Subgroups — C₁₃×#32535 #15718874

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2511	C₂	1	2	yes	maximal
2	C₁₃×#2096	C₂	1	2	yes	maximal
3	C₁₃×#2066	C₂	1	2	yes	maximal
4	C₁₃×#2056	C₂	1	2	yes	maximal
5	C₁₃×#1971	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1948	C₂	1	2	yes	maximal
7	C₁₃×#1944	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1940	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1928	C₂	1	2	yes	maximal
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#103	C₂²	1	2	yes	—
18	C₂×C₁₃×#103	C₂²	1	2	yes	—
19	C₂×C₁₃×#103	C₂²	1	2	yes	—
20	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
21	C₁₃×#400	—	2	2	no	—
22	C₁₃×#400	—	2	2	no	—
23	C₁₃×#391	C₂²	1	2	yes	—
24	C₁₃×#391	C₂²	1	2	yes	—
25	C₁₃×#391	C₂²	1	2	yes	—
26	C₁₃×#389	C₂²	1	2	yes	—
27	C₁₃×#387	—	2	2	no	—
28	C₁₃×#387	—	2	2	no	—
29	C₁₃×#384	C₂²	1	2	yes	—
30	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
31	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
36	C₂×C₁₃×Q₁₆	—	2	2	no	—
37	C₂×C₁₃×Q₁₆	—	4	1	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	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×#107	—	4	1	no	—
53	C₁₃×#106	—	2	2	no	—
54	C₁₃×#106	—	4	2	no	—
55	C₁₃×#106	—	4	1	no	—
56	C₁₃×C₄⋊C₈	—	2	4	no	—
57	C₁₃×C₄⋊C₈	—	4	1	no	—
58	C₁₃×#103	—	2	4	no	—
59	C₁₃×#103	—	4	2	no	—
60	C₁₃×#103	—	4	2	no	—
61	C₁₃×#97	—	4	1	no	—
62	C₁₃×#95	—	2	2	no	—
63	C₁₃×#95	—	2	2	no	—
64	C₁₃×#95	—	2	2	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	D₄	1	1	yes	characteristic
71	C₁₃×#95	D₄	1	1	yes	characteristic
72	#32535	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
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	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×Q₁₆	—	4	4	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	#41	1	2	yes	—
97	C₂×C₈×C₁₃	#41	1	2	yes	—
98	C₁₃×C₄⋊C₄	—	2	4	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	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	4	4	no	—
106	C₁₃×C₄⋊C₄	—	8	2	no	—
107	C₁₃×C₄⋊C₄	—	8	2	no	—
108	C₄²×C₁₃	—	2	2	no	—
109	C₄²×C₁₃	—	2	1	no	—
110	C₄²×C₁₃	—	2	2	no	—
111	C₄²×C₁₃	—	2	1	no	—
112	C₄²×C₁₃	—	8	2	no	—
113	#2511	C₂×C₁₃	1	2	yes	—
114	#2096	C₂×C₁₃	1	2	yes	—
115	#2066	C₂×C₁₃	1	2	yes	—
116	#2056	C₂×C₁₃	1	2	yes	—
117	#1971	C₂×C₁₃	1	1	yes	characteristic
118	#1948	C₂×C₁₃	1	2	yes	—
119	#1944	C₂×C₁₃	1	1	yes	characteristic
120	#1940	C₂×C₁₃	1	1	yes	characteristic
121	#1928	C₂×C₁₃	1	2	yes	—
122	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₁₃	—	4	2	no	—
124	Q₈×C₁₃	—	4	2	no	—
125	Q₈×C₁₃	—	4	2	no	—
126	Q₈×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	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	#137	1	4	yes	—
149	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
155	C₈×C₁₃	—	2	2	no	—
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	—	4	2	no	—
158	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×#106	C₂²×C₁₃	1	2	yes	—
162	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
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	#400	—	2	2	no	—
170	#400	—	2	2	no	—
171	#391	C₂²×C₁₃	1	2	yes	—
172	#391	C₂²×C₁₃	1	2	yes	—
173	#391	C₂²×C₁₃	1	2	yes	—
174	#389	C₂²×C₁₃	1	2	yes	—
175	#387	—	2	2	no	—
176	#387	—	2	2	no	—
177	#384	C₂²×C₁₃	1	2	yes	—
178	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
179	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
180	#340	C₂²×C₁₃	1	2	yes	—
181	#340	C₂²×C₁₃	1	2	yes	—
182	#336	C₂²×C₁₃	1	2	yes	—
183	C₂²×C₁₃	#537	1	1	yes	characteristic
184	C₂²×C₁₃	#547	1	1	yes	characteristic
185	C₂²×C₁₃	#550	1	1	yes	characteristic
186	C₂²×C₁₃	C₂×#137	1	2	yes	—
187	C₂²×C₁₃	#577	1	2	yes	—
188	C₄×C₁₃	—	2	4	no	—
189	C₄×C₁₃	—	4	2	no	—
190	C₄×C₁₃	—	4	1	no	—
191	C₄×C₁₃	—	4	1	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	—	8	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	#538	1	1	yes	characteristic
197	C₄×C₁₃	#539	1	1	yes	characteristic
198	C₄×C₁₃	#556	1	1	yes	characteristic
199	C₄×C₁₃	#563	1	1	yes	characteristic
200	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×Q₁₆	—	4	1	no	—
203	C₂×Q₁₆	—	4	1	no	—
204	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
205	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
216	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
217	#107	—	4	1	no	—
218	#106	—	2	2	no	—
219	#106	—	4	2	no	—
220	#106	—	4	1	no	—
221	C₄⋊C₈	—	2	4	no	—
222	C₄⋊C₈	—	4	1	no	—
223	#103	—	2	4	no	—
224	#103	—	4	2	no	—
225	#103	—	4	2	no	—
226	#97	—	4	1	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	D₄×C₁₃	1	1	yes	characteristic
236	#95	D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₁₃	#2727	1	1	yes	characteristic
238	C₂×C₁₃	#3255	1	1	yes	characteristic
239	C₂×C₁₃	#3263	1	1	yes	characteristic
240	C₂×C₁₃	#3344	1	2	yes	—
241	C₂×C₁₃	#3400	1	2	yes	—
242	C₂×Q₈	—	2	2	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	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
256	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
257	Q₁₆	—	4	4	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₁₃×#41	1	2	yes	—
266	C₂×C₈	C₁₃×#41	1	2	yes	—
267	C₄⋊C₄	—	2	4	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	—	2	2	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	8	2	no	—
276	C₄⋊C₄	—	8	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	8	2	no	—
282	C₁₃	#32535	1	1	yes	characteristic, direct product factor, Sylow
283	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	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	C₁₃×#137	1	4	yes	—
310	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
311	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
312	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
313	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
314	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
315	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	2	no	—
318	C₈	—	4	2	no	—
319	C₂²	C₁₃×#537	1	1	yes	characteristic
320	C₂²	C₁₃×#547	1	1	yes	characteristic
321	C₂²	C₁₃×#550	1	1	yes	characteristic
322	C₂²	C₂×C₁₃×#137	1	2	yes	—
323	C₂²	C₁₃×#577	1	2	yes	—
324	C₄	—	2	4	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	C₁₃×#538	1	1	yes	characteristic
333	C₄	C₁₃×#539	1	1	yes	characteristic
334	C₄	C₁₃×#556	1	1	yes	characteristic
335	C₄	C₁₃×#563	1	1	yes	characteristic
336	C₂	C₁₃×#2727	1	1	yes	characteristic
337	C₂	C₁₃×#3255	1	1	yes	characteristic
338	C₂	C₁₃×#3263	1	1	yes	characteristic
339	C₂	C₁₃×#3344	1	2	yes	—
340	C₂	C₁₃×#3400	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 ◁ C₄×C₁₃×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
72. #32535	C₁₃	direct (N × H)
282. C₁₃	#32535	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×#2511	2
C₁₃×#2096	2
C₁₃×#2066	2
C₁₃×#2056	2
C₁₃×#1971	2
C₁₃×#1948	2
C₁₃×#1944	2
C₁₃×#1940	2
C₁₃×#1928	2
#32535	13

Subgroups — C13×#32535 #15718874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32535 #15718874