Subgroups — #15833598

Order 3328.

560 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	#13125419	C₂	1	1	yes	maximal, characteristic
2	#13123932	C₂	1	2	yes	maximal
3	#13113743	C₂	1	2	yes	maximal, complemented
4	#13113734	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113652	C₂	1	2	yes	maximal, complemented
6	Q₁₆×#1055	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113642	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113598	C₂	1	1	yes	maximal, characteristic
9	#13113307	C₂	1	2	yes	maximal
10	#13113118	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719283	C₂²	1	2	yes	—
13	C₁₃×#446	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	Q₈×#1055	C₂²	1	2	yes	—
19	#11718548	C₄	1	1	yes	characteristic
20	#11718535	C₄	1	1	yes	characteristic
21	#11718501	—	2	2	no	—
22	#11718496	C₂²	1	2	yes	—
23	#11718492	—	2	2	no	—
24	#11718491	C₂²	1	2	yes	—
25	#11718474	C₄	1	2	yes	—
26	#11718453	C₄	1	2	yes	—
27	#11718370	—	2	2	no	—
28	#11718362	—	2	2	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	#11718346	C₂²	1	1	yes	characteristic
31	#11718346	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	#11718340	C₂²	1	2	yes	—
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	#11718046	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×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
53	#88151	C₂×C₄	1	2	yes	—
54	#88150	—	2	2	no	—
55	#88143	—	2	2	no	—
56	#88142	C₂×C₄	1	2	yes	—
57	#88126	—	2	2	no	—
58	C₄²×D₁₃	C₂³	1	1	yes	characteristic
59	C₄×Dic₂₆	—	2	2	no	—
60	C₄×Dic₂₆	C₂³	1	2	yes	—
61	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
62	C₁₃×C₄⋊C₈	—	2	2	no	—
63	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
64	C₁₃×#103	C₂×C₄	1	2	yes	—
65	C₁₃×#97	—	2	2	no	—
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	#88075	C₂×C₄	1	2	yes	—
70	#88060	—	2	2	no	—
71	#88060	—	2	1	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	2	no	—
78	#88054	—	2	2	no	—
79	#88054	—	2	1	no	—
80	#88054	—	2	2	no	—
81	#88054	D₄	1	1	yes	characteristic
82	#88054	D₄	1	1	yes	characteristic
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	C₄×#1055	—	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	#17972	—	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	2	no	—
103	C₂×C₈×C₁₃	—	2	1	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	—	2	4	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	2	no	—
127	C₂×#1055	—	2	1	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	C₈×D₁₃	—	2	2	no	—
134	C₈×D₁₃	—	2	2	no	—
135	#3170	—	13	1	no	—
136	#2985	—	13	2	no	—
137	#1593	—	13	2	no	—
138	#1591	—	13	1	no	—
139	#1591	D₁₃	1	1	yes	characteristic, complemented
140	#1579	—	13	2	no	—
141	C₈×Q₁₆	—	13	1	no	—
142	C₈×Q₁₆	—	13	1	no	—
143	#1568	—	13	1	no	—
144	#1497	—	13	2	no	—
145	#1450	—	13	1	no	—
146	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
147	Q₈×C₁₃	—	2	2	no	—
148	Q₈×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
154	C₂×Dic₁₃	—	2	2	no	—
155	C₂×Dic₁₃	—	2	2	no	—
156	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
157	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
159	C₄×D₁₃	—	2	1	no	—
160	C₄×D₁₃	—	2	1	no	—
161	C₄×D₁₃	—	2	1	no	—
162	C₄×D₁₃	—	2	1	no	—
163	C₄×D₁₃	—	4	1	no	—
164	Dic₂₆	—	2	2	no	—
165	Dic₂₆	—	2	2	no	—
166	Dic₂₆	—	8	2	no	—
167	C₈×C₁₃	—	2	1	no	—
168	C₈×C₁₃	—	2	1	no	—
169	C₈×C₁₃	—	2	1	no	—
170	C₈×C₁₃	—	4	1	no	—
171	C₈×C₁₃	—	4	2	no	—
172	C₈×C₁₃	#131	1	2	yes	—
173	#1055	—	2	1	no	—
174	#1055	—	2	1	no	—
175	#1055	—	2	1	no	—
176	#1055	—	4	1	no	—
177	#1055	—	4	2	no	—
178	#1055	#131	1	2	yes	—
179	#533	—	13	2	no	—
180	#500	—	13	1	no	—
181	#494	—	13	1	no	—
182	#487	—	13	2	no	—
183	#467	—	13	1	no	—
184	#467	—	13	1	no	—
185	#467	—	13	2	no	—
186	#446	—	13	2	no	—
187	#446	—	26	2	no	—
188	#446	—	26	2	no	—
189	#446	C₂×D₁₃	1	2	yes	—
190	C₈×Q₈	—	13	2	no	—
191	C₈×Q₈	—	13	2	no	—
192	C₄×Q₁₆	—	13	2	no	—
193	C₄×Q₁₆	—	13	1	no	—
194	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
195	#433	—	26	2	no	—
196	#433	—	26	2	no	—
197	#432	—	13	2	no	—
198	#431	—	13	1	no	—
199	#431	—	13	1	no	—
200	C₂×C₄×C₈	—	13	1	no	—
201	#335	—	13	1	no	—
202	#335	C₂×D₁₃	1	1	yes	characteristic
203	#334	—	13	2	no	—
204	Q₈⋊C₈	—	13	2	no	—
205	Q₈⋊C₈	—	13	2	no	—
206	Q₈⋊C₈	—	13	2	no	—
207	Q₈⋊C₈	C₂×D₁₃	1	2	yes	—
208	#322	—	13	1	no	—
209	#322	—	13	1	no	—
210	C₈²	—	13	1	no	—
211	C₈²	C₂×D₁₃	1	1	yes	characteristic
212	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂×D₁₃	—	2	1	no	—
214	C₂×D₁₃	—	2	1	no	—
215	C₂×D₁₃	—	2	1	no	—
216	C₄×C₁₃	—	2	1	no	—
217	C₄×C₁₃	—	4	2	no	—
218	C₄×C₁₃	—	4	2	no	—
219	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
220	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
221	C₄×C₁₃	#578	1	1	yes	characteristic
222	C₄×C₁₃	#578	1	1	yes	characteristic
223	Dic₁₃	—	2	1	no	—
224	Dic₁₃	—	2	1	no	—
225	Dic₁₃	—	2	1	no	—
226	Dic₁₃	—	4	2	no	—
227	Dic₁₃	—	4	2	no	—
228	C₂×Q₁₆	—	13	2	no	—
229	C₂×Q₁₆	—	13	1	no	—
230	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
231	C₂²×C₈	—	13	1	no	—
232	C₂²×C₈	—	13	1	no	—
233	C₄⋊Q₈	—	13	2	no	—
234	C₄⋊Q₈	—	13	2	no	—
235	#125	—	26	2	no	—
236	C₂²⋊Q₈	—	13	2	no	—
237	C₂²⋊Q₈	—	13	2	no	—
238	C₂²⋊Q₈	—	26	2	no	—
239	C₄×Q₈	—	13	2	no	—
240	C₄×Q₈	—	26	2	no	—
241	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
242	#117	—	26	2	no	—
243	C₂×C₄²	—	13	1	no	—
244	#107	—	13	1	no	—
245	#107	—	13	2	no	—
246	#107	C₄×D₁₃	1	1	yes	characteristic
247	C₄⋊C₈	—	2	2	no	—
248	C₄⋊C₈	—	13	1	no	—
249	C₄⋊C₈	—	13	1	no	—
250	C₄⋊C₈	—	13	2	no	—
251	C₄⋊C₈	—	26	2	no	—
252	C₄⋊C₈	—	26	1	no	—
253	C₄⋊C₈	—	26	2	no	—
254	C₄⋊C₈	—	26	2	no	—
255	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
256	#103	—	13	2	no	—
257	#103	—	13	2	no	—
258	#103	—	13	2	no	—
259	#103	C₄×D₁₃	1	2	yes	—
260	C₂²⋊C₈	—	13	1	no	—
261	C₂²⋊C₈	—	13	1	no	—
262	C₂²⋊C₈	—	26	1	no	—
263	C₂²⋊C₈	—	26	2	no	—
264	#97	—	2	2	no	—
265	#97	—	26	2	no	—
266	#97	—	26	2	no	—
267	C₄×C₈	—	13	1	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	—	26	2	no	—
273	C₄×C₈	—	26	2	no	—
274	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
277	C₂×C₁₃	#2931	1	1	yes	characteristic
278	C₂×C₁₃	#2950	1	1	yes	characteristic
279	C₂×C₁₃	#2957	1	1	yes	characteristic
280	D₁₃	—	4	1	no	—
281	C₂×Q₈	—	13	2	no	—
282	C₂×Q₈	—	26	2	no	—
283	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
284	C₂²×C₄	—	13	1	no	—
285	C₂²×C₄	—	13	1	no	—
286	C₂²×C₄	—	13	1	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	26	4	no	—
289	Q₁₆	—	26	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	13	1	no	—
293	C₂×C₈	—	13	1	no	—
294	C₂×C₈	—	13	1	no	—
295	C₂×C₈	—	13	1	no	—
296	C₂×C₈	—	26	2	no	—
297	C₂×C₈	—	26	2	no	—
298	C₂×C₈	—	26	2	no	—
299	C₂×C₈	—	26	1	no	—
300	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
302	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
303	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	13	2	no	—
306	C₄⋊C₄	—	26	2	no	—
307	C₄⋊C₄	—	26	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	C₂×C₄×D₁₃	1	2	yes	—
313	C₂²⋊C₄	—	26	2	no	—
314	C₂²⋊C₄	—	26	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	13	1	no	—
317	C₄²	—	13	1	no	—
318	C₄²	—	13	1	no	—
319	C₄²	—	26	2	no	—
320	C₄²	C₂³×D₁₃	1	1	yes	characteristic
321	C₁₃	#17972	1	1	yes	characteristic, complemented, Sylow
322	C₂³	—	13	1	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	26	2	no	—
326	Q₈	—	26	2	no	—
327	Q₈	—	104	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	13	1	no	—
331	C₂×C₄	—	13	1	no	—
332	C₂×C₄	—	13	1	no	—
333	C₂×C₄	—	26	1	no	—
334	C₂×C₄	—	26	1	no	—
335	C₂×C₄	—	26	1	no	—
336	C₂×C₄	—	26	1	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	26	2	no	—
339	C₂×C₄	—	52	1	no	—
340	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
342	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	13	2	no	—
349	C₈	—	26	1	no	—
350	C₈	—	26	1	no	—
351	C₈	—	26	1	no	—
352	C₈	—	52	1	no	—
353	C₈	—	52	2	no	—
354	C₈	#88161	1	2	yes	—
355	C₂²	—	26	1	no	—
356	C₂²	—	26	1	no	—
357	C₂²	—	26	1	no	—
358	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	26	1	no	—
363	C₄	—	26	1	no	—
364	C₄	—	26	1	no	—
365	C₄	—	52	2	no	—
366	C₄	—	52	2	no	—
367	C₄	C₂×#88161	1	1	yes	characteristic
368	C₄	D₁₃×#131	1	1	yes	characteristic
369	C₄	#11719471	1	1	yes	characteristic
370	C₄	#11719476	1	1	yes	characteristic
371	C₂	—	52	1	no	—
372	C₂	#13123545	1	1	yes	characteristic
373	C₂	#13123699	1	1	yes	characteristic
374	C₂	#13123748	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1591	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1591	D₁₃
hypocenter	C₁₃	#17972
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #13113743	C₂	semidirect
4. #13113734	C₂	semidirect
5. #13113652	C₂	semidirect
6. Q₁₆×#1055	C₂	semidirect
7. #13113642	C₂	semidirect
11. C₁₃×#1591	C₂	semidirect
139. #1591	D₁₃	semidirect
321. C₁₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#13125419	2
#13123932	2
#13113743	2
#13113734	2
#13113652	2
Q₁₆×#1055	2
#13113642	2
#13113598	2
#13113307	2
#13113118	2
C₁₃×#1591	2
#17972	13