Subgroups — #15753956

Order 3328.

444 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114522	C₂	1	1	yes	maximal, characteristic, complemented
2	#13113424	C₂	1	1	yes	maximal, characteristic
3	#13113322	C₂	1	1	yes	maximal, characteristic
4	C₈×#6099	C₂	1	1	yes	maximal, characteristic, complemented
5	#13112734	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112728	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2966	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#529	C₂²	1	1	yes	characteristic
9	C₁₃×#446	C₄	1	1	yes	characteristic, complemented
10	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic, complemented
11	C₁₃×#436	C₄	1	1	yes	characteristic, complemented
12	C₁₃×#435	—	2	1	no	—
13	C₈×D₄×C₁₃	—	2	1	no	—
14	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic, complemented
15	C₁₃×#433	—	2	1	no	—
16	C₁₃×#432	C₂²	1	1	yes	characteristic
17	C₁₃×#405	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88055	—	2	1	no	—
20	C₂×C₈×Dic₁₃	—	2	1	no	—
21	#11718593	C₂²	1	1	yes	characteristic
22	#11718586	C₂²	1	1	yes	characteristic
23	C₄×#6099	C₂²	1	1	yes	characteristic
24	C₂×#88043	C₂²	1	1	yes	characteristic
25	#11718135	C₄	1	2	yes	complemented
26	#11718132	C₄	1	2	yes	complemented
27	#11718130	C₄	1	2	yes	—
28	#11718129	C₄	1	2	yes	—
29	#11718046	—	2	1	no	—
30	C₈×#1055	—	2	1	no	—
31	C₂×C₁₃×M₁₆	—	2	1	no	—
32	C₂²×C₈×C₁₃	—	2	1	no	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₁₃×#126	—	2	1	no	—
36	C₁₃×#125	C₂×C₄	1	1	yes	characteristic
37	C₁₃×#124	C₂×C₄	1	1	yes	characteristic
38	C₁₃×#123	—	2	1	no	—
39	C₁₃×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
40	C₁₃×#121	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×D₄×C₁₃	—	2	1	no	—
43	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₁₃×#117	—	2	1	no	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₁₃	—	2	1	no	—
47	C₂×#6099	—	2	1	no	—
48	C₂²×#1055	—	2	1	no	—
49	C₁₃×C₄⋊C₈	—	2	1	no	—
50	C₁₃×C₄⋊C₈	—	2	1	no	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₁₃×C₂²⋊C₈	—	2	1	no	—
54	C₁₃×C₂²⋊C₈	—	2	1	no	—
55	C₁₃×C₂²⋊C₈	—	2	1	no	—
56	C₁₃×#97	—	2	1	no	—
57	C₁₃×#97	—	2	1	no	—
58	C₄×C₈×C₁₃	—	2	1	no	—
59	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	#88071	—	2	1	no	—
64	#88071	C₂×C₄	1	2	yes	—
65	C₂²⋊#1055	C₂×C₄	1	2	yes	—
66	#88055	—	2	4	no	—
67	C₈×Dic₁₃	—	2	4	no	—
68	C₄⋊#1055	—	2	1	no	—
69	#88043	—	2	1	no	—
70	#88043	C₂×C₄	1	2	yes	—
71	#88043	C₂×C₄	1	2	yes	—
72	C₄×#1055	—	2	1	no	—
73	C₄×#1055	—	2	1	no	—
74	#8247	—	13	1	no	maximal, Sylow
75	C₂×Q₈×C₁₃	C₄²	1	1	yes	characteristic
76	C₂×D₄×C₁₃	—	2	1	no	—
77	C₂×D₄×C₁₃	C₄²	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	—	2	1	no	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
82	C₂²×Dic₁₃	—	2	2	no	—
83	C₁₃×M₁₆	—	4	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	4	2	no	—
89	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	1	no	—
94	C₁₃×C₄⋊C₄	—	2	1	no	—
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	C₄²	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	C₄²	1	1	yes	characteristic
99	C₁₃×C₂²⋊C₄	—	2	1	no	—
100	C₁₃×C₂²⋊C₄	—	2	1	no	—
101	C₁₃×C₂²⋊C₄	—	2	1	no	—
102	C₁₃×C₂²⋊C₄	—	2	1	no	—
103	C₁₃×C₂²⋊C₄	—	2	1	no	—
104	C₄²×C₁₃	—	2	1	no	—
105	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	C₄×Dic₁₃	—	2	2	no	—
110	C₄×Dic₁₃	—	2	2	no	—
111	#6099	—	2	4	no	—
112	C₂×#1055	—	2	2	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	—	2	2	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	—	2	2	no	—
117	#2966	D₁₃	1	1	yes	characteristic
118	#1726	—	13	1	no	—
119	#1524	—	13	1	no	—
120	#1500	—	13	1	no	—
121	C₈×M₁₆	—	13	1	no	—
122	#1302	—	13	1	no	—
123	#1300	—	13	1	no	—
124	C₂³×C₁₃	—	2	1	no	—
125	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
126	Q₈×C₁₃	—	4	1	no	—
127	D₄×C₁₃	—	4	1	no	—
128	D₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₁₃	—	2	4	no	—
145	C₂×Dic₁₃	—	4	2	no	—
146	C₂×Dic₁₃	—	4	2	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	4	1	no	—
150	C₈×C₁₃	—	4	1	no	—
151	C₈×C₁₃	#104	1	2	yes	—
152	C₈×C₁₃	#104	1	2	yes	—
153	#1055	—	2	4	no	—
154	#1055	—	2	4	no	—
155	#1055	—	4	2	no	—
156	#529	C₂×D₁₃	1	1	yes	characteristic
157	#446	Dic₁₃	1	1	yes	characteristic, complemented
158	C₈×Q₈	Dic₁₃	1	1	yes	characteristic, complemented
159	#436	Dic₁₃	1	1	yes	characteristic, complemented
160	#435	—	2	1	no	—
161	C₈×D₄	—	2	1	no	—
162	C₈×D₄	Dic₁₃	1	1	yes	characteristic, complemented
163	#433	—	2	1	no	—
164	#432	C₂×D₁₃	1	1	yes	characteristic
165	#431	—	13	1	no	—
166	#405	—	2	1	no	—
167	C₄×M₁₆	—	13	1	no	—
168	C₂×#97	—	13	1	no	—
169	C₂×#97	—	26	1	no	—
170	C₂×C₄×C₈	—	26	1	no	—
171	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
172	#376	—	13	1	no	—
173	#330	—	13	2	no	—
174	#329	—	13	2	no	—
175	#328	—	13	2	no	—
176	#327	—	13	2	no	—
177	#322	—	26	1	no	—
178	C₈²	—	26	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	4	1	no	—
182	C₂²×C₁₃	—	4	1	no	—
183	C₂²×C₁₃	—	4	1	no	—
184	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₁₃	#443	1	2	yes	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	4	1	no	—
192	C₄×C₁₃	—	4	1	no	—
193	C₄×C₁₃	—	4	1	no	—
194	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
195	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
196	Dic₁₃	—	4	4	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	26	1	no	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	26	1	no	—
201	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
203	#126	—	2	1	no	—
204	#125	C₄×D₁₃	1	1	yes	characteristic
205	#124	C₄×D₁₃	1	1	yes	characteristic
206	#123	—	2	1	no	—
207	C₂²⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
208	#121	C₄×D₁₃	1	1	yes	characteristic
209	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
212	#117	—	2	1	no	—
213	C₂×C₄²	—	26	1	no	—
214	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	26	1	no	—
218	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
219	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
220	C₂²⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	13	2	no	—
224	#97	—	2	1	no	—
225	#97	—	2	1	no	—
226	#97	—	13	2	no	—
227	#97	—	13	2	no	—
228	#97	—	26	4	no	—
229	#97	—	26	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	26	4	no	—
232	C₄×C₈	—	26	1	no	—
233	C₄×C₈	—	26	1	no	—
234	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
236	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
238	#95	—	13	2	no	—
239	#95	—	26	1	no	—
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	—	8	1	no	—
242	C₂×C₁₃	#1757	1	1	yes	characteristic
243	C₂×C₁₃	C₄×#104	1	1	yes	characteristic
244	C₂×C₁₃	#1764	1	1	yes	characteristic
245	C₂×Q₈	C₄×Dic₁₃	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	C₄×Dic₁₃	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	26	2	no	—
250	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
251	C₂²×C₄	#6127	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
253	M₁₆	—	4	2	no	—
254	M₁₆	—	26	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	—	26	2	no	—
265	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
266	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
267	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
268	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	C₄×Dic₁₃	1	1	yes	characteristic
274	C₄⋊C₄	C₄×Dic₁₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	26	2	no	—
282	C₄²	—	26	2	no	—
283	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
284	C₄²	#6127	1	1	yes	characteristic
285	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₁₃	#8247	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	2	1	no	—
289	C₂³	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
290	Q₈	—	4	1	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	2	no	—
293	C₂×C₄	—	2	2	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	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	26	4	no	—
303	C₂×C₄	—	52	2	no	—
304	C₂×C₄	—	52	2	no	—
305	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
306	C₂×C₄	#88180	1	1	yes	characteristic
307	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
308	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
309	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
310	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	4	1	no	—
314	C₈	—	4	1	no	—
315	C₈	—	26	4	no	—
316	C₈	—	26	4	no	—
317	C₈	—	52	2	no	—
318	C₈	#88077	1	2	yes	—
319	C₈	#88077	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#11718805	1	2	yes	—
326	C₂²	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	52	4	no	—
336	C₄	C₂×#88077	1	1	yes	characteristic
337	C₄	C₂×#88077	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	8	1	no	—
340	C₂	#13114636	1	1	yes	characteristic
341	C₂	C₄×#88077	1	1	yes	characteristic
342	C₂	#13114671	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2966	C₂
socle	C₂²×C₁₃	C₄×C₂²⋊C₄
prime radical	C₁₃×#121	C₂×C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2966	D₁₃
hypocenter	C₁₃	#8247
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #13114522	C₂	semidirect
4. C₈×#6099	C₂	semidirect
5. #13112734	C₂	semidirect
6. #13112728	C₂	semidirect
9. C₁₃×#446	C₄	semidirect
10. C₈×Q₈×C₁₃	C₄	semidirect
11. C₁₃×#436	C₄	semidirect
14. C₈×D₄×C₁₃	C₄	semidirect
25. #11718135	C₄	semidirect
26. #11718132	C₄	semidirect
157. #446	Dic₁₃	semidirect
158. C₈×Q₈	Dic₁₃	semidirect
159. #436	Dic₁₃	semidirect
162. C₈×D₄	Dic₁₃	semidirect
287. C₁₃	#8247	semidirect

Maximal subgroups

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

subgroup	index
#13114522	2
#13113424	2
#13113322	2
C₈×#6099	2
#13112734	2
#13112728	2
C₁₃×#2966	2
#8247	13