Subgroups — #29650884

Order 15616.

576 conjugacy classes of subgroups.

The subgroup lattice has 366 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	#21454970	C₂	1	2	yes	maximal, complemented
3	#21444699	C₂	1	2	yes	maximal, complemented
4	#21444678	C₂	1	2	yes	maximal, complemented
5	#21444671	C₂	1	2	yes	maximal, complemented
6	#21444650	C₂	1	2	yes	maximal, complemented
7	#21444578	C₂	1	1	yes	maximal, characteristic, complemented
8	#21444285	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	#16840296	C₂²	1	2	yes	—
11	#16840164	C₄	1	1	yes	characteristic
12	#16840146	C₄	1	1	yes	characteristic
13	#16840117	C₄	1	2	yes	—
14	#16840003	C₄	1	1	yes	characteristic
15	#16839998	C₄	1	1	yes	characteristic
16	#16839991	C₄	1	2	yes	—
17	#16839892	C₂²	1	2	yes	—
18	C₈×Dic₁₂₂	C₂²	1	2	yes	—
19	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
20	C₄×#11758190	C₂²	1	2	yes	—
21	C₄×D₄₈₈	C₂²	1	1	yes	characteristic
22	#16839841	C₂²	1	2	yes	—
23	#16839837	—	2	2	no	—
24	#16839836	—	2	2	no	—
25	C₈×D₂₄₄	C₂²	1	2	yes	—
26	#16839822	—	2	2	no	—
27	#16839787	—	2	2	no	—
28	#16839786	C₂²	1	2	yes	—
29	#16839515	C₂²	1	2	yes	—
30	#16839515	C₂²	1	1	yes	characteristic
31	#16839515	C₂²	1	1	yes	characteristic
32	#16839494	C₂²	1	2	yes	—
33	#16839494	C₂²	1	2	yes	—
34	#16839491	C₂²	1	2	yes	—
35	#16839491	C₂²	1	2	yes	—
36	C₆₁×#431	C₂²	1	1	yes	characteristic
37	C₆₁×#431	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
39	C₆₁×#322	C₂²	1	2	yes	—
40	C₆₁×#322	C₂²	1	2	yes	—
41	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
42	C₂×#11758190	C₂×C₄	1	2	yes	—
43	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
44	C₂×#11758194	—	2	2	no	—
45	C₂×#11758189	—	2	2	no	—
46	C₂×C₈×D₆₁	—	2	2	no	—
47	#14519771	—	2	2	no	—
48	#14519769	C₂×C₄	1	2	yes	—
49	#14519765	C₂×C₄	1	2	yes	—
50	#14519760	—	2	2	no	—
51	#14519756	C₂×C₄	1	2	yes	—
52	#14519751	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₂₂	C₂³	1	2	yes	—
54	C₄×D₂₄₄	C₂³	1	2	yes	—
55	C₄×C₂²⋊D₆₁	—	2	2	no	—
56	#14519734	—	2	2	no	—
57	#14519711	C₂×C₄	1	1	yes	characteristic
58	#14519711	C₂×C₄	1	1	yes	characteristic
59	#14519709	C₂×C₄	1	2	yes	—
60	C₄⋊#92418	—	2	2	no	—
61	C₄⋊#92418	C₂³	1	2	yes	—
62	C₄⋊#92418	C₂³	1	2	yes	—
63	#14519706	—	2	2	no	—
64	#14519706	—	2	2	no	—
65	#14519702	C₂×C₄	1	2	yes	—
66	#14519702	C₂×C₄	1	2	yes	—
67	#14519699	C₂×C₄	1	2	yes	—
68	#14519699	C₂×C₄	1	2	yes	—
69	#14519689	—	2	2	no	—
70	#14519689	—	2	2	no	—
71	#14519687	—	2	2	no	—
72	C₈×Dic₆₁	—	2	2	no	—
73	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₆₁	—	2	1	no	—
77	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
78	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
79	C₂²⋊C₈×C₆₁	—	2	1	no	—
80	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
81	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₆₁	C₂³	1	2	yes	—
84	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₁	C₂³	1	2	yes	—
87	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
90	C₂×C₂²⋊D₆₁	—	2	2	no	—
91	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₆₁	—	2	2	no	—
93	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	4	1	no	—
96	C₂×C₈×C₆₁	—	4	1	no	—
97	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
103	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₆₁	#41	1	1	yes	characteristic
106	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₆₁	—	2	2	no	—
108	#11758198	—	2	2	no	—
109	#11758198	—	2	2	no	—
110	#11758198	—	2	2	no	—
111	#11758198	—	2	2	no	—
112	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
113	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
114	#11758196	—	2	2	no	—
115	#11758196	—	2	2	no	—
116	#11758196	—	2	2	no	—
117	#11758196	—	2	2	no	—
118	C₄×Dic₆₁	—	2	2	no	—
119	#11758194	—	8	2	no	—
120	C₂×#92418	—	2	2	no	—
121	C₂×#92418	—	2	2	no	—
122	Dic₂₄₄	—	4	1	no	—
123	D₄₈₈	—	4	1	no	—
124	#11758190	—	4	2	no	—
125	#11758189	—	8	2	no	—
126	C₈×D₆₁	—	8	2	no	—
127	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
128	C₂²×D₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	2	1	no	—
130	C₂×C₄×C₆₁	—	2	1	no	—
131	C₂×C₄×C₆₁	—	2	1	no	—
132	C₂×C₄×C₆₁	—	2	1	no	—
133	C₂×C₄×C₆₁	—	4	1	no	—
134	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
140	C₂²⋊D₆₁	—	8	2	no	—
141	C₂×Dic₆₁	—	2	2	no	—
142	C₂×Dic₆₁	—	2	2	no	—
143	C₂×Dic₆₁	—	2	2	no	—
144	D₂₄₄	—	2	2	no	—
145	D₂₄₄	—	2	2	no	—
146	C₄×D₆₁	—	4	2	no	—
147	C₄×D₆₁	—	4	2	no	—
148	Dic₁₂₂	—	2	2	no	—
149	Dic₁₂₂	—	2	2	no	—
150	C₈×C₆₁	—	2	1	no	—
151	C₈×C₆₁	—	2	1	no	—
152	C₈×C₆₁	—	2	1	no	—
153	C₈×C₆₁	—	2	1	no	—
154	C₈×C₆₁	—	2	2	no	—
155	C₈×C₆₁	—	2	2	no	—
156	C₈×C₆₁	—	4	2	no	—
157	#92418	—	4	2	no	—
158	#92418	—	4	2	no	—
159	#17992	—	61	1	no	maximal, Sylow
160	C₂²×C₆₁	—	2	1	no	—
161	C₂²×C₆₁	—	2	1	no	—
162	C₂²×C₆₁	—	2	1	no	—
163	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×D₆₁	—	4	2	no	—
165	C₂×D₆₁	—	4	2	no	—
166	C₂×D₆₁	—	4	2	no	—
167	C₄×C₆₁	—	2	1	no	—
168	C₄×C₆₁	—	2	1	no	—
169	C₄×C₆₁	—	2	1	no	—
170	C₄×C₆₁	—	2	1	no	—
171	C₄×C₆₁	#568	1	1	yes	characteristic
172	C₄×C₆₁	#568	1	1	yes	characteristic
173	C₄×C₆₁	#576	1	1	yes	characteristic
174	C₄×C₆₁	#578	1	1	yes	characteristic
175	Dic₆₁	—	4	2	no	—
176	Dic₆₁	—	4	2	no	—
177	Dic₆₁	—	4	2	no	—
178	#3168	—	61	1	no	—
179	#2993	—	61	2	no	—
180	#1588	—	61	2	no	—
181	#1585	—	61	2	no	—
182	#1584	—	61	2	no	—
183	#1581	—	61	2	no	—
184	#1567	—	61	1	no	—
185	#1495	—	61	2	no	—
186	#1452	D₆₁	1	1	yes	characteristic, complemented
187	C₂×C₆₁	—	4	1	no	—
188	C₂×C₆₁	#2932	1	1	yes	characteristic
189	C₂×C₆₁	#2951	1	1	yes	characteristic
190	C₂×C₆₁	#2957	1	1	yes	characteristic
191	D₆₁	—	8	2	no	—
192	#529	—	61	2	no	—
193	#499	—	61	1	no	—
194	#495	—	61	1	no	—
195	#488	—	61	2	no	—
196	#467	—	61	1	no	—
197	#466	—	61	1	no	—
198	#465	—	61	2	no	—
199	#446	—	61	2	no	—
200	C₈×Q₈	—	61	2	no	—
201	C₄×Q₁₆	—	61	1	no	—
202	C₄×SD₁₆	—	61	2	no	—
203	C₄×D₈	—	61	1	no	—
204	#436	—	61	2	no	—
205	#436	—	122	2	no	—
206	#435	—	122	2	no	—
207	C₈×D₄	—	61	2	no	—
208	#433	—	122	2	no	—
209	#431	C₂×D₆₁	1	1	yes	characteristic
210	#431	C₂×D₆₁	1	1	yes	characteristic
211	#424	—	122	2	no	—
212	#423	—	61	2	no	—
213	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
214	#335	—	61	2	no	—
215	#335	—	61	1	no	—
216	#335	—	61	1	no	—
217	Q₈⋊C₈	—	61	2	no	—
218	Q₈⋊C₈	—	61	2	no	—
219	D₄⋊C₈	—	61	2	no	—
220	D₄⋊C₈	—	61	2	no	—
221	#322	C₂×D₆₁	1	2	yes	—
222	#322	C₂×D₆₁	1	2	yes	—
223	C₆₁	#17992	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₁₆	—	61	1	no	—
225	C₂×SD₁₆	—	61	2	no	—
226	C₂×D₈	—	61	1	no	—
227	C₂×M₁₆	—	122	2	no	—
228	C₂×M₁₆	—	122	2	no	—
229	C₂²×C₈	—	122	2	no	—
230	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
231	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
232	#126	—	122	2	no	—
233	#125	—	61	2	no	—
234	#124	—	61	2	no	—
235	#123	—	122	2	no	—
236	C₂²⋊Q₈	—	61	2	no	—
237	#121	—	61	2	no	—
238	C₄×Q₈	—	61	2	no	—
239	C₄×D₄	—	61	2	no	—
240	C₄×D₄	—	122	2	no	—
241	#117	—	122	2	no	—
242	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
243	#107	—	61	1	no	—
244	#107	—	61	1	no	—
245	#106	—	61	2	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	61	2	no	—
248	C₄⋊C₈	—	61	2	no	—
249	C₄⋊C₈	—	122	2	no	—
250	C₄⋊C₈	—	122	2	no	—
251	C₄⋊C₈	—	122	2	no	—
252	C₄⋊C₈	D₂₄₄	1	1	yes	characteristic
253	C₄⋊C₈	D₂₄₄	1	1	yes	characteristic
254	#103	—	61	2	no	—
255	#103	—	61	2	no	—
256	#102	—	61	2	no	—
257	#102	—	61	2	no	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	122	2	no	—
260	C₂²⋊C₈	—	122	2	no	—
261	C₂²⋊C₈	D₂₄₄	1	2	yes	—
262	#97	—	122	2	no	—
263	C₄×C₈	—	122	2	no	—
264	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
265	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₆₁	1	2	yes	—
267	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₆₁	1	2	yes	—
270	C₂×Q₈	—	61	2	no	—
271	C₂×D₄	—	61	2	no	—
272	C₂×D₄	—	122	2	no	—
273	C₂²×C₄	—	122	2	no	—
274	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
276	C₂²×C₄	#11758222	1	1	yes	characteristic
277	Q₁₆	—	244	1	no	—
278	SD₁₆	—	244	2	no	—
279	D₈	—	244	1	no	—
280	M₁₆	—	488	2	no	—
281	M₁₆	—	488	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	122	2	no	—
286	C₂×C₈	—	122	2	no	—
287	C₂×C₈	—	488	2	no	—
288	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
292	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
293	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
294	C₄⋊C₄	—	61	2	no	—
295	C₄⋊C₄	—	61	2	no	—
296	C₄⋊C₄	—	122	2	no	—
297	C₄⋊C₄	—	122	2	no	—
298	C₄⋊C₄	—	122	2	no	—
299	C₄⋊C₄	—	122	2	no	—
300	C₂²⋊C₄	—	122	2	no	—
301	C₂²⋊C₄	—	122	2	no	—
302	C₂²⋊C₄	—	122	2	no	—
303	C₂²⋊C₄	—	122	2	no	—
304	C₂²⋊C₄	—	122	2	no	—
305	C₄²	—	122	2	no	—
306	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
307	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
308	C₄²	#11758222	1	1	yes	characteristic
309	C₄²	C₂³×D₆₁	1	1	yes	characteristic
310	C₂³	—	122	2	no	—
311	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
312	Q₈	—	122	2	no	—
313	Q₈	—	122	2	no	—
314	D₄	—	122	2	no	—
315	D₄	—	122	2	no	—
316	D₄	—	488	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	122	2	no	—
323	C₂×C₄	—	122	2	no	—
324	C₂×C₄	—	122	2	no	—
325	C₂×C₄	—	244	2	no	—
326	C₂×C₄	—	244	2	no	—
327	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
328	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
329	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
332	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	2	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	244	2	no	—
341	C₈	—	244	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	244	2	no	—
346	C₂²	—	244	2	no	—
347	C₂²	—	244	2	no	—
348	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	244	2	no	—
354	C₄	—	244	2	no	—
355	C₄	—	244	2	no	—
356	C₄	#16840528	1	1	yes	characteristic
357	C₄	#16840528	1	1	yes	characteristic
358	C₄	#16840567	1	1	yes	characteristic
359	C₄	#16840579	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	488	2	no	—
362	C₂	#21454538	1	1	yes	characteristic
363	C₂	#21454696	1	1	yes	characteristic
364	C₂	#21454737	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1452	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	#16839998	C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1452	D₆₁
hypocenter	C₆₁	#17992
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #21454970	C₂	semidirect
3. #21444699	C₂	semidirect
4. #21444678	C₂	semidirect
5. #21444671	C₂	semidirect
6. #21444650	C₂	semidirect
7. #21444578	C₂	semidirect
8. #21444285	C₂	semidirect
9. C₆₁×#1452	C₂	semidirect
186. #1452	D₆₁	semidirect
223. C₆₁	#17992	semidirect

Maximal subgroups

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

subgroup	index
#21456389	2
#21454970	2
#21444699	2
#21444678	2
#21444671	2
#21444650	2
#21444578	2
#21444285	2
C₆₁×#1452	2
#17992	61