Subgroups — C₁₃×#25872 #15712211

Order 3328.

1214 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3497	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3141	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2040	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1980	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1793	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₃×#101	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×2₋¹⁺⁴	C₂²	1	2	yes	—
9	C₁₃×#550	—	2	2	no	—
10	D₄×Q₈×C₁₃	—	2	2	no	—
11	C₁₃×#548	—	2	4	no	—
12	C₁₃×#537	—	2	2	no	—
13	C₁₃×#537	C₂²	1	2	yes	—
14	C₁₃×#532	—	2	2	no	—
15	C₁₃×#532	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#489	—	2	4	no	—
19	C₁₃×#487	—	2	2	no	—
20	C₁₃×#486	—	2	2	no	—
21	C₂×C₁₃×#104	C₂²	1	8	yes	—
22	C₁₃×#417	C₂²	1	8	yes	—
23	C₂×C₁₃×#101	C₂²	1	2	yes	—
24	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
25	C₁₃×#400	—	2	2	no	—
26	C₁₃×#391	—	2	2	no	—
27	C₁₃×#391	C₂²	1	2	yes	—
28	C₁₃×#382	C₂²	1	2	yes	—
29	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₁₃×#343	C₂²	1	2	yes	—
31	C₁₃×2₋¹⁺⁴	—	2	8	no	—
32	C₁₃×2₋¹⁺⁴	—	4	4	no	—
33	C₂×C₁₃×#41	—	2	4	no	—
34	C₂×C₁₃×#41	—	2	4	no	—
35	C₂×C₁₃×#41	C₂³	1	4	yes	—
36	C₂²×Q₈×C₁₃	—	2	4	no	—
37	C₂²×Q₈×C₁₃	—	2	1	no	—
38	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
40	C₁₃×C₄⋊Q₈	—	2	2	no	—
41	C₁₃×C₄⋊Q₈	—	2	2	no	—
42	C₁₃×C₄⋊Q₈	—	2	2	no	—
43	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
44	C₁₃×#127	—	2	2	no	—
45	C₁₃×#127	D₄	1	2	yes	—
46	C₁₃×#124	—	2	4	no	—
47	C₁₃×#124	D₄	1	4	yes	—
48	C₁₃×C₂²⋊Q₈	—	2	8	no	—
49	C₁₃×C₂²⋊Q₈	—	2	4	no	—
50	C₁₃×#121	—	2	8	no	—
51	C₁₃×#121	—	2	4	no	—
52	C₁₃×#121	—	2	2	no	—
53	C₁₃×#121	—	2	4	no	—
54	C₁₃×#121	—	2	2	no	—
55	C₄×Q₈×C₁₃	—	2	2	no	—
56	C₄×Q₈×C₁₃	—	2	2	no	—
57	C₄×D₄×C₁₃	—	2	4	no	—
58	C₄×D₄×C₁₃	—	2	4	no	—
59	C₄×D₄×C₁₃	—	2	2	no	—
60	C₄×D₄×C₁₃	—	2	2	no	—
61	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
64	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₁₃	D₄	1	2	yes	—
67	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
68	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
69	C₁₃×#104	—	4	8	no	—
70	C₁₃×#103	—	2	8	no	—
71	C₁₃×#102	—	2	8	no	—
72	C₁₃×#101	—	2	4	no	—
73	C₁₃×#97	—	2	2	no	—
74	C₄×C₈×C₁₃	—	2	2	no	—
75	C₁₃×#95	—	2	1	no	—
76	C₁₃×#95	—	2	4	no	—
77	C₁₃×#95	—	2	4	no	—
78	#25872	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₁₃×#41	—	2	8	no	—
80	C₁₃×#41	—	2	8	no	—
81	C₁₃×#41	—	4	8	no	—
82	C₁₃×#41	—	4	16	no	—
83	C₁₃×#41	—	4	8	no	—
84	C₂×Q₈×C₁₃	—	2	4	no	—
85	C₂×Q₈×C₁₃	—	2	2	no	—
86	C₂×Q₈×C₁₃	—	2	2	no	—
87	C₂×Q₈×C₁₃	—	4	4	no	—
88	C₂×Q₈×C₁₃	—	4	16	no	—
89	C₂×Q₈×C₁₃	—	4	1	no	—
90	C₂×Q₈×C₁₃	—	4	1	no	—
91	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
92	C₂×Q₈×C₁₃	#41	1	4	yes	—
93	C₂×D₄×C₁₃	—	2	4	no	—
94	C₂×D₄×C₁₃	—	2	4	no	—
95	C₂×D₄×C₁₃	—	2	4	no	—
96	C₂×D₄×C₁₃	—	2	2	no	—
97	C₂×D₄×C₁₃	—	2	2	no	—
98	C₂×D₄×C₁₃	C₂×D₄	1	4	yes	—
99	C₂²×C₄×C₁₃	—	2	8	no	—
100	C₂²×C₄×C₁₃	—	2	4	no	—
101	C₂²×C₄×C₁₃	—	2	2	no	—
102	C₂²×C₄×C₁₃	—	2	2	no	—
103	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₁₃×M₁₆	—	2	4	no	—
109	C₁₃×M₁₆	—	2	4	no	—
110	C₂×C₈×C₁₃	—	2	4	no	—
111	C₁₃×C₄⋊C₄	—	2	4	no	—
112	C₁₃×C₄⋊C₄	—	2	4	no	—
113	C₁₃×C₄⋊C₄	—	2	2	no	—
114	C₁₃×C₄⋊C₄	—	2	2	no	—
115	C₁₃×C₄⋊C₄	—	2	2	no	—
116	C₁₃×C₄⋊C₄	—	2	4	no	—
117	C₁₃×C₄⋊C₄	—	4	2	no	—
118	C₁₃×C₄⋊C₄	—	4	2	no	—
119	C₁₃×C₂²⋊C₄	—	2	8	no	—
120	C₁₃×C₂²⋊C₄	—	2	4	no	—
121	C₁₃×C₂²⋊C₄	—	2	4	no	—
122	C₄²×C₁₃	—	2	2	no	—
123	C₄²×C₁₃	—	2	2	no	—
124	C₄²×C₁₃	—	2	4	no	—
125	C₄²×C₁₃	C₂×D₄	1	2	yes	—
126	C₄²×C₁₃	C₂×D₄	1	2	yes	—
127	#3497	C₂×C₁₃	1	1	yes	characteristic
128	#3141	C₂×C₁₃	1	2	yes	complemented
129	#2568	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2040	C₂×C₁₃	1	4	yes	complemented
131	#1980	C₂×C₁₃	1	2	yes	complemented
132	#1793	C₂×C₁₃	1	4	yes	complemented
133	C₄×#101	C₂×C₁₃	1	1	yes	characteristic, complemented
134	C₂³×C₁₃	—	2	4	no	—
135	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	Q₈×C₁₃	—	2	8	no	—
137	Q₈×C₁₃	—	2	8	no	—
138	Q₈×C₁₃	—	4	8	no	—
139	Q₈×C₁₃	—	8	2	no	—
140	Q₈×C₁₃	—	8	2	no	—
141	D₄×C₁₃	—	2	8	no	—
142	D₄×C₁₃	—	4	8	no	—
143	D₄×C₁₃	—	4	4	no	—
144	D₄×C₁₃	—	4	4	no	—
145	D₄×C₁₃	—	4	4	no	—
146	D₄×C₁₃	—	8	2	no	—
147	D₄×C₁₃	—	8	4	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	2	4	no	—
153	C₂×C₄×C₁₃	—	4	1	no	—
154	C₂×C₄×C₁₃	—	4	16	no	—
155	C₂×C₄×C₁₃	—	4	8	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	—	4	2	no	—
159	C₂×C₄×C₁₃	—	4	4	no	—
160	C₂×C₄×C₁₃	—	4	2	no	—
161	C₂×C₄×C₁₃	—	4	2	no	—
162	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
163	C₂×C₄×C₁₃	#120	1	2	yes	—
164	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
165	C₂×C₄×C₁₃	#124	1	4	yes	—
166	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
167	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
168	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
169	C₈×C₁₃	—	4	4	no	—
170	C₂×2₋¹⁺⁴	C₂²×C₁₃	1	2	yes	—
171	#550	—	2	2	no	—
172	D₄×Q₈	—	2	2	no	—
173	#548	—	2	4	no	—
174	#537	—	2	2	no	—
175	#537	C₂²×C₁₃	1	2	yes	—
176	#532	—	2	2	no	—
177	#532	C₂²×C₁₃	1	2	yes	—
178	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
179	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
180	#489	—	2	4	no	—
181	#487	—	2	2	no	—
182	#486	—	2	2	no	—
183	C₂×#104	C₂²×C₁₃	1	8	yes	—
184	#417	C₂²×C₁₃	1	8	yes	—
185	C₂×#101	C₂²×C₁₃	1	2	yes	—
186	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
187	#400	—	2	2	no	—
188	#391	—	2	2	no	—
189	#391	C₂²×C₁₃	1	2	yes	—
190	#382	C₂²×C₁₃	1	2	yes	—
191	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
192	#343	C₂²×C₁₃	1	2	yes	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	—	2	1	no	—
195	C₂²×C₁₃	—	4	4	no	—
196	C₂²×C₁₃	—	4	4	no	—
197	C₂²×C₁₃	—	4	4	no	—
198	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
199	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
200	C₂²×C₁₃	#535	1	1	yes	characteristic
201	C₄×C₁₃	—	2	4	no	—
202	C₄×C₁₃	—	2	1	no	—
203	C₄×C₁₃	—	2	2	no	—
204	C₄×C₁₃	—	4	4	no	—
205	C₄×C₁₃	—	4	2	no	—
206	C₄×C₁₃	—	8	2	no	—
207	C₄×C₁₃	—	8	2	no	—
208	C₄×C₁₃	#454	1	2	yes	—
209	C₄×C₁₃	#546	1	4	yes	—
210	2₋¹⁺⁴	—	2	8	no	—
211	2₋¹⁺⁴	—	4	4	no	—
212	C₂×#41	—	2	4	no	—
213	C₂×#41	—	2	4	no	—
214	C₂×#41	C₂³×C₁₃	1	4	yes	—
215	C₂²×Q₈	—	2	4	no	—
216	C₂²×Q₈	—	2	1	no	—
217	C₂²×Q₈	C₂³×C₁₃	1	1	yes	characteristic
218	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
219	C₄⋊Q₈	—	2	2	no	—
220	C₄⋊Q₈	—	2	2	no	—
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
223	#127	—	2	2	no	—
224	#127	D₄×C₁₃	1	2	yes	—
225	#124	—	2	4	no	—
226	#124	D₄×C₁₃	1	4	yes	—
227	C₂²⋊Q₈	—	2	8	no	—
228	C₂²⋊Q₈	—	2	4	no	—
229	#121	—	2	8	no	—
230	#121	—	2	4	no	—
231	#121	—	2	2	no	—
232	#121	—	2	4	no	—
233	#121	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×D₄	—	2	4	no	—
237	C₄×D₄	—	2	4	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	—	2	2	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
243	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
244	C₂×C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
245	C₂×C₄²	D₄×C₁₃	1	2	yes	—
246	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
247	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
248	#104	—	4	8	no	—
249	#103	—	2	8	no	—
250	#102	—	2	8	no	—
251	#101	—	2	4	no	—
252	#97	—	2	2	no	—
253	C₄×C₈	—	2	2	no	—
254	#95	—	2	1	no	—
255	#95	—	2	4	no	—
256	#95	—	2	4	no	—
257	C₂×C₁₃	—	2	1	no	—
258	C₂×C₁₃	—	2	1	no	—
259	C₂×C₁₃	—	8	4	no	—
260	C₂×C₁₃	#2435	1	1	yes	characteristic
261	C₂×C₁₃	C₂×#454	1	1	yes	characteristic
262	C₂×C₁₃	#3022	1	1	yes	characteristic
263	#41	—	2	8	no	—
264	#41	—	2	8	no	—
265	#41	—	4	8	no	—
266	#41	—	4	16	no	—
267	#41	—	4	8	no	—
268	C₂×Q₈	—	2	4	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	4	4	no	—
272	C₂×Q₈	—	4	16	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×Q₈	—	4	1	no	—
275	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
276	C₂×Q₈	C₁₃×#41	1	4	yes	—
277	C₂×D₄	—	2	4	no	—
278	C₂×D₄	—	2	4	no	—
279	C₂×D₄	—	2	4	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	C₂×D₄×C₁₃	1	4	yes	—
283	C₂²×C₄	—	2	8	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
289	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
290	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
291	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
292	M₁₆	—	2	4	no	—
293	M₁₆	—	2	4	no	—
294	C₂×C₈	—	2	4	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	4	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	2	8	no	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₂²⋊C₄	—	2	4	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	4	no	—
309	C₄²	C₂×D₄×C₁₃	1	2	yes	—
310	C₄²	C₂×D₄×C₁₃	1	2	yes	—
311	C₁₃	#25872	1	1	yes	characteristic, direct product factor, Sylow
312	C₂³	—	2	4	no	—
313	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
314	Q₈	—	2	8	no	—
315	Q₈	—	2	8	no	—
316	Q₈	—	4	8	no	—
317	Q₈	—	8	2	no	—
318	Q₈	—	8	2	no	—
319	D₄	—	2	8	no	—
320	D₄	—	4	8	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	4	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	4	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	16	no	—
333	C₂×C₄	—	4	8	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
341	C₂×C₄	C₁₃×#120	1	2	yes	—
342	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
343	C₂×C₄	C₁₃×#124	1	4	yes	—
344	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
345	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
346	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
347	C₈	—	4	4	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	4	4	no	—
351	C₂²	—	4	4	no	—
352	C₂²	—	4	4	no	—
353	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
354	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
355	C₂²	C₁₃×#535	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	4	no	—
359	C₄	—	4	2	no	—
360	C₄	—	4	4	no	—
361	C₄	—	8	2	no	—
362	C₄	—	8	2	no	—
363	C₄	C₁₃×#454	1	2	yes	—
364	C₄	C₁₃×#546	1	4	yes	—
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	—	8	4	no	—
368	C₂	C₁₃×#2435	1	1	yes	characteristic
369	C₂	C₂×C₁₃×#454	1	1	yes	characteristic
370	C₂	C₁₃×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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×#3141	C₂	semidirect
3. C₁₃×#2568	C₂	semidirect
4. C₁₃×#2040	C₂	semidirect
5. C₁₃×#1980	C₂	semidirect
6. C₁₃×#1793	C₂	semidirect
7. C₄×C₁₃×#101	C₂	semidirect
78. #25872	C₁₃	direct (N × H)
128. #3141	C₂×C₁₃	semidirect
129. #2568	C₂×C₁₃	semidirect
130. #2040	C₂×C₁₃	semidirect
131. #1980	C₂×C₁₃	semidirect
132. #1793	C₂×C₁₃	semidirect
133. C₄×#101	C₂×C₁₃	semidirect
311. C₁₃	#25872	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×#3497	2
C₁₃×#3141	2
C₁₃×#2568	2
C₁₃×#2040	2
C₁₃×#1980	2
C₁₃×#1793	2
C₄×C₁₃×#101	2
#25872	13

Subgroups — C13×#25872 #15712211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#25872 #15712211