Subgroups — #16785588

Order 3328.

894 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	C₂×#11718265	C₂	1	2	yes	maximal
2	C₂×#11718265	C₂	1	2	yes	maximal
3	#13130014	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#261	C₂²	1	4	yes	—
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×(C₂×#261)⋊C₄	—	2	2	no	—
10	C₂×(C₂×#261)⋊C₄	—	2	2	no	—
11	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
12	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
15	C₄×C₈×D₁₃	—	2	4	no	—
16	#11718265	—	2	4	no	—
17	#11718265	C₄	1	8	yes	complemented
18	C₂³×#261	—	2	4	no	—
19	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊#261	—	2	4	no	—
23	C₂×C₄⋊#261	—	2	4	no	—
24	C₂×C₄⋊#261	—	2	4	no	—
25	C₂×C₄⋊#261	—	2	4	no	—
26	C₂×C₄⋊#261	—	2	4	no	—
27	C₂×C₄⋊#261	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₁₃	—	2	2	no	—
37	C₂×C₈×D₁₃	—	2	2	no	—
38	C₂×C₈×D₁₃	—	2	2	no	—
39	C₂×C₈×D₁₃	—	2	2	no	—
40	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₈×D₁₃	D₄	1	4	yes	—
42	C₄²×D₁₃	—	2	2	no	—
43	C₄²×D₁₃	—	2	2	no	—
44	(C₂×#261)⋊C₄	—	4	4	no	—
45	(C₂×#261)⋊C₄	—	4	4	no	—
46	C₄×C₈×C₁₃	—	2	2	no	—
47	C₈×Dic₁₃	—	2	2	no	—
48	C₈×Dic₁₃	—	2	2	no	—
49	C₄×#1055	—	2	2	no	—
50	#10820	—	13	1	no	maximal, Sylow
51	C₂³×D₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×#261	—	2	8	no	—
53	C₂²×#261	—	4	4	no	—
54	C₂²×#261	—	4	4	no	—
55	C₂²×#261	—	4	4	no	—
56	C₂²×#261	—	4	4	no	—
57	C₂²×#261	—	4	4	no	—
58	C₂²×#261	—	4	4	no	—
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×D₁₃	—	2	2	no	—
66	C₂×C₄×D₁₃	—	2	2	no	—
67	C₂×C₄×D₁₃	—	2	2	no	—
68	C₂×C₄×D₁₃	—	2	2	no	—
69	C₂×C₄×D₁₃	D₈	1	2	yes	—
70	C₂×C₄×D₁₃	SD₁₆	1	4	yes	—
71	C₂×C₄×D₁₃	Q₁₆	1	2	yes	—
72	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×D₁₃	#41	1	2	yes	—
75	C₂×C₄×D₁₃	#41	1	2	yes	—
76	C₂×C₄×D₁₃	#41	1	2	yes	—
77	C₂×C₄×D₁₃	#41	1	2	yes	—
78	C₄⋊#261	—	2	8	no	—
79	C₄⋊#261	—	2	8	no	—
80	C₄⋊#261	—	4	4	no	—
81	C₄⋊#261	—	4	4	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	2	1	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	C₄²	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₁₃	—	2	1	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	C₄×Dic₁₃	—	2	1	no	—
91	C₄×Dic₁₃	—	2	1	no	—
92	C₄×Dic₁₃	—	2	2	no	—
93	C₄×Dic₁₃	—	2	2	no	—
94	C₂×#1055	—	2	1	no	—
95	C₂×#1055	—	2	1	no	—
96	C₂×#1055	—	2	1	no	—
97	C₂×#1055	—	2	1	no	—
98	C₂×#1055	C₄²	1	2	yes	—
99	C₂×#1055	C₂²⋊C₄	1	2	yes	—
100	C₈×D₁₃	—	2	8	no	—
101	C₈×D₁₃	—	2	8	no	—
102	C₂²×C₄×C₈	—	13	1	no	—
103	C₂×#340	—	13	2	no	—
104	C₂×#340	—	13	2	no	—
105	C₄⋊#95	—	13	2	no	—
106	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×D₁₃	C₄×D₄	1	2	yes	—
108	C₂²×D₁₃	#121	1	2	yes	—
109	C₂²×D₁₃	#124	1	2	yes	—
110	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
111	C₂²×D₁₃	C₂×SD₁₆	1	2	yes	—
112	C₂²×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
113	C₂²×D₁₃	#135	1	2	yes	—
114	C₂²×D₁₃	#135	1	2	yes	—
115	C₂×#261	—	4	8	no	—
116	C₂×#261	—	4	8	no	—
117	C₂×#261	—	4	8	no	—
118	C₂×#261	—	8	4	no	—
119	C₂×#261	—	8	4	no	—
120	C₂×#261	—	8	4	no	—
121	C₂×#261	—	8	4	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	#106	1	2	yes	—
127	C₂×C₄×C₁₃	#107	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
134	C₂×Dic₁₃	—	2	1	no	—
135	C₂×Dic₁₃	—	2	1	no	—
136	C₂×Dic₁₃	—	2	1	no	—
137	C₂×Dic₁₃	—	2	1	no	—
138	C₂×Dic₁₃	#106	1	2	yes	—
139	C₂×Dic₁₃	#107	1	2	yes	—
140	C₂×Dic₁₃	C₄×Q₈	1	2	yes	—
141	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
142	C₂×Dic₁₃	#125	1	2	yes	—
143	C₄×D₁₃	—	2	8	no	—
144	C₄×D₁₃	—	2	4	no	—
145	C₄×D₁₃	—	2	4	no	—
146	C₄×D₁₃	#117	1	4	yes	—
147	C₄×D₁₃	C₄×D₄	1	4	yes	—
148	C₄×D₁₃	#123	1	4	yes	—
149	C₄×D₁₃	#126	1	4	yes	—
150	C₈×C₁₃	—	2	2	no	—
151	C₈×C₁₃	—	2	2	no	—
152	#1055	—	2	2	no	—
153	#1055	—	2	2	no	—
154	C₂³×C₈	—	13	1	no	—
155	C₂³×C₈	—	13	1	no	—
156	C₂²×C₄⋊C₄	—	13	4	no	—
157	C₂²×C₄²	—	13	1	no	—
158	C₂×C₄×C₈	—	13	1	no	—
159	C₂×C₄×C₈	—	13	1	no	—
160	C₂×C₄×C₈	—	13	1	no	—
161	C₂×C₄×C₈	—	26	4	no	—
162	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
163	C₂×#95	—	26	2	no	—
164	C₂×#95	—	26	2	no	—
165	#340	—	13	8	no	—
166	#340	—	26	4	no	—
167	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×C₁₃	#380	1	1	yes	characteristic
169	C₂²×C₁₃	#382	1	1	yes	characteristic
170	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
172	C₂²×C₁₃	#427	1	1	yes	characteristic
173	C₂²×C₁₃	#427	1	1	yes	characteristic
174	C₂×D₁₃	#388	1	4	yes	—
175	C₂×D₁₃	C₄×D₈	1	2	yes	—
176	C₂×D₁₃	C₄×SD₁₆	1	4	yes	—
177	C₂×D₁₃	C₄×Q₁₆	1	2	yes	—
178	C₂×D₁₃	#465	1	4	yes	—
179	C₂×D₁₃	#466	1	2	yes	—
180	C₂×D₁₃	#467	1	2	yes	—
181	C₂×D₁₃	#486	1	2	yes	—
182	C₂×D₁₃	#487	1	2	yes	—
183	C₂×D₁₃	#488	1	4	yes	—
184	#261	—	8	8	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	#376	1	1	yes	characteristic
188	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
189	C₄×C₁₃	#381	1	1	yes	characteristic
190	C₄×C₁₃	#383	1	1	yes	characteristic
191	Dic₁₃	—	2	2	no	—
192	Dic₁₃	—	2	2	no	—
193	Dic₁₃	#387	1	4	yes	—
194	C₂³×C₄	—	13	1	no	—
195	C₂³×C₄	—	13	1	no	—
196	C₂³×C₄	—	13	1	no	—
197	C₂³×C₄	—	26	4	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	13	1	no	—
200	C₂²×C₈	—	13	4	no	—
201	C₂²×C₈	—	13	4	no	—
202	C₂²×C₈	—	26	2	no	—
203	C₂²×C₈	—	26	2	no	—
204	C₂²×C₈	—	26	2	no	—
205	C₂²×C₈	—	26	2	no	—
206	C₂²×C₈	C₂×#261	1	1	yes	characteristic
207	C₂²×C₈	C₂×#261	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	13	8	no	—
209	C₂×C₄⋊C₄	—	26	4	no	—
210	C₂×C₄⋊C₄	—	26	4	no	—
211	C₂×C₄⋊C₄	—	26	4	no	—
212	C₂×C₄⋊C₄	—	26	4	no	—
213	C₂×C₄⋊C₄	—	26	4	no	—
214	C₂×C₄²	—	13	1	no	—
215	C₂×C₄²	—	13	1	no	—
216	C₂×C₄²	—	13	1	no	—
217	C₂×C₄²	—	26	2	no	—
218	C₂×C₄²	—	26	2	no	—
219	C₂×C₄²	C₂×#261	1	1	yes	characteristic
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	—	26	2	no	—
222	C₄×C₈	—	26	2	no	—
223	C₄×C₈	—	26	2	no	—
224	#95	—	52	4	no	—
225	#95	—	52	4	no	—
226	C₂×C₁₃	#1434	1	1	yes	characteristic
227	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
228	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
229	C₂×C₁₃	#1807	1	1	yes	characteristic
230	C₂×C₁₃	#1808	1	1	yes	characteristic
231	C₂×C₁₃	#1834	1	1	yes	characteristic
232	C₂×C₁₃	#1835	1	1	yes	characteristic
233	D₁₃	#1933	1	4	yes	—
234	D₁₃	#1934	1	4	yes	—
235	C₂⁴	—	13	1	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	2	no	—
238	C₂²×C₄	—	13	2	no	—
239	C₂²×C₄	—	13	2	no	—
240	C₂²×C₄	—	13	2	no	—
241	C₂²×C₄	—	13	2	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	13	1	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	4	no	—
246	C₂²×C₄	—	13	2	no	—
247	C₂²×C₄	—	13	2	no	—
248	C₂²×C₄	—	26	2	no	—
249	C₂²×C₄	—	26	2	no	—
250	C₂²×C₄	—	26	2	no	—
251	C₂²×C₄	—	26	2	no	—
252	C₂²×C₄	—	26	8	no	—
253	C₂²×C₄	—	52	4	no	—
254	C₂²×C₄	—	52	4	no	—
255	C₂²×C₄	—	52	4	no	—
256	C₂²×C₄	—	52	4	no	—
257	C₂²×C₄	—	52	4	no	—
258	C₂²×C₄	—	52	4	no	—
259	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
260	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
261	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	13	2	no	—
268	C₂×C₈	—	26	8	no	—
269	C₂×C₈	—	26	8	no	—
270	C₂×C₈	—	26	1	no	—
271	C₂×C₈	—	26	1	no	—
272	C₂×C₈	—	26	1	no	—
273	C₂×C₈	—	26	1	no	—
274	C₂×C₈	C₄×#261	1	2	yes	—
275	C₂×C₈	C₂²⋊#261	1	2	yes	—
276	C₄⋊C₄	—	26	8	no	—
277	C₄⋊C₄	—	26	8	no	—
278	C₄⋊C₄	—	52	4	no	—
279	C₄⋊C₄	—	52	4	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	26	2	no	—
283	C₄²	—	26	2	no	—
284	C₄²	—	26	1	no	—
285	C₄²	—	26	1	no	—
286	C₁₃	#10820	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	13	2	no	—
288	C₂³	—	13	2	no	—
289	C₂³	—	13	2	no	—
290	C₂³	—	13	1	no	—
291	C₂³	—	13	1	no	—
292	C₂³	—	13	2	no	—
293	C₂³	—	13	2	no	—
294	C₂³	—	13	2	no	—
295	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	13	2	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	13	4	no	—
304	C₂×C₄	—	13	4	no	—
305	C₂×C₄	—	13	4	no	—
306	C₂×C₄	—	13	4	no	—
307	C₂×C₄	—	13	2	no	—
308	C₂×C₄	—	13	2	no	—
309	C₂×C₄	—	26	8	no	—
310	C₂×C₄	—	26	4	no	—
311	C₂×C₄	—	26	4	no	—
312	C₂×C₄	—	26	1	no	—
313	C₂×C₄	—	26	1	no	—
314	C₂×C₄	—	26	1	no	—
315	C₂×C₄	—	26	1	no	—
316	C₂×C₄	—	52	8	no	—
317	C₂×C₄	—	52	8	no	—
318	C₂×C₄	—	52	8	no	—
319	C₂×C₄	—	104	4	no	—
320	C₂×C₄	—	104	4	no	—
321	C₂×C₄	—	104	4	no	—
322	C₂×C₄	—	104	4	no	—
323	C₂×C₄	#88101	1	2	yes	—
324	C₂×C₄	#88102	1	2	yes	—
325	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
326	C₂×C₄	#88237	1	1	yes	characteristic
327	C₂×C₄	#88237	1	1	yes	characteristic
328	C₂×C₄	#88237	1	1	yes	characteristic
329	C₂×C₄	#88237	1	1	yes	characteristic
330	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	26	2	no	—
334	C₈	—	26	2	no	—
335	C₂²	—	13	2	no	—
336	C₂²	—	13	2	no	—
337	C₂²	—	13	4	no	—
338	C₂²	—	13	4	no	—
339	C₂²	—	13	4	no	—
340	C₂²	—	13	2	no	—
341	C₂²	—	13	2	no	—
342	C₂²	—	13	4	no	—
343	C₂²	—	13	2	no	—
344	C₂²	—	13	2	no	—
345	C₂²	C₄×C₄⋊#261	1	1	yes	characteristic
346	C₂²	#11719060	1	1	yes	characteristic
347	C₂²	C₂×#88101	1	1	yes	characteristic
348	C₂²	C₂×#88102	1	1	yes	characteristic
349	C₂²	#11719092	1	1	yes	characteristic
350	C₂²	#11719092	1	1	yes	characteristic
351	C₂²	#11719136	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	13	4	no	—
355	C₄	—	26	2	no	—
356	C₄	—	26	2	no	—
357	C₄	—	104	8	no	—
358	C₄	#11719057	1	1	yes	characteristic
359	C₄	#11719061	1	1	yes	characteristic
360	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
361	C₄	#11719135	1	1	yes	characteristic
362	C₂	—	13	4	no	—
363	C₂	—	13	4	no	—
364	C₂	#13130012	1	1	yes	characteristic
365	C₂	C₄×#88101	1	1	yes	characteristic
366	C₂	C₄×#88102	1	1	yes	characteristic
367	C₂	#13130712	1	1	yes	characteristic
368	C₂	#13130713	1	1	yes	characteristic
369	C₂	#13130757	1	1	yes	characteristic
370	C₂	#13130758	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×D₄
cosocle	C₂²×C₄×D₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#10820
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
11. C₂×C₄×C₈×C₁₃	C₄	semidirect
12. C₂×C₈×Dic₁₃	C₄	semidirect
13. C₂×C₈×Dic₁₃	C₄	semidirect
14. C₂×C₄×#1055	C₄	semidirect
17. #11718265	C₄	semidirect
162. C₂×C₄×C₈	#261	semidirect
286. C₁₃	#10820	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718265	2
#13130014	2
C₂×C₄×C₈×D₁₃	2
#10820	13