Subgroups — #15871109

Order 3328.

738 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	#13125182	C₂	1	1	yes	maximal, characteristic
2	#13123773	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123553	C₂	1	1	yes	maximal, characteristic
4	#13115812	C₂	1	1	yes	maximal, characteristic, complemented
5	#13115663	C₂	1	1	yes	maximal, characteristic
6	#13115545	C₂	1	2	yes	maximal, complemented
7	#13115538	C₂	1	2	yes	maximal, complemented
8	#13114679	C₂	1	2	yes	maximal, complemented
9	#13114672	C₂	1	2	yes	maximal, complemented
10	#97×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1838	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88161	C₂²	1	2	yes	—
13	#11719182	C₂²	1	1	yes	characteristic
14	#11719175	C₂²	1	1	yes	characteristic
15	C₁₃×#423	C₂²	1	1	yes	characteristic
16	C₁₃×#423	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₂×#88065	C₂²	1	2	yes	—
19	C₂×#88065	C₂²	1	2	yes	—
20	#11718790	C₂²	1	2	yes	—
21	#11718783	C₂²	1	2	yes	—
22	#11718780	C₂²	1	2	yes	—
23	#11718774	C₂²	1	2	yes	—
24	#11718774	C₂²	1	2	yes	—
25	C₂×#88055	C₂²	1	1	yes	characteristic
26	C₂×#88055	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
29	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
30	C₂×#88043	C₂²	1	1	yes	characteristic
31	#11718497	—	2	4	no	—
32	#11718489	—	2	4	no	—
33	#11718375	C₄	1	4	yes	complemented
34	#11718374	C₄	1	2	yes	complemented
35	#11718364	C₄	1	2	yes	complemented
36	C₁₃×#341	C₂²	1	2	yes	—
37	C₁₃×#341	C₂²	1	2	yes	—
38	#11718147	C₂²	1	2	yes	—
39	#11718147	C₂²	1	2	yes	—
40	C₂×#6127	C₂³	1	2	yes	—
41	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	#88184	—	2	2	no	—
47	#88180	C₂³	1	2	yes	—
48	#88178	—	2	2	no	—
49	C₂×C₄×Dic₁₃	—	2	2	no	—
50	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
52	C₂²×#1055	C₂³	1	1	yes	characteristic
53	C₂²×#1055	C₂³	1	1	yes	characteristic
54	#88161	—	2	4	no	—
55	#88161	—	2	4	no	—
56	C₂×#6094	—	2	2	no	—
57	C₂×C₈×D₁₃	—	2	2	no	—
58	#88129	C₂×C₄	1	2	yes	—
59	#88128	C₂×C₄	1	1	yes	characteristic
60	C₄×D₅₂	—	2	2	no	—
61	#88124	C₂×C₄	1	2	yes	—
62	#88123	C₂×C₄	1	1	yes	characteristic
63	#88123	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₆	—	2	2	no	—
65	C₁₃×C₄⋊C₈	—	2	2	no	—
66	C₁₃×C₄⋊C₈	—	2	2	no	—
67	C₁₃×#97	C₂×C₄	1	2	yes	—
68	C₁₃×#97	C₂×C₄	1	2	yes	—
69	#88065	—	2	4	no	—
70	#88065	—	2	4	no	—
71	#88061	C₂×C₄	1	4	yes	—
72	#88061	C₂×C₄	1	4	yes	—
73	#88060	—	2	4	no	—
74	#88056	C₂×C₄	1	4	yes	—
75	#88056	C₂×C₄	1	4	yes	—
76	#88055	—	2	2	no	—
77	#88055	—	2	2	no	—
78	#88055	—	2	2	no	—
79	#88054	—	2	4	no	—
80	C₈×Dic₁₃	—	2	2	no	—
81	C₈×Dic₁₃	—	2	2	no	—
82	C₈×Dic₁₃	—	2	2	no	—
83	#88043	D₄	1	4	yes	—
84	#21662	—	13	1	no	maximal, Sylow
85	C₂²×C₄×C₁₃	#41	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂×C₂²⋊D₁₃	—	2	2	no	—
88	C₂²×Dic₁₃	—	2	1	no	—
89	C₂²×Dic₁₃	—	2	1	no	—
90	#6127	—	2	2	no	—
91	#6127	—	2	2	no	—
92	#6127	—	2	2	no	—
93	#6127	—	2	2	no	—
94	C₂×D₅₂	—	2	2	no	—
95	C₂×D₅₂	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₁₃	—	2	2	no	—
97	C₂×Dic₂₆	—	2	2	no	—
98	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
99	C₁₃×M₁₆	—	2	2	no	—
100	C₁₃×M₁₆	—	2	2	no	—
101	C₁₃×M₁₆	—	2	2	no	—
102	C₁₃×M₁₆	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	1	no	—
104	C₂×C₈×C₁₃	—	2	1	no	—
105	C₂×C₈×C₁₃	—	2	1	no	—
106	C₂×C₈×C₁₃	—	2	1	no	—
107	C₂×C₈×C₁₃	—	2	2	no	—
108	C₂×C₈×C₁₃	—	2	2	no	—
109	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
111	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₁₃	C₂×D₄	1	2	yes	—
114	C₂²⋊Dic₁₃	—	2	2	no	—
115	#6103	—	2	4	no	—
116	C₄⋊Dic₁₃	—	2	2	no	—
117	C₄⋊Dic₁₃	—	2	2	no	—
118	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
119	#6101	—	2	4	no	—
120	C₄×Dic₁₃	—	2	2	no	—
121	C₄×Dic₁₃	—	2	1	no	—
122	C₄×Dic₁₃	—	2	1	no	—
123	C₄×Dic₁₃	—	4	4	no	—
124	C₄×Dic₁₃	#41	1	2	yes	—
125	C₄×Dic₁₃	#41	1	2	yes	—
126	C₂×#1055	—	2	1	no	—
127	C₂×#1055	—	2	1	no	—
128	C₂×#1055	C₂×D₄	1	2	yes	—
129	C₂×#1055	C₂×D₄	1	2	yes	—
130	C₂×#1055	#41	1	2	yes	—
131	C₂×#1055	#41	1	2	yes	—
132	#6094	—	4	4	no	—
133	C₈×D₁₃	—	4	4	no	—
134	#3135	—	13	1	no	—
135	#2961	—	13	1	no	—
136	#2932	—	13	1	no	—
137	#1994	—	13	1	no	—
138	#1962	—	13	1	no	—
139	#1938	—	13	2	no	—
140	#1937	—	13	2	no	—
141	#1838	D₁₃	1	1	yes	characteristic, complemented
142	#1765	—	13	2	no	—
143	#1764	—	13	2	no	—
144	C₄×#97	—	13	1	no	—
145	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
146	C₂²×D₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₂²⋊D₁₃	—	4	2	no	—
157	C₂²⋊D₁₃	—	4	2	no	—
158	C₂×Dic₁₃	—	2	4	no	—
159	C₂×Dic₁₃	—	2	2	no	—
160	C₂×Dic₁₃	—	2	4	no	—
161	C₂×Dic₁₃	—	2	2	no	—
162	C₂×Dic₁₃	—	2	2	no	—
163	C₂×Dic₁₃	—	4	2	no	—
164	D₅₂	—	2	2	no	—
165	D₅₂	—	2	2	no	—
166	D₅₂	—	8	2	no	—
167	C₄×D₁₃	—	4	2	no	—
168	C₄×D₁₃	—	4	2	no	—
169	Dic₂₆	—	2	2	no	—
170	Dic₂₆	—	2	2	no	—
171	Dic₂₆	—	8	2	no	—
172	C₈×C₁₃	—	2	2	no	—
173	C₈×C₁₃	—	2	2	no	—
174	C₈×C₁₃	—	4	2	no	—
175	C₈×C₁₃	—	4	2	no	—
176	#1055	—	2	2	no	—
177	#1055	#124	1	4	yes	—
178	C₂×#131	—	13	2	no	—
179	#533	—	13	1	no	—
180	#532	—	13	1	no	—
181	#491	—	13	2	no	—
182	#490	—	13	2	no	—
183	#489	—	13	4	no	—
184	#446	—	26	4	no	—
185	#435	—	26	4	no	—
186	#424	—	13	2	no	—
187	#423	C₂×D₁₃	1	1	yes	characteristic
188	#423	C₂×D₁₃	1	1	yes	characteristic
189	C₂×#104	—	13	2	no	—
190	C₂×#104	—	13	2	no	—
191	#416	—	13	2	no	—
192	#416	—	13	2	no	—
193	#408	—	13	2	no	—
194	#405	—	13	2	no	—
195	C₂×#97	—	13	1	no	—
196	C₂×#97	—	13	1	no	—
197	C₂×#97	—	13	1	no	—
198	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
199	C₂×C₄×C₈	—	13	1	no	—
200	C₂×C₄×C₈	—	13	1	no	—
201	C₄³	—	13	1	no	—
202	#341	C₂×D₁₃	1	2	yes	—
203	#341	C₂×D₁₃	1	2	yes	—
204	#339	—	13	2	no	—
205	#339	—	13	2	no	—
206	C₂²×C₁₃	—	2	2	no	—
207	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂²×C₁₃	#529	1	1	yes	characteristic
209	C₂²×C₁₃	#529	1	1	yes	characteristic
210	C₂×D₁₃	—	4	4	no	—
211	C₂×D₁₃	—	4	2	no	—
212	C₄×C₁₃	—	2	2	no	—
213	C₄×C₁₃	—	2	2	no	—
214	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
215	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
216	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
217	C₄×C₁₃	#532	1	1	yes	characteristic
218	Dic₁₃	—	2	4	no	—
219	Dic₁₃	—	4	2	no	—
220	Dic₁₃	—	4	4	no	—
221	Dic₁₃	—	4	2	no	—
222	C₂×#41	—	13	2	no	—
223	#131	—	26	4	no	—
224	#131	—	26	4	no	—
225	C₂×M₁₆	—	26	2	no	—
226	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
227	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
228	C₂²×C₈	—	13	1	no	—
229	C₂²×C₈	—	13	1	no	—
230	C₂²×C₈	—	26	2	no	—
231	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₄⋊Q₈	—	13	2	no	—
234	C₄⋊Q₈	—	13	1	no	—
235	#127	—	13	1	no	—
236	#125	—	13	2	no	—
237	#124	—	13	2	no	—
238	C₂²⋊Q₈	—	26	2	no	—
239	#121	—	26	2	no	—
240	C₄×Q₈	—	26	2	no	—
241	C₄×D₄	—	26	2	no	—
242	#117	—	13	2	no	—
243	C₂×C₄²	—	13	1	no	—
244	C₂×C₄²	—	13	1	no	—
245	C₂×C₄²	—	26	2	no	—
246	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
247	C₄⋊C₈	—	2	2	no	—
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	—	26	4	no	—
250	#104	—	26	4	no	—
251	#104	—	26	4	no	—
252	#103	—	13	4	no	—
253	#103	—	13	4	no	—
254	#102	—	13	4	no	—
255	#102	—	13	4	no	—
256	C₂²⋊C₈	—	26	4	no	—
257	#97	—	13	4	no	—
258	#97	—	26	2	no	—
259	#97	—	26	2	no	—
260	#97	—	26	2	no	—
261	#97	C₄×D₁₃	1	2	yes	—
262	#97	C₄×D₁₃	1	2	yes	—
263	C₄×C₈	—	26	2	no	—
264	C₄×C₈	—	26	2	no	—
265	C₄×C₈	—	26	2	no	—
266	C₂×C₁₃	—	2	1	no	—
267	C₂×C₁₃	—	2	1	no	—
268	C₂×C₁₃	C₄×#124	1	1	yes	characteristic
269	C₂×C₁₃	#2956	1	2	yes	—
270	D₁₃	—	8	2	no	—
271	#41	—	26	2	no	—
272	#41	—	26	2	no	—
273	#41	—	26	2	no	—
274	#41	—	26	2	no	—
275	C₂×Q₈	—	13	2	no	—
276	C₂×Q₈	—	26	2	no	—
277	C₂×D₄	—	13	2	no	—
278	C₂×D₄	—	26	2	no	—
279	C₂×D₄	—	26	2	no	—
280	C₂²×C₄	—	26	1	no	—
281	C₂²×C₄	—	26	1	no	—
282	C₂²×C₄	—	26	2	no	—
283	C₂²×C₄	#6131	1	2	yes	—
284	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	52	4	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	13	2	no	—
297	C₂×C₈	—	13	2	no	—
298	C₂×C₈	—	13	2	no	—
299	C₂×C₈	—	13	2	no	—
300	C₂×C₈	—	26	1	no	—
301	C₂×C₈	—	26	1	no	—
302	C₂×C₈	—	52	4	no	—
303	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
304	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
305	C₄⋊C₄	—	13	4	no	—
306	C₄⋊C₄	—	26	2	no	—
307	C₄⋊C₄	—	26	4	no	—
308	C₄⋊C₄	—	26	2	no	—
309	C₂²⋊C₄	—	26	2	no	—
310	C₂²⋊C₄	—	26	4	no	—
311	C₄²	—	13	2	no	—
312	C₄²	—	13	2	no	—
313	C₄²	—	26	2	no	—
314	C₄²	—	26	1	no	—
315	C₄²	—	26	1	no	—
316	C₄²	—	52	4	no	—
317	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
318	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
319	C₄²	D₄×D₁₃	1	2	yes	—
320	C₁₃	#21662	1	1	yes	characteristic, complemented, Sylow
321	C₂³	—	26	2	no	—
322	C₂³	C₂×#6131	1	1	yes	characteristic
323	Q₈	—	26	2	no	—
324	Q₈	—	26	2	no	—
325	Q₈	—	104	2	no	—
326	D₄	—	26	2	no	—
327	D₄	—	26	2	no	—
328	D₄	—	52	2	no	—
329	D₄	—	52	2	no	—
330	D₄	—	104	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	26	4	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	26	4	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	52	2	no	—
338	C₂×C₄	—	52	2	no	—
339	C₂×C₄	—	52	2	no	—
340	C₂×C₄	#88147	1	2	yes	—
341	C₂×C₄	#88147	1	2	yes	—
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	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
346	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
347	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	13	4	no	—
353	C₈	—	26	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	52	4	no	—
356	C₂²	—	52	2	no	—
357	C₂²	C₂×#88147	1	1	yes	characteristic
358	C₂²	#11719396	1	1	yes	characteristic
359	C₂²	#11719396	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	26	4	no	—
363	C₄	—	52	2	no	—
364	C₄	—	52	4	no	—
365	C₄	—	52	2	no	—
366	C₄	#11719227	1	1	yes	characteristic
367	C₄	#11719227	1	1	yes	characteristic
368	C₄	#11719373	1	1	yes	characteristic
369	C₄	D₁₃×#124	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	104	2	no	—
373	C₂	#13117375	1	1	yes	characteristic
374	C₂	#13123745	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1838	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11719182	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1838	D₁₃
hypocenter	C₁₃	#21662
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1838

Complements

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

N	H ≅ G/N	type
2. #13123773	C₂	semidirect
4. #13115812	C₂	semidirect
6. #13115545	C₂	semidirect
7. #13115538	C₂	semidirect
8. #13114679	C₂	semidirect
9. #13114672	C₂	semidirect
10. #97×Dic₁₃	C₂	semidirect
11. C₁₃×#1838	C₂	semidirect
33. #11718375	C₄	semidirect
34. #11718374	C₄	semidirect
35. #11718364	C₄	semidirect
141. #1838	D₁₃	semidirect
320. C₁₃	#21662	semidirect

Maximal subgroups

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

subgroup	index
#13125182	2
#13123773	2
#13123553	2
#13115812	2
#13115663	2
#13115545	2
#13115538	2
#13114679	2
#13114672	2
#97×Dic₁₃	2
C₁₃×#1838	2
#21662	13