Subgroups — D₁₃×#2151 #16015899

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
3	D₁₃×#405	C₂	1	1	yes	maximal, characteristic
4	#13116596	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116595	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116594	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116593	C₂	1	4	yes	maximal, complemented
8	D₁₃×C₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
9	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₁₃	C₄	1	1	yes	characteristic
11	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
12	D₁₃×#117	C₄	1	1	yes	characteristic
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#405	C₂²	1	1	yes	characteristic
16	#11718783	C₄	1	1	yes	characteristic
17	#11718762	C₂²	1	1	yes	characteristic
18	C₂×#88052	C₂²	1	1	yes	characteristic
19	C₂²×#6090	C₂²	1	1	yes	characteristic
20	#11718658	C₄	1	1	yes	characteristic
21	D₁₃×M₃₂	—	2	4	no	—
22	C₂×#88038	C₂²	1	2	yes	—
23	C₂×C₁₆×D₁₃	—	2	4	no	—
24	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
25	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
26	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
27	#11718369	C₄	1	2	yes	—
28	D₁₃×#97	C₄	1	2	yes	—
29	#11718350	C₂²	1	2	yes	—
30	C₄×C₈×D₁₃	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
32	#11718093	C₂²	1	4	yes	complemented
33	#11718093	C₂²	1	4	yes	complemented
34	C₄⋊#6090	C₂²	1	4	yes	complemented
35	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
39	#88180	C₂×C₄	1	1	yes	characteristic
40	C₂×#6099	C₂×C₄	1	1	yes	characteristic
41	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
42	M₁₆×D₁₃	—	2	4	no	—
43	C₂×#6094	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₁₃	D₄	1	4	yes	—
48	C₂×C₈×D₁₃	Q₈	1	4	yes	—
49	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
51	#88146	C₈	1	2	yes	—
52	C₄⋊C₄×D₁₃	C₈	1	2	yes	—
53	C₂²⋊C₄×D₁₃	C₈	1	2	yes	—
54	#88131	C₈	1	2	yes	—
55	#88126	C₂×C₄	1	2	yes	—
56	C₄²×D₁₃	C₂×C₄	1	2	yes	—
57	C₁₃×M₃₂	—	2	2	no	—
58	C₂×C₁₃×C₁₆	—	2	2	no	—
59	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
60	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
61	C₁₃×#97	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₃	C₂³	1	2	yes	—
63	#88055	C₂×C₄	1	2	yes	—
64	#88055	C₂×C₄	1	2	yes	—
65	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₃	C₂³	1	2	yes	—
67	#88052	—	2	2	no	—
68	C₂×#6090	—	2	2	no	—
69	C₂×#6090	C₂³	1	2	yes	—
70	C₂×#6090	C₂³	1	2	yes	—
71	#88043	C₂×C₄	1	2	yes	—
72	C₄×#1055	C₂×C₄	1	2	yes	—
73	#88038	—	2	4	no	—
74	C₁₆×D₁₃	—	2	4	no	—
75	C₁₆×D₁₃	—	2	4	no	—
76	C₁₆×D₁₃	—	2	4	no	—
77	C₂×#2151	—	13	1	no	maximal, Sylow
78	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
82	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
83	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
86	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₁₃×M₁₆	—	2	2	no	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
95	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
98	#6103	C₂×C₈	1	4	yes	—
99	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
100	#6101	C₂×C₈	1	4	yes	—
101	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
102	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
103	#6099	—	2	2	no	—
104	C₂×#1055	C₄⋊C₄	1	4	yes	—
105	C₂×#1055	C₂²×C₄	1	2	yes	—
106	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
107	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
108	#6094	—	2	4	no	—
109	C₈×D₁₃	—	2	4	no	—
110	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
111	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
112	C₈×D₁₃	C₂×D₄	1	2	yes	—
113	C₈×D₁₃	C₂×D₄	1	2	yes	—
114	C₈×D₁₃	C₂×Q₈	1	2	yes	—
115	C₈×D₁₃	C₂×Q₈	1	2	yes	—
116	C₁₃×C₁₆	—	2	2	no	—
117	C₁₃×C₁₆	—	2	2	no	—
118	#6090	—	2	2	no	—
119	#6090	—	2	2	no	—
120	C₂²×M₃₂	—	13	1	no	—
121	C₂³×C₁₆	—	13	1	no	—
122	C₂×#405	—	13	1	no	—
123	#2151	—	13	1	no	—
124	#2151	—	13	1	no	—
125	#2151	—	13	1	no	—
126	#2151	—	13	4	no	—
127	#2151	D₁₃	1	1	yes	characteristic, direct product factor
128	C₂×C₄⋊C₁₆	—	13	4	no	—
129	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂²×D₁₃	—	2	4	no	—
131	C₂²×D₁₃	C₄⋊C₈	1	4	yes	—
132	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
133	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
136	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
140	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
141	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
143	C₄×D₁₃	—	2	8	no	—
144	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
145	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
146	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
147	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
148	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
149	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
150	C₈×C₁₃	—	2	2	no	—
151	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
152	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
154	#1055	—	2	2	no	—
155	#1055	C₂×C₄⋊C₄	1	2	yes	—
156	#1055	C₂×C₄⋊C₄	1	2	yes	—
157	C₂²×M₁₆	—	13	1	no	—
158	C₂³×C₈	—	13	1	no	—
159	C₂×#117	—	13	1	no	—
160	C₂×M₃₂	—	13	1	no	—
161	C₂×M₃₂	—	13	2	no	—
162	C₂×M₃₂	—	26	4	no	—
163	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
164	C₂²×C₁₆	—	13	1	no	—
165	C₂²×C₁₆	—	13	2	no	—
166	C₂²×C₁₆	—	13	2	no	—
167	C₂²×C₁₆	—	13	2	no	—
168	C₂²×C₁₆	—	26	4	no	—
169	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
170	#405	—	13	1	no	—
171	#405	—	13	1	no	—
172	#405	—	13	1	no	—
173	#405	—	13	2	no	—
174	#405	—	13	2	no	—
175	#405	C₂×D₁₃	1	1	yes	characteristic
176	C₂×#97	—	13	2	no	—
177	C₂×C₄×C₈	—	13	2	no	—
178	C₄⋊C₁₆	—	13	4	no	—
179	C₄⋊C₁₆	—	13	4	no	—
180	C₄⋊C₁₆	—	13	4	no	—
181	C₄⋊C₁₆	C₂×D₁₃	1	4	yes	complemented
182	C₂²×C₁₃	—	2	2	no	—
183	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
184	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
185	C₂×D₁₃	—	2	2	no	—
186	C₂×D₁₃	—	2	8	no	—
187	C₂×D₁₃	C₂×C₄⋊C₈	1	2	yes	—
188	C₂×D₁₃	C₂×C₄⋊C₈	1	2	yes	—
189	C₂×D₁₃	#504	1	4	yes	—
190	C₄×C₁₃	—	2	4	no	—
191	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
192	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
193	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
194	Dic₁₃	—	2	4	no	—
195	Dic₁₃	C₂×C₄⋊C₈	1	2	yes	—
196	Dic₁₃	C₂×C₄⋊C₈	1	2	yes	—
197	C₂³×C₄	—	13	1	no	—
198	C₂×M₁₆	—	13	1	no	—
199	C₂×M₁₆	—	13	2	no	—
200	C₂×M₁₆	—	26	4	no	—
201	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
202	C₂²×C₈	—	13	1	no	—
203	C₂²×C₈	—	13	1	no	—
204	C₂²×C₈	—	13	1	no	—
205	C₂²×C₈	—	13	4	no	—
206	C₂²×C₈	—	13	4	no	—
207	C₂²×C₈	—	13	1	no	—
208	C₂²×C₈	—	13	1	no	—
209	C₂²×C₈	—	13	2	no	—
210	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
211	#117	—	13	1	no	—
212	#117	—	13	2	no	—
213	#117	—	13	2	no	—
214	#117	—	13	2	no	—
215	#117	C₄×D₁₃	1	1	yes	characteristic
216	C₂×C₄⋊C₄	—	13	2	no	—
217	C₂×C₂²⋊C₄	—	13	2	no	—
218	C₂×C₄²	—	13	2	no	—
219	M₃₂	—	2	2	no	—
220	M₃₂	—	26	4	no	—
221	M₃₂	—	26	2	no	—
222	C₂×C₁₆	—	2	2	no	—
223	C₂×C₁₆	—	13	2	no	—
224	C₂×C₁₆	—	13	2	no	—
225	C₂×C₁₆	—	26	4	no	—
226	C₂×C₁₆	—	26	4	no	—
227	C₂×C₁₆	—	26	4	no	—
228	C₂×C₁₆	—	26	2	no	—
229	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
230	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
231	#97	—	13	2	no	—
232	#97	—	13	2	no	—
233	#97	—	13	2	no	—
234	#97	C₄×D₁₃	1	2	yes	—
235	C₄×C₈	—	13	2	no	—
236	C₄×C₈	—	13	2	no	—
237	C₄×C₈	—	13	2	no	—
238	C₄×C₈	C₂²×D₁₃	1	2	yes	—
239	C₂×C₁₃	—	2	2	no	—
240	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
241	C₂×C₁₃	C₂×#504	1	2	yes	—
242	D₁₃	—	2	2	no	—
243	D₁₃	#2151	1	4	yes	direct product factor
244	C₂⁴	—	13	1	no	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	13	4	no	—
247	C₂²×C₄	—	13	1	no	—
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	13	4	no	—
250	C₂²×C₄	—	13	1	no	—
251	C₂²×C₄	—	13	1	no	—
252	C₂²×C₄	—	13	4	no	—
253	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
254	M₁₆	—	2	2	no	—
255	M₁₆	—	26	4	no	—
256	M₁₆	—	26	2	no	—
257	C₂×C₈	—	13	4	no	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	13	1	no	—
260	C₂×C₈	—	13	4	no	—
261	C₂×C₈	—	13	4	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	26	4	no	—
268	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
269	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
270	C₂×C₈	D₄×D₁₃	1	2	yes	—
271	C₂×C₈	Q₈×D₁₃	1	2	yes	—
272	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
273	C₄⋊C₄	—	13	4	no	—
274	C₄⋊C₄	—	13	2	no	—
275	C₄⋊C₄	C₈×D₁₃	1	2	yes	—
276	C₂²⋊C₄	—	13	2	no	—
277	C₂²⋊C₄	—	13	4	no	—
278	C₂²⋊C₄	C₈×D₁₃	1	2	yes	—
279	C₄²	—	13	4	no	—
280	C₄²	—	13	2	no	—
281	C₄²	C₂×C₄×D₁₃	1	2	yes	—
282	C₁₆	—	2	2	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	26	2	no	—
285	C₁₆	—	26	2	no	—
286	C₁₃	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	13	1	no	—
288	C₂³	—	13	1	no	—
289	C₂³	—	13	4	no	—
290	C₂³	—	26	4	no	—
291	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
292	C₂×C₄	—	13	1	no	—
293	C₂×C₄	—	13	1	no	—
294	C₂×C₄	—	13	4	no	—
295	C₂×C₄	—	13	2	no	—
296	C₂×C₄	—	13	2	no	—
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	4	no	—
300	C₂×C₄	—	13	4	no	—
301	C₂×C₄	—	13	4	no	—
302	C₂×C₄	—	26	8	no	—
303	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
304	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
305	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
306	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
307	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	13	2	no	—
310	C₈	—	13	2	no	—
311	C₈	—	26	2	no	—
312	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
313	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
314	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	13	2	no	—
317	C₂²	—	13	4	no	—
318	C₂²	—	13	2	no	—
319	C₂²	—	26	8	no	—
320	C₂²	—	26	2	no	—
321	C₂²	D₁₃×C₄⋊C₈	1	2	yes	—
322	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
323	C₄	—	2	4	no	—
324	C₄	—	13	2	no	—
325	C₄	—	13	2	no	—
326	C₄	—	26	4	no	—
327	C₄	D₁₃×C₄⋊C₈	1	2	yes	—
328	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
329	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	13	4	no	—
332	C₂	—	26	2	no	—
333	C₂	C₂×D₁₃×C₄⋊C₈	1	1	yes	characteristic
334	C₂	D₁₃×#504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2151	C₂
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2151	D₁₃
hypocenter	C₁₃	C₂×#2151
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
4. #13116596	C₂	semidirect
5. #13116595	C₂	semidirect
6. #13116594	C₂	semidirect
7. #13116593	C₂	semidirect
8. D₁₃×C₄⋊C₁₆	C₂	semidirect
9. C₁₃×#2151	C₂	semidirect
31. C₁₃×C₄⋊C₁₆	C₂²	semidirect
32. #11718093	C₂²	semidirect
33. #11718093	C₂²	semidirect
34. C₄⋊#6090	C₂²	semidirect
127. #2151	D₁₃	direct (N × H)
181. C₄⋊C₁₆	C₂×D₁₃	semidirect
243. D₁₃	#2151	direct (N × H)
286. C₁₃	C₂×#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃×M₃₂	2
C₂²×C₁₆×D₁₃	2
D₁₃×#405	2
#13116596	2
#13116595	2
#13116594	2
#13116593	2
D₁₃×C₄⋊C₁₆	2
C₁₃×#2151	2
C₂×#2151	13

Subgroups — D13×#2151 #16015899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃×#2151 #16015899