Subgroups — C₁₃×#29202 #15715541

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2647	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2327	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2068	C₂	1	2	yes	maximal
4	C₁₃×#2066	C₂	1	2	yes	maximal
5	C₁₃×#1960	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1956	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1865	C₂	1	2	yes	maximal
8	C₁₃×#1863	C₂	1	2	yes	maximal
9	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#1836	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1768	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#103	C₂²	1	2	yes	—
19	C₂×C₁₃×#103	C₂²	1	2	yes	—
20	C₂×C₁₃×#103	C₂²	1	2	yes	—
21	C₂×C₁₃×#103	C₂²	1	2	yes	—
22	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#400	—	2	2	no	—
25	C₁₃×#398	—	2	2	no	—
26	C₁₃×#391	C₂²	1	2	yes	—
27	C₁₃×#389	C₂²	1	2	yes	—
28	C₁₃×#389	C₂²	1	2	yes	—
29	C₁₃×#387	—	2	2	no	—
30	C₁₃×#387	—	2	2	no	—
31	C₁₃×#384	C₂²	1	1	yes	characteristic
32	C₁₃×#382	C₂²	1	1	yes	characteristic
33	C₁₃×#382	C₂²	1	1	yes	characteristic
34	C₁₃×#376	C₂²	1	1	yes	characteristic
35	C₁₃×#340	C₂²	1	2	yes	—
36	C₁₃×#340	C₂²	1	2	yes	—
37	C₁₃×#336	C₂²	1	1	yes	characteristic
38	C₁₃×#336	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	—	2	2	no	—
45	C₁₃×C₄⋊Q₈	Q₈	1	4	yes	—
46	C₄×Q₈×C₁₃	—	4	2	no	—
47	C₄×Q₈×C₁₃	—	4	2	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₄²×C₁₃	—	2	2	no	—
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×#107	—	4	2	no	—
61	C₁₃×#106	—	4	2	no	—
62	C₁₃×C₄⋊C₈	—	2	2	no	—
63	C₁₃×C₄⋊C₈	—	2	2	no	—
64	C₁₃×#103	—	2	4	no	—
65	C₁₃×#103	—	2	4	no	—
66	C₁₃×#103	—	4	2	no	—
67	C₁₃×#103	—	4	2	no	—
68	C₁₃×#97	—	2	2	no	—
69	C₄×C₈×C₁₃	—	2	2	no	—
70	C₁₃×#95	—	2	2	no	—
71	C₁₃×#95	—	2	2	no	—
72	C₁₃×#95	—	2	2	no	—
73	C₁₃×#95	—	2	2	no	—
74	C₁₃×#95	—	2	2	no	—
75	C₁₃×#95	—	2	2	no	—
76	C₁₃×#95	D₄	1	2	yes	—
77	C₁₃×#95	D₄	1	2	yes	—
78	#29202	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×Q₈×C₁₃	—	2	2	no	—
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
85	C₂²×C₄×C₁₃	—	2	2	no	—
86	C₂²×C₄×C₁₃	—	2	2	no	—
87	C₂²×C₄×C₁₃	—	2	2	no	—
88	C₂²×C₄×C₁₃	—	2	2	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	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₂×C₈×C₁₃	—	2	1	no	—
95	C₂×C₈×C₁₃	—	2	1	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	2	1	no	—
98	C₂×C₈×C₁₃	—	2	1	no	—
99	C₂×C₈×C₁₃	—	2	1	no	—
100	C₂×C₈×C₁₃	—	2	1	no	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₂×C₈×C₁₃	—	2	1	no	—
104	C₂×C₈×C₁₃	#41	1	2	yes	—
105	C₂×C₈×C₁₃	#41	1	2	yes	—
106	C₁₃×C₄⋊C₄	—	2	8	no	—
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	2	2	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	—	4	4	no	—
114	C₁₃×C₄⋊C₄	—	8	2	no	—
115	C₁₃×C₄⋊C₄	—	8	2	no	—
116	C₁₃×C₄⋊C₄	—	8	2	no	—
117	C₄²×C₁₃	—	2	1	no	—
118	C₄²×C₁₃	—	2	1	no	—
119	C₄²×C₁₃	—	2	1	no	—
120	C₄²×C₁₃	—	2	1	no	—
121	C₄²×C₁₃	—	4	4	no	—
122	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
123	C₄²×C₁₃	#41	1	2	yes	—
124	#2647	C₂×C₁₃	1	1	yes	characteristic
125	#2327	C₂×C₁₃	1	1	yes	characteristic
126	#2068	C₂×C₁₃	1	2	yes	—
127	#2066	C₂×C₁₃	1	2	yes	—
128	#1960	C₂×C₁₃	1	1	yes	characteristic
129	#1956	C₂×C₁₃	1	1	yes	characteristic
130	#1865	C₂×C₁₃	1	2	yes	—
131	#1863	C₂×C₁₃	1	2	yes	—
132	#1846	C₂×C₁₃	1	1	yes	characteristic
133	#1836	C₂×C₁₃	1	1	yes	characteristic
134	#1768	C₂×C₁₃	1	1	yes	characteristic
135	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	Q₈×C₁₃	—	2	4	no	—
137	Q₈×C₁₃	—	2	4	no	—
138	Q₈×C₁₃	—	4	2	no	—
139	Q₈×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	4	no	—
143	C₂×C₄×C₁₃	—	2	1	no	—
144	C₂×C₄×C₁₃	—	2	1	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	—	4	2	no	—
159	C₂×C₄×C₁₃	—	4	2	no	—
160	C₂×C₄×C₁₃	—	4	2	no	—
161	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
162	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
168	C₈×C₁₃	—	2	2	no	—
169	C₈×C₁₃	—	2	2	no	—
170	C₈×C₁₃	—	4	1	no	—
171	C₈×C₁₃	—	4	1	no	—
172	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
173	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
174	C₂×#107	C₂²×C₁₃	1	2	yes	—
175	C₂×#106	C₂²×C₁₃	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
178	C₂×#103	C₂²×C₁₃	1	2	yes	—
179	C₂×#103	C₂²×C₁₃	1	2	yes	—
180	C₂×#103	C₂²×C₁₃	1	2	yes	—
181	C₂×#103	C₂²×C₁₃	1	2	yes	—
182	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
184	#400	—	2	2	no	—
185	#398	—	2	2	no	—
186	#391	C₂²×C₁₃	1	2	yes	—
187	#389	C₂²×C₁₃	1	2	yes	—
188	#389	C₂²×C₁₃	1	2	yes	—
189	#387	—	2	2	no	—
190	#387	—	2	2	no	—
191	#384	C₂²×C₁₃	1	1	yes	characteristic
192	#382	C₂²×C₁₃	1	1	yes	characteristic
193	#382	C₂²×C₁₃	1	1	yes	characteristic
194	#376	C₂²×C₁₃	1	1	yes	characteristic
195	#340	C₂²×C₁₃	1	2	yes	—
196	#340	C₂²×C₁₃	1	2	yes	—
197	#336	C₂²×C₁₃	1	1	yes	characteristic
198	#336	C₂²×C₁₃	1	1	yes	characteristic
199	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
200	C₂²×C₁₃	#532	1	1	yes	characteristic
201	C₂²×C₁₃	#535	1	1	yes	characteristic
202	C₂²×C₁₃	#577	1	2	yes	—
203	C₂²×C₁₃	#578	1	2	yes	—
204	C₄×C₁₃	—	2	4	no	—
205	C₄×C₁₃	—	4	4	no	—
206	C₄×C₁₃	—	4	1	no	—
207	C₄×C₁₃	—	4	1	no	—
208	C₄×C₁₃	—	4	2	no	—
209	C₄×C₁₃	—	8	2	no	—
210	C₄×C₁₃	—	8	2	no	—
211	C₄×C₁₃	—	8	2	no	—
212	C₄×C₁₃	#553	1	1	yes	characteristic
213	C₄×C₁₃	#553	1	1	yes	characteristic
214	C₄×C₁₃	#554	1	1	yes	characteristic
215	C₄×C₁₃	#554	1	1	yes	characteristic
216	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
217	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	Q₈×C₁₃	1	4	yes	—
223	C₄×Q₈	—	4	2	no	—
224	C₄×Q₈	—	4	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
231	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
232	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
237	#107	—	4	2	no	—
238	#106	—	4	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	2	no	—
241	#103	—	2	4	no	—
242	#103	—	2	4	no	—
243	#103	—	4	2	no	—
244	#103	—	4	2	no	—
245	#97	—	2	2	no	—
246	C₄×C₈	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	—	2	2	no	—
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	#95	—	2	2	no	—
252	#95	—	2	2	no	—
253	#95	D₄×C₁₃	1	2	yes	—
254	#95	D₄×C₁₃	1	2	yes	—
255	C₂×C₁₃	#2566	1	1	yes	characteristic
256	C₂×C₁₃	#3082	1	2	yes	—
257	C₂×C₁₃	#3174	1	1	yes	characteristic
258	C₂×C₁₃	#3177	1	1	yes	characteristic
259	C₂×C₁₃	#3225	1	2	yes	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	C₂×Q₈×C₁₃	1	4	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
273	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
274	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	C₁₃×#41	1	2	yes	—
286	C₂×C₈	C₁₃×#41	1	2	yes	—
287	C₄⋊C₄	—	2	8	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	4	4	no	—
295	C₄⋊C₄	—	8	2	no	—
296	C₄⋊C₄	—	8	2	no	—
297	C₄⋊C₄	—	8	2	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	4	4	no	—
303	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
304	C₄²	C₁₃×#41	1	2	yes	—
305	C₁₃	#29202	1	1	yes	characteristic, direct product factor, Sylow
306	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	2	4	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
333	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
334	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
335	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
336	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
337	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
338	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	4	1	no	—
342	C₈	—	4	1	no	—
343	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
344	C₂²	C₁₃×#532	1	1	yes	characteristic
345	C₂²	C₁₃×#535	1	1	yes	characteristic
346	C₂²	C₁₃×#577	1	2	yes	—
347	C₂²	C₁₃×#578	1	2	yes	—
348	C₄	—	2	4	no	—
349	C₄	—	4	4	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	C₁₃×#553	1	1	yes	characteristic
357	C₄	C₁₃×#553	1	1	yes	characteristic
358	C₄	C₁₃×#554	1	1	yes	characteristic
359	C₄	C₁₃×#554	1	1	yes	characteristic
360	C₂	C₁₃×#2566	1	1	yes	characteristic
361	C₂	C₁₃×#3082	1	2	yes	—
362	C₂	C₁₃×#3174	1	1	yes	characteristic
363	C₂	C₁₃×#3177	1	1	yes	characteristic
364	C₂	C₁₃×#3225	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
78. #29202	C₁₃	direct (N × H)
305. C₁₃	#29202	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×#2647	2
C₁₃×#2327	2
C₁₃×#2068	2
C₁₃×#2066	2
C₁₃×#1960	2
C₁₃×#1956	2
C₁₃×#1865	2
C₁₃×#1863	2
C₁₃×#1846	2
C₁₃×#1836	2
C₁₃×#1768	2
#29202	13

Subgroups — C13×#29202 #15715541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29202 #15715541