Subgroups — #29651443

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454832	C₂	1	2	yes	maximal, complemented
2	C₄×#14519800	C₂	1	1	yes	maximal, characteristic
3	#21444716	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444658	C₂	1	2	yes	maximal, complemented
5	#21444653	C₂	1	2	yes	maximal, complemented
6	C₈×D₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#21444613	C₂	1	1	yes	maximal, characteristic, complemented
8	#21444581	C₂	1	1	yes	maximal, characteristic, complemented
9	#21444252	C₂	1	2	yes	maximal, complemented
10	C₈×#11758189	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14519800	C₄	1	1	yes	characteristic
13	C₄×#11758226	C₂²	1	2	yes	—
14	#16839893	C₂²	1	2	yes	—
15	Q₁₆×Dic₆₁	C₄	1	1	yes	characteristic
16	C₄×#11758201	C₂²	1	2	yes	—
17	#16839847	C₄	1	2	yes	—
18	C₄×D₄₈₈	C₂²	1	1	yes	characteristic
19	#16839844	C₄	1	1	yes	characteristic
20	#16839835	—	2	2	no	—
21	#16839832	—	2	2	no	—
22	C₈×D₂₄₄	C₂²	1	2	yes	—
23	#16839827	C₂²	1	2	yes	—
24	#16839818	—	2	2	no	—
25	#16839818	C₂²	1	2	yes	—
26	#16839813	C₂²	1	1	yes	characteristic
27	#16839799	C₄	1	1	yes	characteristic
28	#16839757	C₄	1	2	yes	—
29	C₄×#11758189	C₂²	1	1	yes	characteristic
30	#97×D₆₁	—	2	2	no	—
31	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
32	#16839515	C₂²	1	1	yes	characteristic
33	#16839512	C₂²	1	2	yes	—
34	Q₈⋊#92418	C₂²	1	2	yes	—
35	#16839491	C₂²	1	2	yes	—
36	#16839489	C₂²	1	2	yes	—
37	#16839484	C₂²	1	1	yes	characteristic
38	#16839483	C₂²	1	1	yes	characteristic
39	C₈×#92418	C₂²	1	1	yes	characteristic
40	C₆₁×#446	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
42	C₆₁×#335	C₂²	1	1	yes	characteristic
43	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₆₁	C₂²	1	1	yes	characteristic
45	C₂×#11758226	C₂×C₄	1	2	yes	—
46	#14519800	—	2	4	no	—
47	C₂×#11758201	C₂×C₄	1	2	yes	—
48	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
49	C₂×#11758189	D₄	1	1	yes	characteristic
50	C₂×#11758189	D₄	1	1	yes	characteristic
51	C₂×C₈×D₆₁	—	2	2	no	—
52	C₂×C₈×D₆₁	—	2	2	no	—
53	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
55	Q₈×Dic₆₁	C₂×C₄	1	2	yes	—
56	C₄×D₂₄₄	—	2	2	no	—
57	C₄×D₂₄₄	C₂³	1	2	yes	—
58	#14519739	—	2	2	no	—
59	#14519739	C₂×C₄	1	2	yes	—
60	#14519735	—	2	2	no	—
61	#14519735	C₂×C₄	1	2	yes	—
62	C₄²×D₆₁	—	2	2	no	—
63	C₄²×D₆₁	C₂³	1	1	yes	characteristic
64	#14519711	C₂×C₄	1	1	yes	characteristic
65	#14519708	C₂×C₄	1	2	yes	—
66	C₄⋊#92418	—	2	2	no	—
67	C₄⋊#92418	C₂³	1	2	yes	—
68	#14519706	—	2	1	no	—
69	#14519701	C₂×C₄	1	2	yes	—
70	#14519699	C₂×C₄	1	2	yes	—
71	#14519697	C₂×C₄	1	2	yes	—
72	#14519689	—	2	2	no	—
73	#14519689	—	2	1	no	—
74	#14519689	—	2	2	no	—
75	#14519689	—	2	2	no	—
76	#14519688	—	2	2	no	—
77	#14519687	—	2	2	no	—
78	#14519687	D₄	1	1	yes	characteristic
79	#14519687	D₄	1	1	yes	characteristic
80	C₄×#92418	C₂³	1	1	yes	characteristic
81	C₄×#92418	C₂³	1	1	yes	characteristic
82	C₄×#92418	C₂³	1	1	yes	characteristic
83	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
85	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
87	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₆₁	—	2	2	no	—
89	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
90	#103×C₆₁	C₂×C₄	1	2	yes	—
91	#97×C₆₁	—	2	2	no	—
92	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
96	#11758226	—	2	4	no	—
97	#11758226	—	4	2	no	—
98	C₂×D₂₄₄	—	2	2	no	—
99	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
100	C₂×C₄×D₆₁	—	2	2	no	—
101	C₂×C₄×D₆₁	—	2	2	no	—
102	C₂×C₄×D₆₁	—	2	2	no	—
103	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
106	Q₁₆×C₆₁	—	2	2	no	—
107	C₂×C₈×C₆₁	—	2	2	no	—
108	C₂×C₈×C₆₁	—	2	1	no	—
109	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
113	C₄⋊C₄×C₆₁	—	2	2	no	—
114	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
115	C₄²×C₆₁	—	2	2	no	—
116	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
117	#11758201	—	2	4	no	—
118	#11758198	—	2	2	no	—
119	#11758198	—	2	2	no	—
120	#11758198	—	2	2	no	—
121	C₄⋊Dic₆₁	—	2	2	no	—
122	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
123	C₄×Dic₆₁	—	2	2	no	—
124	C₄×Dic₆₁	—	2	2	no	—
125	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
126	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
127	C₄×Dic₆₁	#41	1	1	yes	characteristic
128	C₂×#92418	—	2	2	no	—
129	C₂×#92418	—	2	1	no	—
130	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
131	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
132	C₂×#92418	C₂×D₄	1	1	yes	characteristic
133	C₂×#92418	C₂×D₄	1	1	yes	characteristic
134	D₄₈₈	M₁₆	1	4	yes	complemented
135	#11758189	—	2	2	no	—
136	#11758189	—	2	2	no	—
137	C₈×D₆₁	—	2	2	no	—
138	C₈×D₆₁	—	4	1	no	—
139	C₈×D₆₁	—	4	4	no	—
140	C₈×D₆₁	—	4	4	no	—
141	C₂²×D₆₁	—	2	2	no	—
142	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
143	Q₈×C₆₁	—	2	2	no	—
144	Q₈×C₆₁	—	2	2	no	—
145	C₂×C₄×C₆₁	—	2	2	no	—
146	C₂×C₄×C₆₁	—	2	2	no	—
147	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
150	C₂×Dic₆₁	—	2	2	no	—
151	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
154	D₂₄₄	—	2	2	no	—
155	D₂₄₄	—	4	4	no	—
156	D₂₄₄	C₂×M₁₆	1	4	yes	—
157	C₄×D₆₁	—	2	1	no	—
158	C₄×D₆₁	—	2	1	no	—
159	C₄×D₆₁	—	2	1	no	—
160	C₄×D₆₁	—	2	1	no	—
161	C₄×D₆₁	—	4	1	no	—
162	C₄×D₆₁	—	4	4	no	—
163	C₄×D₆₁	—	4	2	no	—
164	C₄×D₆₁	—	4	2	no	—
165	C₄×D₆₁	—	8	2	no	—
166	C₈×C₆₁	—	2	1	no	—
167	C₈×C₆₁	—	2	1	no	—
168	C₈×C₆₁	—	4	1	no	—
169	C₈×C₆₁	—	4	2	no	—
170	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
171	C₈×C₆₁	#135	1	2	yes	complemented
172	#92418	—	2	1	no	—
173	#92418	—	2	1	no	—
174	#92418	—	2	1	no	—
175	#92418	—	4	1	no	—
176	#92418	—	4	2	no	—
177	#92418	#135	1	2	yes	complemented
178	#18035	—	61	1	no	maximal, Sylow
179	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂×D₆₁	—	2	1	no	—
181	C₂×D₆₁	—	2	4	no	—
182	C₂×D₆₁	—	2	1	no	—
183	C₂×D₆₁	—	2	1	no	—
184	C₂×D₆₁	—	4	2	no	—
185	C₂×D₆₁	—	4	2	no	—
186	C₄×C₆₁	—	2	1	no	—
187	C₄×C₆₁	—	4	2	no	—
188	C₄×C₆₁	—	4	2	no	—
189	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
190	C₄×C₆₁	#568	1	1	yes	characteristic
191	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
192	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
193	Dic₆₁	—	2	1	no	—
194	Dic₆₁	—	2	1	no	—
195	Dic₆₁	—	2	1	no	—
196	Dic₆₁	—	4	2	no	—
197	#2970	—	61	2	no	—
198	C₄×#135	—	61	1	no	—
199	#1591	—	61	1	no	—
200	#1591	D₆₁	1	1	yes	characteristic, complemented
201	#1582	—	61	2	no	—
202	#1582	—	61	2	no	—
203	C₈×D₈	—	61	1	no	—
204	C₈×D₈	—	61	1	no	—
205	#1568	—	61	1	no	—
206	#1488	—	61	2	no	—
207	C₈×M₁₆	—	61	1	no	—
208	C₂×C₆₁	D₄×M₁₆	1	1	yes	characteristic
209	C₂×C₆₁	C₄×#135	1	1	yes	characteristic
210	C₂×C₆₁	#2957	1	1	yes	characteristic
211	D₆₁	—	4	1	no	—
212	D₆₁	—	4	4	no	—
213	C₂×#135	—	61	1	no	—
214	C₄×#41	—	61	2	no	—
215	#446	—	61	2	no	—
216	#446	C₂×D₆₁	1	2	yes	—
217	C₄×Q₁₆	—	61	1	no	—
218	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
219	C₄×SD₁₆	—	61	2	no	—
220	C₄×SD₁₆	—	61	2	no	—
221	C₄×D₈	—	61	1	no	—
222	C₄×D₈	—	61	1	no	—
223	#435	—	122	2	no	—
224	#435	—	122	2	no	—
225	C₈×D₄	—	61	2	no	—
226	C₈×D₄	—	61	2	no	—
227	#432	—	61	2	no	—
228	#432	—	122	2	no	—
229	#431	—	61	1	no	—
230	#427	—	61	1	no	—
231	#416	—	61	2	no	—
232	C₄×M₁₆	—	61	1	no	—
233	C₂×#97	—	122	2	no	—
234	C₂×C₄×C₈	—	61	1	no	—
235	#335	—	61	1	no	—
236	#335	C₂×D₆₁	1	1	yes	characteristic
237	#334	—	61	2	no	—
238	Q₈⋊C₈	—	61	2	no	—
239	Q₈⋊C₈	C₂×D₆₁	1	2	yes	—
240	D₄⋊C₈	—	61	2	no	—
241	D₄⋊C₈	—	61	2	no	—
242	#322	—	61	1	no	—
243	#322	—	61	1	no	—
244	C₈²	—	61	1	no	—
245	C₈²	C₂×D₆₁	1	1	yes	characteristic
246	C₆₁	#18035	1	1	yes	characteristic, complemented, Sylow
247	C₂×#41	—	61	2	no	—
248	#135	—	122	4	no	—
249	C₂×Q₁₆	C₄×D₆₁	1	1	yes	characteristic
250	C₂×SD₁₆	—	61	2	no	—
251	C₂×D₈	—	61	1	no	—
252	C₂×M₁₆	—	61	1	no	—
253	C₂×M₁₆	—	61	1	no	—
254	C₂²×C₈	—	61	1	no	—
255	C₂²×C₈	—	61	1	no	—
256	C₂²×C₈	—	122	2	no	—
257	C₂²×C₈	—	122	2	no	—
258	C₄×Q₈	—	61	2	no	—
259	C₄×Q₈	C₂²×D₆₁	1	2	yes	—
260	C₄×D₄	—	61	2	no	—
261	C₄×D₄	—	61	2	no	—
262	C₄×D₄	—	122	2	no	—
263	C₄×D₄	—	122	2	no	—
264	#117	—	61	2	no	—
265	#117	—	122	2	no	—
266	C₂×C₄²	—	61	1	no	—
267	C₂×C₄²	—	122	2	no	—
268	#107	—	61	1	no	—
269	#107	C₄×D₆₁	1	1	yes	characteristic
270	#106	—	61	2	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	61	2	no	—
273	C₄⋊C₈	—	122	2	no	—
274	C₄⋊C₈	—	122	1	no	—
275	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
276	#103	—	61	2	no	—
277	#103	C₄×D₆₁	1	2	yes	—
278	#102	—	61	2	no	—
279	#102	—	61	2	no	—
280	C₂²⋊C₈	—	122	2	no	—
281	C₂²⋊C₈	—	122	1	no	—
282	C₂²⋊C₈	—	122	2	no	—
283	C₂²⋊C₈	—	122	2	no	—
284	#97	—	2	2	no	—
285	#97	—	61	1	no	—
286	#97	—	61	1	no	—
287	#97	—	122	2	no	—
288	#97	—	122	2	no	—
289	C₄×C₈	—	61	1	no	—
290	C₄×C₈	—	61	1	no	—
291	C₄×C₈	—	61	1	no	—
292	C₄×C₈	—	61	1	no	—
293	C₄×C₈	—	61	1	no	—
294	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
295	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
297	#41	—	122	4	no	—
298	#41	—	244	2	no	—
299	C₂×Q₈	C₂×C₄×D₆₁	1	2	yes	—
300	C₂×D₄	—	61	2	no	—
301	C₂×D₄	—	122	2	no	—
302	C₂²×C₄	—	61	1	no	—
303	C₂²×C₄	—	61	1	no	—
304	C₂²×C₄	—	61	1	no	—
305	C₂²×C₄	—	122	2	no	—
306	C₂²×C₄	—	122	2	no	—
307	C₂²×C₄	—	122	2	no	—
308	Q₁₆	—	2	2	no	—
309	SD₁₆	—	122	4	no	—
310	D₈	—	61	4	no	—
311	M₁₆	—	122	2	no	—
312	M₁₆	—	122	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	61	1	no	—
316	C₂×C₈	—	61	1	no	—
317	C₂×C₈	—	61	1	no	—
318	C₂×C₈	—	61	1	no	—
319	C₂×C₈	—	122	2	no	—
320	C₂×C₈	—	122	2	no	—
321	C₂×C₈	—	122	1	no	—
322	C₂×C₈	—	244	1	no	—
323	C₂×C₈	—	244	4	no	—
324	C₂×C₈	—	244	4	no	—
325	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
326	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
327	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
328	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	61	2	no	—
331	C₄⋊C₄	—	122	2	no	—
332	C₄⋊C₄	C₂×C₄×D₆₁	1	2	yes	—
333	C₂²⋊C₄	—	122	2	no	—
334	C₂²⋊C₄	—	122	2	no	—
335	C₂²⋊C₄	—	122	2	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	61	1	no	—
338	C₄²	—	61	1	no	—
339	C₄²	—	61	1	no	—
340	C₄²	—	122	2	no	—
341	C₄²	—	122	2	no	—
342	C₄²	C₂³×D₆₁	1	1	yes	characteristic
343	C₂³	—	61	1	no	—
344	C₂³	—	122	2	no	—
345	Q₈	—	2	2	no	—
346	Q₈	—	2	2	no	—
347	D₄	—	61	4	no	—
348	D₄	—	122	2	no	—
349	D₄	—	244	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	61	1	no	—
353	C₂×C₄	—	61	1	no	—
354	C₂×C₄	—	61	1	no	—
355	C₂×C₄	—	122	1	no	—
356	C₂×C₄	—	122	1	no	—
357	C₂×C₄	—	122	1	no	—
358	C₂×C₄	—	122	1	no	—
359	C₂×C₄	—	122	2	no	—
360	C₂×C₄	—	488	2	no	—
361	C₂×C₄	—	244	1	no	—
362	C₂×C₄	—	244	4	no	—
363	C₂×C₄	—	244	2	no	—
364	C₂×C₄	—	244	2	no	—
365	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
366	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
367	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	4	1	no	—
371	C₈	—	4	2	no	—
372	C₈	—	61	2	no	—
373	C₈	—	122	1	no	—
374	C₈	—	122	1	no	—
375	C₈	—	122	1	no	—
376	C₈	—	244	1	no	—
377	C₈	—	244	2	no	—
378	C₈	M₁₆×D₆₁	1	2	yes	—
379	C₈	#14519800	1	2	yes	complemented
380	C₂²	—	122	1	no	—
381	C₂²	—	122	1	no	—
382	C₂²	—	122	4	no	—
383	C₂²	—	122	1	no	—
384	C₂²	—	244	2	no	—
385	C₂²	—	244	2	no	—
386	C₂²	C₄×D₄×D₆₁	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	4	2	no	—
389	C₄	—	4	2	no	—
390	C₄	—	122	1	no	—
391	C₄	—	122	1	no	—
392	C₄	—	122	1	no	—
393	C₄	—	244	2	no	—
394	C₄	C₂×M₁₆×D₆₁	1	1	yes	characteristic
395	C₄	#16840527	1	1	yes	characteristic
396	C₄	#135×D₆₁	1	1	yes	characteristic
397	C₄	C₂×#14519800	1	1	yes	characteristic
398	C₂	—	244	1	no	—
399	C₂	—	244	4	no	—
400	C₂	#21454555	1	1	yes	characteristic
401	C₂	C₄×#14519800	1	1	yes	characteristic
402	C₂	#21454731	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1591	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#14519800	C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1591	D₆₁
hypocenter	C₆₁	#18035
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
1. #21454832	C₂	semidirect
3. #21444716	C₂	semidirect
4. #21444658	C₂	semidirect
5. #21444653	C₂	semidirect
6. C₈×D₄₈₈	C₂	semidirect
7. #21444613	C₂	semidirect
8. #21444581	C₂	semidirect
9. #21444252	C₂	semidirect
10. C₈×#11758189	C₂	semidirect
11. C₆₁×#1591	C₂	semidirect
134. D₄₈₈	M₁₆	semidirect
171. C₈×C₆₁	#135	semidirect
177. #92418	#135	semidirect
200. #1591	D₆₁	semidirect
246. C₆₁	#18035	semidirect
379. C₈	#14519800	semidirect

Maximal subgroups

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

subgroup	index
#21454832	2
C₄×#14519800	2
#21444716	2
#21444658	2
#21444653	2
C₈×D₄₈₈	2
#21444613	2
#21444581	2
#21444252	2
C₈×#11758189	2
C₆₁×#1591	2
#18035	61