Subgroups — C₁₃×#19264 #15705603

Order 3328.

650 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×#3169	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3112	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1678	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1676	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1668	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1551	C₂	1	2	yes	maximal, complemented
9	C₁₃×#532	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#500	C₂²	1	1	yes	characteristic
12	C₁₃×#493	C₂²	1	1	yes	characteristic
13	C₁₃×#486	C₂²	1	2	yes	—
14	C₁₃×#480	—	2	2	no	—
15	C₁₃×#477	C₂²	1	4	yes	—
16	C₁₃×#466	—	2	1	no	—
17	C₁₃×#466	—	2	1	no	—
18	C₁₃×#460	C₂²	1	4	yes	—
19	C₁₃×#448	—	2	2	no	—
20	C₈×Q₈×C₁₃	—	2	1	no	—
21	C₈×Q₈×C₁₃	—	2	1	no	—
22	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
23	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
24	C₁₃×#440	—	2	4	no	—
25	C₄×C₁₃×D₈	—	2	1	no	—
26	C₄×C₁₃×D₈	—	2	1	no	—
27	C₁₃×#432	C₂²	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₁₃	—	2	1	no	—
31	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
32	C₁₃×#335	C₂²	1	2	yes	—
33	C₁₃×#334	C₂²	1	2	yes	—
34	C₁₃×#333	C₂²	1	4	yes	—
35	C₁₃×#331	C₂²	1	4	yes	—
36	C₂×C₁₃×#41	—	2	2	no	—
37	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₁₃×SD₁₆	—	2	4	no	—
39	C₂×C₁₃×D₈	—	2	1	no	—
40	C₂×C₁₃×D₈	—	2	1	no	—
41	C₂²×C₈×C₁₃	—	2	1	no	—
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₁₃×#127	C₂³	1	2	yes	—
46	C₁₃×#124	—	2	2	no	—
47	C₁₃×#121	—	2	2	no	—
48	C₁₃×#121	—	2	2	no	—
49	C₄×Q₈×C₁₃	—	2	1	no	—
50	C₄×Q₈×C₁₃	—	2	1	no	—
51	C₄×Q₈×C₁₃	D₄	1	2	yes	—
52	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
53	C₄×D₄×C₁₃	—	2	2	no	—
54	C₄×D₄×C₁₃	—	2	2	no	—
55	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
56	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₃	—	2	1	no	—
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₁₃×#107	—	2	1	no	—
60	C₁₃×#107	—	2	1	no	—
61	C₁₃×#107	—	2	4	no	—
62	C₁₃×C₄⋊C₈	—	2	2	no	—
63	C₁₃×C₄⋊C₈	—	2	2	no	—
64	C₁₃×C₄⋊C₈	—	2	1	no	—
65	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
66	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
67	C₁₃×#103	—	2	4	no	—
68	C₁₃×#102	—	2	2	no	—
69	C₁₃×#102	—	2	2	no	—
70	C₁₃×#102	—	2	4	no	—
71	C₁₃×C₂²⋊C₈	—	2	2	no	—
72	C₁₃×#97	—	2	2	no	—
73	C₄×C₈×C₁₃	—	2	2	no	—
74	C₄×C₈×C₁₃	—	2	2	no	—
75	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	#19264	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₁₃×#41	—	8	4	no	—
81	C₂×Q₈×C₁₃	—	2	4	no	—
82	C₂×Q₈×C₁₃	—	2	1	no	—
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	2	2	no	—
88	C₂×D₄×C₁₃	—	4	2	no	—
89	C₂²×C₄×C₁₃	—	2	1	no	—
90	C₂²×C₄×C₁₃	—	2	1	no	—
91	C₂²×C₄×C₁₃	—	4	2	no	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
95	C₁₃×SD₁₆	—	4	8	no	—
96	C₁₃×D₈	—	4	2	no	—
97	C₁₃×D₈	—	4	2	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	4	no	—
101	C₂×C₈×C₁₃	—	4	1	no	—
102	C₂×C₈×C₁₃	—	4	2	no	—
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	2	no	—
108	C₁₃×C₄⋊C₄	—	2	1	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	C₂×D₄	1	2	yes	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₄²×C₁₃	—	2	2	no	—
116	C₄²×C₁₃	—	2	2	no	—
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₃	#41	1	1	yes	characteristic
120	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
121	#3169	C₂×C₁₃	1	1	yes	characteristic
122	#3112	C₂×C₁₃	1	2	yes	complemented
123	C₂×C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#1678	C₂×C₁₃	1	4	yes	complemented
125	#1676	C₂×C₁₃	1	2	yes	complemented
126	#1668	C₂×C₁₃	1	2	yes	complemented
127	#1568	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#1551	C₂×C₁₃	1	2	yes	complemented
129	C₂³×C₁₃	—	4	2	no	—
130	C₂³×C₁₃	#121	1	1	yes	characteristic
131	Q₈×C₁₃	—	2	4	no	—
132	Q₈×C₁₃	—	2	2	no	—
133	Q₈×C₁₃	—	8	2	no	—
134	Q₈×C₁₃	#135	1	4	yes	—
135	D₄×C₁₃	—	4	4	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	8	2	no	—
139	D₄×C₁₃	—	8	4	no	—
140	C₂×C₄×C₁₃	—	2	4	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	1	no	—
145	C₂×C₄×C₁₃	—	2	1	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	8	4	no	—
150	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
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	—	2	1	no	—
159	C₈×C₁₃	—	4	4	no	—
160	C₈×C₁₃	—	4	2	no	—
161	C₈×C₁₃	#137	1	2	yes	—
162	#532	C₂²×C₁₃	1	2	yes	—
163	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
164	#500	C₂²×C₁₃	1	1	yes	characteristic
165	#493	C₂²×C₁₃	1	1	yes	characteristic
166	#486	C₂²×C₁₃	1	2	yes	—
167	#480	—	2	2	no	—
168	#477	C₂²×C₁₃	1	4	yes	—
169	#466	—	2	1	no	—
170	#466	—	2	1	no	—
171	#460	C₂²×C₁₃	1	4	yes	—
172	#448	—	2	2	no	—
173	C₈×Q₈	—	2	1	no	—
174	C₈×Q₈	—	2	1	no	—
175	C₈×Q₈	C₂²×C₁₃	1	2	yes	—
176	C₈×Q₈	C₂²×C₁₃	1	2	yes	—
177	#440	—	2	4	no	—
178	C₄×D₈	—	2	1	no	—
179	C₄×D₈	—	2	1	no	—
180	#432	C₂²×C₁₃	1	2	yes	—
181	C₂×C₄⋊C₈	—	2	1	no	—
182	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
183	C₂×C₄×C₈	—	2	1	no	—
184	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
185	#335	C₂²×C₁₃	1	2	yes	—
186	#334	C₂²×C₁₃	1	2	yes	—
187	#333	C₂²×C₁₃	1	4	yes	—
188	#331	C₂²×C₁₃	1	4	yes	—
189	C₂²×C₁₃	—	2	1	no	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	8	2	no	—
192	C₂²×C₁₃	—	8	2	no	—
193	C₂²×C₁₃	—	8	2	no	—
194	C₂²×C₁₃	#472	1	2	yes	—
195	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	4	no	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	C₂×#135	1	2	yes	—
204	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
205	C₄×C₁₃	#575	1	1	yes	characteristic
206	C₂×#41	—	2	2	no	—
207	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
208	C₂×SD₁₆	—	2	4	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
213	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
214	C₄⋊Q₈	C₂³×C₁₃	1	2	yes	—
215	#127	C₂³×C₁₃	1	2	yes	—
216	#124	—	2	2	no	—
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	D₄×C₁₃	1	2	yes	—
222	C₄×Q₈	C₂³×C₁₃	1	2	yes	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
227	C₂×C₄²	—	2	1	no	—
228	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
229	#107	—	2	1	no	—
230	#107	—	2	1	no	—
231	#107	—	2	4	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	—	2	2	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	C₂³×C₁₃	1	2	yes	—
236	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
237	#103	—	2	4	no	—
238	#102	—	2	2	no	—
239	#102	—	2	2	no	—
240	#102	—	2	4	no	—
241	C₂²⋊C₈	—	2	2	no	—
242	#97	—	2	2	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
246	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
249	C₂×C₁₃	—	2	2	no	—
250	C₂×C₁₃	—	16	2	no	—
251	C₂×C₁₃	#3037	1	1	yes	characteristic
252	C₂×C₁₃	#3050	1	1	yes	characteristic
253	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
254	#41	—	8	4	no	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	C₂×D₄×C₁₃	1	2	yes	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
268	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
269	SD₁₆	—	4	8	no	—
270	D₈	—	4	2	no	—
271	D₈	—	4	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	2	no	—
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	2	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	C₂×D₄×C₁₃	1	2	yes	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
292	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
293	C₄²	C₁₃×#41	1	1	yes	characteristic
294	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
295	C₁₃	#19264	1	1	yes	characteristic, direct product factor, Sylow
296	C₂³	—	4	2	no	—
297	C₂³	C₁₃×#121	1	1	yes	characteristic
298	Q₈	—	2	4	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	8	2	no	—
301	Q₈	C₁₃×#135	1	4	yes	—
302	D₄	—	4	4	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
318	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
319	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
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	—	2	1	no	—
326	C₈	—	4	4	no	—
327	C₈	—	4	2	no	—
328	C₈	C₁₃×#137	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	C₁₃×#472	1	2	yes	—
335	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₂×C₁₃×#135	1	2	yes	—
344	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
345	C₄	C₁₃×#575	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₁₃×#3037	1	1	yes	characteristic
349	C₂	C₁₃×#3050	1	1	yes	characteristic
350	C₂	C₂×C₁₃×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₁₃×#3169	C₂
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×#3112	C₂	semidirect
3. C₂×C₈×Q₈×C₁₃	C₂	semidirect
4. C₁₃×#1678	C₂	semidirect
5. C₁₃×#1676	C₂	semidirect
6. C₁₃×#1668	C₂	semidirect
7. C₁₃×#1568	C₂	semidirect
8. C₁₃×#1551	C₂	semidirect
79. #19264	C₁₃	direct (N × H)
122. #3112	C₂×C₁₃	semidirect
123. C₂×C₈×Q₈	C₂×C₁₃	semidirect
124. #1678	C₂×C₁₃	semidirect
125. #1676	C₂×C₁₃	semidirect
126. #1668	C₂×C₁₃	semidirect
127. #1568	C₂×C₁₃	semidirect
128. #1551	C₂×C₁₃	semidirect
295. C₁₃	#19264	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×#3169	2
C₁₃×#3112	2
C₂×C₈×Q₈×C₁₃	2
C₁₃×#1678	2
C₁₃×#1676	2
C₁₃×#1668	2
C₁₃×#1568	2
C₁₃×#1551	2
#19264	13

Subgroups — C13×#19264 #15705603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#19264 #15705603