Subgroups — #15833617

Order 3328.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125406	C₂	1	1	yes	maximal, characteristic
2	#13123981	C₂	1	2	yes	maximal, complemented
3	#13113749	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113744	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113725	C₂	1	2	yes	maximal, complemented
6	#13113659	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113654	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113635	C₂	1	2	yes	maximal, complemented
9	#13113299	C₂	1	2	yes	maximal, complemented
10	#13113119	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719184	C₂²	1	2	yes	—
13	C₁₃×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718789	—	2	2	no	—
16	#11718788	—	2	2	no	—
17	#11718777	C₄	1	1	yes	characteristic
18	#11718776	C₄	1	1	yes	characteristic
19	#11718775	C₄	1	2	yes	—
20	#11718586	C₂²	1	1	yes	characteristic
21	#11718586	C₂²	1	1	yes	characteristic
22	#11718501	—	2	2	no	—
23	#11718500	—	2	2	no	—
24	#11718499	C₂²	1	2	yes	—
25	#11718495	C₂²	1	2	yes	—
26	#11718492	C₂²	1	2	yes	—
27	#11718491	C₂²	1	2	yes	—
28	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
29	#11718358	C₄	1	2	yes	—
30	#11718357	C₄	1	1	yes	characteristic
31	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
32	C₄×#6095	C₂²	1	2	yes	—
33	#11718344	C₄	1	1	yes	characteristic
34	C₁₃×#334	C₂²	1	2	yes	—
35	C₁₃×#334	C₂²	1	1	yes	characteristic
36	C₁₃×#334	C₂²	1	1	yes	characteristic
37	#11718076	C₂²	1	2	yes	—
38	#11718076	C₂²	1	2	yes	—
39	#11718075	C₂²	1	2	yes	—
40	#11718075	C₂²	1	2	yes	—
41	#11718047	C₂²	1	1	yes	characteristic
42	#11718047	C₂²	1	1	yes	characteristic
43	C₈×#1055	C₂²	1	1	yes	characteristic
44	C₈×#1055	C₂²	1	1	yes	characteristic
45	C₂×C₁₃×M₁₆	—	2	2	no	—
46	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	#88184	C₂×C₄	1	2	yes	—
50	#88183	—	2	2	no	—
51	C₄×C₂²⋊D₁₃	—	2	2	no	—
52	#88178	C₂×C₄	1	2	yes	—
53	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
54	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
55	C₂×#6095	C₂×C₄	1	2	yes	—
56	C₂×#6094	—	2	2	no	—
57	C₂×C₈×D₁₃	—	2	2	no	—
58	#88130	—	2	2	no	—
59	#88129	C₂×C₄	1	2	yes	—
60	C₄×D₅₂	C₂³	1	2	yes	—
61	#88126	—	2	2	no	—
62	#88124	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₆	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₈	—	2	2	no	—
65	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
66	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₂²⋊#1055	—	2	1	no	—
72	C₂²⋊#1055	D₄	1	2	yes	—
73	#88061	C₂×C₄	1	2	yes	—
74	#88061	C₂×C₄	1	2	yes	—
75	#88060	—	2	2	no	—
76	#88060	—	2	2	no	—
77	#88058	C₂×C₄	1	1	yes	characteristic
78	#88058	C₂×C₄	1	1	yes	characteristic
79	#88057	C₂×C₄	1	2	yes	—
80	#88056	C₂×C₄	1	2	yes	—
81	#88056	C₂×C₄	1	2	yes	—
82	#88055	—	2	2	no	—
83	#88054	—	2	2	no	—
84	#88054	—	2	2	no	—
85	C₈×Dic₁₃	—	2	2	no	—
86	C₄⋊#1055	—	2	1	no	—
87	C₄⋊#1055	D₄	1	2	yes	—
88	C₄×#1055	C₂³	1	1	yes	characteristic
89	C₄×#1055	C₂³	1	1	yes	characteristic
90	C₄×#1055	C₂³	1	1	yes	characteristic
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	#17974	—	13	1	no	maximal, Sylow
93	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
96	C₂×C₂²⋊D₁₃	—	2	2	no	—
97	C₂×D₅₂	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₁₃	—	2	2	no	—
99	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
100	C₁₃×M₁₆	—	8	2	no	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃	#41	1	1	yes	characteristic
112	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊Dic₁₃	—	2	2	no	—
114	#6103	—	2	2	no	—
115	#6103	—	2	2	no	—
116	#6103	—	2	2	no	—
117	#6103	—	2	2	no	—
118	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
119	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
120	#6101	—	2	2	no	—
121	#6101	—	2	2	no	—
122	#6101	—	2	2	no	—
123	#6101	—	2	2	no	—
124	C₄×Dic₁₃	—	2	2	no	—
125	C₂×#1055	—	2	1	no	—
126	C₂×#1055	—	2	1	no	—
127	C₂×#1055	C₂×D₄	1	2	yes	—
128	C₂×#1055	C₂×D₄	1	2	yes	—
129	Dic₅₂	—	2	2	no	—
130	D₁₀₄	—	2	2	no	—
131	#6095	—	2	4	no	—
132	#6094	—	8	2	no	—
133	C₈×D₁₃	—	8	2	no	—
134	#3168	—	13	1	no	—
135	#2993	—	13	2	no	—
136	#1594	—	13	1	no	—
137	#1593	—	13	1	no	—
138	#1590	—	13	2	no	—
139	#1580	—	13	1	no	—
140	#1579	—	13	1	no	—
141	C₈×SD₁₆	—	13	2	no	—
142	#1566	D₁₃	1	1	yes	characteristic, complemented
143	#1495	—	13	2	no	—
144	#1450	—	13	1	no	—
145	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
146	C₂²×D₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	1	no	—
148	C₂×C₄×C₁₃	—	2	1	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	4	1	no	—
151	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂²⋊D₁₃	—	8	2	no	—
158	C₂×Dic₁₃	—	2	2	no	—
159	C₂×Dic₁₃	—	2	2	no	—
160	C₂×Dic₁₃	—	2	2	no	—
161	D₅₂	—	2	2	no	—
162	D₅₂	—	2	2	no	—
163	C₄×D₁₃	—	4	2	no	—
164	C₄×D₁₃	—	4	2	no	—
165	Dic₂₆	—	2	2	no	—
166	Dic₂₆	—	2	2	no	—
167	C₈×C₁₃	—	2	1	no	—
168	C₈×C₁₃	—	2	1	no	—
169	C₈×C₁₃	—	4	2	no	—
170	C₈×C₁₃	—	4	2	no	—
171	C₈×C₁₃	#131	1	2	yes	—
172	C₈×C₁₃	#131	1	2	yes	—
173	#1055	—	2	2	no	—
174	#1055	—	2	2	no	—
175	#1055	—	4	1	no	—
176	#1055	—	4	1	no	—
177	#529	—	13	2	no	—
178	#499	—	13	1	no	—
179	#495	—	13	1	no	—
180	#488	—	13	2	no	—
181	#467	—	13	1	no	—
182	#466	—	13	1	no	—
183	#465	—	13	2	no	—
184	#446	—	13	2	no	—
185	C₈×Q₈	—	13	2	no	—
186	C₄×Q₁₆	—	13	1	no	—
187	C₄×SD₁₆	—	13	2	no	—
188	C₄×D₈	—	13	1	no	—
189	#436	—	13	2	no	—
190	#436	—	26	2	no	—
191	#435	—	26	2	no	—
192	C₈×D₄	—	13	2	no	—
193	#433	—	26	2	no	—
194	#431	—	13	1	no	—
195	#431	—	13	1	no	—
196	#424	—	26	2	no	—
197	#423	C₂×D₁₃	1	2	yes	—
198	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
199	#334	C₂×D₁₃	1	2	yes	—
200	#334	C₂×D₁₃	1	1	yes	characteristic
201	#334	C₂×D₁₃	1	1	yes	characteristic
202	Q₈⋊C₈	—	13	2	no	—
203	Q₈⋊C₈	—	13	2	no	—
204	D₄⋊C₈	—	13	2	no	—
205	D₄⋊C₈	—	13	2	no	—
206	#322	—	13	1	no	—
207	#322	—	13	1	no	—
208	C₈²	—	13	1	no	—
209	C₈²	—	13	1	no	—
210	C₂²×C₁₃	—	2	1	no	—
211	C₂²×C₁₃	—	2	1	no	—
212	C₂²×C₁₃	—	2	1	no	—
213	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂×D₁₃	—	4	2	no	—
215	C₂×D₁₃	—	4	2	no	—
216	C₂×D₁₃	—	4	2	no	—
217	C₄×C₁₃	—	2	1	no	—
218	C₄×C₁₃	—	2	1	no	—
219	C₄×C₁₃	—	2	1	no	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
222	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
223	C₄×C₁₃	#577	1	2	yes	—
224	Dic₁₃	—	4	2	no	—
225	Dic₁₃	—	4	2	no	—
226	Dic₁₃	—	4	2	no	—
227	C₂×Q₁₆	—	13	1	no	—
228	C₂×SD₁₆	—	13	2	no	—
229	C₂×D₈	—	13	1	no	—
230	C₂×M₁₆	—	2	2	no	—
231	C₂×M₁₆	—	26	2	no	—
232	C₂²×C₈	—	26	2	no	—
233	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
234	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
235	#126	—	26	2	no	—
236	#125	—	13	2	no	—
237	#124	—	13	2	no	—
238	#123	—	26	2	no	—
239	C₂²⋊Q₈	—	13	2	no	—
240	#121	—	13	2	no	—
241	C₄×Q₈	—	13	2	no	—
242	C₄×D₄	—	13	2	no	—
243	C₄×D₄	—	26	2	no	—
244	#117	—	26	2	no	—
245	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
246	#107	—	13	1	no	—
247	#107	—	13	1	no	—
248	#106	—	13	2	no	—
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	—	13	2	no	—
251	C₄⋊C₈	—	26	2	no	—
252	C₄⋊C₈	—	26	2	no	—
253	C₄⋊C₈	—	26	1	no	—
254	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
255	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
256	#103	—	13	2	no	—
257	#103	—	13	2	no	—
258	#102	—	13	2	no	—
259	#102	—	13	2	no	—
260	C₂²⋊C₈	—	13	2	no	—
261	C₂²⋊C₈	—	26	2	no	—
262	C₂²⋊C₈	—	26	2	no	—
263	C₂²⋊C₈	—	26	1	no	—
264	#97	—	26	2	no	—
265	C₄×C₈	—	13	1	no	—
266	C₄×C₈	—	13	1	no	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	13	1	no	—
269	C₄×C₈	—	26	2	no	—
270	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
271	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
274	C₂×C₁₃	—	4	1	no	—
275	C₂×C₁₃	#2931	1	1	yes	characteristic
276	C₂×C₁₃	#2952	1	1	yes	characteristic
277	C₂×C₁₃	#2957	1	1	yes	characteristic
278	D₁₃	—	8	2	no	—
279	C₂×Q₈	—	13	2	no	—
280	C₂×D₄	—	13	2	no	—
281	C₂×D₄	—	26	2	no	—
282	C₂²×C₄	—	26	2	no	—
283	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
285	C₂²×C₄	#6131	1	1	yes	characteristic
286	Q₁₆	—	26	2	no	—
287	SD₁₆	—	26	4	no	—
288	D₈	—	26	2	no	—
289	M₁₆	—	8	2	no	—
290	M₁₆	—	104	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	13	2	no	—
296	C₂×C₈	—	13	2	no	—
297	C₂×C₈	—	26	1	no	—
298	C₂×C₈	—	26	1	no	—
299	C₂×C₈	—	104	2	no	—
300	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
304	C₄⋊C₄	—	13	2	no	—
305	C₄⋊C₄	—	13	2	no	—
306	C₄⋊C₄	—	26	2	no	—
307	C₄⋊C₄	—	26	2	no	—
308	C₄⋊C₄	—	26	2	no	—
309	C₄⋊C₄	—	26	2	no	—
310	C₂²⋊C₄	—	26	2	no	—
311	C₂²⋊C₄	—	26	2	no	—
312	C₂²⋊C₄	—	26	2	no	—
313	C₂²⋊C₄	—	26	2	no	—
314	C₂²⋊C₄	—	26	2	no	—
315	C₄²	—	26	2	no	—
316	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
317	C₄²	D₄×D₁₃	1	1	yes	characteristic
318	C₄²	#6131	1	1	yes	characteristic
319	C₄²	C₂³×D₁₃	1	1	yes	characteristic
320	C₁₃	#17974	1	1	yes	characteristic, complemented, Sylow
321	C₂³	—	26	2	no	—
322	C₂³	#88147	1	1	yes	characteristic
323	Q₈	—	26	2	no	—
324	Q₈	—	26	2	no	—
325	D₄	—	26	2	no	—
326	D₄	—	26	2	no	—
327	D₄	—	104	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	26	2	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	26	2	no	—
335	C₂×C₄	—	52	2	no	—
336	C₂×C₄	—	52	2	no	—
337	C₂×C₄	#88147	1	1	yes	characteristic
338	C₂×C₄	#88147	1	1	yes	characteristic
339	C₂×C₄	#88147	1	1	yes	characteristic
340	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
342	C₂×C₄	C₂×#6131	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	4	2	no	—
346	C₈	—	4	2	no	—
347	C₈	—	26	2	no	—
348	C₈	—	26	2	no	—
349	C₈	—	52	1	no	—
350	C₈	—	52	1	no	—
351	C₈	#88161	1	2	yes	—
352	C₈	#88161	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	52	2	no	—
357	C₂²	—	52	2	no	—
358	C₂²	—	52	2	no	—
359	C₂²	C₂×#88147	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	52	2	no	—
365	C₄	—	52	2	no	—
366	C₄	—	52	2	no	—
367	C₄	C₂×#88161	1	1	yes	characteristic
368	C₄	C₂×#88161	1	1	yes	characteristic
369	C₄	#11719467	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	104	2	no	—
372	C₂	#13123547	1	1	yes	characteristic
373	C₂	#13123719	1	1	yes	characteristic
374	C₂	#13123752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1566	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718776	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1566	D₁₃
hypocenter	C₁₃	#17974
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
2. #13123981	C₂	semidirect
3. #13113749	C₂	semidirect
4. #13113744	C₂	semidirect
5. #13113725	C₂	semidirect
6. #13113659	C₂	semidirect
7. #13113654	C₂	semidirect
8. #13113635	C₂	semidirect
9. #13113299	C₂	semidirect
10. #13113119	C₂	semidirect
11. C₁₃×#1566	C₂	semidirect
142. #1566	D₁₃	semidirect
320. C₁₃	#17974	semidirect

Maximal subgroups

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

subgroup	index
#13125406	2
#13123981	2
#13113749	2
#13113744	2
#13113725	2
#13113659	2
#13113654	2
#13113635	2
#13113299	2
#13113119	2
C₁₃×#1566	2
#17974	13