Subgroups — C₁₃×#10153 #15696492

Order 3328.

620 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3208	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#492	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1965	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×SD₁₆	—	2	2	no	—
9	C₁₃×#533	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
12	C₁₃×#501	—	2	2	no	—
13	C₁₃×#500	—	2	1	no	—
14	C₁₃×#498	—	2	1	no	—
15	C₁₃×#492	—	2	2	no	—
16	C₁₃×#492	—	2	1	no	—
17	C₁₃×#492	—	2	2	no	—
18	C₁₃×#486	—	2	1	no	—
19	C₁₃×#460	—	2	2	no	—
20	C₁₃×#450	—	2	2	no	—
21	C₁₃×#431	C₂²	1	1	yes	characteristic
22	C₁₃×#428	—	2	2	no	—
23	C₁₃×#423	C₂²	1	1	yes	characteristic
24	C₂×C₁₃×#100	—	2	2	no	—
25	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#332	—	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	D₄	1	2	yes	—
33	C₂×C₁₃×SD₁₆	—	2	8	no	—
34	C₂×C₁₃×SD₁₆	—	2	4	no	—
35	C₂×C₁₃×SD₁₆	—	4	8	no	—
36	C₂×C₁₃×M₁₆	—	2	1	no	—
37	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
38	C₂²×C₈×C₁₃	—	2	1	no	—
39	C₁₃×C₄⋊Q₈	—	2	1	no	—
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
43	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
45	C₁₃×#127	—	2	1	no	—
46	C₁₃×#127	—	2	1	no	—
47	C₁₃×#127	—	2	1	no	—
48	C₁₃×#127	—	2	1	no	—
49	C₁₃×#125	—	2	1	no	—
50	C₁₃×C₂²⋊Q₈	—	2	1	no	—
51	C₁₃×C₂²⋊Q₈	—	2	1	no	—
52	C₄×Q₈×C₁₃	—	2	1	no	—
53	C₄×Q₈×C₁₃	—	2	1	no	—
54	C₁₃×#117	—	2	1	no	—
55	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
56	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×#107	—	2	4	no	—
59	C₁₃×#106	—	2	4	no	—
60	C₁₃×C₄⋊C₈	—	2	1	no	—
61	C₁₃×C₄⋊C₈	—	2	1	no	—
62	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
63	C₁₃×#102	—	2	4	no	—
64	C₁₃×#100	—	4	4	no	—
65	C₁₃×C₂²⋊C₈	—	2	1	no	—
66	C₁₃×#97	—	2	1	no	—
67	C₄×C₈×C₁₃	—	2	1	no	—
68	C₄×C₈×C₁₃	—	2	1	no	—
69	C₄×C₈×C₁₃	—	2	1	no	—
70	C₄×C₈×C₁₃	D₄	1	2	yes	—
71	#10153	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂⁴×C₁₃	—	2	2	no	—
73	C₂×Q₈×C₁₃	—	2	1	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	2	1	no	—
76	C₂×Q₈×C₁₃	—	4	4	no	—
77	C₂×Q₈×C₁₃	—	4	1	no	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	4	4	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂×D₄×C₁₃	—	4	1	no	—
84	C₂×D₄×C₁₃	—	4	1	no	—
85	C₂×D₄×C₁₃	—	4	2	no	—
86	C₂×D₄×C₁₃	—	4	2	no	—
87	C₂×D₄×C₁₃	—	8	2	no	—
88	C₂²×C₄×C₁₃	—	4	1	no	—
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
92	C₁₃×SD₁₆	—	4	16	no	—
93	C₁₃×SD₁₆	—	8	4	no	—
94	C₁₃×M₁₆	—	2	2	no	—
95	C₁₃×M₁₆	—	4	2	no	—
96	C₁₃×M₁₆	SD₁₆	1	4	yes	complemented
97	C₂×C₈×C₁₃	—	2	4	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	—	4	1	no	—
102	C₁₃×C₄⋊C₄	—	2	1	no	—
103	C₁₃×C₄⋊C₄	—	2	1	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	2	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×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	1	no	—
113	C₄²×C₁₃	—	4	1	no	—
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	#3208	C₂×C₁₃	1	1	yes	characteristic
119	#3165	C₂×C₁₃	1	1	yes	characteristic, complemented
120	C₂×#492	C₂×C₁₃	1	1	yes	characteristic
121	#1965	C₂×C₁₃	1	1	yes	characteristic
122	#1549	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#1493	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#1278	C₂×C₁₃	1	1	yes	characteristic, complemented
125	C₂³×C₁₃	—	4	2	no	—
126	C₂³×C₁₃	—	4	4	no	—
127	C₂³×C₁₃	—	8	2	no	—
128	C₂³×C₁₃	—	8	2	no	—
129	C₂³×C₁₃	#121	1	1	yes	characteristic
130	Q₈×C₁₃	—	4	4	no	—
131	Q₈×C₁₃	—	4	2	no	—
132	Q₈×C₁₃	—	4	2	no	—
133	Q₈×C₁₃	—	8	2	no	—
134	D₄×C₁₃	—	4	4	no	—
135	D₄×C₁₃	—	4	4	no	—
136	D₄×C₁₃	—	8	2	no	—
137	D₄×C₁₃	—	8	2	no	—
138	D₄×C₁₃	—	8	2	no	—
139	D₄×C₁₃	—	8	1	no	—
140	D₄×C₁₃	—	8	1	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	8	2	no	—
147	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
150	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
154	C₂×C₄×C₁₃	#137	1	2	yes	—
155	C₈×C₁₃	—	2	4	no	—
156	C₈×C₁₃	—	2	4	no	—
157	C₈×C₁₃	—	4	2	no	—
158	C₈×C₁₃	—	8	1	no	—
159	C₂²×SD₁₆	—	2	2	no	—
160	#533	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
163	#501	—	2	2	no	—
164	#500	—	2	1	no	—
165	#498	—	2	1	no	—
166	#492	—	2	2	no	—
167	#492	—	2	1	no	—
168	#492	—	2	2	no	—
169	#486	—	2	1	no	—
170	#460	—	2	2	no	—
171	#450	—	2	2	no	—
172	#431	C₂²×C₁₃	1	1	yes	characteristic
173	#428	—	2	2	no	—
174	#423	C₂²×C₁₃	1	1	yes	characteristic
175	C₂×#100	—	2	2	no	—
176	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
178	#332	—	2	2	no	—
179	Q₈⋊C₈	—	2	2	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	16	4	no	—
186	C₂²×C₁₃	#392	1	1	yes	characteristic
187	C₂²×C₁₃	#461	1	2	yes	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	1	no	—
192	C₄×C₁₃	—	8	1	no	—
193	C₄×C₁₃	#448	1	2	yes	—
194	C₄×C₁₃	#453	1	2	yes	—
195	C₄×C₁₃	#471	1	2	yes	—
196	C₄×C₁₃	#492	1	1	yes	characteristic
197	C₄×C₁₃	#497	1	1	yes	characteristic
198	C₂²×Q₈	—	2	1	no	—
199	C₂²×D₄	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	D₄×C₁₃	1	2	yes	—
202	C₂×SD₁₆	—	2	8	no	—
203	C₂×SD₁₆	—	2	4	no	—
204	C₂×SD₁₆	—	4	8	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
207	C₂²×C₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
212	C₄⋊Q₈	D₄×C₁₃	1	1	yes	characteristic
213	C₄⋊Q₈	D₄×C₁₃	1	1	yes	characteristic
214	#127	—	2	1	no	—
215	#127	—	2	1	no	—
216	#127	—	2	1	no	—
217	#127	—	2	1	no	—
218	#125	—	2	1	no	—
219	C₂²⋊Q₈	—	2	1	no	—
220	C₂²⋊Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	—	2	1	no	—
223	#117	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
227	#107	—	2	4	no	—
228	#106	—	2	4	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
232	#102	—	2	4	no	—
233	#100	—	4	4	no	—
234	C₂²⋊C₈	—	2	1	no	—
235	#97	—	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	2	yes	—
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	—	16	2	no	—
242	C₂×C₁₃	#2002	1	1	yes	characteristic
243	C₂×C₁₃	#2013	1	1	yes	characteristic
244	C₂×C₁₃	#2019	1	1	yes	characteristic
245	C₂⁴	—	2	2	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	8	2	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
264	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
265	SD₁₆	—	4	16	no	—
266	SD₁₆	—	8	4	no	—
267	M₁₆	—	2	2	no	—
268	M₁₆	—	4	2	no	—
269	M₁₆	C₁₃×SD₁₆	1	4	yes	complemented
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₄²	—	4	1	no	—
287	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
291	C₁₃	#10153	1	1	yes	characteristic, direct product factor, Sylow
292	C₂³	—	4	2	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	8	2	no	—
295	C₂³	—	8	2	no	—
296	C₂³	C₁₃×#121	1	1	yes	characteristic
297	Q₈	—	4	4	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	4	2	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	4	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	1	no	—
307	D₄	—	8	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
315	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
316	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
317	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
318	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
319	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
320	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
321	C₂×C₄	C₁₃×#137	1	2	yes	—
322	C₈	—	2	4	no	—
323	C₈	—	2	4	no	—
324	C₈	—	4	2	no	—
325	C₈	—	8	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	16	4	no	—
332	C₂²	C₁₃×#392	1	1	yes	characteristic
333	C₂²	C₁₃×#461	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	C₁₃×#448	1	2	yes	—
340	C₄	C₁₃×#453	1	2	yes	—
341	C₄	C₁₃×#471	1	2	yes	—
342	C₄	C₁₃×#492	1	1	yes	characteristic
343	C₄	C₁₃×#497	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	16	2	no	—
346	C₂	C₁₃×#2002	1	1	yes	characteristic
347	C₂	C₁₃×#2013	1	1	yes	characteristic
348	C₂	C₁₃×#2019	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
2. C₁₃×#3165	C₂	semidirect
5. C₁₃×#1549	C₂	semidirect
6. C₁₃×#1493	C₂	semidirect
7. C₁₃×#1278	C₂	semidirect
71. #10153	C₁₃	direct (N × H)
96. C₁₃×M₁₆	SD₁₆	semidirect
119. #3165	C₂×C₁₃	semidirect
122. #1549	C₂×C₁₃	semidirect
123. #1493	C₂×C₁₃	semidirect
124. #1278	C₂×C₁₃	semidirect
269. M₁₆	C₁₃×SD₁₆	semidirect
291. C₁₃	#10153	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×#3208	2
C₁₃×#3165	2
C₂×C₁₃×#492	2
C₁₃×#1965	2
C₁₃×#1549	2
C₁₃×#1493	2
C₁₃×#1278	2
#10153	13

Subgroups — C13×#10153 #15696492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10153 #15696492