Subgroups — C₁₃×#32528 #15718867

Order 3328.

596 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2514	C₂	1	2	yes	maximal
2	C₁₃×#2097	C₂	1	2	yes	maximal
3	C₁₃×#2066	C₂	1	2	yes	maximal
4	C₁₃×#2056	C₂	1	2	yes	maximal
5	C₁₃×#1969	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1949	C₂	1	2	yes	maximal
7	C₁₃×#1942	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1939	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1928	C₂	1	2	yes	maximal
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	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×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#103	C₂²	1	2	yes	—
18	C₂×C₁₃×#103	C₂²	1	2	yes	—
19	C₂×C₁₃×#103	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#401	—	2	2	no	—
22	C₁₃×#400	—	2	2	no	—
23	C₁₃×#391	C₂²	1	2	yes	—
24	C₁₃×#391	C₂²	1	2	yes	—
25	C₁₃×#391	C₂²	1	2	yes	—
26	C₁₃×#389	C₂²	1	2	yes	—
27	C₁₃×#387	—	2	2	no	—
28	C₁₃×#383	—	2	2	no	—
29	C₁₃×#382	C₂²	1	2	yes	—
30	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
31	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#340	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
36	C₂×C₁₃×Q₁₆	—	2	2	no	—
37	C₂×C₁₃×Q₁₆	—	4	1	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	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₃	—	2	2	no	—
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×#107	—	2	2	no	—
52	C₁₃×#107	—	4	1	no	—
53	C₁₃×#107	—	4	1	no	—
54	C₁₃×#106	—	4	2	no	—
55	C₁₃×C₄⋊C₈	—	2	4	no	—
56	C₁₃×C₄⋊C₈	—	4	1	no	—
57	C₁₃×#103	—	2	4	no	—
58	C₁₃×#103	—	4	2	no	—
59	C₁₃×#103	—	4	2	no	—
60	C₄×C₈×C₁₃	—	4	1	no	—
61	C₁₃×#95	—	2	2	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	C₁₃×#95	—	2	2	no	—
69	C₁₃×#95	—	2	2	no	—
70	C₁₃×#95	—	2	2	no	—
71	C₁₃×#95	D₄	1	2	yes	—
72	#32528	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	—	2	2	no	—
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	—	2	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	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×Q₁₆	—	4	4	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	#41	1	2	yes	—
97	C₂×C₈×C₁₃	#41	1	2	yes	—
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	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	4	no	—
107	C₄²×C₁₃	—	2	2	no	—
108	C₄²×C₁₃	—	2	1	no	—
109	C₄²×C₁₃	—	2	1	no	—
110	C₄²×C₁₃	—	2	2	no	—
111	C₄²×C₁₃	—	4	4	no	—
112	#2514	C₂×C₁₃	1	2	yes	—
113	#2097	C₂×C₁₃	1	2	yes	—
114	#2066	C₂×C₁₃	1	2	yes	—
115	#2056	C₂×C₁₃	1	2	yes	—
116	#1969	C₂×C₁₃	1	1	yes	characteristic
117	#1949	C₂×C₁₃	1	2	yes	—
118	#1942	C₂×C₁₃	1	1	yes	characteristic
119	#1939	C₂×C₁₃	1	1	yes	characteristic
120	#1928	C₂×C₁₃	1	2	yes	—
121	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	Q₈×C₁₃	—	4	2	no	—
123	Q₈×C₁₃	—	4	2	no	—
124	Q₈×C₁₃	—	4	2	no	—
125	Q₈×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	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	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	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	#135	1	2	yes	—
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
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	1	yes	characteristic
155	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	—	2	2	no	—
158	C₈×C₁₃	—	4	2	no	—
159	C₂²×Q₁₆	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×#107	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×#106	C₂²×C₁₃	1	2	yes	—
164	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
165	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
166	C₂×#103	C₂²×C₁₃	1	2	yes	—
167	C₂×#103	C₂²×C₁₃	1	2	yes	—
168	C₂×#103	C₂²×C₁₃	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
170	#401	—	2	2	no	—
171	#400	—	2	2	no	—
172	#391	C₂²×C₁₃	1	2	yes	—
173	#391	C₂²×C₁₃	1	2	yes	—
174	#391	C₂²×C₁₃	1	2	yes	—
175	#389	C₂²×C₁₃	1	2	yes	—
176	#387	—	2	2	no	—
177	#383	—	2	2	no	—
178	#382	C₂²×C₁₃	1	2	yes	—
179	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
180	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
181	#340	C₂²×C₁₃	1	2	yes	—
182	#340	C₂²×C₁₃	1	2	yes	—
183	#336	C₂²×C₁₃	1	2	yes	—
184	C₂²×C₁₃	#537	1	1	yes	characteristic
185	C₂²×C₁₃	#547	1	1	yes	characteristic
186	C₂²×C₁₃	#550	1	1	yes	characteristic
187	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
188	C₂²×C₁₃	#575	1	1	yes	characteristic
189	C₂²×C₁₃	#578	1	1	yes	characteristic
190	C₂²×C₁₃	#578	1	1	yes	characteristic
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	4	2	no	—
193	C₄×C₁₃	—	4	2	no	—
194	C₄×C₁₃	—	4	1	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	#538	1	1	yes	characteristic
200	C₄×C₁₃	#539	1	1	yes	characteristic
201	C₄×C₁₃	#556	1	1	yes	characteristic
202	C₄×C₁₃	#563	1	1	yes	characteristic
203	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	4	1	no	—
206	C₂×Q₁₆	—	4	1	no	—
207	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
208	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
212	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
218	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
219	#107	—	2	2	no	—
220	#107	—	4	1	no	—
221	#107	—	4	1	no	—
222	#106	—	4	2	no	—
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	4	1	no	—
225	#103	—	2	4	no	—
226	#103	—	4	2	no	—
227	#103	—	4	2	no	—
228	C₄×C₈	—	4	1	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	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	D₄×C₁₃	1	2	yes	—
240	C₂×C₁₃	#2727	1	1	yes	characteristic
241	C₂×C₁₃	#3250	1	1	yes	characteristic
242	C₂×C₁₃	#3261	1	1	yes	characteristic
243	C₂×C₁₃	#3342	1	1	yes	characteristic
244	C₂×C₁₃	#3352	1	1	yes	characteristic
245	C₂×C₁₃	#3393	1	1	yes	characteristic
246	C₂×C₁₃	#3402	1	1	yes	characteristic
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
259	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
262	Q₁₆	—	4	4	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₁₃×#41	1	2	yes	—
271	C₂×C₈	C₁₃×#41	1	2	yes	—
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	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	4	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	4	4	no	—
286	C₁₃	#32528	1	1	yes	characteristic, direct product factor, Sylow
287	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	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	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	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	—	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	C₁₃×#135	1	2	yes	—
316	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
317	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
318	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
319	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
320	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
321	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	4	2	no	—
325	C₂²	C₁₃×#537	1	1	yes	characteristic
326	C₂²	C₁₃×#547	1	1	yes	characteristic
327	C₂²	C₁₃×#550	1	1	yes	characteristic
328	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
329	C₂²	C₁₃×#575	1	1	yes	characteristic
330	C₂²	C₁₃×#578	1	1	yes	characteristic
331	C₂²	C₁₃×#578	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	2	no	—
339	C₄	—	8	2	no	—
340	C₄	C₁₃×#538	1	1	yes	characteristic
341	C₄	C₁₃×#539	1	1	yes	characteristic
342	C₄	C₁₃×#556	1	1	yes	characteristic
343	C₄	C₁₃×#563	1	1	yes	characteristic
344	C₂	C₁₃×#2727	1	1	yes	characteristic
345	C₂	C₁₃×#3250	1	1	yes	characteristic
346	C₂	C₁₃×#3261	1	1	yes	characteristic
347	C₂	C₁₃×#3342	1	1	yes	characteristic
348	C₂	C₁₃×#3352	1	1	yes	characteristic
349	C₂	C₁₃×#3393	1	1	yes	characteristic
350	C₂	C₁₃×#3402	1	1	yes	characteristic

	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 ◁ C₄×C₁₃×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
72. #32528	C₁₃	direct (N × H)
286. C₁₃	#32528	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×#2514	2
C₁₃×#2097	2
C₁₃×#2066	2
C₁₃×#2056	2
C₁₃×#1969	2
C₁₃×#1949	2
C₁₃×#1942	2
C₁₃×#1939	2
C₁₃×#1928	2
#32528	13

Subgroups — C13×#32528 #15718867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32528 #15718867