Subgroups — #30613007

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462764	C₂	1	2	yes	maximal, complemented
2	#21461463	C₂	1	2	yes	maximal, complemented
3	#21461459	C₂	1	1	yes	maximal, characteristic, complemented
4	#21461451	C₂	1	1	yes	maximal, characteristic, complemented
5	#21461409	C₂	1	2	yes	maximal, complemented
6	Q₁₆×#92420	C₂	1	1	yes	maximal, characteristic, complemented
7	#21461400	C₂	1	1	yes	maximal, characteristic, complemented
8	#21461376	C₂	1	1	yes	maximal, characteristic, complemented
9	#21461176	C₂	1	2	yes	maximal, complemented
10	#21461022	C₂	1	1	yes	maximal, characteristic, complemented
11	#21456390	C₂	1	1	yes	maximal, characteristic
12	#16840813	C₂²	1	2	yes	—
13	Q₈×#92420	C₂²	1	2	yes	—
14	#16840796	C₂²	1	2	yes	—
15	#16840795	—	2	2	no	—
16	#16840794	—	2	2	no	—
17	#16840790	C₂²	1	2	yes	—
18	#16840787	—	2	2	no	—
19	#16840778	C₂²	1	1	yes	characteristic
20	#16840778	C₂²	1	1	yes	characteristic
21	#16840769	—	2	2	no	—
22	#16840765	C₂²	1	2	yes	—
23	#16840626	C₂²	1	2	yes	—
24	#16840625	C₂²	1	1	yes	characteristic
25	#16840625	C₂²	1	1	yes	characteristic
26	Q₈⋊#92420	C₂²	1	2	yes	—
27	Q₈⋊#92420	C₂²	1	2	yes	—
28	#16840610	C₂²	1	2	yes	—
29	#16840610	C₂²	1	2	yes	—
30	#16840603	C₂²	1	1	yes	characteristic
31	#16840603	C₂²	1	1	yes	characteristic
32	C₈×#92420	C₂²	1	1	yes	characteristic
33	C₈×#92420	C₂²	1	1	yes	characteristic
34	#16840298	C₂²	1	2	yes	—
35	#16840163	C₄	1	1	yes	characteristic
36	#16840147	C₄	1	1	yes	characteristic
37	#16840118	C₄	1	2	yes	—
38	#16839998	C₄	1	1	yes	characteristic
39	#16839993	C₄	1	2	yes	—
40	Q₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
41	#16839847	C₂²	1	2	yes	—
42	#16839844	C₂²	1	1	yes	characteristic
43	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
44	C₆₁×#467	C₄	1	1	yes	characteristic
45	C₂×#11758213	—	2	2	no	—
46	C₂×#11758217	—	2	2	no	—
47	C₂×D₆₁⋊C₈	—	2	2	no	—
48	#14519844	—	2	2	no	—
49	#14519844	—	2	2	no	—
50	C₄⋊#92420	—	2	2	no	—
51	C₄⋊#92420	—	2	1	no	—
52	C₄⋊#92420	D₄	1	1	yes	characteristic
53	C₄⋊#92420	D₄	1	1	yes	characteristic
54	C₄⋊#92420	C₂³	1	2	yes	—
55	C₄⋊#92420	C₂³	1	2	yes	—
56	#14519834	—	2	2	no	—
57	#14519834	—	2	2	no	—
58	C₂²⋊#92420	—	2	1	no	—
59	C₂²⋊#92420	D₄	1	1	yes	characteristic
60	C₂²⋊#92420	D₄	1	1	yes	characteristic
61	#14519832	—	2	2	no	—
62	C₄×#92420	—	2	2	no	—
63	C₄×#92420	C₂³	1	1	yes	characteristic
64	C₄×#92420	C₂³	1	1	yes	characteristic
65	C₄×#92420	C₂³	1	1	yes	characteristic
66	C₄×#92420	C₂³	1	1	yes	characteristic
67	C₂×#11758201	C₂×C₄	1	2	yes	—
68	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
69	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
70	#14519772	—	2	2	no	—
71	#14519768	C₂×C₄	1	2	yes	—
72	#14519766	C₂×C₄	1	2	yes	—
73	#14519761	—	2	2	no	—
74	#14519751	C₂×C₄	1	2	yes	—
75	Q₈×Dic₆₁	C₂³	1	2	yes	—
76	#14519739	C₂³	1	2	yes	—
77	#14519737	—	2	2	no	—
78	#14519735	—	2	2	no	—
79	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
80	#14519711	C₂×C₄	1	1	yes	characteristic
81	#14519708	C₂×C₄	1	2	yes	—
82	#14519701	C₂×C₄	1	2	yes	—
83	#14519699	C₂×C₄	1	2	yes	—
84	#14519697	C₂×C₄	1	2	yes	—
85	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
88	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
89	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
90	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
91	C₂²⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
92	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
93	#103×C₆₁	C₂×C₄	1	2	yes	—
94	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊D₆₁	—	2	2	no	—
97	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×Dic₆₁	#41	1	1	yes	characteristic
99	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
100	C₂×C₄×D₆₁	—	2	2	no	—
101	#11758217	—	8	2	no	—
102	C₂×#92420	—	2	2	no	—
103	C₂×#92420	—	2	2	no	—
104	C₂×#92420	—	2	1	no	—
105	C₂×#92420	—	2	1	no	—
106	C₂×#92420	C₂×D₄	1	1	yes	characteristic
107	C₂×#92420	C₂×D₄	1	1	yes	characteristic
108	C₂×#92420	C₂×D₄	1	1	yes	characteristic
109	C₂×#92420	C₂×D₄	1	1	yes	characteristic
110	#11758213	—	8	2	no	—
111	D₆₁⋊C₈	—	8	2	no	—
112	Q₁₆×C₆₁	—	2	2	no	—
113	C₂×C₈×C₆₁	—	2	2	no	—
114	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
116	C₄⋊C₄×C₆₁	—	2	2	no	—
117	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
118	C₂²⋊C₄×C₆₁	—	2	2	no	—
119	#11758201	—	2	4	no	—
120	#11758198	—	2	2	no	—
121	#11758198	—	2	2	no	—
122	#11758198	—	2	2	no	—
123	#11758198	—	2	2	no	—
124	C₄⋊Dic₆₁	—	2	2	no	—
125	C₄⋊Dic₆₁	—	2	2	no	—
126	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
127	#11758196	—	2	2	no	—
128	C₄×Dic₆₁	—	2	2	no	—
129	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
130	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
131	C₄×Dic₆₁	#41	1	1	yes	characteristic
132	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
133	C₂×#92418	—	2	2	no	—
134	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
135	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
136	D₄₈₈	—	2	2	no	—
137	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
138	C₂²×D₆₁	—	2	2	no	—
139	Q₈×C₆₁	—	2	2	no	—
140	Q₈×C₆₁	—	2	2	no	—
141	C₂×C₄×C₆₁	—	2	1	no	—
142	C₂×C₄×C₆₁	—	2	1	no	—
143	C₂×C₄×C₆₁	—	2	2	no	—
144	C₂×C₄×C₆₁	—	2	2	no	—
145	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
147	C₂²⋊D₆₁	—	8	2	no	—
148	C₂×Dic₆₁	—	2	1	no	—
149	C₂×Dic₆₁	—	2	1	no	—
150	C₂×Dic₆₁	—	2	2	no	—
151	C₂×Dic₆₁	—	4	1	no	—
152	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
154	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
156	D₂₄₄	—	2	2	no	—
157	D₂₄₄	—	2	2	no	—
158	C₄×D₆₁	—	4	2	no	—
159	C₄×D₆₁	—	4	2	no	—
160	#92420	—	2	1	no	—
161	#92420	—	2	1	no	—
162	#92420	—	2	1	no	—
163	#92420	—	2	1	no	—
164	#92420	—	4	1	no	—
165	#92420	—	4	1	no	—
166	#92420	—	4	2	no	—
167	#92420	—	4	2	no	—
168	C₈×C₆₁	—	2	1	no	—
169	C₈×C₆₁	#131	1	2	yes	—
170	#92418	—	2	1	no	—
171	#92418	#131	1	2	yes	—
172	#17973	—	61	1	no	maximal, Sylow
173	C₂²×C₆₁	—	2	1	no	—
174	C₂²×C₆₁	—	2	1	no	—
175	C₂²×C₆₁	—	2	1	no	—
176	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×D₆₁	—	4	2	no	—
178	C₂×D₆₁	—	4	2	no	—
179	C₂×D₆₁	—	4	2	no	—
180	C₄×C₆₁	—	2	1	no	—
181	C₄×C₆₁	—	4	2	no	—
182	C₄×C₆₁	—	4	2	no	—
183	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
184	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
185	Dic₆₁	—	2	1	no	—
186	Dic₆₁	—	2	1	no	—
187	Dic₆₁	—	2	1	no	—
188	Dic₆₁	—	4	2	no	—
189	Dic₆₁	#576	1	1	yes	characteristic
190	Dic₆₁	#578	1	1	yes	characteristic
191	#3168	—	61	1	no	—
192	#2993	—	61	2	no	—
193	#1592	—	61	2	no	—
194	#1591	—	61	1	no	—
195	#1589	—	61	1	no	—
196	SD₁₆⋊C₈	—	61	2	no	—
197	C₈×Q₁₆	—	61	1	no	—
198	C₈×D₈	—	61	1	no	—
199	#1567	—	61	1	no	—
200	#1495	—	61	2	no	—
201	#1450	—	61	1	no	—
202	C₂×C₆₁	—	4	1	no	—
203	C₂×C₆₁	#2931	1	1	yes	characteristic
204	C₂×C₆₁	#2951	1	1	yes	characteristic
205	C₂×C₆₁	#2957	1	1	yes	characteristic
206	D₆₁	—	8	2	no	—
207	#529	—	61	2	no	—
208	#499	—	61	1	no	—
209	#495	—	61	1	no	—
210	#488	—	61	2	no	—
211	#467	#6923	1	1	yes	characteristic
212	#466	—	61	1	no	—
213	#465	—	61	2	no	—
214	#446	—	61	2	no	—
215	C₈×Q₈	—	61	2	no	—
216	C₄×Q₁₆	—	61	1	no	—
217	C₄×SD₁₆	—	61	2	no	—
218	C₄×D₈	—	61	1	no	—
219	#436	—	61	2	no	—
220	#436	—	122	2	no	—
221	#435	—	122	2	no	—
222	C₈×D₄	—	61	2	no	—
223	#433	—	122	2	no	—
224	#431	—	61	1	no	—
225	#431	—	61	1	no	—
226	#424	—	122	2	no	—
227	#423	—	61	2	no	—
228	C₂×C₄×C₈	—	61	1	no	—
229	#335	—	61	1	no	—
230	#335	—	61	1	no	—
231	#335	—	61	2	no	—
232	Q₈⋊C₈	—	61	2	no	—
233	Q₈⋊C₈	—	61	2	no	—
234	D₄⋊C₈	—	61	2	no	—
235	D₄⋊C₈	—	61	2	no	—
236	#322	—	61	1	no	—
237	#322	—	61	1	no	—
238	C₈²	—	61	1	no	—
239	C₈²	—	61	1	no	—
240	C₆₁	#17973	1	1	yes	characteristic, complemented, Sylow
241	C₂×Q₁₆	C₂×#6923	1	1	yes	characteristic
242	C₂×SD₁₆	—	61	2	no	—
243	C₂×D₈	—	61	1	no	—
244	C₂×M₁₆	—	122	2	no	—
245	C₂×M₁₆	—	122	2	no	—
246	C₂²×C₈	—	61	1	no	—
247	C₂²×C₈	—	122	2	no	—
248	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
249	#126	—	122	2	no	—
250	#125	—	61	2	no	—
251	#124	—	61	2	no	—
252	#123	—	122	2	no	—
253	C₂²⋊Q₈	C₂×#6923	1	2	yes	—
254	#121	—	61	2	no	—
255	C₄×Q₈	—	61	2	no	—
256	C₄×D₄	—	61	2	no	—
257	C₄×D₄	—	122	2	no	—
258	#117	—	122	2	no	—
259	C₂×C₄²	—	61	1	no	—
260	#107	—	61	1	no	—
261	#107	C₂×#6923	1	1	yes	characteristic
262	#106	—	61	2	no	—
263	C₄⋊C₈	—	61	1	no	—
264	C₄⋊C₈	—	61	1	no	—
265	C₄⋊C₈	—	61	2	no	—
266	C₄⋊C₈	—	61	2	no	—
267	C₄⋊C₈	—	122	1	no	—
268	C₄⋊C₈	—	122	2	no	—
269	C₄⋊C₈	—	122	2	no	—
270	C₄⋊C₈	—	122	2	no	—
271	#103	—	61	2	no	—
272	#103	C₂×#6923	1	2	yes	—
273	#102	—	61	2	no	—
274	#102	—	61	2	no	—
275	C₂²⋊C₈	—	61	1	no	—
276	C₂²⋊C₈	—	61	1	no	—
277	C₂²⋊C₈	—	122	2	no	—
278	C₂²⋊C₈	—	122	2	no	—
279	C₂²⋊C₈	—	122	1	no	—
280	#97	—	122	2	no	—
281	C₄×C₈	—	61	1	no	—
282	C₄×C₈	—	61	1	no	—
283	C₄×C₈	—	61	1	no	—
284	C₄×C₈	—	61	1	no	—
285	C₄×C₈	—	61	1	no	—
286	C₄×C₈	—	61	1	no	—
287	C₄×C₈	—	61	1	no	—
288	C₄×C₈	—	61	1	no	—
289	C₄×C₈	—	122	2	no	—
290	C₂×Q₈	C₂²×#6923	1	2	yes	—
291	C₂×D₄	—	61	2	no	—
292	C₂×D₄	—	122	2	no	—
293	C₂²×C₄	—	61	1	no	—
294	C₂²×C₄	—	61	1	no	—
295	C₂²×C₄	—	122	2	no	—
296	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
297	Q₁₆	—	2	2	no	—
298	SD₁₆	—	122	4	no	—
299	D₈	—	122	2	no	—
300	M₁₆	—	488	2	no	—
301	M₁₆	—	488	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	61	1	no	—
304	C₂×C₈	—	61	1	no	—
305	C₂×C₈	—	61	1	no	—
306	C₂×C₈	—	61	1	no	—
307	C₂×C₈	—	61	1	no	—
308	C₂×C₈	—	61	1	no	—
309	C₂×C₈	—	122	2	no	—
310	C₂×C₈	—	122	1	no	—
311	C₂×C₈	—	122	1	no	—
312	C₂×C₈	—	122	2	no	—
313	C₂×C₈	—	122	2	no	—
314	C₂×C₈	—	488	2	no	—
315	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
316	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	61	2	no	—
319	C₄⋊C₄	—	122	2	no	—
320	C₄⋊C₄	—	122	2	no	—
321	C₄⋊C₄	—	122	2	no	—
322	C₄⋊C₄	C₂²×#6923	1	2	yes	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₂²⋊C₄	—	122	2	no	—
325	C₂²⋊C₄	—	122	2	no	—
326	C₂²⋊C₄	—	122	2	no	—
327	C₂²⋊C₄	—	122	2	no	—
328	C₄²	—	61	1	no	—
329	C₄²	—	61	1	no	—
330	C₄²	—	61	1	no	—
331	C₄²	—	61	1	no	—
332	C₄²	—	122	2	no	—
333	C₂³	—	122	2	no	—
334	C₂³	D₄×#6923	1	1	yes	characteristic
335	Q₈	—	2	2	no	—
336	Q₈	—	2	2	no	—
337	D₄	—	122	2	no	—
338	D₄	—	122	2	no	—
339	D₄	—	488	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	61	1	no	—
345	C₂×C₄	—	61	1	no	—
346	C₂×C₄	—	61	1	no	—
347	C₂×C₄	—	61	1	no	—
348	C₂×C₄	—	122	1	no	—
349	C₂×C₄	—	122	1	no	—
350	C₂×C₄	—	122	2	no	—
351	C₂×C₄	—	244	2	no	—
352	C₂×C₄	—	244	2	no	—
353	C₂×C₄	—	244	1	no	—
354	C₂×C₄	D₄×#6923	1	1	yes	characteristic
355	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	61	2	no	—
358	C₈	—	122	1	no	—
359	C₈	—	122	1	no	—
360	C₈	—	122	1	no	—
361	C₈	—	122	1	no	—
362	C₈	—	122	1	no	—
363	C₈	—	244	2	no	—
364	C₈	—	244	2	no	—
365	C₈	—	244	1	no	—
366	C₈	—	244	1	no	—
367	C₈	#14519864	1	2	yes	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	244	2	no	—
372	C₂²	—	244	2	no	—
373	C₂²	—	244	2	no	—
374	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	61	1	no	—
379	C₄	—	61	1	no	—
380	C₄	—	122	1	no	—
381	C₄	—	122	1	no	—
382	C₄	—	122	1	no	—
383	C₄	—	244	2	no	—
384	C₄	C₂×#14519864	1	1	yes	characteristic
385	C₄	D₆₁⋊#131	1	1	yes	characteristic
386	C₂	—	4	1	no	—
387	C₂	—	488	2	no	—
388	C₂	#21462542	1	1	yes	characteristic
389	C₂	#21462581	1	1	yes	characteristic
390	C₂	#21462599	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#467	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	#16839998	C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	#467	#6923
hypocenter	C₆₁	#17973
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
1. #21462764	C₂	semidirect
2. #21461463	C₂	semidirect
3. #21461459	C₂	semidirect
4. #21461451	C₂	semidirect
5. #21461409	C₂	semidirect
6. Q₁₆×#92420	C₂	semidirect
7. #21461400	C₂	semidirect
8. #21461376	C₂	semidirect
9. #21461176	C₂	semidirect
10. #21461022	C₂	semidirect
240. C₆₁	#17973	semidirect

Maximal subgroups

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

subgroup	index
#21462764	2
#21461463	2
#21461459	2
#21461451	2
#21461409	2
Q₁₆×#92420	2
#21461400	2
#21461376	2
#21461176	2
#21461022	2
#21456390	2
#17973	61