Subgroups — #15753391

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₄×#88060	C₂	1	1	yes	maximal, characteristic
3	C₄×#88060	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718047	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112663	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₃	C₄	1	2	yes	—
9	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88060	C₄	1	2	yes	—
14	C₂×#88060	C₄	1	2	yes	—
15	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
18	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₁₃	—	2	2	no	—
20	C₄×C₈×D₁₃	—	2	2	no	—
21	#11718142	C₄	1	2	yes	—
22	#11718142	C₄	1	2	yes	—
23	#11718047	—	2	2	no	—
24	#11718046	—	2	2	no	—
25	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	C₂×C₄	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	C₂²×#1055	C₂×C₄	1	2	yes	—
36	C₂²×#1055	C₂×C₄	1	2	yes	—
37	C₂²×#1055	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	—	2	4	no	—
39	C₂×C₈×D₁₃	—	2	4	no	—
40	C₂×C₈×D₁₃	—	2	4	no	—
41	C₄²×D₁₃	—	2	1	no	—
42	C₄²×D₁₃	—	2	1	no	—
43	C₄²×D₁₃	—	2	1	no	—
44	C₄²×D₁₃	—	2	1	no	—
45	C₄×C₈×C₁₃	—	2	1	no	—
46	C₄×C₈×C₁₃	—	2	1	no	—
47	C₄×C₈×C₁₃	—	2	1	no	—
48	C₄×C₈×C₁₃	—	2	1	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	C₄×C₈×C₁₃	—	2	1	no	—
51	#88060	—	2	4	no	—
52	#88060	C₈	1	8	yes	complemented
53	C₈×Dic₁₃	—	2	4	no	—
54	C₄×#1055	—	2	1	no	—
55	C₄×#1055	—	2	1	no	—
56	C₄×#1055	—	2	1	no	—
57	C₄×#1055	—	2	1	no	—
58	C₄×#1055	D₄	1	2	yes	—
59	C₄×#1055	D₄	1	2	yes	—
60	#8145	—	13	1	no	maximal, Sylow
61	C₂³×D₁₃	C₄²	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×Dic₁₃	C₄²	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
67	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
68	C₂×C₄×D₁₃	—	2	1	no	—
69	C₂×C₄×D₁₃	—	2	1	no	—
70	C₂×C₄×D₁₃	—	2	1	no	—
71	C₂×C₄×D₁₃	—	2	1	no	—
72	C₂×C₄×D₁₃	—	2	1	no	—
73	C₂×C₄×D₁₃	—	2	1	no	—
74	C₂×C₄×D₁₃	—	2	2	no	—
75	C₂×C₄×D₁₃	—	2	2	no	—
76	C₂×C₄×D₁₃	—	2	2	no	—
77	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
78	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
79	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	complemented
88	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₃	#41	1	1	yes	characteristic
92	C₄²×C₁₃	#41	1	1	yes	characteristic
93	C₄×Dic₁₃	—	2	1	no	—
94	C₄×Dic₁₃	—	2	1	no	—
95	C₄×Dic₁₃	—	2	1	no	—
96	C₄×Dic₁₃	—	2	1	no	—
97	C₄×Dic₁₃	—	2	1	no	—
98	C₄×Dic₁₃	—	2	1	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	—	2	2	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	C₂²⋊C₄	1	4	yes	—
107	C₂×#1055	C₂×C₈	1	4	yes	complemented
108	C₈×D₁₃	—	2	16	no	—
109	C₂²×C₄×C₈	—	13	1	no	—
110	C₄×C₂²⋊C₈	—	13	1	no	—
111	C₄×C₂²⋊C₈	—	13	1	no	—
112	C₂×#322	—	13	1	no	—
113	C₂×#322	—	13	1	no	—
114	#1273	—	13	1	no	—
115	#1273	D₁₃	1	1	yes	characteristic, complemented
116	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
117	C₂²×D₁₃	—	2	1	no	—
118	C₂²×D₁₃	—	2	1	no	—
119	C₂²×D₁₃	—	2	1	no	—
120	C₂²×D₁₃	—	2	1	no	—
121	C₂²×D₁₃	C₄×C₈	1	2	yes	—
122	C₂²×D₁₃	#97	1	2	yes	—
123	C₂²×D₁₃	#97	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
134	C₂×C₄×C₁₃	#131	1	2	yes	—
135	C₂×C₄×C₁₃	#131	1	2	yes	—
136	C₂×Dic₁₃	—	2	1	no	—
137	C₂×Dic₁₃	—	2	1	no	—
138	C₂×Dic₁₃	—	2	1	no	—
139	C₂×Dic₁₃	—	2	1	no	—
140	C₂×Dic₁₃	—	2	1	no	—
141	C₂×Dic₁₃	—	2	1	no	—
142	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
143	C₂×Dic₁₃	#97	1	2	yes	—
144	C₂×Dic₁₃	#97	1	2	yes	—
145	C₄×D₁₃	—	2	4	no	—
146	C₄×D₁₃	—	2	4	no	—
147	C₄×D₁₃	—	2	4	no	—
148	C₄×D₁₃	—	2	4	no	—
149	C₄×D₁₃	—	2	4	no	—
150	C₄×D₁₃	—	2	4	no	—
151	C₈×C₁₃	—	2	4	no	—
152	C₈×C₁₃	—	2	4	no	—
153	C₈×C₁₃	—	4	2	no	—
154	#1055	—	2	4	no	—
155	#1055	—	2	4	no	—
156	#1055	—	4	2	no	—
157	C₂³×C₈	—	13	2	no	—
158	C₂²×C₄²	—	13	1	no	—
159	C₂×C₂²⋊C₈	—	13	2	no	—
160	C₂×C₂²⋊C₈	—	13	2	no	—
161	C₂×C₄×C₈	—	13	1	no	—
162	C₂×C₄×C₈	—	13	2	no	—
163	C₂×C₄×C₈	—	13	1	no	—
164	C₂×C₄×C₈	—	13	1	no	—
165	C₂×C₄×C₈	—	26	2	no	—
166	C₂×C₄×C₈	—	26	2	no	—
167	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
170	#336	—	13	2	no	—
171	#336	—	13	2	no	—
172	#322	—	26	2	no	—
173	#322	—	26	2	no	—
174	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
178	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
179	C₂²×C₁₃	#405	1	1	yes	characteristic
180	C₂²×C₁₃	#405	1	1	yes	characteristic
181	C₂×D₁₃	—	2	2	no	—
182	C₂×D₁₃	—	2	2	no	—
183	C₂×D₁₃	—	2	2	no	—
184	C₂×D₁₃	—	2	2	no	—
185	C₂×D₁₃	—	2	2	no	—
186	C₂×D₁₃	—	2	2	no	—
187	C₂×D₁₃	#322	1	4	yes	complemented
188	C₄×C₁₃	#407	1	2	yes	—
189	C₄×C₁₃	#432	1	2	yes	—
190	C₄×C₁₃	C₈×D₄	1	2	yes	—
191	C₄×C₁₃	C₈×D₄	1	2	yes	—
192	C₄×C₁₃	#435	1	2	yes	—
193	C₄×C₁₃	#435	1	2	yes	—
194	Dic₁₃	—	2	2	no	—
195	Dic₁₃	—	2	2	no	—
196	Dic₁₃	—	2	2	no	—
197	C₂³×C₄	—	13	1	no	—
198	C₂³×C₄	—	13	1	no	—
199	C₂³×C₄	—	13	1	no	—
200	C₂²×C₈	—	13	2	no	—
201	C₂²×C₈	—	13	2	no	—
202	C₂²×C₈	—	13	2	no	—
203	C₂²×C₈	—	26	4	no	—
204	C₂²×C₈	—	26	4	no	—
205	C₂²×C₈	—	26	4	no	—
206	C₂²×C₈	C₄×D₁₃	1	2	yes	—
207	C₂²×C₈	C₄×D₁₃	1	2	yes	—
208	C₂²×C₈	C₄×D₁₃	1	2	yes	—
209	C₂×C₄²	—	13	1	no	—
210	C₂×C₄²	—	13	1	no	—
211	C₂×C₄²	—	13	1	no	—
212	C₂×C₄²	—	26	1	no	—
213	C₂×C₄²	—	26	1	no	—
214	C₂×C₄²	—	26	1	no	—
215	C₂×C₄²	—	26	1	no	—
216	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
217	C₂²⋊C₈	—	13	8	no	—
218	C₂²⋊C₈	—	26	4	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	13	2	no	—
226	C₄×C₈	—	13	2	no	—
227	C₄×C₈	—	26	4	no	—
228	C₄×C₈	—	26	1	no	—
229	C₄×C₈	—	26	1	no	—
230	C₄×C₈	—	26	1	no	—
231	C₄×C₈	—	26	1	no	—
232	C₂×C₁₃	C₂×#322	1	1	yes	characteristic
233	C₂×C₁₃	C₈×M₁₆	1	1	yes	characteristic
234	C₂×C₁₃	#1451	1	1	yes	characteristic
235	C₂×C₁₃	#1452	1	1	yes	characteristic
236	C₂×C₁₃	#1749	1	1	yes	characteristic
237	C₂×C₁₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₁₃	#1752	1	1	yes	characteristic
239	D₁₃	—	2	4	no	—
240	C₂⁴	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	1	no	—
243	C₂²×C₄	—	13	1	no	—
244	C₂²×C₄	—	13	4	no	—
245	C₂²×C₄	—	13	4	no	—
246	C₂²×C₄	—	13	4	no	—
247	C₂²×C₄	—	26	1	no	—
248	C₂²×C₄	—	26	1	no	—
249	C₂²×C₄	—	26	1	no	—
250	C₂²×C₄	—	26	1	no	—
251	C₂²×C₄	—	26	1	no	—
252	C₂²×C₄	—	26	1	no	—
253	C₂²×C₄	—	26	2	no	—
254	C₂²×C₄	—	26	2	no	—
255	C₂²×C₄	—	26	2	no	—
256	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	13	4	no	—
267	C₂×C₈	—	13	4	no	—
268	C₂×C₈	—	26	16	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	—	26	2	no	—
271	C₂×C₈	—	26	2	no	—
272	C₂×C₈	—	26	2	no	—
273	C₂×C₈	—	26	2	no	—
274	C₂×C₈	—	26	2	no	—
275	C₂×C₈	—	26	2	no	—
276	C₂×C₈	C₈×D₁₃	1	4	yes	complemented
277	C₂×C₈	#6094	1	4	yes	—
278	C₄²	—	26	1	no	—
279	C₄²	—	26	1	no	—
280	C₄²	—	26	1	no	—
281	C₄²	—	26	1	no	—
282	C₄²	—	26	1	no	—
283	C₄²	—	26	1	no	—
284	C₄²	D₄×D₁₃	1	1	yes	characteristic
285	C₄²	D₄×D₁₃	1	1	yes	characteristic
286	C₄²	#6129	1	1	yes	characteristic
287	C₄²	#6131	1	1	yes	characteristic
288	C₁₃	#8145	1	1	yes	characteristic, complemented, Sylow
289	C₂³	—	13	2	no	—
290	C₂³	—	13	2	no	—
291	C₂³	—	13	2	no	—
292	C₂³	—	26	1	no	—
293	C₂³	—	26	1	no	—
294	C₂³	—	26	1	no	—
295	C₂³	—	26	1	no	—
296	C₂³	C₄²×D₁₃	1	1	yes	characteristic
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	2	no	—
300	C₂×C₄	—	26	1	no	—
301	C₂×C₄	—	26	1	no	—
302	C₂×C₄	—	26	1	no	—
303	C₂×C₄	—	26	1	no	—
304	C₂×C₄	—	26	1	no	—
305	C₂×C₄	—	26	1	no	—
306	C₂×C₄	—	26	4	no	—
307	C₂×C₄	—	26	4	no	—
308	C₂×C₄	—	26	4	no	—
309	C₂×C₄	—	26	4	no	—
310	C₂×C₄	—	26	4	no	—
311	C₂×C₄	—	26	4	no	—
312	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
313	C₂×C₄	#88135	1	1	yes	characteristic
314	C₂×C₄	#88135	1	1	yes	characteristic
315	C₂×C₄	#88146	1	1	yes	characteristic
316	C₂×C₄	#88147	1	1	yes	characteristic
317	C₂×C₄	#88147	1	1	yes	characteristic
318	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
319	C₂×C₄	C₂×#6094	1	2	yes	—
320	C₂×C₄	#88155	1	2	yes	—
321	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
322	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
323	C₂×C₄	#88161	1	2	yes	—
324	C₈	—	2	4	no	—
325	C₈	—	2	4	no	—
326	C₈	—	4	2	no	—
327	C₈	—	26	4	no	—
328	C₈	—	26	4	no	—
329	C₈	—	52	2	no	—
330	C₂²	—	13	4	no	—
331	C₂²	—	26	2	no	—
332	C₂²	—	26	2	no	—
333	C₂²	—	26	2	no	—
334	C₂²	—	26	2	no	—
335	C₂²	—	26	2	no	—
336	C₂²	—	26	2	no	—
337	C₂²	#11718327	1	1	yes	characteristic
338	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
339	C₂²	C₄×#6094	1	1	yes	characteristic
340	C₂²	#11718350	1	1	yes	characteristic
341	C₂²	D₁₃×#97	1	1	yes	characteristic
342	C₂²	D₁₃×#97	1	1	yes	characteristic
343	C₂²	#11718369	1	1	yes	characteristic
344	C₄	—	26	2	no	—
345	C₄	—	26	2	no	—
346	C₄	—	26	2	no	—
347	C₄	#11718386	1	2	yes	—
348	C₄	#11718387	1	2	yes	—
349	C₄	#11718389	1	2	yes	—
350	C₄	#11718495	1	2	yes	—
351	C₄	#11718496	1	2	yes	—
352	C₄	#11718497	1	2	yes	—
353	C₂	—	26	4	no	—
354	C₂	D₁₃×#322	1	1	yes	characteristic
355	C₂	C₈×#6094	1	1	yes	characteristic
356	C₂	#13113122	1	1	yes	characteristic
357	C₂	#13113127	1	1	yes	characteristic
358	C₂	#13114597	1	1	yes	characteristic
359	C₂	#13114606	1	1	yes	characteristic
360	C₂	#13114611	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₁₃×#1273	C₂
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂³×D₁₃	C₄²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1273	D₁₃
hypocenter	C₁₃	#8145
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
4. C₂×#11718046	C₂	semidirect
5. C₂×#11718047	C₂	semidirect
6. #13112663	C₂	semidirect
7. C₁₃×#1273	C₂	semidirect
52. #88060	C₈	semidirect
87. C₂×C₈×C₁₃	C₂×C₈	semidirect
107. C₂×#1055	C₂×C₈	semidirect
115. #1273	D₁₃	semidirect
187. C₂×D₁₃	#322	semidirect
276. C₂×C₈	C₈×D₁₃	semidirect
288. C₁₃	#8145	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₁₃	2
C₄×#88060	2
C₂×#11718046	2
C₂×#11718047	2
#13112663	2
C₁₃×#1273	2
#8145	13