Subgroups — C₁₃×#29136 #15715475

Order 3328.

754 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2328	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2101	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2041	C₂	1	2	yes	maximal
5	C₁₃×#1936	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1934	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1804	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1802	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1768	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×#102	C₂²	1	2	yes	—
14	C₂×C₁₃×#102	C₂²	1	2	yes	—
15	C₂×C₁₃×#102	C₂²	1	2	yes	—
16	C₂×C₁₃×#102	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#400	—	2	2	no	—
20	C₁₃×#398	—	2	2	no	—
21	C₁₃×#390	C₂²	1	2	yes	—
22	C₁₃×#389	C₂²	1	2	yes	—
23	C₁₃×#389	C₂²	1	2	yes	—
24	C₁₃×#386	—	2	2	no	—
25	C₁₃×#386	—	2	2	no	—
26	C₁₃×#384	C₂²	1	1	yes	characteristic
27	C₁₃×#382	C₂²	1	1	yes	characteristic
28	C₁₃×#382	C₂²	1	1	yes	characteristic
29	C₁₃×#376	C₂²	1	1	yes	characteristic
30	C₁₃×#340	C₂²	1	2	yes	—
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×D₄×C₁₃	—	2	2	no	—
36	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₁₃	—	2	2	no	—
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₁₃×#127	—	2	4	no	—
41	C₄×D₄×C₁₃	—	2	4	no	—
42	C₄×D₄×C₁₃	—	2	4	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×C₄⋊C₈	—	2	4	no	—
56	C₁₃×#102	—	2	4	no	—
57	C₁₃×#102	—	2	4	no	—
58	C₁₃×#102	—	4	2	no	—
59	C₁₃×#102	—	4	2	no	—
60	C₁₃×#97	—	4	1	no	—
61	C₄×C₈×C₁₃	—	4	1	no	—
62	C₁₃×#95	—	2	2	no	—
63	C₁₃×#95	—	2	2	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	D₄	1	2	yes	—
67	C₁₃×#95	D₄	1	2	yes	—
68	#29136	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂⁴×C₁₃	—	2	2	no	—
70	C₂×D₄×C₁₃	—	2	8	no	—
71	C₂×D₄×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	—	2	2	no	—
73	C₂×D₄×C₁₃	—	2	2	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	—	8	2	no	—
77	C₂×D₄×C₁₃	—	8	2	no	—
78	C₂×D₄×C₁₃	#41	1	4	yes	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	4	4	no	—
83	C₂²×C₄×C₁₃	—	8	2	no	—
84	C₂²×C₄×C₁₃	—	8	2	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	8	2	no	—
103	C₁₃×C₄⋊C₄	—	8	2	no	—
104	C₁₃×C₄⋊C₄	—	8	2	no	—
105	C₁₃×C₂²⋊C₄	—	4	4	no	—
106	C₁₃×C₂²⋊C₄	—	8	2	no	—
107	C₁₃×C₂²⋊C₄	—	8	2	no	—
108	C₄²×C₁₃	—	2	2	no	—
109	C₄²×C₁₃	—	2	1	no	—
110	C₄²×C₁₃	—	2	1	no	—
111	C₄²×C₁₃	#41	1	4	yes	—
112	#2647	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2328	C₂×C₁₃	1	1	yes	characteristic
114	#2101	C₂×C₁₃	1	2	yes	complemented
115	#2041	C₂×C₁₃	1	2	yes	—
116	#1936	C₂×C₁₃	1	2	yes	complemented
117	#1934	C₂×C₁₃	1	2	yes	complemented
118	#1804	C₂×C₁₃	1	2	yes	complemented
119	#1802	C₂×C₁₃	1	2	yes	complemented
120	#1768	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂³×C₁₃	—	2	4	no	—
122	C₂³×C₁₃	—	4	2	no	—
123	C₂³×C₁₃	—	4	2	no	—
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	—	4	2	no	—
126	C₂³×C₁₃	—	4	2	no	—
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	D₄×C₁₃	—	2	4	no	—
130	D₄×C₁₃	—	2	4	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	8	8	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	4	2	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	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	8	8	no	—
153	C₂×C₄×C₁₃	#124	1	4	yes	—
154	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
155	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
156	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
158	C₈×C₁₃	—	4	2	no	—
159	C₈×C₁₃	—	4	2	no	—
160	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
163	C₂×#102	C₂²×C₁₃	1	2	yes	—
164	C₂×#102	C₂²×C₁₃	1	2	yes	—
165	C₂×#102	C₂²×C₁₃	1	2	yes	—
166	C₂×#102	C₂²×C₁₃	1	2	yes	—
167	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
169	#400	—	2	2	no	—
170	#398	—	2	2	no	—
171	#390	C₂²×C₁₃	1	2	yes	—
172	#389	C₂²×C₁₃	1	2	yes	—
173	#389	C₂²×C₁₃	1	2	yes	—
174	#386	—	2	2	no	—
175	#386	—	2	2	no	—
176	#384	C₂²×C₁₃	1	1	yes	characteristic
177	#382	C₂²×C₁₃	1	1	yes	characteristic
178	#382	C₂²×C₁₃	1	1	yes	characteristic
179	#376	C₂²×C₁₃	1	1	yes	characteristic
180	#340	C₂²×C₁₃	1	2	yes	—
181	#340	C₂²×C₁₃	1	2	yes	—
182	#340	C₂²×C₁₃	1	2	yes	—
183	#340	C₂²×C₁₃	1	2	yes	—
184	#336	C₂²×C₁₃	1	2	yes	—
185	C₂²×C₁₃	—	4	4	no	—
186	C₂²×C₁₃	—	4	4	no	—
187	C₂²×C₁₃	—	4	4	no	—
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	—	8	2	no	—
191	C₂²×C₁₃	—	8	2	no	—
192	C₂²×C₁₃	#525	1	1	yes	characteristic
193	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
194	C₂²×C₁₃	#537	1	1	yes	characteristic
195	C₂²×C₁₃	#576	1	2	yes	—
196	C₂²×C₁₃	#577	1	2	yes	—
197	C₄×C₁₃	—	2	4	no	—
198	C₄×C₁₃	—	4	2	no	—
199	C₄×C₁₃	—	4	1	no	—
200	C₄×C₁₃	—	4	1	no	—
201	C₄×C₁₃	—	8	2	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	#552	1	2	yes	—
205	C₄×C₁₃	#553	1	2	yes	—
206	C₂²×D₄	—	2	2	no	—
207	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
208	C₂³×C₄	—	2	2	no	—
209	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
210	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
211	#127	—	2	4	no	—
212	C₄×D₄	—	2	4	no	—
213	C₄×D₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	—	2	2	no	—
223	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
225	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
226	C₄⋊C₈	—	2	4	no	—
227	#102	—	2	4	no	—
228	#102	—	2	4	no	—
229	#102	—	4	2	no	—
230	#102	—	4	2	no	—
231	#97	—	4	1	no	—
232	C₄×C₈	—	4	1	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	D₄×C₁₃	1	2	yes	—
238	#95	D₄×C₁₃	1	2	yes	—
239	C₂×C₁₃	—	8	4	no	—
240	C₂×C₁₃	#2562	1	1	yes	characteristic
241	C₂×C₁₃	#3118	1	2	yes	—
242	C₂×C₁₃	#3136	1	2	yes	—
243	C₂×C₁₃	#3259	1	1	yes	characteristic
244	C₂×C₁₃	#3260	1	1	yes	characteristic
245	C₂⁴	—	2	2	no	—
246	C₂×D₄	—	2	8	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	8	2	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂×D₄	C₁₃×#41	1	4	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	4	4	no	—
259	C₂²×C₄	—	8	2	no	—
260	C₂²×C₄	—	8	2	no	—
261	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
263	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	8	2	no	—
279	C₄⋊C₄	—	8	2	no	—
280	C₄⋊C₄	—	8	2	no	—
281	C₂²⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	8	2	no	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	C₁₃×#41	1	4	yes	—
288	C₁₃	#29136	1	1	yes	characteristic, direct product factor, Sylow
289	C₂³	—	2	4	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
297	D₄	—	2	4	no	—
298	D₄	—	2	4	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	8	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	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	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	8	no	—
321	C₂×C₄	C₁₃×#124	1	4	yes	—
322	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
323	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
324	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
325	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
326	C₈	—	4	2	no	—
327	C₈	—	4	2	no	—
328	C₂²	—	4	4	no	—
329	C₂²	—	4	4	no	—
330	C₂²	—	4	4	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	C₁₃×#525	1	1	yes	characteristic
336	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
337	C₂²	C₁₃×#537	1	1	yes	characteristic
338	C₂²	C₁₃×#576	1	2	yes	—
339	C₂²	C₁₃×#577	1	2	yes	—
340	C₄	—	2	4	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	C₁₃×#552	1	2	yes	—
348	C₄	C₁₃×#553	1	2	yes	—
349	C₂	—	8	4	no	—
350	C₂	C₁₃×#2562	1	1	yes	characteristic
351	C₂	C₁₃×#3118	1	2	yes	—
352	C₂	C₁₃×#3136	1	2	yes	—
353	C₂	C₁₃×#3259	1	1	yes	characteristic
354	C₂	C₁₃×#3260	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2647	C₂	semidirect
3. C₁₃×#2101	C₂	semidirect
5. C₁₃×#1936	C₂	semidirect
6. C₁₃×#1934	C₂	semidirect
7. C₁₃×#1804	C₂	semidirect
8. C₁₃×#1802	C₂	semidirect
9. C₁₃×#1768	C₂	semidirect
68. #29136	C₁₃	direct (N × H)
112. #2647	C₂×C₁₃	semidirect
114. #2101	C₂×C₁₃	semidirect
116. #1936	C₂×C₁₃	semidirect
117. #1934	C₂×C₁₃	semidirect
118. #1804	C₂×C₁₃	semidirect
119. #1802	C₂×C₁₃	semidirect
120. #1768	C₂×C₁₃	semidirect
288. C₁₃	#29136	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×#2647	2
C₁₃×#2328	2
C₁₃×#2101	2
C₁₃×#2041	2
C₁₃×#1936	2
C₁₃×#1934	2
C₁₃×#1804	2
C₁₃×#1802	2
C₁₃×#1768	2
#29136	13

Subgroups — C13×#29136 #15715475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29136 #15715475