Subgroups — D₁₃×#2110 #16015574

Order 3328.

738 conjugacy classes of subgroups.

The subgroup lattice has 362 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	2	yes	maximal
2	D₁₃×#405	C₂	1	1	yes	maximal, characteristic
3	#13116441	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116440	C₂	1	2	yes	maximal, complemented
5	#13116439	C₂	1	4	yes	maximal, complemented
6	D₁₃×#347	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₁₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	D₁₃×#117	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₁₃×#405	C₂²	1	1	yes	characteristic
13	#11718783	C₄	1	1	yes	characteristic
14	#11718762	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	2	yes	—
16	#11718658	C₄	1	1	yes	characteristic
17	D₁₃×M₃₂	C₄	1	4	yes	complemented
18	D₁₃×M₃₂	C₄	1	4	yes	complemented
19	D₁₃×M₃₂	C₂²	1	4	yes	—
20	D₁₃×M₃₂	C₂²	1	4	yes	—
21	C₂×#88038	C₄	1	4	yes	complemented
22	C₂×C₁₆×D₁₃	C₄	1	4	yes	complemented
23	#11718369	C₂²	1	2	yes	—
24	D₁₃×#97	C₂²	1	2	yes	—
25	#11718350	C₄	1	2	yes	—
26	C₄×C₈×D₁₃	C₄	1	2	yes	—
27	C₁₃×#347	C₂²	1	4	yes	complemented
28	#11718104	C₂²	1	8	yes	complemented
29	#11718058	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
34	#88180	C₂×C₄	1	1	yes	characteristic
35	C₂×#6099	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	M₁₆×D₁₃	—	2	2	no	—
38	M₁₆×D₁₃	—	2	2	no	—
39	C₂×#6094	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
49	#88146	—	2	1	no	—
50	C₄⋊C₄×D₁₃	—	2	1	no	—
51	C₂²⋊C₄×D₁₃	—	2	1	no	—
52	#88131	—	2	1	no	—
53	#88126	C₂×C₄	1	2	yes	—
54	C₄²×D₁₃	C₂×C₄	1	2	yes	—
55	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
56	C₁₃×M₃₂	C₂³	1	4	yes	—
57	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
58	C₁₃×#97	C₂³	1	2	yes	—
59	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
60	#88055	C₂×C₄	1	2	yes	—
61	#88055	C₂³	1	2	yes	—
62	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
63	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
64	#88052	C₂×C₄	1	4	yes	complemented
65	#88052	C₂³	1	4	yes	—
66	C₂×#6090	C₂×C₄	1	4	yes	complemented
67	#88043	C₂×C₄	1	2	yes	—
68	C₄×#1055	C₂×C₄	1	2	yes	—
69	#88038	C₂×C₄	1	8	yes	—
70	#88038	C₂×C₄	1	8	yes	complemented
71	C₁₆×D₁₃	C₂×C₄	1	8	yes	—
72	C₁₆×D₁₃	C₂×C₄	1	8	yes	complemented
73	C₂×#2110	—	13	1	no	maximal, Sylow
74	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
77	C₂×C₄×D₁₃	—	2	2	no	—
78	C₂×C₄×D₁₃	C₄²	1	2	yes	—
79	C₂×C₄×D₁₃	C₄²	1	2	yes	—
80	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
81	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
82	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
83	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
84	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₁₃×M₁₆	—	2	2	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	1	no	—
93	C₁₃×C₂²⋊C₄	—	2	1	no	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₂²⋊Dic₁₃	—	2	1	no	—
96	#6103	—	2	2	no	—
97	C₄⋊Dic₁₃	—	2	1	no	—
98	#6101	—	2	2	no	—
99	C₄×Dic₁₃	—	2	2	no	—
100	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
101	#6099	—	2	2	no	—
102	C₂×#1055	C₄²	1	2	yes	—
103	C₂×#1055	C₄²	1	2	yes	—
104	C₂×#1055	C₂²×C₄	1	2	yes	—
105	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
106	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
107	#6094	—	2	4	no	—
108	C₈×D₁₃	—	2	2	no	—
109	C₈×D₁₃	—	2	2	no	—
110	C₈×D₁₃	C₄²	1	1	yes	characteristic
111	C₈×D₁₃	C₄²	1	1	yes	characteristic
112	C₈×D₁₃	C₄²	1	2	yes	—
113	C₈×D₁₃	C₄²	1	1	yes	characteristic
114	C₈×D₁₃	C₄²	1	1	yes	characteristic
115	C₈×D₁₃	C₄²	1	2	yes	—
116	C₈×D₁₃	C₂²×C₄	1	2	yes	—
117	C₈×D₁₃	C₂²×C₄	1	2	yes	—
118	C₈×D₁₃	C₂²×C₄	1	2	yes	—
119	C₈×D₁₃	C₂²×C₄	1	2	yes	—
120	C₁₃×C₁₆	C₂²×C₄	1	8	yes	—
121	#6090	C₂²×C₄	1	8	yes	—
122	C₂²×M₃₂	—	13	2	no	—
123	C₂×#405	—	13	1	no	—
124	#2110	—	13	2	no	—
125	#2110	—	13	1	no	—
126	#2110	—	13	4	no	—
127	#2110	D₁₃	1	1	yes	characteristic, direct product factor
128	C₂×#347	—	13	4	no	—
129	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂²×D₁₃	—	4	2	no	—
131	C₂²×D₁₃	#97	1	2	yes	—
132	C₂²×D₁₃	#97	1	2	yes	—
133	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
134	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	#97	1	2	yes	—
142	C₂×Dic₁₃	#97	1	2	yes	—
143	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
145	C₄×D₁₃	—	4	4	no	—
146	C₄×D₁₃	#97	1	2	yes	—
147	C₄×D₁₃	#97	1	2	yes	—
148	C₄×D₁₃	#97	1	2	yes	—
149	C₄×D₁₃	#97	1	2	yes	—
150	C₄×D₁₃	C₂×C₄²	1	2	yes	—
151	C₄×D₁₃	C₂×C₄²	1	1	yes	characteristic
152	C₄×D₁₃	C₂×C₄²	1	1	yes	characteristic
153	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
154	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
155	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
158	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
159	C₈×C₁₃	C₂³×C₄	1	2	yes	—
160	#1055	—	2	2	no	—
161	#1055	C₂×C₄²	1	2	yes	—
162	#1055	C₂×C₄²	1	1	yes	characteristic
163	#1055	C₂×C₄²	1	1	yes	characteristic
164	C₂²×M₁₆	—	13	1	no	—
165	C₂³×C₈	—	13	1	no	—
166	C₂×#117	—	13	1	no	—
167	C₂×M₃₂	—	13	2	no	—
168	C₂×M₃₂	—	13	4	no	—
169	C₂×M₃₂	—	13	4	no	—
170	C₂×M₃₂	—	13	4	no	—
171	C₂×M₃₂	—	13	4	no	—
172	C₂×M₃₂	—	13	4	no	—
173	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
174	C₂²×C₁₆	—	13	4	no	—
175	#405	—	13	1	no	—
176	#405	—	13	1	no	—
177	#405	—	13	1	no	—
178	#405	—	13	2	no	—
179	#405	—	13	2	no	—
180	#405	C₂×D₁₃	1	1	yes	characteristic
181	C₂×#97	—	13	2	no	—
182	C₂×C₄×C₈	—	13	2	no	—
183	#347	—	13	8	no	—
184	#347	—	13	4	no	—
185	#347	C₂×D₁₃	1	4	yes	complemented
186	C₂²×C₁₃	—	4	1	no	—
187	C₂²×C₁₃	C₂×#97	1	2	yes	—
188	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
189	C₂×D₁₃	—	2	2	no	—
190	C₂×D₁₃	—	2	4	no	—
191	C₂×D₁₃	—	4	2	no	—
192	C₂×D₁₃	—	4	1	no	—
193	C₂×D₁₃	C₂×#97	1	1	yes	characteristic
194	C₂×D₁₃	C₂×#97	1	2	yes	—
195	C₂×D₁₃	C₂×#97	1	1	yes	characteristic
196	C₄×C₁₃	—	4	2	no	—
197	C₄×C₁₃	C₂×#97	1	2	yes	—
198	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
199	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
200	Dic₁₃	—	4	2	no	—
201	Dic₁₃	C₂×#97	1	1	yes	characteristic
202	Dic₁₃	C₂×#97	1	1	yes	characteristic
203	Dic₁₃	C₂×#97	1	2	yes	—
204	C₂³×C₄	—	13	1	no	—
205	C₂×M₁₆	—	13	1	no	—
206	C₂×M₁₆	—	13	2	no	—
207	C₂×M₁₆	—	26	2	no	—
208	C₂×M₁₆	—	26	2	no	—
209	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
210	C₂²×C₈	—	13	1	no	—
211	C₂²×C₈	—	13	1	no	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	—	13	2	no	—
214	C₂²×C₈	—	13	2	no	—
215	C₂²×C₈	—	13	2	no	—
216	C₂²×C₈	—	13	2	no	—
217	C₂²×C₈	—	13	1	no	—
218	C₂²×C₈	—	13	1	no	—
219	C₂²×C₈	—	13	2	no	—
220	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
221	#117	—	13	1	no	—
222	#117	—	13	2	no	—
223	#117	—	26	1	no	—
224	#117	—	26	1	no	—
225	#117	C₄×D₁₃	1	1	yes	characteristic
226	C₂×C₄⋊C₄	—	26	1	no	—
227	C₂×C₂²⋊C₄	—	26	1	no	—
228	C₂×C₄²	—	13	2	no	—
229	M₃₂	—	13	4	no	—
230	M₃₂	—	13	4	no	—
231	M₃₂	—	13	8	no	—
232	M₃₂	—	13	8	no	—
233	M₃₂	C₄×D₁₃	1	4	yes	complemented
234	M₃₂	C₂²×D₁₃	1	4	yes	—
235	C₂×C₁₆	—	13	4	no	—
236	C₂×C₁₆	—	13	8	no	—
237	C₂×C₁₆	—	13	8	no	—
238	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
239	#97	—	13	2	no	—
240	#97	—	13	2	no	—
241	#97	—	13	2	no	—
242	#97	C₂²×D₁₃	1	2	yes	—
243	C₄×C₈	—	13	2	no	—
244	C₄×C₈	—	13	2	no	—
245	C₄×C₈	—	13	2	no	—
246	C₄×C₈	C₄×D₁₃	1	2	yes	—
247	C₂×C₁₃	—	2	1	no	—
248	C₂×C₁₃	—	2	2	no	—
249	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
250	D₁₃	—	2	1	no	—
251	D₁₃	—	2	2	no	—
252	D₁₃	#2110	1	2	yes	direct product factor
253	C₂⁴	—	13	1	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	1	no	—
258	C₂²×C₄	—	13	1	no	—
259	C₂²×C₄	—	13	2	no	—
260	C₂²×C₄	—	13	2	no	—
261	C₂²×C₄	—	13	1	no	—
262	C₂²×C₄	—	13	1	no	—
263	C₂²×C₄	—	26	2	no	—
264	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
265	M₁₆	—	2	2	no	—
266	M₁₆	—	26	4	no	—
267	M₁₆	—	26	2	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	—	13	2	no	—
270	C₂×C₈	—	13	1	no	—
271	C₂×C₈	—	13	1	no	—
272	C₂×C₈	—	13	1	no	—
273	C₂×C₈	—	13	1	no	—
274	C₂×C₈	—	13	2	no	—
275	C₂×C₈	—	13	1	no	—
276	C₂×C₈	—	13	1	no	—
277	C₂×C₈	—	13	2	no	—
278	C₂×C₈	—	13	2	no	—
279	C₂×C₈	—	13	2	no	—
280	C₂×C₈	—	13	2	no	—
281	C₂×C₈	—	13	2	no	—
282	C₂×C₈	—	13	2	no	—
283	C₂×C₈	—	26	2	no	—
284	C₂×C₈	—	26	2	no	—
285	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
286	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
289	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	26	2	no	—
292	C₄⋊C₄	—	26	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	26	2	no	—
295	C₂²⋊C₄	—	26	1	no	—
296	C₄²	—	13	2	no	—
297	C₄²	—	26	2	no	—
298	C₄²	C₂×C₄×D₁₃	1	2	yes	—
299	C₁₆	—	13	8	no	—
300	C₁₆	C₂×C₄×D₁₃	1	8	yes	—
301	C₁₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	13	1	no	—
303	C₂³	—	13	2	no	—
304	C₂³	—	13	1	no	—
305	C₂³	—	13	2	no	—
306	C₂³	—	52	2	no	—
307	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	13	1	no	—
310	C₂×C₄	—	13	1	no	—
311	C₂×C₄	—	13	2	no	—
312	C₂×C₄	—	13	2	no	—
313	C₂×C₄	—	13	1	no	—
314	C₂×C₄	—	13	2	no	—
315	C₂×C₄	—	13	1	no	—
316	C₂×C₄	—	13	1	no	—
317	C₂×C₄	—	13	1	no	—
318	C₂×C₄	—	13	2	no	—
319	C₂×C₄	—	13	2	no	—
320	C₂×C₄	—	13	2	no	—
321	C₂×C₄	—	13	2	no	—
322	C₂×C₄	—	13	2	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	52	4	no	—
325	C₂×C₄	C₄²×D₁₃	1	2	yes	—
326	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
327	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	13	1	no	—
331	C₈	—	13	1	no	—
332	C₈	—	13	2	no	—
333	C₈	—	26	2	no	—
334	C₈	C₄²×D₁₃	1	1	yes	characteristic
335	C₈	C₄²×D₁₃	1	1	yes	characteristic
336	C₈	C₂²×C₄×D₁₃	1	2	yes	—
337	C₂²	—	4	1	no	—
338	C₂²	—	13	1	no	—
339	C₂²	—	13	1	no	—
340	C₂²	—	13	2	no	—
341	C₂²	—	26	2	no	—
342	C₂²	—	26	4	no	—
343	C₂²	—	52	1	no	—
344	C₂²	—	52	2	no	—
345	C₂²	D₁₃×#97	1	2	yes	—
346	C₂²	C₂×M₁₆×D₁₃	1	1	yes	characteristic
347	C₄	—	4	2	no	—
348	C₄	—	13	1	no	—
349	C₄	—	13	1	no	—
350	C₄	—	13	2	no	—
351	C₄	—	52	2	no	—
352	C₄	D₁₃×#97	1	2	yes	—
353	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
354	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	13	2	no	—
358	C₂	—	26	1	no	—
359	C₂	—	26	2	no	—
360	C₂	C₂×D₁₃×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2110	C₂
socle	C₂×C₁₃	C₂²×#97
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2110	D₁₃
hypocenter	C₁₃	C₂×#2110
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #13116441	C₂	semidirect
4. #13116440	C₂	semidirect
5. #13116439	C₂	semidirect
6. D₁₃×#347	C₂	semidirect
7. C₁₃×#2110	C₂	semidirect
17. D₁₃×M₃₂	C₄	semidirect
18. D₁₃×M₃₂	C₄	semidirect
21. C₂×#88038	C₄	semidirect
22. C₂×C₁₆×D₁₃	C₄	semidirect
27. C₁₃×#347	C₂²	semidirect
28. #11718104	C₂²	semidirect
29. #11718058	C₂²	semidirect
55. C₁₃×M₃₂	C₂×C₄	semidirect
57. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
64. #88052	C₂×C₄	semidirect
66. C₂×#6090	C₂×C₄	semidirect
70. #88038	C₂×C₄	semidirect
72. C₁₆×D₁₃	C₂×C₄	semidirect
127. #2110	D₁₃	direct (N × H)
185. #347	C₂×D₁₃	semidirect
233. M₃₂	C₄×D₁₃	semidirect
238. C₂×C₁₆	C₄×D₁₃	semidirect
252. D₁₃	#2110	direct (N × H)
301. C₁₃	C₂×#2110	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
D₁₃×#405	2
#13116441	2
#13116440	2
#13116439	2
D₁₃×#347	2
C₁₃×#2110	2
C₂×#2110	13

Subgroups — D13×#2110 #16015574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃×#2110 #16015574