Subgroups — C₁₃×#3351 #13112409

Order 1664.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#553	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#552	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#485	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#482	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#481	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#468	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#468	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
14	C₁₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
19	C₁₃×#126	—	2	1	no	—
20	C₁₃×#126	—	2	1	no	—
21	C₁₃×#125	—	2	1	no	—
22	C₁₃×#125	C₂²	1	1	yes	characteristic
23	C₁₃×#125	C₂²	1	1	yes	characteristic
24	C₁₃×#123	—	2	1	no	—
25	C₁₃×#123	—	2	1	no	—
26	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
27	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₁₃×#121	—	2	1	no	—
29	C₁₃×#121	C₂²	1	1	yes	characteristic
30	C₁₃×#121	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₁₃	—	2	1	no	—
33	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
35	C₁₃×#117	—	2	1	no	—
36	C₁₃×#117	C₂²	1	1	yes	characteristic
37	C₁₃×#117	C₂²	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
40	C₁₃×#107	C₂²	1	1	yes	characteristic
41	C₁₃×#107	C₂²	1	1	yes	characteristic
42	C₁₃×#107	C₂²	1	1	yes	characteristic
43	C₁₃×#107	C₂²	1	1	yes	characteristic
44	C₁₃×#106	C₂²	1	1	yes	characteristic
45	C₁₃×#106	C₂²	1	1	yes	characteristic
46	C₁₃×#106	C₂²	1	1	yes	characteristic
47	C₁₃×#106	C₂²	1	1	yes	characteristic
48	C₁₃×#106	C₂²	1	1	yes	characteristic
49	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₁₃×#103	C₂²	1	1	yes	characteristic
51	C₁₃×#103	C₂²	1	1	yes	characteristic
52	C₁₃×#103	C₂²	1	1	yes	characteristic
53	C₁₃×#102	C₂²	1	1	yes	characteristic
54	C₁₃×#102	C₂²	1	1	yes	characteristic
55	C₁₃×#102	C₂²	1	1	yes	characteristic
56	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₃	—	2	1	no	—
61	C₂×D₄×C₁₃	—	2	1	no	—
62	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
63	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	—	2	1	no	—
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
69	C₁₃×SD₁₆	—	2	2	no	—
70	C₁₃×M₁₆	—	2	2	no	—
71	C₁₃×M₁₆	—	2	2	no	—
72	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	—	2	1	no	—
77	C₁₃×C₄⋊C₄	—	2	1	no	—
78	C₁₃×C₄⋊C₄	—	2	1	no	—
79	C₁₃×C₄⋊C₄	—	2	1	no	—
80	C₁₃×C₄⋊C₄	—	2	1	no	—
81	C₁₃×C₄⋊C₄	—	2	1	no	—
82	C₁₃×C₄⋊C₄	—	2	1	no	—
83	C₁₃×C₄⋊C₄	—	2	1	no	—
84	C₁₃×C₄⋊C₄	—	2	1	no	—
85	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₁₃×C₂²⋊C₄	—	2	1	no	—
93	C₁₃×C₂²⋊C₄	—	2	1	no	—
94	C₁₃×C₂²⋊C₄	—	2	1	no	—
95	C₁₃×C₂²⋊C₄	—	2	1	no	—
96	C₁₃×C₂²⋊C₄	—	2	1	no	—
97	C₁₃×C₂²⋊C₄	—	2	1	no	—
98	C₁₃×C₂²⋊C₄	—	2	1	no	—
99	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
100	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	C₂³	1	1	yes	characteristic
105	#3351	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₁₃	—	2	1	no	—
107	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₁₃	—	2	1	no	—
110	Q₈×C₁₃	—	2	1	no	—
111	D₄×C₁₃	—	2	1	no	—
112	D₄×C₁₃	—	2	1	no	—
113	D₄×C₁₃	—	4	1	no	—
114	D₄×C₁₃	—	4	2	no	—
115	D₄×C₁₃	—	8	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	8	1	no	—
129	C₂×C₄×C₁₃	—	8	1	no	—
130	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	—	2	1	no	—
138	C₈×C₁₃	#41	1	2	yes	—
139	#553	C₂×C₁₃	1	1	yes	characteristic
140	#552	C₂×C₁₃	1	1	yes	characteristic, complemented
141	#499	C₂×C₁₃	1	1	yes	characteristic, complemented
142	#485	C₂×C₁₃	1	1	yes	characteristic, complemented
143	#483	C₂×C₁₃	1	1	yes	characteristic, complemented
144	#482	C₂×C₁₃	1	1	yes	characteristic, complemented
145	#481	C₂×C₁₃	1	1	yes	characteristic, complemented
146	#479	C₂×C₁₃	1	1	yes	characteristic, complemented
147	#478	C₂×C₁₃	1	1	yes	characteristic, complemented
148	#468	C₂×C₁₃	1	1	yes	characteristic, complemented
149	#468	C₂×C₁₃	1	1	yes	characteristic, complemented
150	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
151	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
152	#428	C₂×C₁₃	1	1	yes	characteristic, complemented
153	#428	C₂×C₁₃	1	1	yes	characteristic, complemented
154	C₂²×C₁₃	—	2	1	no	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	4	1	no	—
161	C₂²×C₁₃	—	4	1	no	—
162	C₂²×C₁₃	—	4	1	no	—
163	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	4	1	no	—
169	C₄×C₁₃	—	4	1	no	—
170	C₄×C₁₃	—	4	1	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	—	4	1	no	—
173	C₄×C₁₃	—	4	1	no	—
174	C₄×C₁₃	—	4	1	no	—
175	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
176	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
177	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
178	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
179	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
180	#126	—	2	1	no	—
181	#126	—	2	1	no	—
182	#125	—	2	1	no	—
183	#125	C₂²×C₁₃	1	1	yes	characteristic
184	#125	C₂²×C₁₃	1	1	yes	characteristic
185	#123	—	2	1	no	—
186	#123	—	2	1	no	—
187	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
188	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
189	#121	—	2	1	no	—
190	#121	C₂²×C₁₃	1	1	yes	characteristic
191	#121	C₂²×C₁₃	1	1	yes	characteristic
192	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
195	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
196	#117	—	2	1	no	—
197	#117	C₂²×C₁₃	1	1	yes	characteristic
198	#117	C₂²×C₁₃	1	1	yes	characteristic
199	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
201	#107	C₂²×C₁₃	1	1	yes	characteristic
202	#107	C₂²×C₁₃	1	1	yes	characteristic
203	#107	C₂²×C₁₃	1	1	yes	characteristic
204	#107	C₂²×C₁₃	1	1	yes	characteristic
205	#106	C₂²×C₁₃	1	1	yes	characteristic
206	#106	C₂²×C₁₃	1	1	yes	characteristic
207	#106	C₂²×C₁₃	1	1	yes	characteristic
208	#106	C₂²×C₁₃	1	1	yes	characteristic
209	#106	C₂²×C₁₃	1	1	yes	characteristic
210	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
211	#103	C₂²×C₁₃	1	1	yes	characteristic
212	#103	C₂²×C₁₃	1	1	yes	characteristic
213	#103	C₂²×C₁₃	1	1	yes	characteristic
214	#102	C₂²×C₁₃	1	1	yes	characteristic
215	#102	C₂²×C₁₃	1	1	yes	characteristic
216	#102	C₂²×C₁₃	1	1	yes	characteristic
217	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
218	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
220	C₂×C₁₃	—	4	1	no	—
221	C₂×C₁₃	—	4	1	no	—
222	C₂×C₁₃	—	8	1	no	—
223	C₂×C₁₃	#547	1	1	yes	characteristic
224	C₂×C₁₃	#575	1	1	yes	characteristic
225	C₂×C₁₃	#577	1	1	yes	characteristic
226	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
230	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
236	SD₁₆	—	2	2	no	—
237	M₁₆	—	2	2	no	—
238	M₁₆	—	2	2	no	—
239	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
240	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
253	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
254	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
258	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
267	C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	C₂³×C₁₃	1	1	yes	characteristic
272	C₁₃	#3351	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³	—	2	1	no	—
274	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	4	1	no	—
281	D₄	—	4	2	no	—
282	D₄	—	8	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
301	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	C₁₃×#41	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
328	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
329	C₂	—	4	1	no	—
330	C₂	—	4	1	no	—
331	C₂	—	8	1	no	—
332	C₂	C₁₃×#547	1	1	yes	characteristic
333	C₂	C₁₃×#575	1	1	yes	characteristic
334	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#553	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#552	C₂	semidirect
3. C₁₃×#499	C₂	semidirect
4. C₁₃×#485	C₂	semidirect
5. C₁₃×#483	C₂	semidirect
6. C₁₃×#482	C₂	semidirect
7. C₁₃×#481	C₂	semidirect
8. C₁₃×#479	C₂	semidirect
9. C₁₃×#478	C₂	semidirect
10. C₁₃×#468	C₂	semidirect
11. C₁₃×#468	C₂	semidirect
12. C₄×C₁₃×SD₁₆	C₂	semidirect
13. C₁₃×#436	C₂	semidirect
14. C₁₃×#428	C₂	semidirect
15. C₁₃×#428	C₂	semidirect
105. #3351	C₁₃	direct (N × H)
140. #552	C₂×C₁₃	semidirect
141. #499	C₂×C₁₃	semidirect
142. #485	C₂×C₁₃	semidirect
143. #483	C₂×C₁₃	semidirect
144. #482	C₂×C₁₃	semidirect
145. #481	C₂×C₁₃	semidirect
146. #479	C₂×C₁₃	semidirect
147. #478	C₂×C₁₃	semidirect
148. #468	C₂×C₁₃	semidirect
149. #468	C₂×C₁₃	semidirect
150. C₄×SD₁₆	C₂×C₁₃	semidirect
151. #436	C₂×C₁₃	semidirect
152. #428	C₂×C₁₃	semidirect
153. #428	C₂×C₁₃	semidirect
272. C₁₃	#3351	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×#553	2
C₁₃×#552	2
C₁₃×#499	2
C₁₃×#485	2
C₁₃×#483	2
C₁₃×#482	2
C₁₃×#481	2
C₁₃×#479	2
C₁₃×#478	2
C₁₃×#468	2
C₄×C₁₃×SD₁₆	2
C₁₃×#436	2
C₁₃×#428	2
#3351	13

Subgroups — C13×#3351 #13112409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3351 #13112409