Subgroups — D₁₃⋊#2153 #16807011

Order 3328.

698 conjugacy classes of subgroups.

The subgroup lattice has 348 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	#13131179	C₂	1	2	yes	maximal, complemented
3	#13131178	C₂	1	4	yes	maximal, complemented
4	#13131177	C₂	1	2	yes	maximal, complemented
5	D₁₃⋊#364	C₂	1	4	yes	maximal, complemented
6	D₁₃×#405	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	D₁₃×#117	C₄	1	1	yes	characteristic
10	C₂×#88106	C₂²	1	4	yes	—
11	D₁₃⋊M₃₂	—	2	2	no	—
12	D₁₃⋊M₃₂	—	2	2	no	—
13	D₁₃⋊M₃₂	C₂²	1	4	yes	—
14	D₁₃⋊M₃₂	C₂²	1	4	yes	—
15	C₂×#88098	—	2	2	no	—
16	C₂×D₁₃⋊C₁₆	—	2	2	no	—
17	C₁₃×#405	C₄	1	1	yes	characteristic
18	#11718783	C₂²	1	1	yes	characteristic
19	#11718762	C₄	1	1	yes	characteristic
20	#11718658	C₂²	1	1	yes	characteristic
21	#11718369	C₂²	1	2	yes	—
22	D₁₃×#97	C₄	1	2	yes	—
23	#11718350	C₄	1	2	yes	—
24	C₄×C₈×D₁₃	C₂²	1	2	yes	—
25	#11718261	C₂²	1	8	yes	complemented
26	#11718233	C₂²	1	8	yes	complemented
27	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
31	#88180	C₂×C₄	1	1	yes	characteristic
32	C₂×#6099	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂³	1	1	yes	characteristic
34	M₁₆×D₁₃	C₈	1	4	yes	—
35	M₁₆×D₁₃	C₈	1	4	yes	—
36	C₂×#6094	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	D₄	1	2	yes	—
41	C₂×C₈×D₁₃	D₄	1	2	yes	—
42	C₂×C₈×D₁₃	Q₈	1	2	yes	—
43	C₂×C₈×D₁₃	Q₈	1	2	yes	—
44	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
46	#88146	—	2	1	no	—
47	C₄⋊C₄×D₁₃	—	2	1	no	—
48	C₂²⋊C₄×D₁₃	—	2	1	no	—
49	#88131	—	2	1	no	—
50	#88126	C₂×C₄	1	2	yes	—
51	C₄²×D₁₃	C₂×C₄	1	2	yes	—
52	#88106	—	2	4	no	—
53	#88106	C₂³	1	8	yes	—
54	C₂×#6092	—	2	4	no	—
55	#88098	—	2	4	no	—
56	#88098	—	2	4	no	—
57	D₁₃⋊C₁₆	—	2	4	no	—
58	D₁₃⋊C₁₆	—	2	4	no	—
59	C₁₃×#97	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
61	#88055	C₂×C₄	1	2	yes	—
62	#88055	C₂×C₄	1	2	yes	—
63	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
64	C₈×Dic₁₃	C₂³	1	2	yes	—
65	#88043	C₂×C₄	1	2	yes	—
66	C₄×#1055	C₂³	1	2	yes	—
67	C₂×#2153	—	13	1	no	maximal, Sylow
68	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
71	C₂×C₄×D₁₃	—	2	2	no	—
72	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
75	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
77	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
78	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
81	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₁₃×C₄⋊C₄	—	2	1	no	—
87	C₁₃×C₂²⋊C₄	—	2	1	no	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂²⋊Dic₁₃	—	2	1	no	—
90	#6103	—	2	2	no	—
91	C₄⋊Dic₁₃	—	2	1	no	—
92	#6101	—	2	2	no	—
93	C₄×Dic₁₃	—	2	2	no	—
94	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
95	#6099	C₂×C₈	1	4	yes	—
96	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
97	C₂×#1055	C₂²×C₄	1	2	yes	—
98	C₂×#1055	C₂×D₄	1	2	yes	—
99	C₂×#1055	C₂×Q₈	1	2	yes	—
100	C₂×#1055	C₂⁴	1	1	yes	characteristic
101	#6094	C₂×C₈	1	8	yes	—
102	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
103	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
104	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
105	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
106	C₈×D₁₃	C₂×C₈	1	4	yes	—
107	C₈×D₁₃	C₂×C₈	1	4	yes	—
108	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
109	C₈×D₁₃	C₂×D₄	1	2	yes	—
110	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
111	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
112	C₈×D₁₃	C₂×Q₈	1	2	yes	—
113	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
114	#6092	—	2	8	no	—
115	C₂²×M₃₂	—	13	2	no	—
116	C₂×#405	—	13	1	no	—
117	#2153	—	13	2	no	—
118	#2153	—	13	4	no	—
119	#2153	—	13	2	no	—
120	C₂×#364	—	13	4	no	—
121	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂²×D₁₃	—	4	2	no	—
123	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
124	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
125	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
126	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₁₃	—	2	2	no	—
133	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
134	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
135	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
136	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
137	C₄×D₁₃	—	4	4	no	—
138	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
139	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
140	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
141	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
142	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₄×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₄×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
146	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
147	C₄×D₁₃	C₂²×C₈	1	2	yes	—
148	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
149	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₈×C₁₃	C₂²×C₈	1	4	yes	—
152	#1055	C₂×C₄⋊C₄	1	2	yes	—
153	#1055	C₂²×C₈	1	4	yes	—
154	#1055	C₂²×D₄	1	1	yes	characteristic
155	#1055	C₂²×Q₈	1	1	yes	characteristic
156	C₂²×M₁₆	—	13	1	no	—
157	C₂³×C₈	—	13	1	no	—
158	C₂×#117	—	13	1	no	—
159	C₂×M₃₂	—	13	4	no	—
160	C₂×M₃₂	—	13	4	no	—
161	C₂×M₃₂	—	13	4	no	—
162	C₂×M₃₂	—	26	2	no	—
163	C₂×M₃₂	—	26	2	no	—
164	C₂×M₃₂	—	26	2	no	—
165	C₂²×C₁₆	—	26	2	no	—
166	#405	—	13	1	no	—
167	#405	—	13	1	no	—
168	#405	—	13	1	no	—
169	#405	—	13	2	no	—
170	#405	—	13	2	no	—
171	#405	#261	1	1	yes	characteristic
172	C₂×#97	—	13	2	no	—
173	C₂×C₄×C₈	—	13	2	no	—
174	#364	—	13	8	no	—
175	#364	—	13	8	no	—
176	C₂²×C₁₃	—	4	1	no	—
177	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
178	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
179	C₂×D₁₃	—	2	2	no	—
180	C₂×D₁₃	—	2	4	no	—
181	C₂×D₁₃	—	4	2	no	—
182	C₂×D₁₃	—	4	1	no	—
183	C₂×D₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
184	C₂×D₁₃	C₂×C₄⋊C₈	1	2	yes	—
185	C₂×D₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
188	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
189	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
190	Dic₁₃	—	4	2	no	—
191	Dic₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
192	Dic₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
193	Dic₁₃	C₂×C₄⋊C₈	1	2	yes	—
194	C₂³×C₄	—	13	1	no	—
195	C₂×M₁₆	—	13	1	no	—
196	C₂×M₁₆	—	13	4	no	—
197	C₂×M₁₆	—	13	4	no	—
198	C₂×M₁₆	—	13	2	no	—
199	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	2	no	—
202	C₂²×C₈	—	13	2	no	—
203	C₂²×C₈	—	13	2	no	—
204	C₂²×C₈	—	13	2	no	—
205	C₂²×C₈	—	13	1	no	—
206	C₂²×C₈	—	13	1	no	—
207	C₂²×C₈	—	13	1	no	—
208	C₂²×C₈	—	13	1	no	—
209	C₂²×C₈	—	13	2	no	—
210	C₂²×C₈	C₂×#261	1	1	yes	characteristic
211	#117	—	13	1	no	—
212	#117	—	13	2	no	—
213	#117	—	26	1	no	—
214	#117	—	26	1	no	—
215	#117	C₂×#261	1	1	yes	characteristic
216	C₂×C₄⋊C₄	—	26	1	no	—
217	C₂×C₂²⋊C₄	—	26	1	no	—
218	C₂×C₄²	—	13	2	no	—
219	M₃₂	—	13	8	no	—
220	M₃₂	—	26	4	no	—
221	M₃₂	—	26	4	no	—
222	M₃₂	—	26	4	no	—
223	C₂×C₁₆	—	26	4	no	—
224	C₂×C₁₆	—	26	4	no	—
225	C₂×C₁₆	—	26	4	no	—
226	#97	—	13	2	no	—
227	#97	—	13	2	no	—
228	#97	—	13	2	no	—
229	#97	C₂×#261	1	2	yes	—
230	C₄×C₈	—	13	2	no	—
231	C₄×C₈	—	13	2	no	—
232	C₄×C₈	—	13	2	no	—
233	C₄×C₈	C₂×#261	1	2	yes	—
234	C₂×C₁₃	—	2	1	no	—
235	C₂×C₁₃	—	2	2	no	—
236	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
237	D₁₃	—	2	1	no	—
238	D₁₃	—	2	2	no	—
239	D₁₃	#2153	1	2	yes	complemented
240	C₂⁴	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	13	2	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	13	2	no	—
247	C₂²×C₄	—	13	2	no	—
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	26	2	no	—
251	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
252	M₁₆	—	13	4	no	—
253	M₁₆	—	13	8	no	—
254	M₁₆	D₁₃⋊C₈	1	4	yes	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	—	13	2	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	1	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	13	1	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	—	13	1	no	—
270	C₂×C₈	—	13	2	no	—
271	C₂×C₈	—	13	1	no	—
272	C₂×C₈	C₄⋊#261	1	2	yes	—
273	C₂×C₈	C₄⋊#261	1	2	yes	—
274	C₂×C₈	C₂²×#261	1	2	yes	—
275	C₂×C₈	C₂²×#261	1	1	yes	characteristic
276	C₂×C₈	C₂²×#261	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	26	2	no	—
279	C₄⋊C₄	—	26	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	26	2	no	—
282	C₂²⋊C₄	—	26	1	no	—
283	C₄²	—	13	2	no	—
284	C₄²	—	26	2	no	—
285	C₄²	C₂²×#261	1	2	yes	—
286	C₁₆	—	26	8	no	—
287	C₁₃	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	13	2	no	—
289	C₂³	—	13	1	no	—
290	C₂³	—	13	2	no	—
291	C₂³	—	13	1	no	—
292	C₂³	—	52	2	no	—
293	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	13	2	no	—
296	C₂×C₄	—	13	2	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	2	no	—
300	C₂×C₄	—	13	2	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	13	1	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	13	1	no	—
307	C₂×C₄	—	13	2	no	—
308	C₂×C₄	—	13	1	no	—
309	C₂×C₄	—	26	2	no	—
310	C₂×C₄	—	52	4	no	—
311	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
312	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
314	C₂×C₄	C₂³×#261	1	1	yes	characteristic
315	C₈	—	13	4	no	—
316	C₈	—	13	1	no	—
317	C₈	—	13	2	no	—
318	C₈	—	13	1	no	—
319	C₈	C₂×D₁₃⋊C₈	1	4	yes	—
320	C₈	C₂×C₄⋊#261	1	2	yes	—
321	C₈	C₂×C₄⋊#261	1	1	yes	characteristic
322	C₈	C₂×C₄⋊#261	1	1	yes	characteristic
323	C₂²	—	4	1	no	—
324	C₂²	—	13	1	no	—
325	C₂²	—	13	1	no	—
326	C₂²	—	13	2	no	—
327	C₂²	—	26	2	no	—
328	C₂²	—	26	4	no	—
329	C₂²	—	52	1	no	—
330	C₂²	—	52	2	no	—
331	C₂²	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
332	C₂²	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
333	C₄	—	4	2	no	—
334	C₄	—	13	1	no	—
335	C₄	—	13	1	no	—
336	C₄	—	13	2	no	—
337	C₄	—	52	2	no	—
338	C₄	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
339	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
340	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	2	no	—
343	C₂	—	13	2	no	—
344	C₂	—	26	1	no	—
345	C₂	—	26	2	no	—
346	C₂	C₂×D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#405	C₄
socle	C₂×C₁₃	C₂²×C₄⋊C₈
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂×#1055	C₂⁴
hypercenter	#405	#261
hypocenter	C₁₃	C₂×#2153
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #13131179	C₂	semidirect
3. #13131178	C₂	semidirect
4. #13131177	C₂	semidirect
5. D₁₃⋊#364	C₂	semidirect
25. #11718261	C₂²	semidirect
26. #11718233	C₂²	semidirect
239. D₁₃	#2153	semidirect
287. C₁₃	C₂×#2153	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
#13131179	2
#13131178	2
#13131177	2
D₁₃⋊#364	2
D₁₃×#405	2
C₂×#2153	13

Subgroups — D13⋊#2153 #16807011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃⋊#2153 #16807011