Subgroups — #15755628

Order 3328.

492 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117313	C₂	1	1	yes	maximal, characteristic
2	#13114988	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718048	C₂	1	2	yes	maximal
4	#13112676	C₂	1	2	yes	maximal
5	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
11	#11718594	—	2	1	no	—
12	#11718594	—	2	1	no	—
13	#11718593	—	2	1	no	—
14	C₄×#6101	—	2	1	no	—
15	#11718590	C₄	1	1	yes	characteristic
16	#11718590	C₄	1	1	yes	characteristic
17	#11718589	C₄	1	1	yes	characteristic
18	#11718589	C₄	1	1	yes	characteristic
19	C₂×C₄⋊#1055	—	2	1	no	—
20	C₂×C₄⋊#1055	—	2	1	no	—
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	C₁₃×#336	—	2	1	no	—
23	#11718048	—	2	4	no	—
24	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#6101	—	2	1	no	—
37	C₂×#6101	—	2	1	no	—
38	C₂×#6101	—	2	1	no	—
39	C₂×C₄×Dic₁₃	—	2	1	no	—
40	C₂×C₄×Dic₁₃	—	2	1	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₄×Dic₂₆	—	2	1	no	—
43	C₄×Dic₂₆	—	2	1	no	—
44	C₄×Dic₂₆	D₄	1	2	yes	—
45	C₄×Dic₂₆	Q₈	1	2	yes	—
46	C₁₃×C₄⋊C₈	—	2	4	no	—
47	C₄×C₈×C₁₃	—	2	2	no	—
48	C₄×C₈×C₁₃	—	2	2	no	—
49	#88071	—	2	1	no	—
50	#88071	—	2	1	no	—
51	#88071	—	2	1	no	—
52	#88071	—	2	1	no	—
53	#88071	—	2	1	no	—
54	#88071	—	2	1	no	—
55	#88071	—	2	1	no	—
56	C₄⋊#1055	—	2	2	no	—
57	C₄⋊#1055	—	2	2	no	—
58	C₄⋊#1055	—	4	2	no	—
59	C₄⋊#1055	—	4	2	no	—
60	#8568	—	13	1	no	maximal, Sylow
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	2	1	no	—
69	C₂²×Dic₁₃	—	2	1	no	—
70	C₂²×Dic₁₃	—	2	1	no	—
71	C₂²×Dic₁₃	—	2	1	no	—
72	C₂×Dic₂₆	—	2	2	no	—
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	—	2	1	no	—
75	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	2	1	no	—
83	C₄²×C₁₃	—	2	1	no	—
84	C₄²×C₁₃	—	4	1	no	—
85	C₄²×C₁₃	SD₁₆	1	2	yes	—
86	C₄²×C₁₃	Q₁₆	1	2	yes	—
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
89	C₄²×C₁₃	#41	1	1	yes	characteristic
90	C₄²×C₁₃	#41	1	1	yes	characteristic
91	C₄⋊Dic₁₃	—	2	1	no	—
92	C₄⋊Dic₁₃	—	2	2	no	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
95	#6101	—	2	2	no	—
96	#6101	—	2	2	no	—
97	#6101	—	4	2	no	—
98	#6101	—	4	2	no	—
99	C₄×Dic₁₃	—	2	2	no	—
100	C₄×Dic₁₃	—	2	2	no	—
101	C₄×Dic₁₃	—	4	1	no	—
102	C₄×Dic₁₃	—	4	1	no	—
103	C₂×#1055	—	4	2	no	—
104	C₂×#1055	—	4	2	no	—
105	C₂×#1055	—	4	2	no	—
106	#2295	—	13	1	no	—
107	#1842	—	13	1	no	—
108	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic
109	C₂×Q₈⋊C₈	—	13	2	no	—
110	#1279	—	13	2	no	—
111	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	#103	1	2	yes	—
124	C₂×C₄×C₁₃	#103	1	2	yes	—
125	C₂×C₄×C₁₃	#104	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#131	1	2	yes	—
133	C₂×C₄×C₁₃	#131	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
138	C₂×Dic₁₃	—	2	2	no	—
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	4	1	no	—
141	C₂×Dic₁₃	—	4	1	no	—
142	C₂×Dic₁₃	—	4	1	no	—
143	C₂×Dic₁₃	—	4	1	no	—
144	C₂×Dic₁₃	—	4	1	no	—
145	C₂×Dic₁₃	—	4	1	no	—
146	C₂×Dic₁₃	—	4	1	no	—
147	C₂×Dic₁₃	—	4	1	no	—
148	C₂×Dic₁₃	—	4	1	no	—
149	C₂×Dic₁₃	—	4	1	no	—
150	Dic₂₆	—	2	4	no	—
151	Dic₂₆	—	4	2	no	—
152	C₈×C₁₃	—	4	4	no	—
153	#1055	—	8	2	no	—
154	C₂×C₄×Q₈	—	13	1	no	—
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₄⋊C₈	—	13	2	no	—
157	C₂×C₄⋊C₈	—	26	1	no	—
158	C₂×C₄⋊C₈	—	26	1	no	—
159	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
160	#391	—	13	1	no	—
161	#391	—	13	1	no	—
162	#389	—	13	1	no	—
163	#389	—	13	1	no	—
164	#387	—	26	1	no	—
165	#387	—	26	1	no	—
166	C₄×C₄⋊C₄	—	13	1	no	—
167	C₄×C₄⋊C₄	—	26	1	no	—
168	#376	—	26	1	no	—
169	C₄³	C₂×D₁₃	1	1	yes	characteristic
170	#336	—	2	1	no	—
171	Q₈⋊C₈	—	26	4	no	—
172	C₂²×C₁₃	#380	1	1	yes	characteristic
173	C₂²×C₁₃	#408	1	1	yes	characteristic
174	C₂²×C₁₃	#408	1	1	yes	characteristic
175	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
176	C₂²×C₁₃	#417	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
178	C₂²×C₁₃	#421	1	1	yes	characteristic
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	—	2	2	no	—
183	C₄×C₁₃	—	2	2	no	—
184	C₄×C₁₃	—	4	1	no	—
185	C₄×C₁₃	#424	1	2	yes	—
186	C₄×C₁₃	#433	1	2	yes	—
187	C₄×C₁₃	#460	1	1	yes	characteristic
188	C₄×C₁₃	#462	1	1	yes	characteristic
189	C₄×C₁₃	#475	1	1	yes	characteristic
190	C₄×C₁₃	#477	1	1	yes	characteristic
191	Dic₁₃	—	4	2	no	—
192	Dic₁₃	—	4	2	no	—
193	Dic₁₃	—	8	1	no	—
194	Dic₁₃	—	8	1	no	—
195	C₂²×Q₈	—	13	1	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	26	2	no	—
198	C₄×Q₈	—	13	2	no	—
199	C₄×Q₈	—	13	2	no	—
200	C₄×Q₈	—	26	1	no	—
201	C₄×Q₈	—	26	1	no	—
202	C₂×C₄⋊C₄	—	13	1	no	—
203	C₂×C₄⋊C₄	—	13	1	no	—
204	C₂×C₄⋊C₄	—	13	1	no	—
205	C₂×C₄⋊C₄	—	26	1	no	—
206	C₂×C₄⋊C₄	—	26	1	no	—
207	C₂×C₄⋊C₄	—	26	1	no	—
208	C₂×C₄²	—	26	1	no	—
209	C₂×C₄²	—	26	1	no	—
210	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₄²	D₅₂	1	1	yes	characteristic
213	C₂×C₄²	D₅₂	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	26	2	no	—
219	C₄⋊C₈	—	26	2	no	—
220	C₄⋊C₈	—	52	2	no	—
221	C₄⋊C₈	—	52	2	no	—
222	C₄×C₈	—	2	2	no	—
223	C₄×C₈	—	2	2	no	—
224	#95	—	26	1	no	—
225	#95	—	26	1	no	—
226	#95	—	26	1	no	—
227	#95	—	26	1	no	—
228	#95	—	26	1	no	—
229	#95	—	26	1	no	—
230	#95	—	26	1	no	—
231	C₂×C₁₃	#1481	1	1	yes	characteristic
232	C₂×C₁₃	#1483	1	1	yes	characteristic
233	C₂×C₁₃	#1493	1	1	yes	characteristic
234	C₂×C₁₃	#1495	1	1	yes	characteristic
235	C₂×C₁₃	#1798	1	1	yes	characteristic
236	C₂×C₁₃	#1803	1	1	yes	characteristic
237	C₂×C₁₃	#1806	1	1	yes	characteristic
238	C₂×Q₈	—	13	2	no	—
239	C₂×Q₈	—	26	1	no	—
240	C₂×Q₈	—	26	2	no	—
241	C₂×Q₈	—	26	2	no	—
242	C₂²×C₄	—	26	1	no	—
243	C₂²×C₄	—	26	1	no	—
244	C₂²×C₄	—	26	1	no	—
245	C₂²×C₄	—	26	1	no	—
246	C₂²×C₄	#6103	1	1	yes	characteristic
247	C₂²×C₄	#6103	1	1	yes	characteristic
248	C₂²×C₄	#6103	1	1	yes	characteristic
249	C₂²×C₄	#6103	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	52	2	no	—
257	C₂×C₈	—	52	2	no	—
258	C₂×C₈	—	52	2	no	—
259	C₄⋊C₄	—	13	2	no	—
260	C₄⋊C₄	—	26	1	no	—
261	C₄⋊C₄	—	26	2	no	—
262	C₄⋊C₄	—	26	2	no	—
263	C₄⋊C₄	—	26	2	no	—
264	C₄⋊C₄	—	26	2	no	—
265	C₄⋊C₄	—	52	2	no	—
266	C₄⋊C₄	—	52	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	4	1	no	—
273	C₄²	—	26	2	no	—
274	C₄²	—	26	2	no	—
275	C₄²	—	52	1	no	—
276	C₄²	—	52	1	no	—
277	C₄²	#6095	1	2	yes	—
278	C₄²	Dic₅₂	1	2	yes	—
279	C₄²	D₄×D₁₃	1	1	yes	characteristic
280	C₄²	#6129	1	1	yes	characteristic
281	C₄²	Q₈×D₁₃	1	1	yes	characteristic
282	C₄²	#6131	1	1	yes	characteristic
283	C₁₃	#8568	1	1	yes	characteristic, complemented, Sylow
284	C₂³	C₂×#6103	1	1	yes	characteristic
285	Q₈	—	26	4	no	—
286	Q₈	—	52	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	26	2	no	—
299	C₂×C₄	—	26	2	no	—
300	C₂×C₄	—	52	1	no	—
301	C₂×C₄	—	52	1	no	—
302	C₂×C₄	—	52	1	no	—
303	C₂×C₄	—	52	1	no	—
304	C₂×C₄	—	52	1	no	—
305	C₂×C₄	—	52	1	no	—
306	C₂×C₄	—	52	1	no	—
307	C₂×C₄	—	52	1	no	—
308	C₂×C₄	—	52	1	no	—
309	C₂×C₄	—	52	1	no	—
310	C₂×C₄	#88045	1	2	yes	—
311	C₂×C₄	#88056	1	2	yes	—
312	C₂×C₄	#88056	1	2	yes	—
313	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
314	C₂×C₄	#88146	1	1	yes	characteristic
315	C₂×C₄	#88149	1	1	yes	characteristic
316	C₂×C₄	#88151	1	1	yes	characteristic
317	C₂×C₄	#88155	1	2	yes	—
318	C₂×C₄	C₂×#6095	1	1	yes	characteristic
319	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
320	C₂×C₄	#88161	1	2	yes	—
321	C₂×C₄	#88162	1	1	yes	characteristic
322	C₂×C₄	#88163	1	1	yes	characteristic
323	C₂×C₄	#88192	1	1	yes	characteristic
324	C₂×C₄	#88200	1	1	yes	characteristic
325	C₈	—	4	4	no	—
326	C₈	—	104	2	no	—
327	C₂²	C₂×#88045	1	1	yes	characteristic
328	C₂²	#11718648	1	1	yes	characteristic
329	C₂²	#11718666	1	1	yes	characteristic
330	C₂²	C₂×#88056	1	1	yes	characteristic
331	C₂²	#11718771	1	1	yes	characteristic
332	C₂²	#11718790	1	1	yes	characteristic
333	C₂²	#11718794	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	52	2	no	—
341	C₄	—	52	2	no	—
342	C₄	—	104	1	no	—
343	C₄	—	104	1	no	—
344	C₄	#11718500	1	2	yes	—
345	C₄	#11718501	1	2	yes	—
346	C₄	#11718509	1	1	yes	characteristic
347	C₄	#11718510	1	1	yes	characteristic
348	C₄	#11718511	1	1	yes	characteristic
349	C₄	#11718512	1	1	yes	characteristic
350	C₂	#13113240	1	1	yes	characteristic
351	C₂	#13113250	1	1	yes	characteristic
352	C₂	#13113292	1	1	yes	characteristic
353	C₂	#13113302	1	1	yes	characteristic
354	C₂	#13114817	1	1	yes	characteristic
355	C₂	#13114842	1	1	yes	characteristic
356	C₂	#13114857	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#8568
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
283. C₁₃	#8568	semidirect

Maximal subgroups

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

subgroup	index
#13117313	2
#13114988	2
C₂×#11718048	2
#13112676	2
C₄×C₁₃×C₄⋊C₈	2
#8568	13