Subgroups — C₁₃×#26925 #15713264

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3082	C₂	1	2	yes	maximal
2	C₁₃×#2950	C₂	1	2	yes	maximal
3	C₁₃×#2047	C₂	1	2	yes	maximal
4	C₁₃×#2028	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2016	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1930	C₂	1	2	yes	maximal
7	C₁₃×#1922	C₂	1	2	yes	maximal
8	C₁₃×#1886	C₂	1	2	yes	maximal
9	C₁₃×#1857	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
11	C₁₃×#544	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₁₃×#519	C₂²	1	2	yes	—
14	C₁₃×#519	C₂²	1	1	yes	characteristic
15	C₁₃×#479	—	2	2	no	—
16	C₁₃×#479	—	2	2	no	—
17	C₁₃×#477	—	2	2	no	—
18	C₁₃×#475	—	2	2	no	—
19	C₁₃×#441	—	2	2	no	—
20	C₁₃×#441	—	2	2	no	—
21	C₄×C₁₃×Q₁₆	—	2	2	no	—
22	C₄×C₁₃×Q₁₆	—	2	2	no	—
23	C₁₃×#428	C₂²	1	2	yes	—
24	C₁₃×#427	C₂²	1	2	yes	—
25	C₁₃×#424	C₂²	1	2	yes	—
26	C₁₃×#419	C₂²	1	2	yes	—
27	C₁₃×#419	C₂²	1	2	yes	—
28	C₁₃×#419	C₂²	1	2	yes	—
29	C₂×C₁₃×#103	C₂²	1	2	yes	—
30	C₂×C₁₃×#101	C₂²	1	1	yes	characteristic
31	C₂×C₁₃×#101	C₂²	1	2	yes	—
32	C₁₃×#405	C₂²	1	2	yes	—
33	C₁₃×#391	C₂²	1	1	yes	characteristic
34	C₁₃×#343	C₂²	1	1	yes	characteristic
35	C₁₃×#341	C₂²	1	2	yes	—
36	C₁₃×#339	C₂²	1	2	yes	—
37	C₁₃×#339	C₂²	1	2	yes	—
38	C₁₃×#336	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
41	C₂×C₁₃×Q₁₆	—	2	4	no	—
42	C₂×C₁₃×Q₁₆	—	2	4	no	—
43	C₂×C₁₃×Q₁₆	—	2	2	no	—
44	C₂×C₁₃×Q₁₆	—	2	2	no	—
45	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
46	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
48	C₁₃×C₄⋊Q₈	—	2	2	no	—
49	C₁₃×C₄⋊Q₈	—	2	4	no	—
50	C₁₃×#125	—	2	2	no	—
51	C₁₃×C₂²⋊Q₈	—	4	1	no	—
52	C₄×Q₈×C₁₃	—	2	2	no	—
53	C₄×Q₈×C₁₃	—	2	2	no	—
54	C₄×Q₈×C₁₃	—	2	2	no	—
55	C₄×Q₈×C₁₃	—	2	2	no	—
56	C₄×Q₈×C₁₃	—	2	2	no	—
57	C₄×Q₈×C₁₃	—	2	2	no	—
58	C₁₃×#117	—	4	1	no	—
59	C₁₃×#117	—	4	2	no	—
60	C₁₃×#117	C₂³	1	2	yes	—
61	C₁₃×#117	C₂³	1	2	yes	—
62	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
63	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
65	C₁₃×#107	—	2	2	no	—
66	C₁₃×#107	D₄	1	2	yes	—
67	C₁₃×#107	D₄	1	2	yes	—
68	C₁₃×#106	—	2	2	no	—
69	C₁₃×#106	D₄	1	2	yes	—
70	C₁₃×#106	D₄	1	2	yes	—
71	C₁₃×C₄⋊C₈	—	2	2	no	—
72	C₁₃×C₄⋊C₈	—	2	2	no	—
73	C₁₃×#103	—	2	2	no	—
74	C₁₃×#103	—	2	2	no	—
75	C₁₃×#103	—	2	2	no	—
76	C₁₃×#103	—	2	2	no	—
77	C₁₃×#103	—	2	2	no	—
78	C₁₃×#103	—	2	2	no	—
79	C₁₃×#103	—	2	2	no	—
80	C₁₃×#103	—	2	2	no	—
81	C₁₃×#101	—	2	4	no	—
82	C₁₃×#101	—	2	2	no	—
83	C₁₃×#97	—	2	2	no	—
84	C₄×C₈×C₁₃	—	2	2	no	—
85	C₁₃×#95	—	4	1	no	—
86	#26925	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×Q₈×C₁₃	—	2	2	no	—
88	C₂×Q₈×C₁₃	—	2	1	no	—
89	C₂×Q₈×C₁₃	—	2	2	no	—
90	C₂×Q₈×C₁₃	—	2	2	no	—
91	C₂×Q₈×C₁₃	—	2	2	no	—
92	C₂×Q₈×C₁₃	—	2	2	no	—
93	C₂×Q₈×C₁₃	—	2	2	no	—
94	C₂×Q₈×C₁₃	—	2	2	no	—
95	C₂×Q₈×C₁₃	—	4	1	no	—
96	C₂×Q₈×C₁₃	#41	1	4	yes	—
97	C₂×Q₈×C₁₃	#41	1	2	yes	—
98	C₂²×C₄×C₁₃	—	2	1	no	—
99	C₂²×C₄×C₁₃	—	2	1	no	—
100	C₂²×C₄×C₁₃	—	2	2	no	—
101	C₂²×C₄×C₁₃	—	2	1	no	—
102	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₁₃×Q₁₆	—	2	8	no	—
104	C₁₃×Q₁₆	—	2	8	no	—
105	C₁₃×M₁₆	—	2	2	no	—
106	C₁₃×M₁₆	—	2	2	no	—
107	C₁₃×M₁₆	—	2	2	no	—
108	C₁₃×M₁₆	—	2	2	no	—
109	C₂×C₈×C₁₃	—	2	2	no	—
110	C₂×C₈×C₁₃	—	2	2	no	—
111	C₂×C₈×C₁₃	—	2	2	no	—
112	C₂×C₈×C₁₃	—	2	2	no	—
113	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
115	C₁₃×C₄⋊C₄	—	2	2	no	—
116	C₁₃×C₄⋊C₄	—	4	4	no	—
117	C₁₃×C₄⋊C₄	—	4	1	no	—
118	C₁₃×C₄⋊C₄	—	4	1	no	—
119	C₁₃×C₄⋊C₄	—	4	2	no	—
120	C₁₃×C₄⋊C₄	—	4	2	no	—
121	C₁₃×C₄⋊C₄	—	4	1	no	—
122	C₁₃×C₄⋊C₄	—	4	2	no	—
123	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
124	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
125	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
126	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
127	C₁₃×C₂²⋊C₄	—	2	2	no	—
128	C₁₃×C₂²⋊C₄	—	2	2	no	—
129	C₄²×C₁₃	—	2	2	no	—
130	C₄²×C₁₃	—	2	2	no	—
131	C₄²×C₁₃	—	2	2	no	—
132	C₄²×C₁₃	—	4	2	no	—
133	C₄²×C₁₃	—	4	2	no	—
134	C₄²×C₁₃	—	4	2	no	—
135	#3082	C₂×C₁₃	1	2	yes	—
136	#2950	C₂×C₁₃	1	2	yes	—
137	#2047	C₂×C₁₃	1	2	yes	—
138	#2028	C₂×C₁₃	1	1	yes	characteristic
139	#2016	C₂×C₁₃	1	1	yes	characteristic
140	#1930	C₂×C₁₃	1	2	yes	—
141	#1922	C₂×C₁₃	1	2	yes	—
142	#1886	C₂×C₁₃	1	2	yes	—
143	#1857	C₂×C₁₃	1	1	yes	characteristic
144	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
145	Q₈×C₁₃	—	2	4	no	—
146	Q₈×C₁₃	—	2	4	no	—
147	Q₈×C₁₃	—	2	4	no	—
148	Q₈×C₁₃	—	4	2	no	—
149	Q₈×C₁₃	—	4	2	no	—
150	Q₈×C₁₃	—	4	2	no	—
151	Q₈×C₁₃	—	16	1	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	—	2	2	no	—
158	C₂×C₄×C₁₃	—	2	4	no	—
159	C₂×C₄×C₁₃	—	4	1	no	—
160	C₂×C₄×C₁₃	—	4	2	no	—
161	C₂×C₄×C₁₃	—	4	1	no	—
162	C₂×C₄×C₁₃	—	4	2	no	—
163	C₂×C₄×C₁₃	—	4	1	no	—
164	C₂×C₄×C₁₃	—	4	1	no	—
165	C₂×C₄×C₁₃	—	4	1	no	—
166	C₂×C₄×C₁₃	—	8	1	no	—
167	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
168	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
170	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
171	C₈×C₁₃	—	2	2	no	—
172	C₈×C₁₃	—	2	2	no	—
173	C₈×C₁₃	—	4	2	no	—
174	C₈×C₁₃	—	4	2	no	—
175	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
176	#544	C₂²×C₁₃	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
178	#519	C₂²×C₁₃	1	2	yes	—
179	#519	C₂²×C₁₃	1	1	yes	characteristic
180	#479	—	2	2	no	—
181	#479	—	2	2	no	—
182	#477	—	2	2	no	—
183	#475	—	2	2	no	—
184	#441	—	2	2	no	—
185	#441	—	2	2	no	—
186	C₄×Q₁₆	—	2	2	no	—
187	C₄×Q₁₆	—	2	2	no	—
188	#428	C₂²×C₁₃	1	2	yes	—
189	#427	C₂²×C₁₃	1	2	yes	—
190	#424	C₂²×C₁₃	1	2	yes	—
191	#419	C₂²×C₁₃	1	2	yes	—
192	#419	C₂²×C₁₃	1	2	yes	—
193	#419	C₂²×C₁₃	1	2	yes	—
194	C₂×#103	C₂²×C₁₃	1	2	yes	—
195	C₂×#101	C₂²×C₁₃	1	1	yes	characteristic
196	C₂×#101	C₂²×C₁₃	1	2	yes	—
197	#405	C₂²×C₁₃	1	2	yes	—
198	#391	C₂²×C₁₃	1	1	yes	characteristic
199	#343	C₂²×C₁₃	1	1	yes	characteristic
200	#341	C₂²×C₁₃	1	2	yes	—
201	#339	C₂²×C₁₃	1	2	yes	—
202	#339	C₂²×C₁₃	1	2	yes	—
203	#336	C₂²×C₁₃	1	1	yes	characteristic
204	C₂²×C₁₃	—	2	1	no	—
205	C₂²×C₁₃	—	2	1	no	—
206	C₂²×C₁₃	#539	1	1	yes	characteristic
207	C₂²×C₁₃	D₄²	1	1	yes	characteristic
208	C₂²×C₁₃	#551	1	1	yes	characteristic
209	C₄×C₁₃	—	4	2	no	—
210	C₄×C₁₃	—	4	2	no	—
211	C₄×C₁₃	—	4	4	no	—
212	C₄×C₁₃	—	4	2	no	—
213	C₄×C₁₃	—	4	2	no	—
214	C₄×C₁₃	—	4	2	no	—
215	C₄×C₁₃	—	4	2	no	—
216	C₄×C₁₃	—	8	1	no	—
217	C₄×C₁₃	#525	1	2	yes	—
218	C₄×C₁₃	#548	1	2	yes	—
219	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
220	C₂²×Q₈	C₂³×C₁₃	1	1	yes	characteristic
221	C₂×Q₁₆	—	2	4	no	—
222	C₂×Q₁₆	—	2	4	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
226	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
227	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
228	C₄⋊Q₈	—	2	2	no	—
229	C₄⋊Q₈	—	2	4	no	—
230	#125	—	2	2	no	—
231	C₂²⋊Q₈	—	4	1	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	#117	—	4	1	no	—
239	#117	—	4	2	no	—
240	#117	C₂³×C₁₃	1	2	yes	—
241	#117	C₂³×C₁₃	1	2	yes	—
242	C₂×C₄⋊C₄	—	4	1	no	—
243	C₂×C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
244	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
245	#107	—	2	2	no	—
246	#107	D₄×C₁₃	1	2	yes	—
247	#107	D₄×C₁₃	1	2	yes	—
248	#106	—	2	2	no	—
249	#106	D₄×C₁₃	1	2	yes	—
250	#106	D₄×C₁₃	1	2	yes	—
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	2	no	—
253	#103	—	2	2	no	—
254	#103	—	2	2	no	—
255	#103	—	2	2	no	—
256	#103	—	2	2	no	—
257	#103	—	2	2	no	—
258	#103	—	2	2	no	—
259	#103	—	2	2	no	—
260	#103	—	2	2	no	—
261	#101	—	2	4	no	—
262	#101	—	2	2	no	—
263	#97	—	2	2	no	—
264	C₄×C₈	—	2	2	no	—
265	#95	—	4	1	no	—
266	C₂×C₁₃	—	2	1	no	—
267	C₂×C₁₃	—	2	1	no	—
268	C₂×C₁₃	#2463	1	1	yes	characteristic
269	C₂×C₁₃	#3289	1	1	yes	characteristic
270	C₂×C₁₃	#3293	1	1	yes	characteristic
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	4	1	no	—
280	C₂×Q₈	C₁₃×#41	1	4	yes	—
281	C₂×Q₈	C₁₃×#41	1	2	yes	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
287	Q₁₆	—	2	8	no	—
288	Q₁₆	—	2	8	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	2	2	no	—
292	M₁₆	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
298	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	4	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	C₂×D₄×C₁₃	1	2	yes	—
308	C₄⋊C₄	C₂×D₄×C₁₃	1	2	yes	—
309	C₄⋊C₄	C₂×D₄×C₁₃	1	2	yes	—
310	C₄⋊C₄	C₂×D₄×C₁₃	1	2	yes	—
311	C₂²⋊C₄	—	2	2	no	—
312	C₂²⋊C₄	—	2	2	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	4	2	no	—
317	C₄²	—	4	2	no	—
318	C₄²	—	4	2	no	—
319	C₁₃	#26925	1	1	yes	characteristic, direct product factor, Sylow
320	C₂³	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
321	Q₈	—	2	4	no	—
322	Q₈	—	2	4	no	—
323	Q₈	—	2	4	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	4	2	no	—
326	Q₈	—	4	2	no	—
327	Q₈	—	16	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	8	1	no	—
343	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
344	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
345	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
346	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	2	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	4	2	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	C₁₃×#539	1	1	yes	characteristic
354	C₂²	D₄²×C₁₃	1	1	yes	characteristic
355	C₂²	C₁₃×#551	1	1	yes	characteristic
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	4	no	—
363	C₄	—	8	1	no	—
364	C₄	C₁₃×#525	1	2	yes	—
365	C₄	C₁₃×#548	1	2	yes	—
366	C₂	—	2	1	no	—
367	C₂	—	2	1	no	—
368	C₂	C₁₃×#2463	1	1	yes	characteristic
369	C₂	C₁₃×#3289	1	1	yes	characteristic
370	C₂	C₁₃×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂³×C₁₃	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
86. #26925	C₁₃	direct (N × H)
319. C₁₃	#26925	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×#3082	2
C₁₃×#2950	2
C₁₃×#2047	2
C₁₃×#2028	2
C₁₃×#2016	2
C₁₃×#1930	2
C₁₃×#1922	2
C₁₃×#1886	2
C₁₃×#1857	2
#26925	13

Subgroups — C13×#26925 #15713264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#26925 #15713264