Subgroups — C₁₃×#33805 #15720144

Order 3328.

496 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×#3168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2255	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2243	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2236	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2233	C₂	1	2	yes	maximal
7	C₁₃×#2225	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2224	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
11	C₁₃×#529	—	2	1	no	—
12	C₁₃×#500	—	2	1	no	—
13	C₁₃×#500	—	2	1	no	—
14	C₁₃×#500	C₂²	1	2	yes	—
15	C₁₃×#500	C₂²	1	2	yes	—
16	C₁₃×#499	C₂²	1	1	yes	characteristic
17	C₁₃×#495	C₂²	1	1	yes	characteristic
18	C₁₃×#488	—	2	1	no	—
19	C₁₃×#467	C₂²	1	1	yes	characteristic
20	C₁₃×#466	C₂²	1	1	yes	characteristic
21	C₁₃×#465	—	2	1	no	—
22	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
23	C₄×C₁₃×SD₁₆	—	2	1	no	—
24	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
25	C₂×C₁₃×#107	—	2	1	no	—
26	C₂×C₁₃×#107	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#367	C₂²	1	4	yes	—
29	C₁₃×#366	C₂²	1	4	yes	—
30	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₁₃×#358	C₂²	1	4	yes	—
32	C₁₃×#357	C₂²	1	4	yes	—
33	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₁₃×#346	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	—	4	1	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
37	C₂×C₁₃×SD₁₆	—	2	1	no	—
38	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊Q₈	—	2	2	no	—
42	C₁₃×C₄⋊Q₈	—	2	2	no	—
43	C₁₃×C₄⋊Q₈	—	4	1	no	—
44	C₁₃×C₄⋊Q₈	—	4	1	no	—
45	C₁₃×#126	—	4	1	no	—
46	C₁₃×#125	—	2	1	no	—
47	C₁₃×#124	—	2	1	no	—
48	C₁₃×#123	—	4	1	no	—
49	C₁₃×C₂²⋊Q₈	—	2	1	no	—
50	C₁₃×#121	—	2	1	no	—
51	C₄×Q₈×C₁₃	—	2	1	no	—
52	C₄×D₄×C₁₃	—	2	1	no	—
53	C₄×D₄×C₁₃	—	4	1	no	—
54	C₁₃×#117	—	4	1	no	—
55	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
57	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
59	C₁₃×#107	—	2	2	no	—
60	C₁₃×#107	—	2	2	no	—
61	C₁₃×#107	—	2	2	no	—
62	C₁₃×#107	C₂³	1	4	yes	—
63	C₁₃×#107	C₂³	1	1	yes	characteristic
64	C₁₃×#107	C₂³	1	1	yes	characteristic
65	C₁₃×#106	—	2	1	no	—
66	C₁₃×#103	—	2	1	no	—
67	C₁₃×#103	—	2	1	no	—
68	C₁₃×#102	—	2	1	no	—
69	C₁₃×#102	—	2	1	no	—
70	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	#33805	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×Q₈×C₁₃	—	2	1	no	—
76	C₂×Q₈×C₁₃	—	4	2	no	—
77	C₂×Q₈×C₁₃	—	4	2	no	—
78	C₂×Q₈×C₁₃	—	8	4	no	—
79	C₂×D₄×C₁₃	—	2	1	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	—	4	2	no	—
82	C₂²×C₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₁₃×Q₁₆	—	2	2	no	—
87	C₁₃×SD₁₆	—	4	2	no	—
88	C₁₃×D₈	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	2	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	4	1	no	—
106	C₁₃×C₂²⋊C₄	—	4	1	no	—
107	C₁₃×C₂²⋊C₄	—	4	1	no	—
108	C₁₃×C₂²⋊C₄	—	4	1	no	—
109	C₁₃×C₂²⋊C₄	—	4	1	no	—
110	C₄²×C₁₃	—	4	1	no	—
111	C₄²×C₁₃	D₈	1	1	yes	characteristic
112	C₄²×C₁₃	D₈	1	1	yes	characteristic
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₁₃×C₁₆	—	2	4	no	—
116	#3168	C₂×C₁₃	1	1	yes	characteristic
117	C₂×#500	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#2255	C₂×C₁₃	1	2	yes	complemented
119	#2243	C₂×C₁₃	1	2	yes	complemented
120	#2236	C₂×C₁₃	1	2	yes	complemented
121	#2233	C₂×C₁₃	1	2	yes	—
122	#2225	C₂×C₁₃	1	2	yes	complemented
123	#2224	C₂×C₁₃	1	2	yes	complemented
124	#2163	C₂×C₁₃	1	1	yes	characteristic, complemented
125	C₂³×C₁₃	—	4	1	no	—
126	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₁₃	—	4	1	no	—
128	Q₈×C₁₃	—	4	1	no	—
129	Q₈×C₁₃	—	8	4	no	—
130	Q₈×C₁₃	—	8	4	no	—
131	D₄×C₁₃	—	4	1	no	—
132	D₄×C₁₃	—	4	1	no	—
133	D₄×C₁₃	—	8	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	8	4	no	—
143	C₂×C₄×C₁₃	—	8	2	no	—
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	#137	1	2	yes	—
153	C₈×C₁₃	2₋¹⁺⁴	1	2	yes	—
154	C₂×C₄⋊Q₈	—	2	1	no	—
155	#529	—	2	1	no	—
156	#500	—	2	1	no	—
157	#500	—	2	1	no	—
158	#500	C₂²×C₁₃	1	2	yes	—
159	#500	C₂²×C₁₃	1	2	yes	—
160	#499	C₂²×C₁₃	1	1	yes	characteristic
161	#495	C₂²×C₁₃	1	1	yes	characteristic
162	#488	—	2	1	no	—
163	#467	C₂²×C₁₃	1	1	yes	characteristic
164	#466	C₂²×C₁₃	1	1	yes	characteristic
165	#465	—	2	1	no	—
166	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
167	C₄×SD₁₆	—	2	1	no	—
168	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×#107	—	2	1	no	—
170	C₂×#107	C₂²×C₁₃	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
172	#367	C₂²×C₁₃	1	4	yes	—
173	#366	C₂²×C₁₃	1	4	yes	—
174	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
175	#358	C₂²×C₁₃	1	4	yes	—
176	#357	C₂²×C₁₃	1	4	yes	—
177	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
178	#346	C₂²×C₁₃	1	2	yes	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	8	1	no	—
182	C₂²×C₁₃	—	8	1	no	—
183	C₂²×C₁₃	—	8	1	no	—
184	C₂²×C₁₃	#510	1	2	yes	—
185	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	8	1	no	—
191	C₄×C₁₃	—	8	4	no	—
192	C₄×C₁₃	—	8	1	no	—
193	C₄×C₁₃	—	8	1	no	—
194	C₄×C₁₃	#509	1	2	yes	—
195	C₄×C₁₃	#536	1	1	yes	characteristic
196	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
197	C₂²×Q₈	—	4	1	no	—
198	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
199	C₂×SD₁₆	—	2	1	no	—
200	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	4	1	no	—
206	C₄⋊Q₈	—	4	1	no	—
207	#126	—	4	1	no	—
208	#125	—	2	1	no	—
209	#124	—	2	1	no	—
210	#123	—	4	1	no	—
211	C₂²⋊Q₈	—	2	1	no	—
212	#121	—	2	1	no	—
213	C₄×Q₈	—	2	1	no	—
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	4	1	no	—
216	#117	—	4	1	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	4	1	no	—
219	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
220	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
221	#107	—	2	2	no	—
222	#107	—	2	2	no	—
223	#107	—	2	2	no	—
224	#107	C₂³×C₁₃	1	4	yes	—
225	#107	C₂³×C₁₃	1	1	yes	characteristic
226	#107	C₂³×C₁₃	1	1	yes	characteristic
227	#106	—	2	1	no	—
228	#103	—	2	1	no	—
229	#103	—	2	1	no	—
230	#102	—	2	1	no	—
231	#102	—	2	1	no	—
232	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
233	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
236	C₂×C₁₃	—	2	2	no	—
237	C₂×C₁₃	—	16	1	no	—
238	C₂×C₁₃	#3226	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
240	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	8	4	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
250	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
252	Q₁₆	—	2	2	no	—
253	SD₁₆	—	4	2	no	—
254	D₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	C₁₃×D₈	1	1	yes	characteristic
278	C₄²	C₁₃×D₈	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
281	C₁₆	—	2	4	no	—
282	C₁₃	#33805	1	1	yes	characteristic, direct product factor, Sylow
283	C₂³	—	4	1	no	—
284	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
285	Q₈	—	4	1	no	—
286	Q₈	—	4	1	no	—
287	Q₈	—	8	4	no	—
288	Q₈	—	8	4	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	1	no	—
291	D₄	—	8	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
304	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
305	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
306	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	C₁₃×#137	1	2	yes	—
311	C₈	C₁₃×2₋¹⁺⁴	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	C₁₃×#510	1	2	yes	—
318	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	4	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	1	no	—
327	C₄	C₁₃×#509	1	2	yes	—
328	C₄	C₁₃×#536	1	1	yes	characteristic
329	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	16	1	no	—
332	C₂	C₁₃×#3226	1	1	yes	characteristic
333	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
334	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#500	C₂	semidirect
3. C₁₃×#2255	C₂	semidirect
4. C₁₃×#2243	C₂	semidirect
5. C₁₃×#2236	C₂	semidirect
7. C₁₃×#2225	C₂	semidirect
8. C₁₃×#2224	C₂	semidirect
9. C₁₃×#2163	C₂	semidirect
74. #33805	C₁₃	direct (N × H)
117. C₂×#500	C₂×C₁₃	semidirect
118. #2255	C₂×C₁₃	semidirect
119. #2243	C₂×C₁₃	semidirect
120. #2236	C₂×C₁₃	semidirect
122. #2225	C₂×C₁₃	semidirect
123. #2224	C₂×C₁₃	semidirect
124. #2163	C₂×C₁₃	semidirect
282. C₁₃	#33805	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×#3168	2
C₂×C₁₃×#500	2
C₁₃×#2255	2
C₁₃×#2243	2
C₁₃×#2236	2
C₁₃×#2233	2
C₁₃×#2225	2
C₁₃×#2224	2
C₁₃×#2163	2
#33805	13

Subgroups — C13×#33805 #15720144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33805 #15720144