Subgroups — C₁₃×#10149 #15696488

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3138	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#486	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1545	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#533	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	C₁₃×#500	—	2	1	no	—
12	C₁₃×#490	—	2	2	no	—
13	C₁₃×#487	—	2	1	no	—
14	C₁₃×#486	—	2	1	no	—
15	C₁₃×#486	—	2	1	no	—
16	C₁₃×#486	—	2	2	no	—
17	C₁₃×#486	—	2	2	no	—
18	C₁₃×#484	—	2	2	no	—
19	C₁₃×#477	—	2	2	no	—
20	C₁₃×#431	C₂²	1	1	yes	characteristic
21	C₁₃×#423	C₂²	1	1	yes	characteristic
22	C₁₃×#418	—	2	2	no	—
23	C₂×C₁₃×#102	—	2	2	no	—
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×Q₈⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₁₃	—	2	1	no	—
30	C₂²×D₄×C₁₃	—	2	1	no	—
31	C₂²×D₄×C₁₃	—	2	1	no	—
32	C₂²×D₄×C₁₃	—	2	1	no	—
33	C₂×C₁₃×M₁₆	—	2	1	no	—
34	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
35	C₂²×C₈×C₁₃	—	2	1	no	—
36	C₁₃×C₄⋊Q₈	—	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	D₄	1	1	yes	characteristic
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	D₄	1	2	yes	—
46	C₁₃×#125	—	2	1	no	—
47	C₁₃×C₂²⋊Q₈	—	2	1	no	—
48	C₁₃×C₂²⋊Q₈	—	2	1	no	—
49	C₄×Q₈×C₁₃	—	2	1	no	—
50	C₄×Q₈×C₁₃	—	2	1	no	—
51	C₁₃×#117	—	2	1	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×#107	—	2	4	no	—
56	C₁₃×C₄⋊C₈	—	2	1	no	—
57	C₁₃×C₄⋊C₈	—	2	1	no	—
58	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
59	C₁₃×#103	—	2	4	no	—
60	C₁₃×#102	—	2	8	no	—
61	C₁₃×#102	—	2	4	no	—
62	C₁₃×#102	—	2	4	no	—
63	C₁₃×C₂²⋊C₈	—	2	1	no	—
64	C₁₃×#97	—	2	1	no	—
65	C₄×C₈×C₁₃	—	2	1	no	—
66	C₄×C₈×C₁₃	—	2	1	no	—
67	C₄×C₈×C₁₃	—	2	1	no	—
68	C₄×C₈×C₁₃	D₄	1	2	yes	—
69	#10149	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂⁴×C₁₃	—	4	1	no	—
71	C₂×Q₈×C₁₃	—	2	1	no	—
72	C₂×Q₈×C₁₃	—	2	1	no	—
73	C₂×Q₈×C₁₃	—	4	1	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×Q₈×C₁₃	—	8	2	no	—
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	4	4	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂×D₄×C₁₃	—	4	2	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	—	8	2	no	—
86	C₂²×C₄×C₁₃	—	2	2	no	—
87	C₂²×C₄×C₁₃	C₂×D₄	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	2	no	—
91	C₁₃×M₁₆	—	2	2	no	—
92	C₁₃×M₁₆	—	4	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	4	2	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	1	no	—
100	C₁₃×C₄⋊C₄	—	2	1	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	2	1	no	—
103	C₁₃×C₄⋊C₄	—	2	1	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₂²⋊C₄	—	4	1	no	—
109	C₄²×C₁₃	—	4	1	no	—
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	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	#3230	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#3138	C₂×C₁₃	1	1	yes	characteristic
116	C₂×#486	C₂×C₁₃	1	1	yes	characteristic
117	#1997	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#1545	C₂×C₁₃	1	1	yes	characteristic
119	#1493	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#1278	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂³×C₁₃	—	4	2	no	—
122	C₂³×C₁₃	—	8	2	no	—
123	C₂³×C₁₃	—	8	2	no	—
124	C₂³×C₁₃	—	8	2	no	—
125	C₂³×C₁₃	#121	1	1	yes	characteristic
126	Q₈×C₁₃	—	4	2	no	—
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	8	2	no	—
129	Q₈×C₁₃	—	8	2	no	—
130	D₄×C₁₃	—	4	4	no	—
131	D₄×C₁₃	—	4	4	no	—
132	D₄×C₁₃	—	8	2	no	—
133	D₄×C₁₃	—	8	2	no	—
134	D₄×C₁₃	—	8	2	no	—
135	D₄×C₁₃	—	8	1	no	—
136	D₄×C₁₃	—	8	1	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	8	2	no	—
143	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
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	#127	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#135	1	2	yes	—
150	C₂×C₄×C₁₃	#137	1	2	yes	—
151	C₈×C₁₃	—	4	2	no	—
152	C₈×C₁₃	—	4	2	no	—
153	C₈×C₁₃	—	4	2	no	—
154	C₈×C₁₃	—	8	1	no	—
155	#533	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
158	#500	—	2	1	no	—
159	#490	—	2	2	no	—
160	#487	—	2	1	no	—
161	#486	—	2	1	no	—
162	#486	—	2	1	no	—
163	#486	—	2	2	no	—
164	#486	—	2	2	no	—
165	#484	—	2	2	no	—
166	#477	—	2	2	no	—
167	#431	C₂²×C₁₃	1	1	yes	characteristic
168	#423	C₂²×C₁₃	1	1	yes	characteristic
169	#418	—	2	2	no	—
170	C₂×#102	—	2	2	no	—
171	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
173	#343	—	2	2	no	—
174	#331	—	2	2	no	—
175	Q₈⋊C₈	—	2	2	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	16	4	no	—
182	C₂²×C₁₃	#392	1	1	yes	characteristic
183	C₂²×C₁₃	#464	1	2	yes	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	8	1	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	—	8	1	no	—
189	C₄×C₁₃	#452	1	2	yes	—
190	C₄×C₁₃	#454	1	2	yes	—
191	C₄×C₁₃	#472	1	2	yes	—
192	C₄×C₁₃	#495	1	1	yes	characteristic
193	C₄×C₁₃	#497	1	1	yes	characteristic
194	C₂²×Q₈	—	2	1	no	—
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	—	2	1	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
200	C₂²×C₈	—	2	1	no	—
201	C₄⋊Q₈	—	2	1	no	—
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	D₄×C₁₃	1	1	yes	characteristic
206	C₄⋊Q₈	D₄×C₁₃	1	1	yes	characteristic
207	#127	—	2	1	no	—
208	#127	—	2	1	no	—
209	#127	—	2	1	no	—
210	#127	D₄×C₁₃	1	2	yes	—
211	#125	—	2	1	no	—
212	C₂²⋊Q₈	—	2	1	no	—
213	C₂²⋊Q₈	—	2	1	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	2	1	no	—
216	#117	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
219	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
220	#107	—	2	4	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
224	#103	—	2	4	no	—
225	#102	—	2	8	no	—
226	#102	—	2	4	no	—
227	#102	—	2	4	no	—
228	C₂²⋊C₈	—	2	1	no	—
229	#97	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	D₄×C₁₃	1	2	yes	—
234	C₂×C₁₃	—	2	2	no	—
235	C₂×C₁₃	—	16	2	no	—
236	C₂×C₁₃	#2003	1	1	yes	characteristic
237	C₂×C₁₃	#2015	1	1	yes	characteristic
238	C₂×C₁₃	#2018	1	1	yes	characteristic
239	C₂⁴	—	4	1	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	8	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	4	4	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	8	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	4	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₂²⋊C₄	—	4	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
283	C₁₃	#10149	1	1	yes	characteristic, direct product factor, Sylow
284	C₂³	—	4	2	no	—
285	C₂³	—	8	2	no	—
286	C₂³	—	8	2	no	—
287	C₂³	—	8	2	no	—
288	C₂³	C₁₃×#121	1	1	yes	characteristic
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	8	2	no	—
292	Q₈	—	8	2	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	1	no	—
299	D₄	—	8	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
307	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
308	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
309	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
310	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
311	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
312	C₂×C₄	C₁₃×#135	1	2	yes	—
313	C₂×C₄	C₁₃×#137	1	2	yes	—
314	C₈	—	4	2	no	—
315	C₈	—	4	2	no	—
316	C₈	—	4	2	no	—
317	C₈	—	8	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	4	no	—
324	C₂²	C₁₃×#392	1	1	yes	characteristic
325	C₂²	C₁₃×#464	1	2	yes	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	1	no	—
331	C₄	C₁₃×#452	1	2	yes	—
332	C₄	C₁₃×#454	1	2	yes	—
333	C₄	C₁₃×#472	1	2	yes	—
334	C₄	C₁₃×#495	1	1	yes	characteristic
335	C₄	C₁₃×#497	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	16	2	no	—
338	C₂	C₁₃×#2003	1	1	yes	characteristic
339	C₂	C₁₃×#2015	1	1	yes	characteristic
340	C₂	C₁₃×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂×C₁₃×#127	C₂²
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×#3230	C₂	semidirect
4. C₁₃×#1997	C₂	semidirect
6. C₁₃×#1493	C₂	semidirect
7. C₁₃×#1278	C₂	semidirect
69. #10149	C₁₃	direct (N × H)
114. #3230	C₂×C₁₃	semidirect
117. #1997	C₂×C₁₃	semidirect
119. #1493	C₂×C₁₃	semidirect
120. #1278	C₂×C₁₃	semidirect
283. C₁₃	#10149	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×#3230	2
C₁₃×#3138	2
C₂×C₁₃×#486	2
C₁₃×#1997	2
C₁₃×#1545	2
C₁₃×#1493	2
C₁₃×#1278	2
#10149	13

Subgroups — C13×#10149 #15696488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10149 #15696488