230	#124	—	2	1	no	—
231	#124	—	4	8	no	—
232	#121	—	2	1	no	—
233	#121	—	2	1	no	—
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₂²⋊C₄	—	2	8	no	—
239	C₂×C₄²	—	2	1	no	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	D₄×C₁₃	1	2	yes	—
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	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
247	#104	—	4	4	no	—
248	#103	—	2	4	no	—
249	#102	—	2	4	no	—
250	C₂²⋊C₈	—	2	1	no	—
251	C₂²⋊C₈	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₂×C₁₃	—	2	2	no	—
255	C₂×C₁₃	—	16	1	no	—
256	C₂×C₁₃	—	16	2	no	—
257	C₂×C₁₃	#2002	1	1	yes	characteristic
258	C₂×C₁₃	#2004	1	1	yes	characteristic
259	C₂×C₁₃	#2012	1	1	yes	characteristic
260	C₂⁴	—	2	2	no	—
261	#41	—	4	2	no	—
262	#41	—	4	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	4	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	4	4	no	—
273	C₂×D₄	—	4	4	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
287	SD₁₆	—	8	4	no	—
288	D₈	—	8	4	no	—
289	M₁₆	—	4	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	4	1	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	4	16	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₄²	—	2	4	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	4	2	no	—
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	C₄²	C₁₃×#41	1	1	yes	characteristic
310	C₁₃	#10281	1	1	yes	characteristic, direct product factor, Sylow
311	C₂³	—	4	4	no	—
312	C₂³	—	4	4	no	—
313	C₂³	—	4	1	no	—
314	C₂³	—	4	2	no	—
315	C₂³	—	4	4	no	—
316	C₂³	C₁₃×#120	1	1	yes	characteristic
317	Q₈	—	4	4	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	8	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	4	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	2	no	—
329	D₄	—	8	1	no	—
330	D₄	—	8	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	4	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	8	1	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
342	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
343	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
344	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
345	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
346	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
347	C₂×C₄	C₁₃×#135	1	2	yes	—
348	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
349	C₈	—	4	2	no	—
350	C₈	—	4	2	no	—
351	C₈	—	8	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	8	8	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	8	1	no	—
360	C₂²	—	8	2	no	—
361	C₂²	C₁₃×#392	1	1	yes	characteristic
362	C₂²	C₁₃×#449	1	2	yes	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	4	4	no	—
366	C₄	—	8	2	no	—
367	C₄	—	8	1	no	—
368	C₄	C₁₃×#453	1	2	yes	—
369	C₄	C₁₃×#455	1	2	yes	—
370	C₄	C₁₃×#463	1	2	yes	—
371	C₄	C₁₃×#495	1	1	yes	characteristic
372	C₄	C₁₃×#496	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	16	2	no	—
375	C₂	—	16	1	no	—
376	C₂	C₁₃×#2002	1	1	yes	characteristic
377	C₂	C₁₃×#2004	1	1	yes	characteristic
378	C₂	C₁₃×#2012	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₁₃×#3205	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. C₁₃×#2858	C₂	semidirect
4. C₁₃×#1962	C₂	semidirect
5. C₁₃×#1548	C₂	semidirect
6. C₁₃×#1545	C₂	semidirect
7. C₁₃×#1276	C₂	semidirect
71. #10281	C₁₃	direct (N × H)
122. #3206	C₂×C₁₃	semidirect
124. #2858	C₂×C₁₃	semidirect
125. #1962	C₂×C₁₃	semidirect
126. #1548	C₂×C₁₃	semidirect
127. #1545	C₂×C₁₃	semidirect
128. #1276	C₂×C₁₃	semidirect
310. C₁₃	#10281	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×#3205	2
C₁₃×#2858	2
C₁₃×#1962	2
C₁₃×#1548	2
C₁₃×#1545	2
C₁₃×#1276	2
#10281	13

Subgroups — C13×#10281 #15696620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10281 #15696620