Subgroups — C₁₃×#32549 #15718888

Order 3328.

640 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2581	C₂	1	2	yes	maximal
2	C₁₃×#2101	C₂	1	2	yes	maximal
3	C₁₃×#2086	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2074	C₂	1	2	yes	maximal
5	C₁₃×#2056	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1984	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1933	C₂	1	2	yes	maximal
8	C₁₃×#1878	C₂	1	2	yes	maximal
9	C₁₃×#1863	C₂	1	2	yes	maximal
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#103	C₂²	1	2	yes	—
16	C₂×C₁₃×#103	C₂²	1	2	yes	—
17	C₂×C₁₃×#103	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#400	—	2	2	no	—
20	C₁₃×#395	—	2	2	no	—
21	C₁₃×#391	C₂²	1	2	yes	—
22	C₁₃×#391	C₂²	1	2	yes	—
23	C₁₃×#389	C₂²	1	2	yes	—
24	C₁₃×#389	C₂²	1	2	yes	—
25	C₁₃×#387	—	2	2	no	—
26	C₁₃×#387	—	2	2	no	—
27	C₁₃×#382	C₂²	1	2	yes	—
28	C₁₃×#340	C₂²	1	1	yes	characteristic
29	C₁₃×#340	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×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
37	C₂×C₁₃×Q₁₆	—	2	4	no	—
38	C₂×C₁₃×Q₁₆	—	4	1	no	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₁₃	—	4	1	no	—
43	C₄×Q₈×C₁₃	—	4	1	no	—
44	C₄×Q₈×C₁₃	—	4	2	no	—
45	C₄×Q₈×C₁₃	—	4	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
51	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₂×C₄²×C₁₃	—	2	2	no	—
55	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₁₃×#107	—	4	1	no	—
58	C₁₃×C₄⋊C₈	—	4	1	no	—
59	C₁₃×#103	—	2	4	no	—
60	C₁₃×#103	—	2	4	no	—
61	C₁₃×#103	—	4	2	no	—
62	C₄×C₈×C₁₃	—	4	1	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	C₁₃×#95	—	2	2	no	—
69	C₁₃×#95	—	2	2	no	—
70	C₁₃×#95	—	2	2	no	—
71	#32549	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×Q₈×C₁₃	—	2	2	no	—
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	2	2	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×Q₈×C₁₃	—	4	1	no	—
80	C₂×Q₈×C₁₃	#41	1	4	yes	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	—	2	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	2	yes	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×Q₁₆	—	4	4	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	1	no	—
96	C₂×C₈×C₁₃	—	2	1	no	—
97	C₂×C₈×C₁₃	—	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	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	4	2	no	—
106	C₁₃×C₄⋊C₄	—	4	4	no	—
107	C₁₃×C₄⋊C₄	—	4	4	no	—
108	C₁₃×C₄⋊C₄	—	4	4	no	—
109	C₄²×C₁₃	—	2	1	no	—
110	C₄²×C₁₃	—	2	1	no	—
111	C₄²×C₁₃	—	2	2	no	—
112	C₄²×C₁₃	—	2	2	no	—
113	C₄²×C₁₃	—	4	4	no	—
114	#2581	C₂×C₁₃	1	2	yes	—
115	#2101	C₂×C₁₃	1	2	yes	—
116	#2086	C₂×C₁₃	1	1	yes	characteristic
117	#2074	C₂×C₁₃	1	2	yes	—
118	#2056	C₂×C₁₃	1	1	yes	characteristic
119	#1984	C₂×C₁₃	1	1	yes	characteristic
120	#1933	C₂×C₁₃	1	2	yes	—
121	#1878	C₂×C₁₃	1	2	yes	—
122	#1863	C₂×C₁₃	1	2	yes	—
123	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₁₃	—	2	4	no	—
125	Q₈×C₁₃	—	2	4	no	—
126	Q₈×C₁₃	—	4	4	no	—
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	4	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	#135	1	2	yes	—
151	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
152	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
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	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
160	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×#103	C₂²×C₁₃	1	2	yes	—
164	C₂×#103	C₂²×C₁₃	1	2	yes	—
165	C₂×#103	C₂²×C₁₃	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
167	#400	—	2	2	no	—
168	#395	—	2	2	no	—
169	#391	C₂²×C₁₃	1	2	yes	—
170	#391	C₂²×C₁₃	1	2	yes	—
171	#389	C₂²×C₁₃	1	2	yes	—
172	#389	C₂²×C₁₃	1	2	yes	—
173	#387	—	2	2	no	—
174	#387	—	2	2	no	—
175	#382	C₂²×C₁₃	1	2	yes	—
176	#340	C₂²×C₁₃	1	1	yes	characteristic
177	#340	C₂²×C₁₃	1	1	yes	characteristic
178	#340	C₂²×C₁₃	1	2	yes	—
179	#340	C₂²×C₁₃	1	2	yes	—
180	#340	C₂²×C₁₃	1	2	yes	—
181	#336	C₂²×C₁₃	1	2	yes	—
182	#336	C₂²×C₁₃	1	2	yes	—
183	C₂²×C₁₃	#534	1	1	yes	characteristic
184	C₂²×C₁₃	#548	1	2	yes	—
185	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
186	C₂²×C₁₃	#575	1	1	yes	characteristic
187	C₂²×C₁₃	#578	1	2	yes	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	4	2	no	—
190	C₄×C₁₃	—	4	2	no	—
191	C₄×C₁₃	—	4	1	no	—
192	C₄×C₁₃	—	4	4	no	—
193	C₄×C₁₃	—	8	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	#539	1	2	yes	—
197	C₄×C₁₃	#551	1	1	yes	characteristic
198	C₄×C₁₃	#563	1	1	yes	characteristic
199	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
200	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
201	C₂×Q₁₆	—	2	4	no	—
202	C₂×Q₁₆	—	4	1	no	—
203	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
205	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
206	C₄×Q₈	—	4	1	no	—
207	C₄×Q₈	—	4	1	no	—
208	C₄×Q₈	—	4	2	no	—
209	C₄×Q₈	—	4	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
214	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
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×C₄²	—	2	2	no	—
219	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
220	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
221	#107	—	4	1	no	—
222	C₄⋊C₈	—	4	1	no	—
223	#103	—	2	4	no	—
224	#103	—	2	4	no	—
225	#103	—	4	2	no	—
226	C₄×C₈	—	4	1	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	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	C₂×C₁₃	#2728	1	1	yes	characteristic
236	C₂×C₁₃	#3191	1	1	yes	characteristic
237	C₂×C₁₃	#3202	1	1	yes	characteristic
238	C₂×C₁₃	#3364	1	2	yes	—
239	C₂×C₁₃	#3377	1	2	yes	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	C₁₃×#41	1	4	yes	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
256	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
257	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
258	Q₁₆	—	4	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₄⋊C₄	—	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	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	4	4	no	—
282	C₁₃	#32549	1	1	yes	characteristic, direct product factor, Sylow
283	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
284	Q₈	—	2	4	no	—
285	Q₈	—	2	4	no	—
286	Q₈	—	4	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	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	C₁₃×#135	1	2	yes	—
311	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
312	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
313	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
314	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
315	C₈	—	4	2	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₂²	C₁₃×#534	1	1	yes	characteristic
319	C₂²	C₁₃×#548	1	2	yes	—
320	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
321	C₂²	C₁₃×#575	1	1	yes	characteristic
322	C₂²	C₁₃×#578	1	2	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	4	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	C₁₃×#539	1	2	yes	—
332	C₄	C₁₃×#551	1	1	yes	characteristic
333	C₄	C₁₃×#563	1	1	yes	characteristic
334	C₂	C₁₃×#2728	1	1	yes	characteristic
335	C₂	C₁₃×#3191	1	1	yes	characteristic
336	C₂	C₁₃×#3202	1	1	yes	characteristic
337	C₂	C₁₃×#3364	1	2	yes	—
338	C₂	C₁₃×#3377	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
71. #32549	C₁₃	direct (N × H)
282. C₁₃	#32549	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×#2581	2
C₁₃×#2101	2
C₁₃×#2086	2
C₁₃×#2074	2
C₁₃×#2056	2
C₁₃×#1984	2
C₁₃×#1933	2
C₁₃×#1878	2
C₁₃×#1863	2
#32549	13

Subgroups — C13×#32549 #15718888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32549 #15718888