Subgroups — #15834905

Order 3328.

572 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125419	C₂	1	1	yes	maximal, characteristic
2	#13123936	C₂	1	2	yes	maximal
3	#13113734	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113733	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113710	C₂	1	2	yes	maximal, complemented
6	#13113707	C₂	1	2	yes	maximal, complemented
7	#13113642	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113583	C₂	1	1	yes	maximal, characteristic
9	#13113255	C₂	1	2	yes	maximal
10	C₈×#6094	C₂	1	1	yes	maximal, characteristic
11	C₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719283	C₂²	1	2	yes	—
13	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
15	#11718849	C₄	1	1	yes	characteristic
16	#11718842	C₄	1	1	yes	characteristic
17	C₄×#6107	C₂²	1	2	yes	—
18	#11718685	C₂²	1	2	yes	—
19	#11718548	C₄	1	1	yes	characteristic
20	#11718535	C₄	1	1	yes	characteristic
21	#11718501	—	2	2	no	—
22	#11718494	C₂²	1	2	yes	—
23	#11718491	C₂²	1	2	yes	—
24	#11718489	—	2	2	no	—
25	#11718474	C₄	1	2	yes	—
26	#11718453	C₄	1	2	yes	—
27	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
28	#11718351	—	2	2	no	—
29	C₄×#6094	C₂²	1	1	yes	characteristic
30	#11718346	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
32	#11718340	C₂²	1	2	yes	—
33	C₈×Dic₂₆	—	2	2	no	—
34	C₁₃×#335	C₂²	1	1	yes	characteristic
35	C₁₃×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₈²×C₁₃	C₂²	1	1	yes	characteristic
37	Q₈⋊#1055	C₂²	1	2	yes	—
38	#11718076	C₂²	1	2	yes	—
39	#11718072	C₂²	1	2	yes	—
40	#11718050	C₂²	1	1	yes	characteristic
41	#11718048	C₂²	1	2	yes	—
42	#11718047	C₂²	1	1	yes	characteristic
43	#11718046	C₂²	1	1	yes	characteristic
44	C₈×#1055	C₂²	1	1	yes	characteristic
45	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
47	#88200	C₂×C₄	1	2	yes	—
48	#88198	C₂×C₄	1	2	yes	—
49	C₂×#6107	C₂×C₄	1	2	yes	—
50	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
51	C₂×#6094	—	2	1	no	—
52	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
54	#88151	C₂×C₄	1	2	yes	—
55	#88150	—	2	2	no	—
56	#88143	—	2	2	no	—
57	#88142	C₂×C₄	1	2	yes	—
58	#88126	—	2	2	no	—
59	C₄²×D₁₃	C₂³	1	1	yes	characteristic
60	C₄×Dic₂₆	—	2	2	no	—
61	C₄×Dic₂₆	C₂³	1	2	yes	—
62	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₈	—	2	2	no	—
64	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
65	C₁₃×#103	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₃	—	2	2	no	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#88075	C₂×C₄	1	2	yes	—
71	#88060	—	2	2	no	—
72	#88060	D₄	1	1	yes	characteristic
73	#88060	D₄	1	1	yes	characteristic
74	#88058	C₂×C₄	1	1	yes	characteristic
75	#88056	C₂×C₄	1	2	yes	—
76	#88055	—	2	2	no	—
77	#88055	—	2	1	no	—
78	#88054	—	2	2	no	—
79	#88054	—	2	2	no	—
80	#88054	D₄	1	1	yes	characteristic
81	#88054	D₄	1	1	yes	characteristic
82	C₈×Dic₁₃	—	2	2	no	—
83	C₈×Dic₁₃	—	2	2	no	—
84	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
86	#88050	C₂×C₄	1	2	yes	—
87	#88047	C₂×C₄	1	2	yes	—
88	C₄⋊#1055	—	2	2	no	—
89	C₄⋊#1055	C₂³	1	2	yes	—
90	#88043	—	2	2	no	—
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	C₄×#1055	C₂³	1	1	yes	characteristic
94	#18086	—	13	1	no	maximal, Sylow
95	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
99	C₂×Dic₂₆	—	2	2	no	—
100	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
101	C₁₃×Q₁₆	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₁₃×C₄⋊C₄	—	2	2	no	—
109	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
110	C₄²×C₁₃	—	2	2	no	—
111	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
112	#6107	—	4	2	no	—
113	#6103	—	2	2	no	—
114	#6103	—	2	2	no	—
115	C₄⋊Dic₁₃	—	2	2	no	—
116	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
117	#6101	—	2	2	no	—
118	#6101	—	2	2	no	—
119	#6101	—	2	2	no	—
120	#6101	—	2	2	no	—
121	#6101	—	2	2	no	—
122	C₄×Dic₁₃	—	2	2	no	—
123	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₁₃	#41	1	1	yes	characteristic
126	C₂×#1055	—	2	1	no	—
127	C₂×#1055	—	2	2	no	—
128	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
129	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
130	C₂×#1055	C₂×D₄	1	1	yes	characteristic
131	C₂×#1055	C₂×D₄	1	1	yes	characteristic
132	Dic₅₂	—	2	2	no	—
133	#6094	—	2	4	no	—
134	C₈×D₁₃	—	2	2	no	—
135	C₈×D₁₃	—	4	1	no	—
136	#3170	—	13	1	no	—
137	#2986	—	13	2	no	—
138	#1591	—	13	1	no	—
139	#1591	—	13	1	no	—
140	#1587	—	13	2	no	—
141	#1587	—	13	2	no	—
142	C₈×Q₁₆	—	13	1	no	—
143	C₈×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
144	#1564	—	13	1	no	—
145	#1485	—	13	2	no	—
146	C₈×M₁₆	—	13	1	no	—
147	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
148	Q₈×C₁₃	—	2	2	no	—
149	Q₈×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₁₃	—	2	2	no	—
156	C₂×Dic₁₃	—	2	2	no	—
157	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
159	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
160	C₄×D₁₃	—	2	1	no	—
161	C₄×D₁₃	—	2	1	no	—
162	C₄×D₁₃	—	2	1	no	—
163	C₄×D₁₃	—	2	1	no	—
164	C₄×D₁₃	—	2	2	no	—
165	Dic₂₆	—	2	2	no	—
166	Dic₂₆	—	2	2	no	—
167	Dic₂₆	—	4	4	no	—
168	C₈×C₁₃	—	2	2	no	—
169	C₈×C₁₃	—	2	1	no	—
170	C₈×C₁₃	—	2	1	no	—
171	C₈×C₁₃	—	2	1	no	—
172	C₈×C₁₃	—	4	2	no	—
173	C₈×C₁₃	#131	1	2	yes	—
174	#1055	—	2	2	no	—
175	#1055	—	2	1	no	—
176	#1055	—	2	1	no	—
177	#1055	—	2	1	no	—
178	#1055	—	2	1	no	—
179	#1055	—	4	2	no	—
180	#533	—	13	2	no	—
181	#500	—	13	1	no	—
182	#494	—	13	1	no	—
183	#487	—	13	2	no	—
184	#467	—	13	1	no	—
185	#467	—	13	1	no	—
186	#467	—	13	2	no	—
187	#446	—	13	2	no	—
188	#446	—	13	2	no	—
189	#446	—	26	2	no	—
190	C₈×Q₈	—	13	2	no	—
191	C₈×Q₈	—	26	2	no	—
192	C₈×Q₈	C₂×D₁₃	1	2	yes	—
193	C₄×Q₁₆	—	13	2	no	—
194	C₄×Q₁₆	—	13	1	no	—
195	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
196	#433	—	26	2	no	—
197	#433	—	26	2	no	—
198	#431	—	13	1	no	—
199	#431	—	13	2	no	—
200	C₄×M₁₆	—	13	1	no	—
201	C₂×C₄×C₈	—	13	1	no	—
202	#335	—	13	1	no	—
203	#335	C₂×D₁₃	1	1	yes	characteristic
204	#334	—	13	2	no	—
205	Q₈⋊C₈	—	13	2	no	—
206	Q₈⋊C₈	—	13	2	no	—
207	Q₈⋊C₈	—	13	2	no	—
208	Q₈⋊C₈	C₂×D₁₃	1	2	yes	—
209	#322	—	13	1	no	—
210	#322	—	13	1	no	—
211	C₈²	—	13	1	no	—
212	C₈²	C₂×D₁₃	1	1	yes	characteristic
213	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂×D₁₃	—	2	1	no	—
215	C₂×D₁₃	—	2	1	no	—
216	C₂×D₁₃	—	2	1	no	—
217	C₄×C₁₃	—	2	1	no	—
218	C₄×C₁₃	—	4	2	no	—
219	C₄×C₁₃	—	4	2	no	—
220	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
221	C₄×C₁₃	#568	1	1	yes	characteristic
222	C₄×C₁₃	#578	1	1	yes	characteristic
223	C₄×C₁₃	#578	1	1	yes	characteristic
224	Dic₁₃	—	2	1	no	—
225	Dic₁₃	—	2	1	no	—
226	Dic₁₃	—	4	2	no	—
227	Dic₁₃	—	4	2	no	—
228	Dic₁₃	#443	1	2	yes	—
229	C₂×Q₁₆	—	13	2	no	—
230	C₂×Q₁₆	—	13	1	no	—
231	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
232	C₂×M₁₆	—	26	1	no	—
233	C₂²×C₈	—	13	1	no	—
234	C₂²×C₈	—	13	1	no	—
235	C₄⋊Q₈	—	13	2	no	—
236	C₄⋊Q₈	—	13	2	no	—
237	#125	—	26	2	no	—
238	C₂²⋊Q₈	—	13	2	no	—
239	C₂²⋊Q₈	—	13	2	no	—
240	C₂²⋊Q₈	—	26	2	no	—
241	C₄×Q₈	—	13	2	no	—
242	C₄×Q₈	—	26	2	no	—
243	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
244	#117	—	26	2	no	—
245	C₂×C₄²	—	13	1	no	—
246	#107	—	13	1	no	—
247	#107	—	13	2	no	—
248	#107	C₄×D₁₃	1	1	yes	characteristic
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	—	13	1	no	—
251	C₄⋊C₈	—	13	1	no	—
252	C₄⋊C₈	—	13	2	no	—
253	C₄⋊C₈	—	26	2	no	—
254	C₄⋊C₈	—	26	2	no	—
255	C₄⋊C₈	—	26	2	no	—
256	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
257	#103	—	13	2	no	—
258	#103	—	13	2	no	—
259	#103	—	13	2	no	—
260	#103	C₄×D₁₃	1	2	yes	—
261	C₂²⋊C₈	—	13	1	no	—
262	C₂²⋊C₈	—	13	1	no	—
263	C₂²⋊C₈	—	26	2	no	—
264	#97	—	26	2	no	—
265	#97	—	26	1	no	—
266	#97	—	26	2	no	—
267	C₄×C₈	—	2	2	no	—
268	C₄×C₈	—	13	1	no	—
269	C₄×C₈	—	13	1	no	—
270	C₄×C₈	—	13	1	no	—
271	C₄×C₈	—	13	1	no	—
272	C₄×C₈	—	13	1	no	—
273	C₄×C₈	—	26	2	no	—
274	C₄×C₈	—	26	2	no	—
275	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
278	C₂×C₁₃	#2935	1	1	yes	characteristic
279	C₂×C₁₃	#2950	1	1	yes	characteristic
280	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
281	D₁₃	—	4	1	no	—
282	C₂×Q₈	—	13	2	no	—
283	C₂×Q₈	—	26	2	no	—
284	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
285	C₂²×C₄	—	13	1	no	—
286	C₂²×C₄	—	13	1	no	—
287	C₂²×C₄	—	13	1	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	26	2	no	—
290	Q₁₆	—	52	2	no	—
291	M₁₆	—	26	4	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	13	1	no	—
295	C₂×C₈	—	13	1	no	—
296	C₂×C₈	—	13	1	no	—
297	C₂×C₈	—	13	1	no	—
298	C₂×C₈	—	26	1	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	—	26	2	no	—
301	C₂×C₈	—	52	1	no	—
302	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
304	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
305	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	13	2	no	—
308	C₄⋊C₄	—	26	2	no	—
309	C₄⋊C₄	—	26	2	no	—
310	C₄⋊C₄	—	26	2	no	—
311	C₄⋊C₄	—	26	2	no	—
312	C₄⋊C₄	—	26	2	no	—
313	C₄⋊C₄	—	26	2	no	—
314	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
315	C₂²⋊C₄	—	26	2	no	—
316	C₂²⋊C₄	—	26	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	13	1	no	—
319	C₄²	—	13	1	no	—
320	C₄²	—	13	1	no	—
321	C₄²	—	26	2	no	—
322	C₄²	C₂³×D₁₃	1	1	yes	characteristic
323	C₁₃	#18086	1	1	yes	characteristic, complemented, Sylow
324	C₂³	—	13	1	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	2	2	no	—
327	Q₈	—	26	2	no	—
328	Q₈	—	26	2	no	—
329	Q₈	—	52	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	13	1	no	—
333	C₂×C₄	—	13	1	no	—
334	C₂×C₄	—	13	1	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	26	1	no	—
337	C₂×C₄	—	26	1	no	—
338	C₂×C₄	—	26	1	no	—
339	C₂×C₄	—	26	1	no	—
340	C₂×C₄	—	26	2	no	—
341	C₂×C₄	—	26	2	no	—
342	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
344	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₈	—	4	2	no	—
350	C₈	—	26	2	no	—
351	C₈	—	26	1	no	—
352	C₈	—	26	1	no	—
353	C₈	—	26	1	no	—
354	C₈	—	26	1	no	—
355	C₈	—	52	2	no	—
356	C₈	#88155	1	2	yes	—
357	C₂²	—	26	1	no	—
358	C₂²	—	26	1	no	—
359	C₂²	—	26	1	no	—
360	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	13	2	no	—
365	C₄	—	26	1	no	—
366	C₄	—	26	1	no	—
367	C₄	—	52	2	no	—
368	C₄	—	52	2	no	—
369	C₄	C₂×#88155	1	1	yes	characteristic
370	C₄	#11719455	1	1	yes	characteristic
371	C₄	#11719471	1	1	yes	characteristic
372	C₄	#11719476	1	1	yes	characteristic
373	C₂	—	52	1	no	—
374	C₂	#13123575	1	1	yes	characteristic
375	C₂	#13123699	1	1	yes	characteristic
376	C₂	D₁₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×Q₁₆	D₁₃
hypocenter	C₁₃	#18086
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #13113734	C₂	semidirect
4. #13113733	C₂	semidirect
5. #13113710	C₂	semidirect
6. #13113707	C₂	semidirect
7. #13113642	C₂	semidirect
11. C₈×C₁₃×Q₁₆	C₂	semidirect
143. C₈×Q₁₆	D₁₃	semidirect
323. C₁₃	#18086	semidirect

Maximal subgroups

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

subgroup	index
#13125419	2
#13123936	2
#13113734	2
#13113733	2
#13113710	2
#13113707	2
#13113642	2
#13113583	2
#13113255	2
C₈×#6094	2
C₈×C₁₃×Q₁₆	2
#18086	13