Subgroups — C₁₃×#9285 #15695624

Order 3328.

426 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3210	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2981	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1531	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1530	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1497	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1287	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1275	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#533	C₂²	1	1	yes	characteristic
9	C₁₃×#532	C₂²	1	1	yes	characteristic
10	C₁₃×#494	C₄	1	1	yes	characteristic
11	C₁₃×#492	C₄	1	1	yes	characteristic
12	C₁₃×#487	C₄	1	1	yes	characteristic
13	C₁₃×#486	C₄	1	1	yes	characteristic
14	C₁₃×#464	—	2	2	no	—
15	C₁₃×#452	—	2	2	no	—
16	C₁₃×#436	—	2	1	no	—
17	C₁₃×#435	—	2	2	no	—
18	C₈×D₄×C₁₃	—	2	1	no	—
19	C₁₃×#432	C₂²	1	1	yes	characteristic
20	C₁₃×#432	C₂²	1	1	yes	characteristic
21	C₁₃×#431	C₂²	1	1	yes	characteristic
22	C₁₃×#431	C₂²	1	1	yes	characteristic
23	C₁₃×#408	—	2	2	no	—
24	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
25	C₁₃×#330	—	2	2	no	—
26	C₁₃×#329	—	2	2	no	—
27	C₁₃×Q₈⋊C₈	—	2	1	no	—
28	C₁₃×Q₈⋊C₈	—	2	1	no	—
29	C₂×C₁₃×#41	D₄	1	2	yes	—
30	C₂×C₁₃×Q₁₆	—	2	2	no	—
31	C₂×C₁₃×SD₁₆	—	2	2	no	—
32	C₂×C₁₃×M₁₆	—	2	2	no	—
33	C₂²×C₈×C₁₃	—	2	1	no	—
34	C₂²×C₈×C₁₃	—	2	2	no	—
35	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
39	C₁₃×#125	—	2	1	no	—
40	C₁₃×#124	D₄	1	2	yes	—
41	C₁₃×C₂²⋊Q₈	—	2	1	no	—
42	C₁₃×C₂²⋊Q₈	—	2	1	no	—
43	C₁₃×#121	—	2	1	no	—
44	C₁₃×#121	—	2	1	no	—
45	C₄×Q₈×C₁₃	—	2	1	no	—
46	C₄×Q₈×C₁₃	—	2	1	no	—
47	C₄×D₄×C₁₃	—	2	1	no	—
48	C₄×D₄×C₁₃	—	2	1	no	—
49	C₁₃×#117	—	2	1	no	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×C₄⋊C₈	—	2	1	no	—
52	C₁₃×C₄⋊C₈	—	2	1	no	—
53	C₁₃×C₄⋊C₈	—	2	1	no	—
54	C₁₃×C₄⋊C₈	—	2	1	no	—
55	C₁₃×#103	—	2	2	no	—
56	C₁₃×#102	—	2	2	no	—
57	C₁₃×C₂²⋊C₈	—	2	2	no	—
58	C₁₃×C₂²⋊C₈	—	2	2	no	—
59	C₁₃×C₂²⋊C₈	—	2	1	no	—
60	C₁₃×C₂²⋊C₈	—	2	1	no	—
61	C₁₃×C₂²⋊C₈	—	2	1	no	—
62	C₁₃×C₂²⋊C₈	—	2	1	no	—
63	C₁₃×#97	—	2	1	no	—
64	C₁₃×#97	—	2	1	no	—
65	C₁₃×#97	—	2	1	no	—
66	C₁₃×#97	D₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
71	#9285	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃×#41	—	8	2	no	—
73	C₂×Q₈×C₁₃	—	2	1	no	—
74	C₂×Q₈×C₁₃	—	2	1	no	—
75	C₂×Q₈×C₁₃	C₂²⋊C₄	1	2	yes	—
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	2	1	no	—
78	C₂×D₄×C₁₃	—	2	1	no	—
79	C₂×D₄×C₁₃	C₂²⋊C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
84	C₁₃×Q₁₆	—	8	2	no	—
85	C₁₃×SD₁₆	—	8	2	no	—
86	C₁₃×M₁₆	—	8	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
94	C₂×C₈×C₁₃	—	4	1	no	—
95	C₂×C₈×C₁₃	—	4	1	no	—
96	C₂×C₈×C₁₃	—	8	2	no	—
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	1	no	—
99	C₁₃×C₄⋊C₄	—	2	1	no	—
100	C₁₃×C₄⋊C₄	—	2	1	no	—
101	C₁₃×C₄⋊C₄	—	2	1	no	—
102	C₁₃×C₄⋊C₄	—	2	1	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	2	2	no	—
106	C₁₃×C₂²⋊C₄	—	2	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	1	no	—
108	C₄²×C₁₃	—	4	1	no	—
109	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	#3210	C₂×C₁₃	1	1	yes	characteristic
114	#2981	C₂×C₁₃	1	1	yes	characteristic
115	#1531	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#1530	C₂×C₁₃	1	1	yes	characteristic
117	#1497	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#1287	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#1275	C₂×C₁₃	1	1	yes	characteristic, complemented
120	C₂³×C₁₃	—	2	2	no	—
121	C₂³×C₁₃	#123	1	1	yes	characteristic
122	Q₈×C₁₃	—	4	2	no	—
123	Q₈×C₁₃	—	4	2	no	—
124	Q₈×C₁₃	—	4	1	no	—
125	Q₈×C₁₃	—	4	1	no	—
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	4	1	no	—
128	D₄×C₁₃	—	4	1	no	—
129	D₄×C₁₃	—	8	1	no	—
130	D₄×C₁₃	—	8	2	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
146	C₈×C₁₃	—	4	2	no	—
147	C₈×C₁₃	—	4	1	no	—
148	C₈×C₁₃	—	4	2	no	—
149	C₈×C₁₃	—	4	1	no	—
150	C₈×C₁₃	—	4	1	no	—
151	C₈×C₁₃	—	4	1	no	—
152	C₈×C₁₃	—	8	1	no	—
153	#533	C₂²×C₁₃	1	1	yes	characteristic
154	#532	C₂²×C₁₃	1	1	yes	characteristic
155	#494	C₄×C₁₃	1	1	yes	characteristic
156	#492	C₄×C₁₃	1	1	yes	characteristic
157	#487	C₄×C₁₃	1	1	yes	characteristic
158	#486	C₄×C₁₃	1	1	yes	characteristic
159	#464	—	2	2	no	—
160	#452	—	2	2	no	—
161	#436	—	2	1	no	—
162	#435	—	2	2	no	—
163	C₈×D₄	—	2	1	no	—
164	#432	C₂²×C₁₃	1	1	yes	characteristic
165	#432	C₂²×C₁₃	1	1	yes	characteristic
166	#431	C₂²×C₁₃	1	1	yes	characteristic
167	#431	C₂²×C₁₃	1	1	yes	characteristic
168	#408	—	2	2	no	—
169	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
170	#330	—	2	2	no	—
171	#329	—	2	2	no	—
172	Q₈⋊C₈	—	2	1	no	—
173	Q₈⋊C₈	—	2	1	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	4	2	no	—
178	C₂²×C₁₃	—	4	2	no	—
179	C₂²×C₁₃	—	4	2	no	—
180	C₂²×C₁₃	#386	1	1	yes	characteristic
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	—	8	1	no	—
188	C₄×C₁₃	—	8	1	no	—
189	C₄×C₁₃	#443	1	1	yes	characteristic
190	C₄×C₁₃	#444	1	1	yes	characteristic
191	C₂×#41	D₄×C₁₃	1	2	yes	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₄⋊Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
198	C₄⋊Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
200	#127	C₂×C₄×C₁₃	1	1	yes	characteristic
201	#125	—	2	1	no	—
202	#124	D₄×C₁₃	1	2	yes	—
203	C₂²⋊Q₈	—	2	1	no	—
204	C₂²⋊Q₈	—	2	1	no	—
205	#121	—	2	1	no	—
206	#121	—	2	1	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×D₄	—	2	1	no	—
210	C₄×D₄	—	2	1	no	—
211	#117	—	2	1	no	—
212	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	#103	—	2	2	no	—
218	#102	—	2	2	no	—
219	C₂²⋊C₈	—	2	2	no	—
220	C₂²⋊C₈	—	2	2	no	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	#97	—	2	1	no	—
226	#97	—	2	1	no	—
227	#97	—	2	1	no	—
228	#97	D₄×C₁₃	1	2	yes	—
229	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
230	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
231	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
232	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
233	C₂×C₁₃	—	4	1	no	—
234	C₂×C₁₃	—	8	2	no	—
235	C₂×C₁₃	#1876	1	1	yes	characteristic
236	C₂×C₁₃	#1902	1	1	yes	characteristic
237	C₂×C₁₃	#1911	1	1	yes	characteristic
238	#41	—	8	2	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	C₁₃×C₂²⋊C₄	1	2	yes	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	C₁₃×C₂²⋊C₄	1	2	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
249	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
250	Q₁₆	—	8	2	no	—
251	SD₁₆	—	8	2	no	—
252	M₁₆	—	8	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	8	2	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
279	C₁₃	#9285	1	1	yes	characteristic, direct product factor, Sylow
280	C₂³	—	2	2	no	—
281	C₂³	C₁₃×#123	1	1	yes	characteristic
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	1	no	—
288	D₄	—	4	1	no	—
289	D₄	—	8	1	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
301	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
302	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
303	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
304	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
305	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	—	4	1	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	1	no	—
310	C₈	—	4	1	no	—
311	C₈	—	4	1	no	—
312	C₈	—	8	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	C₁₃×#386	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	C₁₃×#443	1	1	yes	characteristic
329	C₄	C₁₃×#444	1	1	yes	characteristic
330	C₂	—	4	1	no	—
331	C₂	—	8	2	no	—
332	C₂	C₁₃×#1876	1	1	yes	characteristic
333	C₂	C₁₃×#1902	1	1	yes	characteristic
334	C₂	C₁₃×#1911	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#386
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₁₃×#532	C₂²
LCS	G ▷ 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
3. C₁₃×#1531	C₂	semidirect
5. C₁₃×#1497	C₂	semidirect
6. C₁₃×#1287	C₂	semidirect
7. C₁₃×#1275	C₂	semidirect
71. #9285	C₁₃	direct (N × H)
115. #1531	C₂×C₁₃	semidirect
117. #1497	C₂×C₁₃	semidirect
118. #1287	C₂×C₁₃	semidirect
119. #1275	C₂×C₁₃	semidirect
279. C₁₃	#9285	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×#3210	2
C₁₃×#2981	2
C₁₃×#1531	2
C₁₃×#1530	2
C₁₃×#1497	2
C₁₃×#1287	2
C₁₃×#1275	2
#9285	13

Subgroups — C13×#9285 #15695624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9285 #15695624