Subgroups — D₁₃×#2227 #16017544

Order 3328.

574 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
3	#13116989	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116988	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116987	C₂	1	2	yes	maximal, complemented
6	#13116986	C₂	1	2	yes	maximal, complemented
7	#13116985	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃×#366	C₂	1	2	yes	maximal
9	D₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
10	D₁₃×#358	C₂	1	2	yes	maximal
11	C₁₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₁₃	C₂²	1	1	yes	characteristic
13	D₁₃×C₄⋊Q₈	—	2	1	no	—
14	C₄×Q₈×D₁₃	—	2	1	no	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
18	#11718739	—	2	1	no	—
19	C₄×#6107	—	2	1	no	—
20	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
21	D₁₃×#107	—	2	1	no	—
22	D₁₃×#107	C₂²	1	1	yes	characteristic
23	D₁₃×#107	C₂²	1	2	yes	—
24	#11718535	—	2	1	no	—
25	#11718535	C₂²	1	2	yes	—
26	#11718534	C₂²	1	1	yes	characteristic
27	D₁₃×#103	—	2	1	no	—
28	#11718451	—	2	1	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	#11718343	C₂²	1	1	yes	characteristic
32	C₁₃×#366	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₃×#358	C₂²	1	2	yes	—
35	#11718156	C₂²	1	2	yes	—
36	#11718096	C₂²	1	2	yes	—
37	#11718095	C₂²	1	2	yes	—
38	#11718093	C₂²	1	2	yes	—
39	#11718084	C₂²	1	2	yes	—
40	#11718081	C₂²	1	2	yes	—
41	#11718081	C₂²	1	2	yes	—
42	C₄⋊#6090	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₁₃	—	2	1	no	—
44	C₂×Q₈×D₁₃	—	4	1	no	—
45	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₄×Q₈×C₁₃	—	2	1	no	—
48	Q₈×Dic₁₃	—	2	1	no	—
49	#88198	—	4	1	no	—
50	C₂×#6107	—	2	1	no	—
51	Q₁₆×D₁₃	—	2	2	no	—
52	Q₁₆×D₁₃	Q₈	1	4	yes	complemented
53	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
54	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
55	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×D₁₃	—	2	2	no	—
57	C₄⋊C₄×D₁₃	—	2	1	no	—
58	C₄⋊C₄×D₁₃	—	4	1	no	—
59	C₄⋊C₄×D₁₃	—	4	1	no	—
60	#88142	—	2	2	no	—
61	#88142	—	4	1	no	—
62	#88141	—	2	1	no	—
63	#88141	—	4	1	no	—
64	C₄²×D₁₃	—	4	1	no	—
65	C₄²×D₁₃	D₄	1	1	yes	characteristic
66	#88123	—	2	1	no	—
67	C₄×Dic₂₆	—	2	1	no	—
68	C₄×Dic₂₆	—	4	1	no	—
69	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
70	C₁₃×#107	—	2	1	no	—
71	C₁₃×#107	C₂³	1	2	yes	—
72	C₁₃×#107	C₂³	1	1	yes	characteristic
73	C₁₃×#103	—	2	1	no	—
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#88075	—	2	1	no	—
76	#88058	—	2	1	no	—
77	#88058	C₂³	1	2	yes	—
78	#88058	C₂³	1	1	yes	characteristic
79	#88056	—	2	1	no	—
80	C₈×Dic₁₃	D₄	1	1	yes	characteristic
81	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
82	C₂×#6090	C₂³	1	2	yes	—
83	#88050	—	2	1	no	—
84	#88047	—	2	1	no	—
85	#88047	—	2	2	no	—
86	#88047	—	2	2	no	—
87	C₄×#1055	D₄	1	1	yes	characteristic
88	C₁₆×D₁₃	—	2	4	no	—
89	C₂×#2227	—	13	1	no	maximal, Sylow
90	C₂×Q₈×C₁₃	—	2	1	no	—
91	C₂×Q₈×C₁₃	—	4	1	no	—
92	Q₈×D₁₃	—	2	4	no	—
93	Q₈×D₁₃	—	4	2	no	—
94	Q₈×D₁₃	—	8	4	no	—
95	C₂×C₄×D₁₃	—	4	1	no	—
96	C₂×C₄×D₁₃	—	4	1	no	—
97	C₂×C₄×D₁₃	—	4	2	no	—
98	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
99	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
100	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×Dic₂₆	—	2	1	no	—
102	C₂×Dic₂₆	—	4	1	no	—
103	C₂×Dic₂₆	—	4	1	no	—
104	C₂×Dic₂₆	—	4	2	no	—
105	C₁₃×Q₁₆	—	2	1	no	—
106	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	complemented
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	—	2	1	no	—
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₄²×C₁₃	—	4	1	no	—
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	#6107	—	2	4	no	—
116	C₄⋊Dic₁₃	—	2	1	no	—
117	C₄⋊Dic₁₃	—	2	2	no	—
118	C₄⋊Dic₁₃	—	4	1	no	—
119	C₄⋊Dic₁₃	—	4	1	no	—
120	#6101	—	4	1	no	—
121	#6101	—	4	1	no	—
122	#6101	—	4	1	no	—
123	#6101	—	4	2	no	—
124	#6101	—	4	2	no	—
125	C₄×Dic₁₃	—	4	1	no	—
126	C₄×Dic₁₃	—	4	1	no	—
127	C₄×Dic₁₃	—	4	1	no	—
128	C₄×Dic₁₃	D₈	1	1	yes	characteristic
129	C₄×Dic₁₃	D₈	1	1	yes	characteristic
130	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
131	C₂×#1055	C₂×D₄	1	1	yes	characteristic
132	C₂×#1055	C₂×D₄	1	1	yes	characteristic
133	Dic₅₂	—	2	1	no	—
134	Dic₅₂	C₂×Q₈	1	2	yes	complemented
135	C₈×D₁₃	—	2	2	no	—
136	C₈×D₁₃	C₂×Q₈	1	2	yes	—
137	C₈×D₁₃	#41	1	2	yes	—
138	C₁₃×C₁₆	—	2	2	no	—
139	#6090	—	2	2	no	—
140	C₂×#500	—	13	1	no	—
141	C₂×C₄×Q₁₆	—	13	1	no	—
142	#2227	—	13	1	no	—
143	#2227	—	13	1	no	—
144	#2227	—	13	2	no	—
145	#2227	—	13	1	no	—
146	#2227	—	13	2	no	—
147	#2227	D₁₃	1	1	yes	characteristic, direct product factor
148	C₂×#366	—	13	2	no	—
149	C₂×C₄⋊C₁₆	—	13	1	no	—
150	C₂×#358	—	13	2	no	—
151	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
152	Q₈×C₁₃	—	2	2	no	—
153	Q₈×C₁₃	—	4	1	no	—
154	Q₈×C₁₃	—	8	2	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	1	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×Dic₁₃	—	4	1	no	—
162	C₂×Dic₁₃	—	4	1	no	—
163	C₂×Dic₁₃	—	4	2	no	—
164	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
165	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
166	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
167	C₄×D₁₃	—	2	2	no	—
168	C₄×D₁₃	—	4	4	no	—
169	C₄×D₁₃	—	8	2	no	—
170	C₄×D₁₃	—	8	4	no	—
171	C₄×D₁₃	Q₃₂	1	4	yes	—
172	C₄×D₁₃	C₂²⋊Q₈	1	2	yes	—
173	C₄×D₁₃	#137	1	2	yes	—
174	Dic₂₆	—	2	2	no	—
175	Dic₂₆	—	4	1	no	—
176	Dic₂₆	—	4	4	no	—
177	Dic₂₆	—	8	4	no	—
178	Dic₂₆	—	8	2	no	—
179	C₈×C₁₃	—	2	1	no	—
180	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
181	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
182	#1055	—	2	1	no	—
183	#1055	C₂²⋊Q₈	1	2	yes	—
184	C₂²×Q₁₆	—	13	1	no	—
185	C₂×C₄⋊Q₈	—	26	1	no	—
186	C₂×C₄×Q₈	—	26	1	no	—
187	C₂²×C₁₆	—	13	2	no	—
188	#500	—	13	2	no	—
189	#500	—	13	1	no	—
190	#500	—	26	1	no	—
191	#500	—	26	1	no	—
192	#500	C₂×D₁₃	1	1	yes	characteristic
193	C₄×Q₁₆	—	13	1	no	—
194	C₄×Q₁₆	—	13	1	no	—
195	C₄×Q₁₆	—	13	1	no	—
196	C₄×Q₁₆	—	26	1	no	—
197	C₄×Q₁₆	—	26	1	no	—
198	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
199	C₂×#107	—	13	2	no	—
200	C₂×#107	—	13	1	no	—
201	C₂×#107	—	26	1	no	—
202	C₂×#103	—	26	1	no	—
203	C₂×C₄×C₈	—	13	1	no	—
204	#366	—	13	2	no	—
205	#366	—	13	2	no	—
206	#366	—	13	2	no	—
207	#366	C₂×D₁₃	1	2	yes	—
208	C₄⋊C₁₆	—	13	2	no	—
209	C₄⋊C₁₆	—	13	1	no	—
210	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
211	#358	—	13	2	no	—
212	#358	—	13	2	no	—
213	#358	—	13	2	no	—
214	#358	C₂×D₁₃	1	2	yes	—
215	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
216	C₂×D₁₃	#474	1	2	yes	—
217	C₂×D₁₃	C₂×Q₃₂	1	2	yes	—
218	C₂×D₁₃	#509	1	2	yes	—
219	C₄×C₁₃	—	2	1	no	—
220	C₄×C₁₃	—	4	2	no	—
221	C₄×C₁₃	—	8	2	no	—
222	C₄×C₁₃	—	8	1	no	—
223	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
224	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
225	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
226	Dic₁₃	—	2	1	no	—
227	Dic₁₃	—	4	2	no	—
228	Dic₁₃	—	8	2	no	—
229	Dic₁₃	—	8	1	no	—
230	Dic₁₃	#474	1	2	yes	—
231	Dic₁₃	C₂×Q₃₂	1	2	yes	—
232	C₂²×Q₈	—	26	1	no	—
233	C₂²×Q₈	—	52	1	no	—
234	C₂×Q₁₆	—	13	4	no	—
235	C₂×Q₁₆	—	13	1	no	—
236	C₂×Q₁₆	—	26	2	no	—
237	C₂×Q₁₆	—	26	1	no	—
238	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
239	C₂²×C₈	—	13	1	no	—
240	C₂²×C₈	—	13	1	no	—
241	C₄⋊Q₈	—	2	1	no	—
242	C₄⋊Q₈	—	26	2	no	—
243	C₄⋊Q₈	—	26	1	no	—
244	C₄⋊Q₈	—	52	1	no	—
245	C₄⋊Q₈	—	52	1	no	—
246	C₄×Q₈	—	2	1	no	—
247	C₄×Q₈	—	26	1	no	—
248	C₄×Q₈	—	26	1	no	—
249	C₄×Q₈	—	26	1	no	—
250	C₄×Q₈	—	52	1	no	—
251	C₄×Q₈	—	52	1	no	—
252	C₂×C₄⋊C₄	—	26	1	no	—
253	C₂×C₄⋊C₄	—	26	2	no	—
254	C₂×C₄⋊C₄	—	52	1	no	—
255	C₂×C₄⋊C₄	—	52	1	no	—
256	C₂×C₄²	—	13	1	no	—
257	C₂×C₄²	—	52	1	no	—
258	C₂×C₁₆	—	13	2	no	—
259	C₂×C₁₆	—	26	4	no	—
260	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
261	#107	—	2	1	no	—
262	#107	—	13	2	no	—
263	#107	—	13	1	no	—
264	#107	—	26	1	no	—
265	#107	—	26	2	no	—
266	#107	—	26	2	no	—
267	#107	—	26	1	no	—
268	#107	C₂²×D₁₃	1	2	yes	—
269	#107	C₂²×D₁₃	1	1	yes	characteristic
270	#103	—	2	1	no	—
271	#103	—	26	1	no	—
272	#103	—	26	1	no	—
273	#103	—	26	1	no	—
274	C₄×C₈	—	13	1	no	—
275	C₄×C₈	—	13	1	no	—
276	C₄×C₈	—	13	1	no	—
277	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
278	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
279	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
280	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
281	D₁₃	#2227	1	4	yes	direct product factor
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	—	4	1	no	—
284	C₂×Q₈	—	26	4	no	—
285	C₂×Q₈	—	26	1	no	—
286	C₂×Q₈	—	52	1	no	—
287	C₂×Q₈	—	52	2	no	—
288	C₂×Q₈	—	52	2	no	—
289	C₂×Q₈	—	52	1	no	—
290	C₂×Q₈	—	104	4	no	—
291	C₂²×C₄	—	13	1	no	—
292	C₂²×C₄	—	13	1	no	—
293	C₂²×C₄	—	13	1	no	—
294	C₂²×C₄	—	52	1	no	—
295	C₂²×C₄	—	52	1	no	—
296	C₂²×C₄	—	52	2	no	—
297	Q₁₆	—	2	1	no	—
298	Q₁₆	—	13	2	no	—
299	Q₁₆	—	26	4	no	—
300	Q₁₆	—	26	1	no	—
301	Q₁₆	Q₈×D₁₃	1	2	yes	complemented
302	C₂×C₈	—	13	1	no	—
303	C₂×C₈	—	13	2	no	—
304	C₂×C₈	—	13	2	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	26	2	no	—
307	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	26	1	no	—
314	C₄⋊C₄	—	26	2	no	—
315	C₄⋊C₄	—	52	1	no	—
316	C₄⋊C₄	—	52	2	no	—
317	C₄⋊C₄	—	52	1	no	—
318	C₄⋊C₄	—	52	1	no	—
319	C₄⋊C₄	—	52	2	no	—
320	C₄⋊C₄	—	52	1	no	—
321	C₄⋊C₄	—	52	1	no	—
322	C₄²	—	4	1	no	—
323	C₄²	—	13	1	no	—
324	C₄²	—	13	1	no	—
325	C₄²	—	13	1	no	—
326	C₄²	—	52	1	no	—
327	C₄²	—	52	1	no	—
328	C₄²	—	52	1	no	—
329	C₄²	D₄×D₁₃	1	1	yes	characteristic
330	C₁₆	—	2	2	no	—
331	C₁₆	—	26	2	no	—
332	C₁₃	C₂×#2227	1	1	yes	characteristic, complemented, Sylow
333	C₂³	—	13	1	no	—
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	8	2	no	—
337	Q₈	—	26	2	no	—
338	Q₈	—	52	4	no	—
339	Q₈	—	52	1	no	—
340	Q₈	—	104	2	no	—
341	Q₈	—	104	4	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	13	1	no	—
346	C₂×C₄	—	13	1	no	—
347	C₂×C₄	—	13	2	no	—
348	C₂×C₄	—	13	2	no	—
349	C₂×C₄	—	13	4	no	—
350	C₂×C₄	—	13	1	no	—
351	C₂×C₄	—	26	2	no	—
352	C₂×C₄	—	52	4	no	—
353	C₂×C₄	—	52	1	no	—
354	C₂×C₄	—	52	1	no	—
355	C₂×C₄	—	52	2	no	—
356	C₂×C₄	—	104	2	no	—
357	C₂×C₄	—	104	4	no	—
358	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
359	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
360	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	13	2	no	—
363	C₈	—	26	1	no	—
364	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
365	C₈	#41×D₁₃	1	1	yes	characteristic
366	C₂²	—	13	2	no	—
367	C₂²	—	13	2	no	—
368	C₂²	—	13	2	no	—
369	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	8	1	no	—
373	C₄	—	8	2	no	—
374	C₄	—	13	2	no	—
375	C₄	—	13	2	no	—
376	C₄	—	26	1	no	—
377	C₄	—	52	2	no	—
378	C₄	—	104	1	no	—
379	C₄	—	104	2	no	—
380	C₄	D₁₃×Q₃₂	1	2	yes	—
381	C₄	D₁₃×C₂²⋊Q₈	1	1	yes	characteristic
382	C₄	D₁₃×#137	1	1	yes	characteristic
383	C₂	—	13	4	no	—
384	C₂	D₁₃×#474	1	1	yes	characteristic
385	C₂	C₂×D₁₃×Q₃₂	1	1	yes	characteristic
386	C₂	D₁₃×#509	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2227	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
hypercenter	#2227	D₁₃
hypocenter	C₁₃	C₂×#2227
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
3. #13116989	C₂	semidirect
4. #13116988	C₂	semidirect
5. #13116987	C₂	semidirect
6. #13116986	C₂	semidirect
7. #13116985	C₂	semidirect
11. C₁₃×#2227	C₂	semidirect
52. Q₁₆×D₁₃	Q₈	semidirect
106. C₁₃×Q₁₆	C₂×Q₈	semidirect
134. Dic₅₂	C₂×Q₈	semidirect
147. #2227	D₁₃	direct (N × H)
281. D₁₃	#2227	direct (N × H)
301. Q₁₆	Q₈×D₁₃	semidirect
332. C₁₃	C₂×#2227	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₁₃×#500	2
C₄×Q₁₆×D₁₃	2
#13116989	2
#13116988	2
#13116987	2
#13116986	2
#13116985	2
D₁₃×#366	2
D₁₃×C₄⋊C₁₆	2
D₁₃×#358	2
C₁₃×#2227	2
C₂×#2227	13

Subgroups — D13×#2227 #16017544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃×#2227 #16017544