Subgroups — #29650590

Order 15616.

614 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456389	C₂	1	1	yes	maximal, characteristic
2	#21454794	C₂	1	2	yes	maximal, complemented
3	#21444734	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444729	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444723	C₂	1	2	yes	maximal, complemented
6	#21444644	C₂	1	1	yes	maximal, characteristic, complemented
7	#21444639	C₂	1	1	yes	maximal, characteristic, complemented
8	#21444633	C₂	1	2	yes	maximal, complemented
9	#21444582	C₂	1	1	yes	maximal, characteristic, complemented
10	#21444280	C₂	1	2	yes	maximal, complemented
11	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840296	C₂²	1	2	yes	—
13	#16840164	C₄	1	1	yes	characteristic
14	#16840146	C₄	1	1	yes	characteristic
15	#16840117	C₄	1	2	yes	—
16	#16840003	C₄	1	1	yes	characteristic
17	#16839998	C₄	1	1	yes	characteristic
18	#16839991	C₄	1	2	yes	—
19	#16839892	C₂²	1	2	yes	—
20	C₈×Dic₁₂₂	C₂²	1	2	yes	—
21	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
22	C₄×#11758190	C₂²	1	2	yes	—
23	C₄×D₄₈₈	C₂²	1	1	yes	characteristic
24	#16839841	C₂²	1	2	yes	—
25	#16839833	—	2	2	no	—
26	C₈×D₂₄₄	C₂²	1	2	yes	—
27	C₈×C₂²⋊D₆₁	—	2	2	no	—
28	#16839821	—	2	2	no	—
29	#16839819	C₂²	1	2	yes	—
30	#16839716	—	2	2	no	—
31	#16839515	C₂²	1	2	yes	—
32	#16839513	C₂²	1	1	yes	characteristic
33	#16839513	C₂²	1	1	yes	characteristic
34	#16839494	C₂²	1	2	yes	—
35	#16839494	C₂²	1	2	yes	—
36	#16839491	C₂²	1	2	yes	—
37	#16839491	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
41	C₆₁×#322	C₂²	1	1	yes	characteristic
42	C₆₁×#322	C₂²	1	1	yes	characteristic
43	C₆₁×#322	C₂²	1	1	yes	characteristic
44	C₆₁×#322	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
46	C₂×#11758190	C₂×C₄	1	2	yes	—
47	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
48	C₂×#11758189	—	2	2	no	—
49	C₂×C₈×D₆₁	—	2	2	no	—
50	#14519771	—	2	2	no	—
51	#14519769	C₂×C₄	1	2	yes	—
52	#14519765	C₂×C₄	1	2	yes	—
53	#14519760	—	2	2	no	—
54	#14519756	C₂×C₄	1	2	yes	—
55	#14519751	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₂₂	C₂³	1	2	yes	—
57	C₄×D₂₄₄	C₂³	1	2	yes	—
58	C₄×C₂²⋊D₆₁	—	2	2	no	—
59	#14519734	—	2	2	no	—
60	#14519711	C₂×C₄	1	1	yes	characteristic
61	#14519711	C₂×C₄	1	1	yes	characteristic
62	#14519709	C₂×C₄	1	2	yes	—
63	C₄⋊#92418	C₂³	1	2	yes	—
64	C₄⋊#92418	C₂³	1	2	yes	—
65	#14519706	—	2	2	no	—
66	#14519706	—	2	2	no	—
67	#14519702	C₂×C₄	1	2	yes	—
68	#14519702	C₂×C₄	1	2	yes	—
69	#14519699	C₂×C₄	1	2	yes	—
70	#14519699	C₂×C₄	1	2	yes	—
71	C₂²⋊#92418	—	2	2	no	—
72	#14519689	—	2	2	no	—
73	#14519689	—	2	2	no	—
74	#14519688	—	2	2	no	—
75	#14519687	—	2	2	no	—
76	C₈×Dic₆₁	—	2	2	no	—
77	C₂²×C₈×C₆₁	—	2	1	no	—
78	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
80	C₂²×C₈×C₆₁	D₄	1	2	yes	—
81	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₆₁	—	2	1	no	—
83	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₆₁	D₄	1	2	yes	—
86	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₆₁	—	2	2	no	—
96	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₆₁	—	2	2	no	—
98	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₆₁	—	2	2	no	—
100	C₂×C₈×C₆₁	—	2	2	no	—
101	C₂×C₈×C₆₁	—	2	1	no	—
102	C₂×C₈×C₆₁	—	2	1	no	—
103	C₂×C₈×C₆₁	—	4	2	no	—
104	C₂×C₈×C₆₁	—	4	2	no	—
105	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
111	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₁	#41	1	1	yes	characteristic
114	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
115	C₂²⋊Dic₆₁	—	2	2	no	—
116	#11758198	—	2	2	no	—
117	#11758198	—	2	2	no	—
118	#11758198	—	2	2	no	—
119	#11758198	—	2	2	no	—
120	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
121	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
122	#11758196	—	2	2	no	—
123	#11758196	—	2	2	no	—
124	#11758196	—	2	2	no	—
125	#11758196	—	2	2	no	—
126	C₄×Dic₆₁	—	2	2	no	—
127	C₂×#92418	—	2	2	no	—
128	C₂×#92418	—	2	2	no	—
129	Dic₂₄₄	—	2	2	no	—
130	D₄₈₈	—	2	2	no	—
131	#11758190	—	2	4	no	—
132	#11758189	—	4	4	no	—
133	C₈×D₆₁	—	4	4	no	—
134	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
135	C₂²×D₆₁	—	2	2	no	—
136	C₂×C₄×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	—	2	1	no	—
138	C₂×C₄×C₆₁	—	2	1	no	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
146	C₂²⋊D₆₁	—	4	4	no	—
147	C₂×Dic₆₁	—	2	2	no	—
148	C₂×Dic₆₁	—	2	2	no	—
149	C₂×Dic₆₁	—	2	2	no	—
150	D₂₄₄	—	2	2	no	—
151	D₂₄₄	—	2	2	no	—
152	C₄×D₆₁	—	4	2	no	—
153	C₄×D₆₁	—	4	2	no	—
154	Dic₁₂₂	—	2	2	no	—
155	Dic₁₂₂	—	2	2	no	—
156	C₈×C₆₁	—	2	1	no	—
157	C₈×C₆₁	—	2	1	no	—
158	C₈×C₆₁	—	2	2	no	—
159	C₈×C₆₁	—	2	2	no	—
160	C₈×C₆₁	—	4	1	no	—
161	C₈×C₆₁	—	4	1	no	—
162	C₈×C₆₁	#131	1	2	yes	—
163	C₈×C₆₁	#131	1	2	yes	—
164	#92418	—	4	2	no	—
165	#92418	—	4	2	no	—
166	#17963	—	61	1	no	maximal, Sylow
167	C₂²×C₆₁	—	2	1	no	—
168	C₂²×C₆₁	—	2	1	no	—
169	C₂²×C₆₁	#444	1	2	yes	—
170	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₆₁	—	4	2	no	—
172	C₂×D₆₁	—	4	2	no	—
173	C₂×D₆₁	—	4	2	no	—
174	C₄×C₆₁	—	2	1	no	—
175	C₄×C₆₁	—	2	1	no	—
176	C₄×C₆₁	—	2	1	no	—
177	C₄×C₆₁	—	2	1	no	—
178	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
179	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
180	C₄×C₆₁	#575	1	2	yes	—
181	Dic₆₁	—	4	2	no	—
182	Dic₆₁	—	4	2	no	—
183	Dic₆₁	—	4	2	no	—
184	#3168	—	61	1	no	—
185	#2966	—	61	2	no	—
186	#1594	—	61	1	no	—
187	#1593	—	61	1	no	—
188	#1592	—	61	2	no	—
189	#1580	—	61	1	no	—
190	#1579	—	61	1	no	—
191	SD₁₆⋊C₈	—	61	2	no	—
192	#1568	—	61	1	no	—
193	#1494	—	61	2	no	—
194	C₂×#322	D₆₁	1	1	yes	characteristic, complemented
195	C₂×C₆₁	—	2	2	no	—
196	C₂×C₆₁	#2931	1	1	yes	characteristic
197	C₂×C₆₁	#2943	1	1	yes	characteristic
198	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
199	D₆₁	—	8	2	no	—
200	#529	—	61	2	no	—
201	#499	—	61	1	no	—
202	#495	—	61	1	no	—
203	#488	—	61	2	no	—
204	#467	—	61	1	no	—
205	#466	—	61	1	no	—
206	#465	—	61	2	no	—
207	#446	—	61	2	no	—
208	C₈×Q₈	—	61	2	no	—
209	C₄×Q₁₆	—	61	1	no	—
210	C₄×SD₁₆	—	61	2	no	—
211	C₄×D₈	—	61	1	no	—
212	#436	—	61	2	no	—
213	#435	—	122	2	no	—
214	C₈×D₄	—	61	2	no	—
215	C₈×D₄	—	122	2	no	—
216	#433	—	122	2	no	—
217	#432	—	61	2	no	—
218	#405	—	122	2	no	—
219	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
220	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
221	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
222	#335	—	61	2	no	—
223	#334	—	61	1	no	—
224	#334	—	61	1	no	—
225	Q₈⋊C₈	—	61	2	no	—
226	Q₈⋊C₈	—	61	2	no	—
227	D₄⋊C₈	—	61	2	no	—
228	D₄⋊C₈	—	61	2	no	—
229	#322	C₂×D₆₁	1	1	yes	characteristic
230	#322	C₂×D₆₁	1	1	yes	characteristic
231	#322	C₂×D₆₁	1	1	yes	characteristic
232	#322	C₂×D₆₁	1	1	yes	characteristic
233	C₆₁	#17963	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₁₆	—	61	1	no	—
235	C₂×SD₁₆	—	61	2	no	—
236	C₂×D₈	—	61	1	no	—
237	C₂×M₁₆	—	122	2	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	122	2	no	—
240	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
241	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
242	C₂²×C₈	D₂₄₄	1	2	yes	—
243	#126	—	122	2	no	—
244	#125	—	61	2	no	—
245	#124	—	61	2	no	—
246	#123	—	122	2	no	—
247	C₂²⋊Q₈	—	61	2	no	—
248	#121	—	61	2	no	—
249	C₄×Q₈	—	61	2	no	—
250	C₄×D₄	—	61	2	no	—
251	C₄×D₄	—	122	2	no	—
252	#117	—	122	2	no	—
253	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
254	#107	—	61	1	no	—
255	#107	—	61	1	no	—
256	#106	—	61	2	no	—
257	C₄⋊C₈	—	61	2	no	—
258	C₄⋊C₈	—	61	2	no	—
259	C₄⋊C₈	—	122	2	no	—
260	C₄⋊C₈	—	122	2	no	—
261	#103	—	61	2	no	—
262	#103	—	61	2	no	—
263	#102	—	61	2	no	—
264	#102	—	61	2	no	—
265	C₂²⋊C₈	—	122	2	no	—
266	C₂²⋊C₈	—	122	2	no	—
267	C₂²⋊C₈	—	122	2	no	—
268	#97	—	122	2	no	—
269	#97	—	122	2	no	—
270	C₄×C₈	—	2	1	no	—
271	C₄×C₈	—	122	2	no	—
272	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
273	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
274	C₄×C₈	D₂₄₄	1	2	yes	—
275	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
281	C₂×Q₈	—	61	2	no	—
282	C₂×D₄	—	61	2	no	—
283	C₂×D₄	—	122	2	no	—
284	C₂²×C₄	—	122	2	no	—
285	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
287	C₂²×C₄	#11758222	1	1	yes	characteristic
288	Q₁₆	—	122	2	no	—
289	SD₁₆	—	122	4	no	—
290	D₈	—	122	2	no	—
291	M₁₆	—	244	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	4	2	no	—
298	C₂×C₈	—	122	2	no	—
299	C₂×C₈	—	122	2	no	—
300	C₂×C₈	—	244	4	no	—
301	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
306	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
307	C₄⋊C₄	—	61	2	no	—
308	C₄⋊C₄	—	61	2	no	—
309	C₄⋊C₄	—	122	2	no	—
310	C₄⋊C₄	—	122	2	no	—
311	C₄⋊C₄	—	122	2	no	—
312	C₄⋊C₄	—	122	2	no	—
313	C₂²⋊C₄	—	122	2	no	—
314	C₂²⋊C₄	—	122	2	no	—
315	C₂²⋊C₄	—	122	2	no	—
316	C₂²⋊C₄	—	122	2	no	—
317	C₂²⋊C₄	—	122	2	no	—
318	C₄²	—	122	2	no	—
319	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
320	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
321	C₄²	#11758222	1	1	yes	characteristic
322	C₄²	C₂³×D₆₁	1	1	yes	characteristic
323	C₂³	—	122	2	no	—
324	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
325	Q₈	—	122	2	no	—
326	Q₈	—	122	2	no	—
327	D₄	—	122	2	no	—
328	D₄	—	122	2	no	—
329	D₄	—	244	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	122	2	no	—
335	C₂×C₄	—	122	2	no	—
336	C₂×C₄	—	122	2	no	—
337	C₂×C₄	—	244	2	no	—
338	C₂×C₄	—	244	2	no	—
339	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
340	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
341	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
344	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	2	no	—
348	C₈	—	2	2	no	—
349	C₈	—	4	1	no	—
350	C₈	—	4	1	no	—
351	C₈	—	244	2	no	—
352	C₈	—	244	2	no	—
353	C₈	#14519783	1	2	yes	—
354	C₈	#14519783	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	244	2	no	—
358	C₂²	—	244	2	no	—
359	C₂²	—	244	2	no	—
360	C₂²	#16839885	1	2	yes	—
361	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	244	2	no	—
367	C₄	—	244	2	no	—
368	C₄	—	244	2	no	—
369	C₄	C₂×#14519783	1	1	yes	characteristic
370	C₄	C₂×#14519783	1	1	yes	characteristic
371	C₄	#16840562	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	488	2	no	—
374	C₂	#21454532	1	1	yes	characteristic
375	C₂	#21454621	1	1	yes	characteristic
376	C₂	C₂×#16839885	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₆₁×#322	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	#16839998	C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂×#322	D₆₁
hypocenter	C₆₁	#17963
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #21454794	C₂	semidirect
3. #21444734	C₂	semidirect
4. #21444729	C₂	semidirect
5. #21444723	C₂	semidirect
6. #21444644	C₂	semidirect
7. #21444639	C₂	semidirect
8. #21444633	C₂	semidirect
9. #21444582	C₂	semidirect
10. #21444280	C₂	semidirect
11. C₂×C₆₁×#322	C₂	semidirect
194. C₂×#322	D₆₁	semidirect
233. C₆₁	#17963	semidirect

Maximal subgroups

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

subgroup	index
#21456389	2
#21454794	2
#21444734	2
#21444729	2
#21444723	2
#21444644	2
#21444639	2
#21444633	2
#21444582	2
#21444280	2
C₂×C₆₁×#322	2
#17963	61