Subgroups — #15846069

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125959	C₂	1	1	yes	maximal, characteristic
2	#13125763	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718342	C₂	1	1	yes	maximal, characteristic, complemented
4	#13123549	C₂	1	1	yes	maximal, characteristic
5	#13114384	C₂	1	4	yes	maximal, complemented
6	#13114260	C₂	1	4	yes	maximal, complemented
7	#13113520	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113507	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88155	—	2	1	no	—
11	#11719182	C₂²	1	1	yes	characteristic
12	C₂×#88123	C₂²	1	1	yes	characteristic
13	C₂×#88123	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₁₃×#431	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718771	—	2	1	no	—
18	C₂×#88056	—	2	1	no	—
19	C₂×#88056	—	2	1	no	—
20	#11718585	C₂²	1	1	yes	characteristic
21	#11718507	C₂²	1	4	yes	—
22	#11718502	C₂²	1	4	yes	—
23	#11718399	—	2	2	no	—
24	#11718397	—	2	2	no	—
25	#11718355	—	2	1	no	—
26	#11718354	—	2	1	no	—
27	#11718349	—	2	1	no	—
28	#11718349	C₂²	1	2	yes	—
29	C₈×D₅₂	—	2	1	no	—
30	C₈×D₅₂	C₂²	1	2	yes	—
31	#11718343	—	2	1	no	—
32	#11718342	—	2	1	no	—
33	#11718342	—	2	1	no	—
34	#11718342	—	2	1	no	—
35	#11718342	C₂²	1	2	yes	—
36	#11718342	C₂²	1	2	yes	—
37	C₁₃×#335	C₂²	1	2	yes	—
38	C₁₃×#334	C₂²	1	2	yes	—
39	#11718078	C₂²	1	4	yes	—
40	#11718074	C₂²	1	4	yes	—
41	C₂×#6127	—	2	1	no	—
42	C₂²×Dic₂₆	—	2	1	no	—
43	C₂²×Dic₂₆	—	2	1	no	—
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	#88184	—	2	1	no	—
48	#88184	D₄	1	2	yes	—
49	C₂×C₄⋊Dic₁₃	—	2	1	no	—
50	C₂×C₄⋊Dic₁₃	—	2	1	no	—
51	C₂×C₄⋊Dic₁₃	—	2	1	no	—
52	C₂×C₄⋊Dic₁₃	—	2	1	no	—
53	C₂×#6099	—	2	1	no	—
54	C₂×#6095	—	2	4	no	—
55	#88155	—	4	4	no	—
56	C₂×#6094	—	2	2	no	—
57	C₂×C₈×D₁₃	—	2	2	no	—
58	#88129	—	2	1	no	—
59	#88128	D₄	1	1	yes	characteristic
60	C₄×D₅₂	—	2	1	no	—
61	C₄×D₅₂	C₂³	1	2	yes	—
62	#88123	—	2	1	no	—
63	#88123	—	2	1	no	—
64	#88123	—	2	1	no	—
65	#88123	—	2	1	no	—
66	#88123	—	2	1	no	—
67	#88123	—	2	1	no	—
68	#88123	D₄	1	1	yes	characteristic
69	#88123	C₂³	1	2	yes	—
70	#88123	C₂³	1	2	yes	—
71	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
72	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
73	C₁₃×C₂²⋊C₈	—	2	1	no	—
74	C₁₃×C₂²⋊C₈	—	2	1	no	—
75	C₄×C₈×C₁₃	—	2	1	no	—
76	C₄×C₈×C₁₃	—	2	1	no	—
77	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
79	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	#88061	—	2	4	no	—
82	#88060	—	2	2	no	—
83	#88060	—	2	2	no	—
84	#88058	—	2	4	no	—
85	#88056	—	2	2	no	—
86	#88056	—	2	2	no	—
87	#88056	—	2	2	no	—
88	#88056	—	2	2	no	—
89	#88056	—	2	4	no	—
90	C₄⋊#1055	—	2	1	no	—
91	C₄⋊#1055	C₂³	1	2	yes	—
92	#19197	—	13	1	no	maximal, Sylow
93	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
96	C₂×C₂²⋊D₁₃	—	2	2	no	—
97	C₂²×Dic₁₃	—	4	1	no	—
98	C₂²×Dic₁₃	—	4	1	no	—
99	#6127	—	4	2	no	—
100	#6127	—	4	2	no	—
101	C₂×D₅₂	—	2	1	no	—
102	C₂×D₅₂	—	2	1	no	—
103	C₂×D₅₂	C₂×D₄	1	2	yes	—
104	C₂×C₄×D₁₃	—	2	2	no	—
105	C₂×Dic₂₆	—	2	1	no	—
106	C₂×Dic₂₆	—	2	2	no	—
107	C₂×Dic₂₆	—	2	2	no	—
108	C₂×Dic₂₆	—	4	2	no	—
109	C₂×Dic₂₆	—	4	2	no	—
110	C₂×Dic₂₆	—	4	1	no	—
111	C₂×Dic₂₆	—	4	1	no	—
112	C₂×Dic₂₆	—	8	2	no	—
113	C₂×C₈×C₁₃	—	2	2	no	—
114	C₂×C₈×C₁₃	—	2	2	no	—
115	C₂×C₈×C₁₃	—	2	2	no	—
116	C₂×C₈×C₁₃	—	2	2	no	—
117	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₁₃	#41	1	1	yes	characteristic
124	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
125	#6103	—	2	2	no	—
126	#6103	—	2	2	no	—
127	C₄⋊Dic₁₃	—	2	1	no	—
128	C₄⋊Dic₁₃	—	2	2	no	—
129	C₄⋊Dic₁₃	—	2	2	no	—
130	C₄⋊Dic₁₃	—	2	2	no	—
131	C₄⋊Dic₁₃	—	2	2	no	—
132	C₄⋊Dic₁₃	—	4	1	no	—
133	C₄⋊Dic₁₃	—	4	1	no	—
134	C₄⋊Dic₁₃	—	4	1	no	—
135	C₄⋊Dic₁₃	—	4	1	no	—
136	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
137	#6099	—	4	2	no	—
138	C₂×#1055	—	2	2	no	—
139	#6095	—	8	4	no	—
140	#6094	—	4	4	no	—
141	C₈×D₁₃	—	4	4	no	—
142	#3226	—	13	1	no	—
143	#3206	—	13	1	no	—
144	C₂×#487	—	13	1	no	—
145	#2931	—	13	1	no	—
146	#1699	—	13	4	no	—
147	#1679	—	13	4	no	—
148	#1564	D₁₃	1	1	yes	characteristic, complemented
149	#1549	—	13	1	no	—
150	#1546	—	13	1	no	—
151	C₂³×C₁₃	#121	1	1	yes	characteristic
152	C₂²×D₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
156	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
157	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
158	C₂×C₄×C₁₃	#135	1	2	yes	—
159	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
160	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
163	C₂²⋊D₁₃	—	4	4	no	—
164	C₂×Dic₁₃	—	2	2	no	—
165	C₂×Dic₁₃	—	4	2	no	—
166	C₂×Dic₁₃	—	4	2	no	—
167	C₂×Dic₁₃	—	8	2	no	—
168	C₂×Dic₁₃	—	8	2	no	—
169	D₅₂	—	2	4	no	—
170	D₅₂	—	4	2	no	—
171	D₅₂	—	4	2	no	—
172	C₄×D₁₃	—	4	2	no	—
173	C₄×D₁₃	—	4	2	no	—
174	Dic₂₆	—	4	2	no	—
175	Dic₂₆	—	4	2	no	—
176	Dic₂₆	—	4	2	no	—
177	Dic₂₆	—	4	4	no	—
178	Dic₂₆	—	8	2	no	—
179	Dic₂₆	—	8	2	no	—
180	C₈×C₁₃	—	2	1	no	—
181	C₈×C₁₃	—	2	1	no	—
182	C₈×C₁₃	—	2	1	no	—
183	C₈×C₁₃	—	2	1	no	—
184	C₈×C₁₃	—	4	2	no	—
185	C₈×C₁₃	—	4	2	no	—
186	#1055	—	4	2	no	—
187	C₂×#131	—	26	1	no	—
188	#532	—	13	1	no	—
189	C₂×C₄⋊Q₈	—	13	1	no	—
190	C₂×C₄⋊Q₈	—	13	1	no	—
191	#500	—	26	1	no	—
192	#492	—	26	1	no	—
193	#487	—	13	2	no	—
194	#487	—	13	2	no	—
195	#487	—	26	1	no	—
196	#487	—	26	1	no	—
197	#487	—	26	1	no	—
198	#486	—	26	1	no	—
199	#480	—	26	2	no	—
200	#474	—	13	4	no	—
201	#461	—	13	4	no	—
202	#448	—	26	2	no	—
203	#436	—	13	2	no	—
204	#436	—	26	1	no	—
205	C₈×D₄	—	13	2	no	—
206	C₈×D₄	—	26	1	no	—
207	#431	C₂×D₁₃	1	1	yes	characteristic
208	#431	C₂×D₁₃	1	1	yes	characteristic
209	#423	—	13	1	no	—
210	C₂×#103	—	26	1	no	—
211	C₂×#103	—	26	1	no	—
212	#408	—	26	1	no	—
213	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
214	#335	C₂×D₁₃	1	2	yes	—
215	#334	C₂×D₁₃	1	2	yes	—
216	#332	—	13	4	no	—
217	#332	—	13	4	no	—
218	C₂²×C₁₃	—	2	1	no	—
219	C₂²×C₁₃	—	2	1	no	—
220	C₂²×C₁₃	#463	1	2	yes	—
221	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
222	C₂×D₁₃	—	4	2	no	—
223	C₂×D₁₃	—	4	2	no	—
224	C₂×D₁₃	—	4	2	no	—
225	C₄×C₁₃	—	2	1	no	—
226	C₄×C₁₃	—	2	1	no	—
227	C₄×C₁₃	#463	1	2	yes	—
228	C₄×C₁₃	#473	1	2	yes	—
229	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
230	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₄×C₁₃	#577	1	2	yes	—
232	Dic₁₃	—	4	2	no	—
233	Dic₁₃	—	8	2	no	—
234	Dic₁₃	—	8	2	no	—
235	C₂×#41	—	26	1	no	—
236	C₂²×Q₈	—	26	1	no	—
237	C₂²×Q₈	—	26	1	no	—
238	C₂×SD₁₆	—	26	4	no	—
239	#131	—	52	4	no	—
240	C₂×M₁₆	—	26	1	no	—
241	C₂×M₁₆	—	26	2	no	—
242	C₂²×C₈	—	26	2	no	—
243	C₂²×C₈	D₅₂	1	1	yes	characteristic
244	C₂²×C₈	D₅₂	1	1	yes	characteristic
245	C₄⋊Q₈	—	13	1	no	—
246	C₄⋊Q₈	—	13	2	no	—
247	C₄⋊Q₈	—	13	2	no	—
248	C₄⋊Q₈	—	26	1	no	—
249	C₄⋊Q₈	—	26	1	no	—
250	C₄⋊Q₈	—	26	1	no	—
251	C₄⋊Q₈	—	26	1	no	—
252	C₄⋊Q₈	—	26	1	no	—
253	C₄⋊Q₈	—	26	1	no	—
254	#127	—	13	1	no	—
255	#124	—	26	1	no	—
256	#121	—	13	2	no	—
257	#121	—	26	1	no	—
258	C₄×D₄	—	13	2	no	—
259	C₄×D₄	—	26	1	no	—
260	C₂×C₄⋊C₄	—	26	1	no	—
261	C₂×C₄⋊C₄	—	26	1	no	—
262	C₂×C₄⋊C₄	—	26	1	no	—
263	C₂×C₄⋊C₄	—	26	1	no	—
264	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
265	#107	—	26	4	no	—
266	C₄⋊C₈	—	13	2	no	—
267	C₄⋊C₈	—	26	1	no	—
268	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
269	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
270	#103	—	26	4	no	—
271	#103	—	26	2	no	—
272	#103	—	26	2	no	—
273	#103	—	26	2	no	—
274	#103	—	26	2	no	—
275	#102	—	26	4	no	—
276	C₂²⋊C₈	—	2	1	no	—
277	C₂²⋊C₈	—	2	1	no	—
278	C₂²⋊C₈	—	26	2	no	—
279	C₂²⋊C₈	—	26	2	no	—
280	C₄×C₈	—	2	1	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	D₅₂	1	1	yes	characteristic
283	C₄×C₈	D₅₂	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
286	C₂×C₁₃	—	2	2	no	—
287	C₂×C₁₃	C₂×#463	1	1	yes	characteristic
288	C₂×C₁₃	#3057	1	1	yes	characteristic
289	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
290	D₁₃	—	8	2	no	—
291	#41	—	52	2	no	—
292	#41	—	52	2	no	—
293	C₂×Q₈	—	26	1	no	—
294	C₂×Q₈	—	26	2	no	—
295	C₂×Q₈	—	26	2	no	—
296	C₂×Q₈	—	52	1	no	—
297	C₂×Q₈	—	52	1	no	—
298	C₂×Q₈	—	52	2	no	—
299	C₂×Q₈	—	52	2	no	—
300	C₂×Q₈	—	104	2	no	—
301	C₂×D₄	—	13	2	no	—
302	C₂×D₄	—	26	2	no	—
303	C₂×D₄	—	26	1	no	—
304	C₂×D₄	—	26	1	no	—
305	C₂²×C₄	—	26	2	no	—
306	C₂²×C₄	—	52	1	no	—
307	C₂²×C₄	—	52	1	no	—
308	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
309	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
310	C₂²×C₄	#6131	1	1	yes	characteristic
311	SD₁₆	—	104	4	no	—
312	M₁₆	—	52	4	no	—
313	M₁₆	—	52	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	26	2	no	—
319	C₂×C₈	—	52	4	no	—
320	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
321	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
322	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
323	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
324	C₄⋊C₄	—	13	2	no	—
325	C₄⋊C₄	—	26	1	no	—
326	C₄⋊C₄	—	26	2	no	—
327	C₄⋊C₄	—	26	2	no	—
328	C₄⋊C₄	—	26	2	no	—
329	C₄⋊C₄	—	26	2	no	—
330	C₄⋊C₄	—	52	1	no	—
331	C₄⋊C₄	—	52	1	no	—
332	C₄⋊C₄	—	52	1	no	—
333	C₄⋊C₄	—	52	1	no	—
334	C₂²⋊C₄	—	26	2	no	—
335	C₂²⋊C₄	—	26	2	no	—
336	C₄²	C₂×D₅₂	1	1	yes	characteristic
337	C₄²	D₄×D₁₃	1	1	yes	characteristic
338	C₄²	#6131	1	1	yes	characteristic
339	C₄²	C₂³×D₁₃	1	1	yes	characteristic
340	C₁₃	#19197	1	1	yes	characteristic, complemented, Sylow
341	C₂³	—	26	2	no	—
342	C₂³	#88149	1	1	yes	characteristic
343	Q₈	—	52	2	no	—
344	Q₈	—	52	2	no	—
345	Q₈	—	52	2	no	—
346	Q₈	—	52	4	no	—
347	Q₈	—	104	2	no	—
348	Q₈	—	104	2	no	—
349	D₄	—	26	4	no	—
350	D₄	—	52	4	no	—
351	D₄	—	52	2	no	—
352	D₄	—	52	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	26	2	no	—
356	C₂×C₄	—	52	2	no	—
357	C₂×C₄	—	52	2	no	—
358	C₂×C₄	—	52	2	no	—
359	C₂×C₄	—	52	2	no	—
360	C₂×C₄	—	104	2	no	—
361	C₂×C₄	—	104	2	no	—
362	C₂×C₄	#88149	1	1	yes	characteristic
363	C₂×C₄	#88149	1	1	yes	characteristic
364	C₂×C₄	#88149	1	1	yes	characteristic
365	C₂×C₄	#88158	1	2	yes	—
366	C₂×C₄	#88162	1	2	yes	—
367	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
368	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
369	C₂×C₄	C₂×#6131	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₈	—	52	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	52	2	no	—
380	C₂²	—	52	2	no	—
381	C₂²	—	52	2	no	—
382	C₂²	#11718505	1	2	yes	—
383	C₂²	C₂×#88149	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	52	2	no	—
387	C₄	—	104	2	no	—
388	C₄	—	104	2	no	—
389	C₄	#11718505	1	2	yes	—
390	C₄	#11718557	1	2	yes	—
391	C₄	C₂×#88158	1	1	yes	characteristic
392	C₄	C₂×#88162	1	1	yes	characteristic
393	C₄	#11719475	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	104	2	no	—
396	C₂	C₂×#11718505	1	1	yes	characteristic
397	C₂	#13124476	1	1	yes	characteristic
398	C₂	C₂×#11718557	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88149
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	#11719182	C₂²
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#19197
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. C₂×#11718342	C₂	semidirect
5. #13114384	C₂	semidirect
6. #13114260	C₂	semidirect
7. #13113520	C₂	semidirect
8. #13113507	C₂	semidirect
9. C₁₃×#1564	C₂	semidirect
148. #1564	D₁₃	semidirect
340. C₁₃	#19197	semidirect

Maximal subgroups

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

subgroup	index
#13125959	2
#13125763	2
C₂×#11718342	2
#13123549	2
#13114384	2
#13114260	2
#13113520	2
#13113507	2
C₁₃×#1564	2
#19197	13