Subgroups — C₁₃×#32563 #15718902

Order 3328.

650 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2582	C₂	1	2	yes	maximal
2	C₁₃×#2099	C₂	1	2	yes	maximal
3	C₁₃×#2086	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2072	C₂	1	2	yes	maximal
5	C₁₃×#2056	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1985	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1935	C₂	1	2	yes	maximal
8	C₁₃×#1878	C₂	1	2	yes	maximal
9	C₁₃×#1863	C₂	1	2	yes	maximal
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#103	C₂²	1	2	yes	—
16	C₂×C₁₃×#103	C₂²	1	2	yes	—
17	C₂×C₁₃×#103	C₂²	1	2	yes	—
18	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
19	C₁₃×#398	—	2	2	no	—
20	C₁₃×#398	—	2	2	no	—
21	C₁₃×#391	—	2	2	no	—
22	C₁₃×#391	C₂²	1	2	yes	—
23	C₁₃×#391	C₂²	1	2	yes	—
24	C₁₃×#389	—	2	2	no	—
25	C₁₃×#389	C₂²	1	2	yes	—
26	C₁₃×#389	C₂²	1	2	yes	—
27	C₁₃×#384	C₂²	1	2	yes	—
28	C₁₃×#340	C₂²	1	1	yes	characteristic
29	C₁₃×#340	C₂²	1	1	yes	characteristic
30	C₁₃×#340	C₂²	1	2	yes	—
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
37	C₂×C₁₃×Q₁₆	—	2	4	no	—
38	C₂×C₁₃×Q₁₆	—	4	1	no	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₁₃	—	4	1	no	—
43	C₄×Q₈×C₁₃	—	4	1	no	—
44	C₄×Q₈×C₁₃	—	4	2	no	—
45	C₄×Q₈×C₁₃	—	4	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	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	D₄	1	2	yes	—
52	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
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	2	yes	—
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₁₃×#107	—	4	1	no	—
60	C₁₃×C₄⋊C₈	—	4	1	no	—
61	C₁₃×#103	—	2	4	no	—
62	C₁₃×#103	—	2	4	no	—
63	C₁₃×#103	—	4	2	no	—
64	C₁₃×#97	—	4	1	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	—	2	2	no	—
68	C₁₃×#95	—	2	2	no	—
69	C₁₃×#95	—	2	2	no	—
70	C₁₃×#95	—	2	2	no	—
71	#32563	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×Q₈×C₁₃	—	2	4	no	—
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	2	2	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×Q₈×C₁₃	#41	1	4	yes	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×Q₁₆	—	4	4	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	1	no	—
95	C₂×C₈×C₁₃	—	2	1	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	4	4	no	—
106	C₁₃×C₄⋊C₄	—	4	4	no	—
107	C₁₃×C₄⋊C₄	—	8	2	no	—
108	C₁₃×C₄⋊C₄	—	8	2	no	—
109	C₁₃×C₄⋊C₄	—	8	2	no	—
110	C₄²×C₁₃	—	2	1	no	—
111	C₄²×C₁₃	—	2	1	no	—
112	C₄²×C₁₃	—	2	2	no	—
113	C₄²×C₁₃	—	2	2	no	—
114	C₄²×C₁₃	—	4	4	no	—
115	#2582	C₂×C₁₃	1	2	yes	—
116	#2099	C₂×C₁₃	1	2	yes	—
117	#2086	C₂×C₁₃	1	1	yes	characteristic
118	#2072	C₂×C₁₃	1	2	yes	—
119	#2056	C₂×C₁₃	1	1	yes	characteristic
120	#1985	C₂×C₁₃	1	1	yes	characteristic
121	#1935	C₂×C₁₃	1	2	yes	—
122	#1878	C₂×C₁₃	1	2	yes	—
123	#1863	C₂×C₁₃	1	2	yes	—
124	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	Q₈×C₁₃	—	2	4	no	—
126	Q₈×C₁₃	—	2	4	no	—
127	Q₈×C₁₃	—	4	4	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	#137	1	4	yes	—
153	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
154	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
157	C₈×C₁₃	—	4	2	no	—
158	C₈×C₁₃	—	4	1	no	—
159	C₈×C₁₃	—	4	1	no	—
160	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
162	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
165	C₂×#103	C₂²×C₁₃	1	2	yes	—
166	C₂×#103	C₂²×C₁₃	1	2	yes	—
167	C₂×#103	C₂²×C₁₃	1	2	yes	—
168	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
169	#398	—	2	2	no	—
170	#398	—	2	2	no	—
171	#391	—	2	2	no	—
172	#391	C₂²×C₁₃	1	2	yes	—
173	#391	C₂²×C₁₃	1	2	yes	—
174	#389	—	2	2	no	—
175	#389	C₂²×C₁₃	1	2	yes	—
176	#389	C₂²×C₁₃	1	2	yes	—
177	#384	C₂²×C₁₃	1	2	yes	—
178	#340	C₂²×C₁₃	1	1	yes	characteristic
179	#340	C₂²×C₁₃	1	1	yes	characteristic
180	#340	C₂²×C₁₃	1	2	yes	—
181	#340	C₂²×C₁₃	1	2	yes	—
182	#340	C₂²×C₁₃	1	2	yes	—
183	#336	C₂²×C₁₃	1	2	yes	—
184	#336	C₂²×C₁₃	1	2	yes	—
185	C₂²×C₁₃	#534	1	1	yes	characteristic
186	C₂²×C₁₃	#548	1	2	yes	—
187	C₂²×C₁₃	C₂×#137	1	2	yes	—
188	C₂²×C₁₃	#576	1	1	yes	characteristic
189	C₂²×C₁₃	#578	1	1	yes	characteristic
190	C₄×C₁₃	—	2	4	no	—
191	C₄×C₁₃	—	4	1	no	—
192	C₄×C₁₃	—	4	2	no	—
193	C₄×C₁₃	—	4	1	no	—
194	C₄×C₁₃	—	4	4	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	#539	1	2	yes	—
199	C₄×C₁₃	#551	1	1	yes	characteristic
200	C₄×C₁₃	#563	1	1	yes	characteristic
201	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
202	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
203	C₂×Q₁₆	—	2	4	no	—
204	C₂×Q₁₆	—	4	1	no	—
205	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
207	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
208	C₄×Q₈	—	4	1	no	—
209	C₄×Q₈	—	4	1	no	—
210	C₄×Q₈	—	4	2	no	—
211	C₄×Q₈	—	4	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
218	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
224	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
225	#107	—	4	1	no	—
226	C₄⋊C₈	—	4	1	no	—
227	#103	—	2	4	no	—
228	#103	—	2	4	no	—
229	#103	—	4	2	no	—
230	#97	—	4	1	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	C₂×C₁₃	#2728	1	1	yes	characteristic
238	C₂×C₁₃	#3195	1	1	yes	characteristic
239	C₂×C₁₃	#3203	1	1	yes	characteristic
240	C₂×C₁₃	#3374	1	2	yes	—
241	C₂×C₁₃	#3375	1	2	yes	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	C₁₃×#41	1	4	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
258	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
259	Q₁₆	—	4	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	8	2	no	—
278	C₄⋊C₄	—	8	2	no	—
279	C₄⋊C₄	—	8	2	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	4	4	no	—
285	C₁₃	#32563	1	1	yes	characteristic, direct product factor, Sylow
286	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
287	Q₈	—	2	4	no	—
288	Q₈	—	2	4	no	—
289	Q₈	—	4	4	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	C₁₃×#137	1	4	yes	—
315	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
316	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
317	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
318	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
319	C₈	—	4	2	no	—
320	C₈	—	4	1	no	—
321	C₈	—	4	1	no	—
322	C₂²	C₁₃×#534	1	1	yes	characteristic
323	C₂²	C₁₃×#548	1	2	yes	—
324	C₂²	C₂×C₁₃×#137	1	2	yes	—
325	C₂²	C₁₃×#576	1	1	yes	characteristic
326	C₂²	C₁₃×#578	1	1	yes	characteristic
327	C₄	—	2	4	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	4	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	2	no	—
335	C₄	C₁₃×#539	1	2	yes	—
336	C₄	C₁₃×#551	1	1	yes	characteristic
337	C₄	C₁₃×#563	1	1	yes	characteristic
338	C₂	C₁₃×#2728	1	1	yes	characteristic
339	C₂	C₁₃×#3195	1	1	yes	characteristic
340	C₂	C₁₃×#3203	1	1	yes	characteristic
341	C₂	C₁₃×#3374	1	2	yes	—
342	C₂	C₁₃×#3375	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 ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
71. #32563	C₁₃	direct (N × H)
285. C₁₃	#32563	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×#2582	2
C₁₃×#2099	2
C₁₃×#2086	2
C₁₃×#2072	2
C₁₃×#2056	2
C₁₃×#1985	2
C₁₃×#1935	2
C₁₃×#1878	2
C₁₃×#1863	2
#32563	13

Subgroups — C13×#32563 #15718902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32563 #15718902