Subgroups — C₁₃×#32791 #15719130

Order 3328.

628 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2851	C₂	1	2	yes	maximal
2	C₁₃×#2101	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2101	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2077	C₂	1	2	yes	maximal
5	C₁₃×#2076	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#2075	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2059	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2059	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1817	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#125	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₁₃×#389	C₂²	1	2	yes	—
18	C₁₃×#389	C₂²	1	2	yes	—
19	C₁₃×#387	—	2	2	no	—
20	C₁₃×#387	—	2	2	no	—
21	C₁₃×#385	—	2	2	no	—
22	C₁₃×#385	—	2	2	no	—
23	C₁₃×#382	C₂²	1	2	yes	—
24	C₁₃×#380	C₂²	1	2	yes	—
25	C₁₃×#380	C₂²	1	2	yes	—
26	C₁₃×#340	C₂²	1	2	yes	—
27	C₁₃×#340	C₂²	1	2	yes	—
28	C₁₃×#340	C₂²	1	2	yes	—
29	C₁₃×#340	C₂²	1	2	yes	—
30	C₁₃×#340	C₂²	1	1	yes	characteristic
31	C₁₃×#340	C₂²	1	1	yes	characteristic
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×#125	—	4	2	no	—
42	C₁₃×#125	—	4	2	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	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
52	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
53	C₂×C₄²×C₁₃	—	4	1	no	—
54	C₂×C₄²×C₁₃	—	4	1	no	—
55	C₂×C₄²×C₁₃	D₄	1	2	yes	—
56	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
57	C₁₃×#107	—	4	1	no	—
58	C₁₃×#107	—	4	1	no	—
59	C₁₃×#106	—	4	2	no	—
60	C₁₃×C₂²⋊C₈	—	4	1	no	—
61	C₁₃×C₂²⋊C₈	—	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	—	2	2	no	—
67	C₁₃×#95	—	2	2	no	—
68	#32791	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	4	2	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
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	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	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	—	4	4	no	—
103	C₁₃×C₄⋊C₄	—	4	4	no	—
104	C₁₃×C₄⋊C₄	—	4	4	no	—
105	C₁₃×C₄⋊C₄	—	4	4	no	—
106	C₁₃×C₂²⋊C₄	—	4	4	no	—
107	C₁₃×C₂²⋊C₄	—	4	4	no	—
108	C₄²×C₁₃	—	2	2	no	—
109	C₄²×C₁₃	—	2	2	no	—
110	C₄²×C₁₃	—	4	2	no	—
111	#2851	C₂×C₁₃	1	2	yes	—
112	#2101	C₂×C₁₃	1	2	yes	complemented
113	#2101	C₂×C₁₃	1	2	yes	complemented
114	#2077	C₂×C₁₃	1	2	yes	—
115	#2076	C₂×C₁₃	1	1	yes	characteristic
116	#2075	C₂×C₁₃	1	1	yes	characteristic
117	#2059	C₂×C₁₃	1	2	yes	complemented
118	#2059	C₂×C₁₃	1	2	yes	complemented
119	#1817	C₂×C₁₃	1	1	yes	characteristic
120	C₂³×C₁₃	—	2	1	no	—
121	C₂³×C₁₃	—	2	2	no	—
122	C₂³×C₁₃	—	2	2	no	—
123	C₂³×C₁₃	—	2	1	no	—
124	C₂³×C₁₃	#135	1	2	yes	—
125	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	4	4	no	—
135	C₂×C₄×C₁₃	—	4	4	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	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	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
154	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
155	C₈×C₁₃	—	4	2	no	—
156	C₈×C₁₃	—	4	1	no	—
157	C₈×C₁₃	—	4	1	no	—
158	C₂×#125	C₂²×C₁₃	1	2	yes	—
159	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×#106	C₂²×C₁₃	1	2	yes	—
163	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
165	#389	C₂²×C₁₃	1	2	yes	—
166	#389	C₂²×C₁₃	1	2	yes	—
167	#387	—	2	2	no	—
168	#387	—	2	2	no	—
169	#385	—	2	2	no	—
170	#385	—	2	2	no	—
171	#382	C₂²×C₁₃	1	2	yes	—
172	#380	C₂²×C₁₃	1	2	yes	—
173	#380	C₂²×C₁₃	1	2	yes	—
174	#340	C₂²×C₁₃	1	2	yes	—
175	#340	C₂²×C₁₃	1	2	yes	—
176	#340	C₂²×C₁₃	1	2	yes	—
177	#340	C₂²×C₁₃	1	2	yes	—
178	#340	C₂²×C₁₃	1	1	yes	characteristic
179	#340	C₂²×C₁₃	1	1	yes	characteristic
180	#340	C₂²×C₁₃	1	2	yes	—
181	#336	C₂²×C₁₃	1	2	yes	—
182	#336	C₂²×C₁₃	1	2	yes	—
183	C₂²×C₁₃	—	2	2	no	—
184	C₂²×C₁₃	—	2	4	no	—
185	C₂²×C₁₃	—	4	1	no	—
186	C₂²×C₁₃	—	4	1	no	—
187	C₂²×C₁₃	—	4	2	no	—
188	C₂²×C₁₃	#534	1	1	yes	characteristic
189	C₂²×C₁₃	#536	1	2	yes	—
190	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
191	C₂²×C₁₃	#575	1	1	yes	characteristic
192	C₂²×C₁₃	#575	1	2	yes	—
193	C₄×C₁₃	—	4	1	no	—
194	C₄×C₁₃	—	4	2	no	—
195	C₄×C₁₃	—	4	2	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	—	8	2	no	—
200	C₄×C₁₃	#563	1	2	yes	—
201	C₄×C₁₃	#563	1	1	yes	characteristic
202	C₄×C₁₃	#563	1	1	yes	characteristic
203	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
204	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
205	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
207	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
209	#125	—	4	2	no	—
210	#125	—	4	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₄²	—	4	1	no	—
222	C₂×C₄²	—	4	1	no	—
223	C₂×C₄²	D₄×C₁₃	1	2	yes	—
224	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
225	#107	—	4	1	no	—
226	#107	—	4	1	no	—
227	#106	—	4	2	no	—
228	C₂²⋊C₈	—	4	1	no	—
229	C₂²⋊C₈	—	4	1	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	C₂×C₁₃	—	4	2	no	—
237	C₂×C₁₃	#2817	1	1	yes	characteristic
238	C₂×C₁₃	#3191	1	1	yes	characteristic
239	C₂×C₁₃	#3197	1	1	yes	characteristic
240	C₂×C₁₃	#3232	1	2	yes	—
241	C₂×C₁₃	#3240	1	2	yes	—
242	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
258	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
259	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₂²⋊C₄	—	4	4	no	—
280	C₂²⋊C₄	—	4	4	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	2	no	—
284	C₁₃	#32791	1	1	yes	characteristic, direct product factor, Sylow
285	C₂³	—	2	1	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	1	no	—
289	C₂³	C₁₃×#135	1	2	yes	—
290	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	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	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
317	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
318	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
319	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
320	C₈	—	4	2	no	—
321	C₈	—	4	1	no	—
322	C₈	—	4	1	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	2	no	—
328	C₂²	C₁₃×#534	1	1	yes	characteristic
329	C₂²	C₁₃×#536	1	2	yes	—
330	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
331	C₂²	C₁₃×#575	1	1	yes	characteristic
332	C₂²	C₁₃×#575	1	2	yes	—
333	C₄	—	4	1	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	2	no	—
339	C₄	—	8	2	no	—
340	C₄	C₁₃×#563	1	2	yes	—
341	C₄	C₁₃×#563	1	1	yes	characteristic
342	C₄	C₁₃×#563	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	C₁₃×#2817	1	1	yes	characteristic
345	C₂	C₁₃×#3191	1	1	yes	characteristic
346	C₂	C₁₃×#3197	1	1	yes	characteristic
347	C₂	C₁₃×#3232	1	2	yes	—
348	C₂	C₁₃×#3240	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2101	C₂	semidirect
3. C₁₃×#2101	C₂	semidirect
7. C₁₃×#2059	C₂	semidirect
8. C₁₃×#2059	C₂	semidirect
68. #32791	C₁₃	direct (N × H)
112. #2101	C₂×C₁₃	semidirect
113. #2101	C₂×C₁₃	semidirect
117. #2059	C₂×C₁₃	semidirect
118. #2059	C₂×C₁₃	semidirect
284. C₁₃	#32791	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×#2851	2
C₁₃×#2101	2
C₁₃×#2077	2
C₁₃×#2076	2
C₁₃×#2075	2
C₁₃×#2059	2
C₁₃×#1817	2
#32791	13

Subgroups — C13×#32791 #15719130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32791 #15719130