Subgroups — C₁₃×#15185 #15701524

Order 3328.

1374 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#391	C₂	1	2	yes	maximal
2	C₂×C₁₃×#391	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#384	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1445	C₂	1	4	yes	maximal
5	C₁₃×#1442	C₂	1	4	yes	maximal
6	C₁₃×C₄⋊#95	C₂	1	2	yes	maximal
7	C₄×C₁₃×#95	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
14	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#391	—	2	8	no	—
16	C₁₃×#391	—	2	4	no	—
17	C₁₃×#384	—	2	4	no	—
18	C₂×C₁₃×#95	C₂²	1	2	yes	—
19	C₂×C₁₃×#95	C₂²	1	8	yes	—
20	C₂×C₁₃×#95	C₂²	1	2	yes	—
21	C₂×C₁₃×#95	C₂²	1	2	yes	—
22	C₂×C₁₃×#95	C₂²	1	8	yes	—
23	C₂×C₁₃×#95	C₂²	1	2	yes	—
24	C₂²×Q₈×C₁₃	—	2	2	no	—
25	C₂²×Q₈×C₁₃	—	2	2	no	—
26	C₂²×Q₈×C₁₃	—	2	4	no	—
27	C₂²×Q₈×C₁₃	—	2	4	no	—
28	C₂²×Q₈×C₁₃	D₄	1	4	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₂×C₁₃×C₄⋊C₄	—	4	4	no	—
45	C₂×C₄²×C₁₃	—	2	4	no	—
46	C₂×C₄²×C₁₃	—	2	4	no	—
47	C₂×C₄²×C₁₃	—	2	4	no	—
48	C₂×C₄²×C₁₃	—	2	4	no	—
49	C₂×C₄²×C₁₃	D₄	1	4	yes	—
50	C₁₃×#95	—	2	16	no	—
51	C₁₃×#95	—	2	4	no	—
52	C₁₃×#95	—	2	4	no	—
53	C₁₃×#95	—	2	4	no	—
54	C₁₃×#95	—	4	8	no	—
55	C₁₃×#95	Q₈	1	8	yes	complemented
56	#15185	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂×Q₈×C₁₃	—	2	8	no	—
59	C₂×Q₈×C₁₃	—	2	16	no	—
60	C₂×Q₈×C₁₃	—	4	4	no	—
61	C₂×Q₈×C₁₃	—	4	4	no	—
62	C₂×Q₈×C₁₃	—	4	8	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	4	no	—
74	C₂²×C₄×C₁₃	—	2	4	no	—
75	C₂²×C₄×C₁₃	—	2	4	no	—
76	C₂²×C₄×C₁₃	—	2	4	no	—
77	C₂²×C₄×C₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	2	4	no	—
80	C₂²×C₄×C₁₃	—	2	4	no	—
81	C₂²×C₄×C₁₃	—	2	4	no	—
82	C₂²×C₄×C₁₃	—	2	8	no	—
83	C₂²×C₄×C₁₃	—	2	4	no	—
84	C₂²×C₄×C₁₃	—	2	8	no	—
85	C₂²×C₄×C₁₃	—	2	4	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
92	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
93	C₂²×C₄×C₁₃	#41	1	4	yes	—
94	C₂²×C₄×C₁₃	#41	1	2	yes	—
95	C₂²×C₄×C₁₃	#41	1	4	yes	—
96	C₂²×C₄×C₁₃	#41	1	4	yes	—
97	C₁₃×C₄⋊C₄	—	4	8	no	—
98	C₁₃×C₄⋊C₄	—	4	8	no	—
99	C₁₃×C₄⋊C₄	—	4	8	no	—
100	C₁₃×C₄⋊C₄	—	4	4	no	—
101	C₄²×C₁₃	—	2	8	no	—
102	C₄²×C₁₃	—	4	8	no	—
103	C₂×#391	C₂×C₁₃	1	1	yes	characteristic
104	C₂×#391	C₂×C₁₃	1	2	yes	—
105	C₂×#384	C₂×C₁₃	1	1	yes	characteristic
106	#1445	C₂×C₁₃	1	4	yes	—
107	#1442	C₂×C₁₃	1	4	yes	—
108	C₄⋊#95	C₂×C₁₃	1	2	yes	—
109	C₄×#95	C₂×C₁₃	1	1	yes	characteristic
110	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
113	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂³×C₁₃	C₂×#41	1	2	yes	—
115	C₂³×C₁₃	C₂×#41	1	2	yes	—
116	C₂³×C₁₃	C₂×#41	1	2	yes	—
117	C₂³×C₁₃	H(5,𝔽₂)	1	2	yes	—
118	C₂³×C₁₃	2₋¹⁺⁴	1	2	yes	—
119	Q₈×C₁₃	—	4	16	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	8	no	—
125	C₂×C₄×C₁₃	—	2	8	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	8	no	—
128	C₂×C₄×C₁₃	—	2	8	no	—
129	C₂×C₄×C₁₃	—	2	8	no	—
130	C₂×C₄×C₁₃	—	2	8	no	—
131	C₂×C₄×C₁₃	—	4	4	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	4	no	—
135	C₂×C₄×C₁₃	—	4	4	no	—
136	C₂×C₄×C₁₃	—	4	4	no	—
137	C₂×C₄×C₁₃	—	4	4	no	—
138	C₂×C₄×C₁₃	—	4	4	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	4	no	—
141	C₂×C₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	8	no	—
145	C₂×C₄×C₁₃	—	4	8	no	—
146	C₂×C₄×C₁₃	—	4	8	no	—
147	C₂×C₄×C₁₃	#121	1	4	yes	—
148	C₂×C₄×C₁₃	#123	1	4	yes	—
149	C₂×C₄×C₁₃	#123	1	8	yes	—
150	C₂×C₄×C₁₃	#124	1	4	yes	—
151	C₂×C₄×C₁₃	#124	1	4	yes	—
152	C₂×C₄×C₁₃	#126	1	8	yes	—
153	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
154	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₂²×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
156	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
157	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
158	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
159	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
160	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
161	#391	—	2	8	no	—
162	#391	—	2	4	no	—
163	#384	—	2	4	no	—
164	C₂×#95	C₂²×C₁₃	1	2	yes	—
165	C₂×#95	C₂²×C₁₃	1	2	yes	—
166	C₂×#95	C₂²×C₁₃	1	8	yes	—
167	C₂×#95	C₂²×C₁₃	1	2	yes	—
168	C₂×#95	C₂²×C₁₃	1	8	yes	—
169	C₂×#95	C₂²×C₁₃	1	2	yes	—
170	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×#123	1	2	yes	—
172	C₂²×C₁₃	C₂×#123	1	1	yes	characteristic
173	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
175	C₂²×C₁₃	C₂×#126	1	2	yes	—
176	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
177	C₂²×C₁₃	#533	1	4	yes	—
178	C₂²×C₁₃	#535	1	1	yes	characteristic
179	C₂²×C₁₃	#536	1	1	yes	characteristic
180	C₂²×C₁₃	#542	1	4	yes	—
181	C₂²×C₁₃	D₄×Q₈	1	2	yes	—
182	C₂²×C₁₃	#550	1	2	yes	—
183	C₂²×C₁₃	#552	1	4	yes	—
184	C₂²×C₁₃	#554	1	4	yes	—
185	C₂²×C₁₃	#557	1	2	yes	—
186	C₂²×C₁₃	#559	1	2	yes	—
187	C₄×C₁₃	—	2	4	no	—
188	C₄×C₁₃	—	2	4	no	—
189	C₄×C₁₃	—	4	4	no	—
190	C₄×C₁₃	—	4	8	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	—	8	4	no	—
193	C₄×C₁₃	#397	1	8	yes	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×Q₈	—	2	2	no	—
196	C₂²×Q₈	—	2	4	no	—
197	C₂²×Q₈	—	2	4	no	—
198	C₂²×Q₈	D₄×C₁₃	1	4	yes	—
199	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
200	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
201	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
202	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
203	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
204	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
205	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	8	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	4	4	no	—
215	C₂×C₄²	—	2	4	no	—
216	C₂×C₄²	—	2	4	no	—
217	C₂×C₄²	—	2	4	no	—
218	C₂×C₄²	—	2	4	no	—
219	C₂×C₄²	D₄×C₁₃	1	4	yes	—
220	#95	—	2	16	no	—
221	#95	—	2	4	no	—
222	#95	—	2	4	no	—
223	#95	—	2	4	no	—
224	#95	—	4	8	no	—
225	#95	Q₈×C₁₃	1	8	yes	complemented
226	C₂×C₁₃	C₂×#397	1	1	yes	characteristic
227	C₂×C₁₃	#2423	1	4	yes	—
228	C₂×C₁₃	#2502	1	1	yes	characteristic
229	C₂×C₁₃	#2503	1	1	yes	characteristic
230	C₂×C₁₃	#2511	1	4	yes	—
231	C₂×C₁₃	#2546	1	1	yes	characteristic
232	C₂×C₁₃	#2548	1	1	yes	characteristic
233	C₂×C₁₃	#2640	1	1	yes	characteristic
234	C₂×C₁₃	#2862	1	1	yes	characteristic
235	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
236	C₂×Q₈	—	2	8	no	—
237	C₂×Q₈	—	2	16	no	—
238	C₂×Q₈	—	4	4	no	—
239	C₂×Q₈	—	4	4	no	—
240	C₂×Q₈	—	4	8	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	2	8	no	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	2	8	no	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
265	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
268	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
269	C₂²×C₄	C₂×Q₈×C₁₃	1	8	yes	—
270	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
271	C₂²×C₄	C₁₃×#41	1	4	yes	—
272	C₂²×C₄	C₁₃×#41	1	2	yes	—
273	C₂²×C₄	C₁₃×#41	1	4	yes	—
274	C₂²×C₄	C₁₃×#41	1	4	yes	—
275	C₄⋊C₄	—	4	8	no	—
276	C₄⋊C₄	—	4	8	no	—
277	C₄⋊C₄	—	4	8	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄²	—	2	8	no	—
280	C₄²	—	4	8	no	—
281	C₁₃	#15185	1	1	yes	characteristic, direct product factor, Sylow
282	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
283	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
284	C₂³	C₂²×Q₈×C₁₃	1	2	yes	—
285	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
286	C₂³	C₂×C₁₃×#41	1	2	yes	—
287	C₂³	C₂×C₁₃×#41	1	2	yes	—
288	C₂³	C₂×C₁₃×#41	1	2	yes	—
289	C₂³	C₁₃×H(5,𝔽₂)	1	2	yes	—
290	C₂³	C₁₃×2₋¹⁺⁴	1	2	yes	—
291	Q₈	—	4	16	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	8	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	8	no	—
317	C₂×C₄	—	4	8	no	—
318	C₂×C₄	—	4	8	no	—
319	C₂×C₄	C₁₃×#121	1	4	yes	—
320	C₂×C₄	C₁₃×#123	1	4	yes	—
321	C₂×C₄	C₁₃×#123	1	8	yes	—
322	C₂×C₄	C₁₃×#124	1	4	yes	—
323	C₂×C₄	C₁₃×#124	1	4	yes	—
324	C₂×C₄	C₁₃×#126	1	8	yes	—
325	C₂×C₄	C₁₃×C₄⋊Q₈	1	4	yes	—
326	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
327	C₂²	C₂×C₁₃×#123	1	1	yes	characteristic
328	C₂²	C₂×C₁₃×#123	1	2	yes	—
329	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
330	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
331	C₂²	C₂×C₁₃×#126	1	2	yes	—
332	C₂²	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
333	C₂²	C₁₃×#533	1	4	yes	—
334	C₂²	C₁₃×#535	1	1	yes	characteristic
335	C₂²	C₁₃×#536	1	1	yes	characteristic
336	C₂²	C₁₃×#542	1	4	yes	—
337	C₂²	D₄×Q₈×C₁₃	1	2	yes	—
338	C₂²	C₁₃×#550	1	2	yes	—
339	C₂²	C₁₃×#552	1	4	yes	—
340	C₂²	C₁₃×#554	1	4	yes	—
341	C₂²	C₁₃×#557	1	2	yes	—
342	C₂²	C₁₃×#559	1	2	yes	—
343	C₄	—	2	4	no	—
344	C₄	—	2	4	no	—
345	C₄	—	4	4	no	—
346	C₄	—	4	8	no	—
347	C₄	—	8	4	no	—
348	C₄	—	8	2	no	—
349	C₄	C₁₃×#397	1	8	yes	—
350	C₂	C₂×C₁₃×#397	1	1	yes	characteristic
351	C₂	C₁₃×#2423	1	4	yes	—
352	C₂	C₁₃×#2502	1	1	yes	characteristic
353	C₂	C₁₃×#2503	1	1	yes	characteristic
354	C₂	C₁₃×#2511	1	4	yes	—
355	C₂	C₁₃×#2546	1	1	yes	characteristic
356	C₂	C₁₃×#2548	1	1	yes	characteristic
357	C₂	C₁₃×#2640	1	1	yes	characteristic
358	C₂	C₁₃×#2862	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₃
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
55. C₁₃×#95	Q₈	semidirect
56. #15185	C₁₃	direct (N × H)
225. #95	Q₈×C₁₃	semidirect
281. C₁₃	#15185	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×#391	2
C₂×C₁₃×#384	2
C₁₃×#1445	2
C₁₃×#1442	2
C₁₃×C₄⋊#95	2
C₄×C₁₃×#95	2
#15185	13

Subgroups — C13×#15185 #15701524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#15185 #15701524