Subgroups — C₁₃×#13073 #15699412

Order 3328.

478 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#472	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1816	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×AGL(1,ℤ/8)	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	C₁₃×#504	—	2	2	no	—
12	C₁₃×#504	—	2	2	no	—
13	C₂×C₁₃×M₃₂	—	2	1	no	—
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	—	2	1	no	—
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#472	—	2	2	no	—
18	C₁₃×#472	C₄	1	4	yes	complemented
19	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
22	C₁₃×#417	—	2	1	no	—
23	C₂×C₁₃×#100	—	2	1	no	—
24	C₁₃×#361	—	2	2	no	—
25	C₁₃×#357	—	2	1	no	—
26	C₁₃×#357	—	2	1	no	—
27	C₁₃×#343	—	2	1	no	—
28	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₁₃	—	2	1	no	—
30	C₁₃×AGL(1,ℤ/8)	—	2	4	no	—
31	C₁₃×AGL(1,ℤ/8)	—	4	2	no	—
32	C₂×C₁₃×SD₁₆	—	4	1	no	—
33	C₂×C₁₃×D₈	—	2	1	no	—
34	C₂×C₁₃×D₈	—	2	1	no	—
35	C₂×C₁₃×D₈	—	2	1	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	—	4	1	no	—
39	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
40	C₁₃×#131	—	2	1	no	—
41	C₁₃×#131	—	2	1	no	—
42	C₁₃×#131	C₂×C₄	1	2	yes	—
43	C₁₃×#131	D₄	1	2	yes	—
44	C₂×C₁₃×M₁₆	—	2	1	no	—
45	C₂×C₁₃×M₁₆	—	2	1	no	—
46	C₂×C₁₃×M₁₆	—	2	1	no	—
47	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	—	2	1	no	—
49	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
51	C₁₃×M₃₂	—	2	4	no	—
52	C₁₃×M₃₂	—	2	1	no	—
53	C₁₃×M₃₂	D₄	1	2	yes	—
54	C₂×C₁₃×C₁₆	—	2	4	no	—
55	C₂×C₁₃×C₁₆	—	2	1	no	—
56	C₂×C₁₃×C₁₆	—	2	1	no	—
57	C₂×C₁₃×C₁₆	—	2	1	no	—
58	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
60	C₁₃×#108	—	2	1	no	—
61	C₁₃×#108	—	2	1	no	—
62	C₁₃×#108	—	2	1	no	—
63	C₁₃×#108	C₂×C₄	1	2	yes	—
64	C₁₃×#107	—	2	1	no	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#103	—	4	1	no	—
67	C₁₃×#102	—	4	1	no	—
68	C₁₃×#100	—	2	4	no	—
69	#13073	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂⁴×C₁₃	—	4	1	no	—
71	C₁₃×#41	—	2	1	no	—
72	C₁₃×#41	—	2	1	no	—
73	C₁₃×#41	C₂²⋊C₄	1	2	yes	—
74	C₁₃×#41	SD₁₆	1	2	yes	—
75	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
76	C₂×D₄×C₁₃	—	2	1	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	4	1	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	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	—	2	1	no	—
85	C₂²×C₄×C₁₃	—	4	1	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₁₃×SD₁₆	—	4	4	no	—
88	C₁₃×D₈	—	2	2	no	—
89	C₁₃×D₈	—	2	2	no	—
90	C₁₃×D₈	—	4	4	no	—
91	C₁₃×D₈	—	4	1	no	—
92	C₁₃×D₈	—	4	1	no	—
93	C₁₃×M₁₆	—	2	1	no	—
94	C₁₃×M₁₆	—	2	2	no	—
95	C₁₃×M₁₆	—	2	2	no	—
96	C₁₃×M₁₆	—	2	2	no	—
97	C₁₃×M₁₆	—	4	1	no	—
98	C₁₃×M₁₆	—	4	1	no	—
99	C₁₃×M₁₆	—	4	1	no	—
100	C₁₃×M₁₆	C₂²⋊C₄	1	2	yes	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	4	1	no	—
105	C₂×C₈×C₁₃	—	4	1	no	—
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
111	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₄⋊C₄	—	4	1	no	—
114	C₁₃×C₁₆	—	2	1	no	—
115	C₁₃×C₁₆	—	2	2	no	—
116	C₁₃×C₁₆	—	2	1	no	—
117	C₂×#504	C₂×C₁₃	1	1	yes	characteristic, complemented
118	C₂×#472	C₂×C₁₃	1	1	yes	characteristic
119	C₂×#361	C₂×C₁₃	1	1	yes	characteristic, complemented
120	C₂×#357	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#1816	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#1387	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#1381	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂³×C₁₃	—	2	1	no	—
125	C₂³×C₁₃	—	4	2	no	—
126	C₂³×C₁₃	—	8	1	no	—
127	C₂³×C₁₃	—	8	1	no	—
128	C₂³×C₁₃	—	16	1	no	—
129	C₂³×C₁₃	—	16	1	no	—
130	C₂³×C₁₃	#135	1	1	yes	characteristic
131	Q₈×C₁₃	—	2	1	no	—
132	Q₈×C₁₃	#102	1	2	yes	—
133	D₄×C₁₃	—	2	2	no	—
134	D₄×C₁₃	—	2	2	no	—
135	D₄×C₁₃	—	2	1	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	8	1	no	—
139	D₄×C₁₃	—	8	1	no	—
140	D₄×C₁₃	#102	1	2	yes	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	8	1	no	—
145	C₂×C₄×C₁₃	—	16	1	no	—
146	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
152	C₈×C₁₃	—	2	2	no	—
153	C₈×C₁₃	—	4	2	no	—
154	C₈×C₁₃	—	8	1	no	—
155	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
156	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
157	C₈×C₁₃	#120	1	1	yes	characteristic
158	C₈×C₁₃	#123	1	1	yes	characteristic
159	C₂×AGL(1,ℤ/8)	—	2	1	no	—
160	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
162	#504	—	2	2	no	—
163	#504	—	2	2	no	—
164	C₂×M₃₂	—	2	1	no	—
165	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
166	C₂²×C₁₆	—	2	1	no	—
167	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
168	#472	—	2	2	no	—
169	#472	C₄×C₁₃	1	4	yes	complemented
170	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
171	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
173	#417	—	2	1	no	—
174	C₂×#100	—	2	1	no	—
175	#361	—	2	2	no	—
176	#357	—	2	1	no	—
177	#357	—	2	1	no	—
178	#343	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	4	1	no	—
183	C₂²×C₁₃	—	4	1	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	16	2	no	—
186	C₂²×C₁₃	—	16	2	no	—
187	C₂²×C₁₃	—	16	1	no	—
188	C₂²×C₁₃	—	16	1	no	—
189	C₂²×C₁₃	#416	1	1	yes	characteristic
190	C₂²×C₁₃	#465	1	1	yes	characteristic
191	C₂²×C₁₃	#466	1	1	yes	characteristic
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	16	1	no	—
194	C₄×C₁₃	#386	1	1	yes	characteristic
195	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
196	C₄×C₁₃	#465	1	1	yes	characteristic
197	C₄×C₁₃	#466	1	1	yes	characteristic
198	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×D₄	—	2	1	no	—
200	AGL(1,ℤ/8)	—	2	4	no	—
201	AGL(1,ℤ/8)	—	4	2	no	—
202	C₂×SD₁₆	—	4	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	—	4	1	no	—
209	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
210	#131	—	2	1	no	—
211	#131	—	2	1	no	—
212	#131	C₂×C₄×C₁₃	1	2	yes	—
213	#131	D₄×C₁₃	1	2	yes	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
218	C₂²×C₈	—	2	1	no	—
219	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
220	C₂×C₄⋊C₄	—	2	1	no	—
221	M₃₂	—	2	4	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	D₄×C₁₃	1	2	yes	—
224	C₂×C₁₆	—	2	4	no	—
225	C₂×C₁₆	—	2	1	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
229	C₂×C₁₆	D₄×C₁₃	1	1	yes	characteristic
230	#108	—	2	1	no	—
231	#108	—	2	1	no	—
232	#108	—	2	1	no	—
233	#108	C₂×C₄×C₁₃	1	2	yes	—
234	#107	—	2	1	no	—
235	#107	—	2	1	no	—
236	#103	—	4	1	no	—
237	#102	—	4	1	no	—
238	#100	—	2	4	no	—
239	C₂×C₁₃	—	2	1	no	—
240	C₂×C₁₃	—	2	1	no	—
241	C₂×C₁₃	—	4	2	no	—
242	C₂×C₁₃	—	16	2	no	—
243	C₂×C₁₃	#1893	1	1	yes	characteristic
244	C₂×C₁₃	#2221	1	1	yes	characteristic
245	C₂×C₁₃	#2223	1	1	yes	characteristic
246	C₂⁴	—	4	1	no	—
247	#41	—	2	1	no	—
248	#41	—	2	1	no	—
249	#41	C₁₃×C₂²⋊C₄	1	2	yes	—
250	#41	C₁₃×SD₁₆	1	2	yes	—
251	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	2	2	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	1	no	—
259	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
263	SD₁₆	—	4	4	no	—
264	D₈	—	2	2	no	—
265	D₈	—	2	2	no	—
266	D₈	—	4	4	no	—
267	D₈	—	4	1	no	—
268	D₈	—	4	1	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	4	1	no	—
276	M₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
285	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
286	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
287	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	2	2	no	—
292	C₁₆	—	2	1	no	—
293	C₁₃	#13073	1	1	yes	characteristic, direct product factor, Sylow
294	C₂³	—	2	1	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	8	1	no	—
297	C₂³	—	8	1	no	—
298	C₂³	—	16	1	no	—
299	C₂³	—	16	1	no	—
300	C₂³	C₁₃×#135	1	1	yes	characteristic
301	Q₈	—	2	1	no	—
302	Q₈	C₁₃×#102	1	2	yes	—
303	D₄	—	2	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	2	1	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	1	no	—
309	D₄	—	8	1	no	—
310	D₄	C₁₃×#102	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	16	1	no	—
316	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	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	C₂×C₁₃×D₈	1	1	yes	characteristic
320	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
321	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	4	2	no	—
324	C₈	—	8	1	no	—
325	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
326	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
327	C₈	C₁₃×#120	1	1	yes	characteristic
328	C₈	C₁₃×#123	1	1	yes	characteristic
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	16	2	no	—
336	C₂²	—	16	2	no	—
337	C₂²	—	16	1	no	—
338	C₂²	—	16	1	no	—
339	C₂²	C₁₃×#416	1	1	yes	characteristic
340	C₂²	C₁₃×#465	1	1	yes	characteristic
341	C₂²	C₁₃×#466	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	16	1	no	—
344	C₄	C₁₃×#386	1	1	yes	characteristic
345	C₄	C₂×C₁₃×#102	1	1	yes	characteristic
346	C₄	C₁₃×#465	1	1	yes	characteristic
347	C₄	C₁₃×#466	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	4	2	no	—
351	C₂	—	16	2	no	—
352	C₂	C₁₃×#1893	1	1	yes	characteristic
353	C₂	C₁₃×#2221	1	1	yes	characteristic
354	C₂	C₁₃×#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₁₃×#504	C₂	semidirect
3. C₂×C₁₃×#361	C₂	semidirect
4. C₂×C₁₃×#357	C₂	semidirect
5. C₁₃×#1816	C₂	semidirect
6. C₁₃×#1387	C₂	semidirect
7. C₁₃×#1381	C₂	semidirect
18. C₁₃×#472	C₄	semidirect
69. #13073	C₁₃	direct (N × H)
117. C₂×#504	C₂×C₁₃	semidirect
119. C₂×#361	C₂×C₁₃	semidirect
120. C₂×#357	C₂×C₁₃	semidirect
121. #1816	C₂×C₁₃	semidirect
122. #1387	C₂×C₁₃	semidirect
123. #1381	C₂×C₁₃	semidirect
169. #472	C₄×C₁₃	semidirect
293. C₁₃	#13073	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#504	2
C₂×C₁₃×#472	2
C₂×C₁₃×#361	2
C₂×C₁₃×#357	2
C₁₃×#1816	2
C₁₃×#1387	2
C₁₃×#1381	2
#13073	13

Subgroups — C13×#13073 #15699412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13073 #15699412