Subgroups — #15833052

Order 3328.

626 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124868	C₂	1	1	yes	maximal, characteristic
2	#13123916	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123911	C₂	1	1	yes	maximal, characteristic
4	#13113716	C₂	1	2	yes	maximal, complemented
5	#13113681	C₂	1	2	yes	maximal, complemented
6	#13113650	C₂	1	2	yes	maximal, complemented
7	C₈×#6095	C₂	1	2	yes	maximal, complemented
8	#13113616	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113284	C₂	1	1	yes	maximal, characteristic, complemented
10	#13113254	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719182	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₁₃×#431	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
17	#11718587	C₂²	1	1	yes	characteristic
18	#11718586	C₂²	1	1	yes	characteristic
19	#11718512	C₄	1	2	yes	—
20	#11718509	C₄	1	2	yes	—
21	#11718497	—	2	2	no	—
22	#11718491	—	2	2	no	—
23	#11718399	C₄	1	2	yes	—
24	#11718396	C₄	1	2	yes	—
25	#11718388	—	2	2	no	—
26	#11718379	—	2	2	no	—
27	C₄×#6095	C₂²	1	2	yes	—
28	C₄×#6095	C₂²	1	2	yes	—
29	#11718349	C₂²	1	2	yes	—
30	C₈×D₅₂	C₂²	1	2	yes	—
31	C₈×Dic₂₆	C₂²	1	2	yes	—
32	C₈×Dic₂₆	C₂²	1	2	yes	—
33	C₁₃×#322	C₂²	1	1	yes	characteristic
34	C₁₃×#322	C₂²	1	1	yes	characteristic
35	C₁₃×#322	C₂²	1	1	yes	characteristic
36	C₈²×C₁₃	C₂²	1	1	yes	characteristic
37	#11718051	C₂²	1	2	yes	—
38	#11718051	C₂²	1	2	yes	—
39	#11718050	C₂²	1	2	yes	—
40	#11718049	C₂²	1	2	yes	—
41	#11718048	C₂²	1	2	yes	—
42	#11718048	C₂²	1	2	yes	—
43	C₂×#6127	—	2	1	no	—
44	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×M₁₆	—	2	1	no	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	#88184	—	2	1	no	—
48	#88184	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×#6101	—	2	1	no	—
51	C₂×C₄×Dic₁₃	—	2	1	no	—
52	C₂×#6095	C₂×C₄	1	2	yes	—
53	C₂×#6095	C₂×C₄	1	2	yes	—
54	C₂×#6094	—	2	2	no	—
55	C₂×C₈×D₁₃	—	2	2	no	—
56	#88129	—	2	1	no	—
57	#88128	C₂×C₄	1	1	yes	characteristic
58	C₄×D₅₂	—	2	1	no	—
59	C₄×D₅₂	C₂³	1	2	yes	—
60	#88123	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₂₆	—	2	1	no	—
62	C₄×Dic₂₆	—	2	1	no	—
63	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₆	C₂³	1	2	yes	—
65	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
66	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
67	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
68	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
69	C₁₃×#97	—	2	1	no	—
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	C₂²⋊#1055	—	2	1	no	—
77	C₂²⋊#1055	—	2	1	no	—
78	#88061	C₂×C₄	1	2	yes	—
79	#88061	C₂×C₄	1	2	yes	—
80	#88060	—	2	2	no	—
81	#88060	—	2	2	no	—
82	#88057	C₂×C₄	1	2	yes	—
83	#88057	C₂×C₄	1	2	yes	—
84	#88056	C₂×C₄	1	2	yes	—
85	#88056	C₂×C₄	1	2	yes	—
86	#88054	—	2	2	no	—
87	#88054	—	2	2	no	—
88	C₈×Dic₁₃	—	2	2	no	—
89	C₈×Dic₁₃	—	2	2	no	—
90	C₄⋊#1055	C₂³	1	2	yes	—
91	C₄⋊#1055	C₂³	1	2	yes	—
92	#88043	—	2	1	no	—
93	C₄×#1055	—	2	1	no	—
94	#17923	—	13	1	no	maximal, Sylow
95	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
98	C₂×C₂²⋊D₁₃	—	2	2	no	—
99	C₂²×Dic₁₃	—	2	1	no	—
100	C₂²×Dic₁₃	—	2	1	no	—
101	#6127	—	8	2	no	—
102	C₂×D₅₂	—	2	1	no	—
103	C₂×D₅₂	—	2	1	no	—
104	C₂×D₅₂	C₂²×C₄	1	2	yes	—
105	C₂×C₄×D₁₃	—	2	2	no	—
106	C₂×Dic₂₆	—	2	1	no	—
107	C₂×Dic₂₆	—	2	4	no	—
108	C₂×Dic₂₆	—	2	1	no	—
109	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
110	C₁₃×M₁₆	—	4	2	no	—
111	C₂×C₈×C₁₃	—	2	1	no	—
112	C₂×C₈×C₁₃	—	2	1	no	—
113	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
117	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₃	#41	1	1	yes	characteristic
120	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
121	#6103	—	2	2	no	—
122	#6103	—	2	2	no	—
123	C₄⋊Dic₁₃	—	2	1	no	—
124	C₄⋊Dic₁₃	—	2	1	no	—
125	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
126	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
127	#6101	—	2	2	no	—
128	#6101	—	2	2	no	—
129	C₄×Dic₁₃	—	2	2	no	—
130	C₄×Dic₁₃	—	2	2	no	—
131	C₂×#1055	—	2	2	no	—
132	C₂×#1055	—	2	2	no	—
133	#6095	—	2	4	no	—
134	#6095	—	2	4	no	—
135	#6094	—	8	2	no	—
136	C₈×D₁₃	—	8	2	no	—
137	#3103	—	13	1	no	—
138	#2982	—	13	1	no	—
139	#2981	—	13	1	no	—
140	#1588	—	13	2	no	—
141	#1583	—	13	2	no	—
142	SD₁₆⋊C₈	—	13	2	no	—
143	C₈×SD₁₆	—	13	2	no	—
144	#1572	—	13	1	no	—
145	#1491	—	13	1	no	—
146	#1484	—	13	1	no	—
147	#1453	D₁₃	1	1	yes	characteristic, complemented
148	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
149	C₂²×D₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₂²⋊D₁₃	—	8	2	no	—
160	C₂×Dic₁₃	—	2	4	no	—
161	C₂×Dic₁₃	—	2	2	no	—
162	C₂×Dic₁₃	—	2	2	no	—
163	C₂×Dic₁₃	—	2	2	no	—
164	C₂×Dic₁₃	—	8	1	no	—
165	D₅₂	—	2	2	no	—
166	D₅₂	—	2	2	no	—
167	D₅₂	—	8	1	no	—
168	D₅₂	—	8	1	no	—
169	C₄×D₁₃	—	4	2	no	—
170	C₄×D₁₃	—	4	2	no	—
171	Dic₂₆	—	2	4	no	—
172	Dic₂₆	—	2	2	no	—
173	Dic₂₆	—	8	1	no	—
174	Dic₂₆	#131	1	4	yes	—
175	C₈×C₁₃	—	2	2	no	—
176	C₈×C₁₃	—	2	2	no	—
177	C₈×C₁₃	—	2	2	no	—
178	C₈×C₁₃	—	2	2	no	—
179	C₈×C₁₃	—	2	2	no	—
180	C₈×C₁₃	—	4	1	no	—
181	#1055	—	4	2	no	—
182	#1055	—	4	2	no	—
183	#532	—	13	1	no	—
184	C₂×C₄×Q₈	—	13	1	no	—
185	#481	—	13	2	no	—
186	#475	—	13	2	no	—
187	#460	—	13	2	no	—
188	#448	—	13	2	no	—
189	C₈×Q₈	—	13	2	no	—
190	C₈×Q₈	—	13	2	no	—
191	C₈×Q₈	—	26	2	no	—
192	C₄×SD₁₆	—	13	2	no	—
193	C₄×SD₁₆	—	13	2	no	—
194	#436	—	13	2	no	—
195	#435	—	26	2	no	—
196	C₈×D₄	—	13	2	no	—
197	#432	—	13	1	no	—
198	#431	—	13	1	no	—
199	#431	—	26	2	no	—
200	#431	C₂×D₁₃	1	1	yes	characteristic
201	#431	C₂×D₁₃	1	1	yes	characteristic
202	#408	—	26	2	no	—
203	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
204	#335	—	13	2	no	—
205	#334	—	13	2	no	—
206	Q₈⋊C₈	—	13	2	no	—
207	Q₈⋊C₈	—	13	2	no	—
208	D₄⋊C₈	—	13	2	no	—
209	D₄⋊C₈	—	13	2	no	—
210	#322	C₂×D₁₃	1	1	yes	characteristic
211	#322	C₂×D₁₃	1	1	yes	characteristic
212	#322	C₂×D₁₃	1	1	yes	characteristic
213	C₈²	C₂×D₁₃	1	1	yes	characteristic
214	C₂²×C₁₃	—	2	1	no	—
215	C₂²×C₁₃	—	2	1	no	—
216	C₂²×C₁₃	—	2	1	no	—
217	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂×D₁₃	—	4	2	no	—
219	C₂×D₁₃	—	4	2	no	—
220	C₂×D₁₃	—	4	2	no	—
221	C₄×C₁₃	—	2	1	no	—
222	C₄×C₁₃	—	2	1	no	—
223	C₄×C₁₃	—	2	1	no	—
224	C₄×C₁₃	—	2	1	no	—
225	C₄×C₁₃	C₂×#131	1	2	yes	—
226	C₄×C₁₃	#577	1	2	yes	—
227	Dic₁₃	—	2	4	no	—
228	Dic₁₃	—	4	2	no	—
229	Dic₁₃	—	4	2	no	—
230	C₂×#41	—	26	1	no	—
231	C₂²×Q₈	—	13	1	no	—
232	C₂×SD₁₆	—	13	2	no	—
233	C₂×SD₁₆	—	13	2	no	—
234	C₂×M₁₆	—	2	1	no	—
235	C₂×M₁₆	—	26	2	no	—
236	C₂²×C₈	—	26	2	no	—
237	C₄⋊Q₈	—	13	1	no	—
238	#127	—	13	1	no	—
239	#124	—	26	1	no	—
240	#121	—	13	2	no	—
241	#121	—	26	1	no	—
242	C₄×Q₈	—	13	2	no	—
243	C₄×Q₈	—	13	2	no	—
244	C₄×Q₈	—	26	1	no	—
245	C₄×Q₈	—	26	1	no	—
246	C₄×D₄	—	13	2	no	—
247	C₄×D₄	—	26	1	no	—
248	C₂×C₄⋊C₄	—	13	1	no	—
249	C₂×C₄⋊C₄	—	26	1	no	—
250	C₂×C₄²	—	26	1	no	—
251	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
252	#106	—	13	2	no	—
253	#106	—	13	2	no	—
254	C₄⋊C₈	—	13	2	no	—
255	C₄⋊C₈	—	13	2	no	—
256	C₄⋊C₈	—	26	2	no	—
257	C₄⋊C₈	—	26	2	no	—
258	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
259	C₄⋊C₈	D₅₂	1	2	yes	—
260	#103	—	13	2	no	—
261	#103	—	13	2	no	—
262	#102	—	13	2	no	—
263	#102	—	13	2	no	—
264	C₂²⋊C₈	—	26	1	no	—
265	C₂²⋊C₈	—	26	1	no	—
266	C₂²⋊C₈	—	26	2	no	—
267	C₂²⋊C₈	—	26	2	no	—
268	C₂²⋊C₈	C₄×D₁₃	1	2	yes	—
269	C₂²⋊C₈	D₅₂	1	2	yes	—
270	#97	—	2	1	no	—
271	#97	—	26	1	no	—
272	C₄×C₈	—	26	2	no	—
273	C₄×C₈	—	26	2	no	—
274	C₄×C₈	—	26	1	no	—
275	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₂×C₁₃	—	4	1	no	—
282	C₂×C₁₃	#2929	1	1	yes	characteristic
283	C₂×C₁₃	#2949	1	1	yes	characteristic
284	C₂×C₁₃	#2956	1	1	yes	characteristic
285	D₁₃	—	8	2	no	—
286	#41	—	104	2	no	—
287	C₂×Q₈	—	13	2	no	—
288	C₂×Q₈	—	26	4	no	—
289	C₂×Q₈	—	26	1	no	—
290	C₂×Q₈	—	26	1	no	—
291	C₂×D₄	—	13	2	no	—
292	C₂×D₄	—	26	2	no	—
293	C₂×D₄	—	26	1	no	—
294	C₂×D₄	—	26	1	no	—
295	C₂²×C₄	—	26	1	no	—
296	C₂²×C₄	—	26	1	no	—
297	C₂²×C₄	—	26	2	no	—
298	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
300	C₂²×C₄	#6127	1	1	yes	characteristic
301	SD₁₆	—	26	4	no	—
302	SD₁₆	—	26	4	no	—
303	M₁₆	—	4	2	no	—
304	M₁₆	—	104	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	26	2	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	—	104	2	no	—
310	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
311	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
312	C₂×C₈	C₂×D₅₂	1	2	yes	—
313	C₂×C₈	C₂×D₅₂	1	2	yes	—
314	C₄⋊C₄	—	13	2	no	—
315	C₄⋊C₄	—	13	2	no	—
316	C₄⋊C₄	—	26	1	no	—
317	C₄⋊C₄	—	26	2	no	—
318	C₄⋊C₄	—	26	1	no	—
319	C₄⋊C₄	—	26	2	no	—
320	C₂²⋊C₄	—	26	2	no	—
321	C₂²⋊C₄	—	26	2	no	—
322	C₄²	—	26	2	no	—
323	C₄²	—	26	2	no	—
324	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
325	C₄²	C₂×D₅₂	1	1	yes	characteristic
326	C₄²	#6127	1	1	yes	characteristic
327	C₄²	C₂³×D₁₃	1	1	yes	characteristic
328	C₁₃	#17923	1	1	yes	characteristic, complemented, Sylow
329	C₂³	—	26	2	no	—
330	C₂³	C₄×D₅₂	1	1	yes	characteristic
331	Q₈	—	13	4	no	—
332	Q₈	—	26	4	no	—
333	Q₈	—	26	2	no	—
334	Q₈	—	104	1	no	—
335	D₄	—	26	2	no	—
336	D₄	—	26	2	no	—
337	D₄	—	104	1	no	—
338	D₄	—	104	1	no	—
339	D₄	—	104	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	26	4	no	—
344	C₂×C₄	—	26	2	no	—
345	C₂×C₄	—	26	2	no	—
346	C₂×C₄	—	26	2	no	—
347	C₂×C₄	—	52	2	no	—
348	C₂×C₄	—	52	2	no	—
349	C₂×C₄	—	104	1	no	—
350	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
351	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
352	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
355	C₂×C₄	C₂×#6127	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	4	1	no	—
362	C₈	—	52	2	no	—
363	C₈	—	52	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	52	2	no	—
368	C₂²	—	52	2	no	—
369	C₂²	—	52	2	no	—
370	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	26	4	no	—
376	C₄	—	52	2	no	—
377	C₄	—	52	2	no	—
378	C₄	D₁₃×#131	1	2	yes	—
379	C₄	#11719456	1	2	yes	—
380	C₂	—	4	1	no	—
381	C₂	—	104	2	no	—
382	C₂	#13123532	1	1	yes	characteristic
383	C₂	#13123697	1	1	yes	characteristic
384	C₂	#13123747	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1453	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11719182	C₂²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1453	D₁₃
hypocenter	C₁₃	#17923
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1453

Complements

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

N	H ≅ G/N	type
2. #13123916	C₂	semidirect
4. #13113716	C₂	semidirect
5. #13113681	C₂	semidirect
6. #13113650	C₂	semidirect
7. C₈×#6095	C₂	semidirect
8. #13113616	C₂	semidirect
9. #13113284	C₂	semidirect
11. C₁₃×#1453	C₂	semidirect
147. #1453	D₁₃	semidirect
328. C₁₃	#17923	semidirect

Maximal subgroups

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

subgroup	index
#13124868	2
#13123916	2
#13123911	2
#13113716	2
#13113681	2
#13113650	2
C₈×#6095	2
#13113616	2
#13113284	2
#13113254	2
C₁₃×#1453	2
#17923	13