Subgroups — C₁₃×#19194 #15705533

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3246	C₂	1	2	yes	maximal
2	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1695	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1693	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1671	C₂	1	4	yes	maximal, complemented
7	C₂×C₁₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1550	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×#131	—	2	1	no	—
10	C₁₃×#533	C₂²	1	2	yes	—
11	C₁₃×#532	C₂²	1	1	yes	characteristic
12	C₁₃×#500	C₂²	1	1	yes	characteristic
13	C₁₃×#500	C₂²	1	1	yes	characteristic
14	C₁₃×#499	—	2	1	no	—
15	C₁₃×#499	—	2	1	no	—
16	C₁₃×#498	C₂²	1	2	yes	—
17	C₁₃×#476	C₂²	1	4	yes	—
18	C₁₃×#475	—	2	2	no	—
19	C₁₃×#465	—	2	4	no	—
20	C₁₃×#461	C₂²	1	4	yes	—
21	C₁₃×#460	—	2	2	no	—
22	C₁₃×#436	—	2	1	no	—
23	C₁₃×#436	C₂²	1	2	yes	—
24	C₈×D₄×C₁₃	—	2	1	no	—
25	C₈×D₄×C₁₃	C₂²	1	2	yes	—
26	C₁₃×#427	—	2	1	no	—
27	C₁₃×#427	—	2	1	no	—
28	C₁₃×#423	C₂²	1	1	yes	characteristic
29	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₁₃×#408	—	2	1	no	—
31	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
32	C₁₃×#334	C₂²	1	2	yes	—
33	C₁₃×#334	C₂²	1	2	yes	—
34	C₁₃×#333	C₂²	1	4	yes	—
35	C₁₃×#332	C₂²	1	4	yes	—
36	C₂×C₁₃×#41	—	2	1	no	—
37	C₂×C₁₃×SD₁₆	—	2	4	no	—
38	C₁₃×#131	—	4	4	no	—
39	C₂×C₁₃×M₁₆	—	2	1	no	—
40	C₂×C₁₃×M₁₆	—	2	2	no	—
41	C₂²×C₈×C₁₃	—	2	2	no	—
42	C₂²×C₈×C₁₃	—	2	2	no	—
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₁₃×#127	D₄	1	1	yes	characteristic
49	C₁₃×#125	—	2	2	no	—
50	C₁₃×#124	—	2	1	no	—
51	C₁₃×C₂²⋊Q₈	—	2	2	no	—
52	C₁₃×C₂²⋊Q₈	—	2	2	no	—
53	C₁₃×#121	—	2	1	no	—
54	C₁₃×#121	D₄	1	2	yes	—
55	C₄×Q₈×C₁₃	—	2	2	no	—
56	C₄×Q₈×C₁₃	—	2	2	no	—
57	C₄×D₄×C₁₃	—	2	1	no	—
58	C₄×D₄×C₁₃	C₂³	1	2	yes	—
59	C₁₃×#117	—	2	2	no	—
60	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
61	C₁₃×#107	—	2	1	no	—
62	C₁₃×#107	—	2	1	no	—
63	C₁₃×#107	—	2	1	no	—
64	C₁₃×#107	—	2	1	no	—
65	C₁₃×#106	—	2	2	no	—
66	C₁₃×#106	—	2	2	no	—
67	C₁₃×#106	—	2	4	no	—
68	C₁₃×C₄⋊C₈	—	2	2	no	—
69	C₁₃×C₄⋊C₈	—	2	1	no	—
70	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
71	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
72	C₁₃×#103	—	2	4	no	—
73	C₁₃×#102	—	2	4	no	—
74	C₁₃×C₂²⋊C₈	—	2	2	no	—
75	C₁₃×C₂²⋊C₈	—	2	2	no	—
76	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
80	#19194	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₁₃×#41	—	4	2	no	—
82	C₁₃×#41	—	4	2	no	—
83	C₂×Q₈×C₁₃	—	2	1	no	—
84	C₂×Q₈×C₁₃	—	2	2	no	—
85	C₂×Q₈×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	2	1	no	—
88	C₂×D₄×C₁₃	—	2	1	no	—
89	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₃	—	2	2	no	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
94	C₁₃×SD₁₆	—	8	4	no	—
95	C₁₃×M₁₆	—	4	2	no	—
96	C₁₃×M₁₆	—	4	4	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	4	no	—
100	C₂×C₈×C₁₃	—	4	4	no	—
101	C₂×C₈×C₁₃	—	8	2	no	—
102	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	—	2	1	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	—	4	2	no	—
113	C₁₃×C₄⋊C₄	—	4	2	no	—
114	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
115	C₁₃×C₂²⋊C₄	—	2	2	no	—
116	C₁₃×C₂²⋊C₄	—	2	2	no	—
117	C₁₃×C₂²⋊C₄	—	4	2	no	—
118	C₄²×C₁₃	—	4	2	no	—
119	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₃	#41	1	1	yes	characteristic
122	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
123	#3246	C₂×C₁₃	1	2	yes	—
124	#3160	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2931	C₂×C₁₃	1	1	yes	characteristic
126	#1695	C₂×C₁₃	1	2	yes	complemented
127	#1693	C₂×C₁₃	1	2	yes	complemented
128	#1671	C₂×C₁₃	1	4	yes	complemented
129	C₂×#334	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#1550	C₂×C₁₃	1	2	yes	complemented
131	C₂³×C₁₃	—	2	2	no	—
132	C₂³×C₁₃	#121	1	1	yes	characteristic
133	Q₈×C₁₃	—	4	2	no	—
134	Q₈×C₁₃	—	4	2	no	—
135	Q₈×C₁₃	—	4	2	no	—
136	Q₈×C₁₃	—	8	2	no	—
137	D₄×C₁₃	—	2	4	no	—
138	D₄×C₁₃	—	4	2	no	—
139	D₄×C₁₃	—	4	4	no	—
140	D₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	4	1	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	#121	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
150	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
152	C₂×C₄×C₁₃	#137	1	2	yes	—
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₁₃	—	2	1	no	—
157	C₈×C₁₃	—	2	1	no	—
158	C₈×C₁₃	—	4	2	no	—
159	C₈×C₁₃	—	4	4	no	—
160	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
161	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
162	C₂×#131	—	2	1	no	—
163	#533	C₂²×C₁₃	1	2	yes	—
164	#532	C₂²×C₁₃	1	1	yes	characteristic
165	#500	C₂²×C₁₃	1	1	yes	characteristic
166	#500	C₂²×C₁₃	1	1	yes	characteristic
167	#499	—	2	1	no	—
168	#499	—	2	1	no	—
169	#498	C₂²×C₁₃	1	2	yes	—
170	#476	C₂²×C₁₃	1	4	yes	—
171	#475	—	2	2	no	—
172	#465	—	2	4	no	—
173	#461	C₂²×C₁₃	1	4	yes	—
174	#460	—	2	2	no	—
175	#436	—	2	1	no	—
176	#436	C₂²×C₁₃	1	2	yes	—
177	C₈×D₄	—	2	1	no	—
178	C₈×D₄	C₂²×C₁₃	1	2	yes	—
179	#427	—	2	1	no	—
180	#427	—	2	1	no	—
181	#423	C₂²×C₁₃	1	1	yes	characteristic
182	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
183	#408	—	2	1	no	—
184	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
185	#334	C₂²×C₁₃	1	2	yes	—
186	#334	C₂²×C₁₃	1	2	yes	—
187	#333	C₂²×C₁₃	1	4	yes	—
188	#332	C₂²×C₁₃	1	4	yes	—
189	C₂²×C₁₃	—	2	1	no	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	—	4	2	no	—
193	C₂²×C₁₃	—	4	2	no	—
194	C₂²×C₁₃	#461	1	2	yes	—
195	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	—	4	2	no	—
200	C₄×C₁₃	—	8	2	no	—
201	C₄×C₁₃	—	8	2	no	—
202	C₄×C₁₃	#461	1	2	yes	—
203	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
204	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
205	C₄×C₁₃	#577	1	2	yes	—
206	C₂×#41	—	2	1	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	#131	—	4	4	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	2	no	—
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
214	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
215	C₄⋊Q₈	D₄×C₁₃	1	1	yes	characteristic
216	C₄⋊Q₈	C₂³×C₁₃	1	2	yes	—
217	C₄⋊Q₈	C₂³×C₁₃	1	2	yes	—
218	#127	D₄×C₁₃	1	1	yes	characteristic
219	#125	—	2	2	no	—
220	#124	—	2	1	no	—
221	C₂²⋊Q₈	—	2	2	no	—
222	C₂²⋊Q₈	—	2	2	no	—
223	#121	—	2	1	no	—
224	#121	D₄×C₁₃	1	2	yes	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	C₂³×C₁₃	1	2	yes	—
229	#117	—	2	2	no	—
230	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
231	#107	—	2	1	no	—
232	#107	—	2	1	no	—
233	#107	—	2	1	no	—
234	#107	—	2	1	no	—
235	#106	—	2	2	no	—
236	#106	—	2	2	no	—
237	#106	—	2	4	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	C₂³×C₁₃	1	2	yes	—
241	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
242	#103	—	2	4	no	—
243	#102	—	2	4	no	—
244	C₂²⋊C₈	—	2	2	no	—
245	C₂²⋊C₈	—	2	2	no	—
246	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
247	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
250	C₂×C₁₃	—	2	2	no	—
251	C₂×C₁₃	—	8	2	no	—
252	C₂×C₁₃	C₂×#461	1	1	yes	characteristic
253	C₂×C₁₃	#3060	1	1	yes	characteristic
254	C₂×C₁₃	#3067	1	1	yes	characteristic
255	#41	—	4	2	no	—
256	#41	—	4	2	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	C₂×D₄×C₁₃	1	2	yes	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
268	SD₁₆	—	8	4	no	—
269	M₁₆	—	4	2	no	—
270	M₁₆	—	4	4	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	—	8	2	no	—
276	C₂×C₈	C₁₃×SD₁₆	1	4	yes	complemented
277	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
278	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	C₂×D₄×C₁₃	1	2	yes	—
289	C₂²⋊C₄	—	2	2	no	—
290	C₂²⋊C₄	—	2	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₄²	—	4	2	no	—
293	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
294	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
295	C₄²	C₁₃×#41	1	1	yes	characteristic
296	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
297	C₁₃	#19194	1	1	yes	characteristic, direct product factor, Sylow
298	C₂³	—	2	2	no	—
299	C₂³	C₁₃×#121	1	1	yes	characteristic
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
316	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
317	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
318	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
319	C₂×C₄	C₁₃×#137	1	2	yes	—
320	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
322	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	4	2	no	—
326	C₈	—	4	4	no	—
327	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
328	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	C₁₃×#461	1	2	yes	—
335	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	C₁₃×#461	1	2	yes	—
343	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
344	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
345	C₄	C₁₃×#577	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	8	2	no	—
348	C₂	C₂×C₁₃×#461	1	1	yes	characteristic
349	C₂	C₁₃×#3060	1	1	yes	characteristic
350	C₂	C₁₃×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₁₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#3160	C₂	semidirect
4. C₁₃×#1695	C₂	semidirect
5. C₁₃×#1693	C₂	semidirect
6. C₁₃×#1671	C₂	semidirect
7. C₂×C₁₃×#334	C₂	semidirect
8. C₁₃×#1550	C₂	semidirect
80. #19194	C₁₃	direct (N × H)
102. C₂×C₈×C₁₃	SD₁₆	semidirect
124. #3160	C₂×C₁₃	semidirect
126. #1695	C₂×C₁₃	semidirect
127. #1693	C₂×C₁₃	semidirect
128. #1671	C₂×C₁₃	semidirect
129. C₂×#334	C₂×C₁₃	semidirect
130. #1550	C₂×C₁₃	semidirect
276. C₂×C₈	C₁₃×SD₁₆	semidirect
297. C₁₃	#19194	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×#3246	2
C₁₃×#3160	2
C₁₃×#2931	2
C₁₃×#1695	2
C₁₃×#1693	2
C₁₃×#1671	2
C₂×C₁₃×#334	2
C₁₃×#1550	2
#19194	13

Subgroups — C13×#19194 #15705533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#19194 #15705533