Subgroups — C₁₃×#2445 #13111503

Order 1664.

696 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#123	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#394	C₂	1	2	yes	maximal, complemented
5	C₁₃×#394	C₂	1	2	yes	maximal, complemented
6	C₁₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#386	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
14	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
15	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
17	C₁₃×#127	—	2	4	no	—
18	C₁₃×#123	—	2	8	no	—
19	C₄×D₄×C₁₃	—	2	4	no	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
35	C₁₃×#95	C₂²	1	1	yes	characteristic
36	C₁₃×#95	C₂²	1	1	yes	characteristic
37	C₁₃×#95	C₂²	1	2	yes	—
38	C₂⁴×C₁₃	C₂³	1	2	yes	—
39	C₂⁴×C₁₃	C₂³	1	2	yes	—
40	C₂×D₄×C₁₃	—	2	4	no	—
41	C₂×D₄×C₁₃	—	2	4	no	—
42	C₂×D₄×C₁₃	—	2	4	no	—
43	C₂×D₄×C₁₃	—	2	2	no	—
44	C₂×D₄×C₁₃	—	2	2	no	—
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	—	2	4	no	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	—	2	2	no	—
49	C₂×D₄×C₁₃	—	4	2	no	—
50	C₂×D₄×C₁₃	—	4	2	no	—
51	C₂×D₄×C₁₃	D₄	1	4	yes	—
52	C₂²×C₄×C₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	2	4	no	—
66	C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₁₃×C₄⋊C₄	—	2	2	no	—
68	C₁₃×C₄⋊C₄	D₄	1	4	yes	complemented
69	C₁₃×C₂²⋊C₄	—	2	4	no	—
70	C₁₃×C₂²⋊C₄	—	2	4	no	—
71	C₁₃×C₂²⋊C₄	—	2	4	no	—
72	C₁₃×C₂²⋊C₄	—	2	4	no	—
73	C₁₃×C₂²⋊C₄	—	4	2	no	—
74	C₁₃×C₂²⋊C₄	—	4	2	no	—
75	C₁₃×C₂²⋊C₄	—	4	2	no	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	#2445	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₁₃	—	2	2	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	—	2	2	no	—
83	C₂³×C₁₃	—	2	2	no	—
84	C₂³×C₁₃	—	2	2	no	—
85	C₂³×C₁₃	—	2	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	2	2	no	—
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	—	2	2	no	—
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	C₂×D₄	1	4	yes	—
94	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
95	D₄×C₁₃	—	2	8	no	—
96	D₄×C₁₃	—	4	8	no	—
97	D₄×C₁₃	—	4	8	no	—
98	D₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	8	no	—
120	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	#41	1	2	yes	—
125	C₂×C₄×C₁₃	#41	1	2	yes	—
126	C₂×C₄×C₁₃	#41	1	2	yes	—
127	C₂×#127	C₂×C₁₃	1	1	yes	characteristic
128	C₂×#123	C₂×C₁₃	1	2	yes	complemented
129	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#394	C₂×C₁₃	1	2	yes	complemented
131	#394	C₂×C₁₃	1	2	yes	complemented
132	#390	C₂×C₁₃	1	1	yes	characteristic, complemented
133	#390	C₂×C₁₃	1	1	yes	characteristic, complemented
134	#390	C₂×C₁₃	1	1	yes	characteristic, complemented
135	#389	C₂×C₁₃	1	1	yes	characteristic, complemented
136	#386	C₂×C₁₃	1	2	yes	complemented
137	C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	—	2	4	no	—
141	C₂²×C₁₃	—	4	2	no	—
142	C₂²×C₁₃	—	4	2	no	—
143	C₂²×C₁₃	—	4	2	no	—
144	C₂²×C₁₃	—	4	2	no	—
145	C₂²×C₁₃	—	4	2	no	—
146	C₂²×C₁₃	—	4	2	no	—
147	C₂²×C₁₃	—	4	2	no	—
148	C₂²×C₁₃	—	4	2	no	—
149	C₂²×C₁₃	—	4	2	no	—
150	C₂²×C₁₃	—	4	2	no	—
151	C₂²×C₁₃	—	4	2	no	—
152	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	4	1	no	—
164	C₄×C₁₃	—	4	2	no	—
165	C₄×C₁₃	—	4	1	no	—
166	C₄×C₁₃	—	4	2	no	—
167	C₄×C₁₃	#123	1	4	yes	complemented
168	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
169	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
170	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
171	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
172	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
173	#127	—	2	4	no	—
174	#123	—	2	8	no	—
175	C₄×D₄	—	2	4	no	—
176	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
178	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
179	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
182	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
183	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
184	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
185	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
188	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
189	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
191	#95	C₂²×C₁₃	1	1	yes	characteristic
192	#95	C₂²×C₁₃	1	1	yes	characteristic
193	#95	C₂²×C₁₃	1	2	yes	—
194	C₂×C₁₃	—	4	4	no	—
195	C₂×C₁₃	—	8	2	no	—
196	C₂×C₁₃	C₂×#123	1	1	yes	characteristic
197	C₂×C₁₃	#532	1	1	yes	characteristic
198	C₂×C₁₃	#540	1	1	yes	characteristic
199	C₂×C₁₃	D₄²	1	1	yes	characteristic
200	C₂×C₁₃	#546	1	1	yes	characteristic
201	C₂×C₁₃	#550	1	1	yes	characteristic
202	C₂×C₁₃	#559	1	1	yes	characteristic
203	C₂⁴	C₂³×C₁₃	1	2	yes	—
204	C₂⁴	C₂³×C₁₃	1	2	yes	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	D₄×C₁₃	1	4	yes	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
223	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
224	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	D₄×C₁₃	1	4	yes	complemented
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₁₃	#2445	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×D₄×C₁₃	1	4	yes	—
259	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
260	D₄	—	2	8	no	—
261	D₄	—	4	8	no	—
262	D₄	—	4	8	no	—
263	D₄	—	4	4	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
286	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
287	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
288	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
289	C₂×C₄	C₁₃×#41	1	2	yes	—
290	C₂×C₄	C₁₃×#41	1	2	yes	—
291	C₂×C₄	C₁₃×#41	1	2	yes	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
307	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
308	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
309	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
310	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
311	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
312	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	2	no	—
321	C₄	C₁₃×#123	1	4	yes	complemented
322	C₂	—	4	4	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×C₁₃×#123	1	1	yes	characteristic
325	C₂	C₁₃×#532	1	1	yes	characteristic
326	C₂	C₁₃×#540	1	1	yes	characteristic
327	C₂	D₄²×C₁₃	1	1	yes	characteristic
328	C₂	C₁₃×#546	1	1	yes	characteristic
329	C₂	C₁₃×#550	1	1	yes	characteristic
330	C₂	C₁₃×#559	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#123	C₂	semidirect
3. C₂×C₄×D₄×C₁₃	C₂	semidirect
4. C₁₃×#394	C₂	semidirect
5. C₁₃×#394	C₂	semidirect
6. C₁₃×#390	C₂	semidirect
7. C₁₃×#390	C₂	semidirect
8. C₁₃×#390	C₂	semidirect
9. C₁₃×#389	C₂	semidirect
10. C₁₃×#386	C₂	semidirect
11. C₄×C₁₃×C₄⋊C₄	C₂	semidirect
68. C₁₃×C₄⋊C₄	D₄	semidirect
79. #2445	C₁₃	direct (N × H)
128. C₂×#123	C₂×C₁₃	semidirect
129. C₂×C₄×D₄	C₂×C₁₃	semidirect
130. #394	C₂×C₁₃	semidirect
131. #394	C₂×C₁₃	semidirect
132. #390	C₂×C₁₃	semidirect
133. #390	C₂×C₁₃	semidirect
134. #390	C₂×C₁₃	semidirect
135. #389	C₂×C₁₃	semidirect
136. #386	C₂×C₁₃	semidirect
137. C₄×C₄⋊C₄	C₂×C₁₃	semidirect
167. C₄×C₁₃	#123	semidirect
233. C₄⋊C₄	D₄×C₁₃	semidirect
244. C₁₃	#2445	direct (N × H)
321. C₄	C₁₃×#123	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#127	2
C₂×C₁₃×#123	2
C₂×C₄×D₄×C₁₃	2
C₁₃×#394	2
C₁₃×#390	2
C₁₃×#389	2
C₁₃×#386	2
C₄×C₁₃×C₄⋊C₄	2
#2445	13

Subgroups — C13×#2445 #13111503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2445 #13111503