Subgroups — #30612013

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462736	C₂	1	2	yes	maximal
2	#21461372	C₂	1	2	yes	maximal, complemented
3	#21461371	C₂	1	4	yes	maximal, complemented
4	#21461367	C₂	1	2	yes	maximal, complemented
5	#21461366	C₂	1	1	yes	maximal, characteristic, complemented
6	D₆₁⋊#334	C₂	1	2	yes	maximal, complemented
7	C₂×#16840623	C₂	1	1	yes	maximal, characteristic, complemented
8	#21454241	C₂	1	1	yes	maximal, characteristic
9	#16840811	—	2	2	no	—
10	#16840790	—	2	2	no	—
11	#16840789	—	2	4	no	—
12	#16840781	C₂²	1	4	yes	—
13	#16840778	C₂²	1	2	yes	—
14	D₆₁⋊(C₄⋊C₈)	C₂²	1	4	yes	—
15	C₂×C₄⋊#92420	C₂²	1	2	yes	—
16	#16840626	C₂²	1	4	yes	—
17	#16840625	C₂²	1	2	yes	—
18	#16840624	C₂²	1	4	yes	—
19	#16840623	C₂²	1	2	yes	—
20	#16840623	C₂²	1	2	yes	—
21	#16840623	C₂²	1	1	yes	characteristic
22	#16840623	C₂²	1	1	yes	characteristic
23	C₂×#14519785	C₄	1	1	yes	characteristic
24	C₂×#14519783	C₄	1	1	yes	characteristic
25	#16840230	C₂²	1	1	yes	characteristic
26	#16839719	C₄	1	2	yes	—
27	#16839718	C₄	1	1	yes	characteristic
28	#16839716	C₄	1	2	yes	—
29	#16839714	C₂²	1	2	yes	—
30	M₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₆₁	C₂²	1	2	yes	—
33	C₆₁×#405	C₄	1	1	yes	characteristic
34	C₂×D₆₁⋊C₈	—	2	4	no	—
35	C₂²×#92420	—	2	2	no	—
36	#14519844	—	2	4	no	—
37	C₄⋊#92420	—	2	2	no	—
38	C₄⋊#92420	C₂³	1	4	yes	—
39	C₄⋊#92420	C₂³	1	2	yes	—
40	C₄⋊#92420	C₂³	1	2	yes	—
41	#14519834	—	2	4	no	—
42	C₂²⋊#92420	—	2	2	no	—
43	C₄×#92420	—	2	2	no	—
44	C₄×#92420	—	2	4	no	—
45	C₂×#11758222	C₂×C₄	1	1	yes	characteristic
46	#14519785	—	2	4	no	—
47	#14519783	C₈	1	8	yes	complemented
48	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
49	C₂×#11758189	C₂×C₄	1	2	yes	—
50	C₂×#11758189	C₂×C₄	1	2	yes	—
51	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
54	#14519742	D₄	1	1	yes	characteristic
55	#14519742	Q₈	1	1	yes	characteristic
56	#14519739	D₄	1	1	yes	characteristic
57	#14519739	Q₈	1	1	yes	characteristic
58	#14519737	D₄	1	2	yes	—
59	#14519737	Q₈	1	2	yes	—
60	#14519734	C₂×C₄	1	2	yes	—
61	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
62	C₄²×D₆₁	C₂³	1	2	yes	—
63	#14519688	C₂×C₄	1	2	yes	—
64	#14519687	C₂×C₄	1	2	yes	—
65	C₄×#92418	C₂×C₄	1	2	yes	—
66	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
69	C₈×Dic₆₁	C₂³	1	2	yes	—
70	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
71	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
73	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
74	#97×C₆₁	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
76	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₆₁	C₄⋊C₄	1	2	yes	—
78	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
80	#11758222	C₂×C₈	1	4	yes	—
81	#11758222	M₁₆	1	4	yes	—
82	C₂×D₂₄₄	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
85	C₂×C₄×D₆₁	C₂×Q₈	1	2	yes	—
86	C₂×Dic₁₂₂	C₄⋊C₄	1	1	yes	characteristic
87	C₂×#92420	—	2	2	no	—
88	C₂×#92420	—	2	2	no	—
89	C₂×#92420	—	2	4	no	—
90	C₂×#92420	—	8	2	no	—
91	D₆₁⋊C₈	—	8	4	no	—
92	M₁₆×C₆₁	—	2	2	no	—
93	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	complemented
94	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
99	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
100	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
101	#11758198	C₄⋊C₄	1	2	yes	—
102	#11758198	C₄⋊C₄	1	2	yes	—
103	#11758196	C₄⋊C₄	1	2	yes	—
104	#11758196	C₄⋊C₄	1	2	yes	—
105	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
107	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
108	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
109	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
110	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
111	#11758194	C₂×C₈	1	4	yes	complemented
112	C₂×#92418	—	2	2	no	—
113	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
114	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
115	C₂×#92418	C₂²×C₄	1	2	yes	—
116	#11758189	—	2	4	no	—
117	#11758189	C₂×C₈	1	8	yes	complemented
118	C₈×D₆₁	—	2	4	no	—
119	C₈×D₆₁	C₂×C₈	1	8	yes	complemented
120	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×D₆₁	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₆₁	C₄⋊C₈	1	8	yes	—
129	C₂×Dic₆₁	—	2	2	no	—
130	C₂×Dic₆₁	—	4	1	no	—
131	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
136	D₂₄₄	C₄⋊C₈	1	4	yes	—
137	C₄×D₆₁	—	2	4	no	—
138	C₄×D₆₁	—	4	2	no	—
139	C₄×D₆₁	C₂²×C₈	1	4	yes	—
140	C₄×D₆₁	C₂×M₁₆	1	4	yes	—
141	Dic₁₂₂	C₄⋊C₈	1	4	yes	—
142	#92420	—	4	2	no	—
143	#92420	—	4	4	no	—
144	#92420	—	4	2	no	—
145	C₈×C₆₁	—	2	2	no	—
146	C₈×C₆₁	C₂²×C₈	1	2	yes	—
147	C₈×C₆₁	C₂²×C₈	1	2	yes	—
148	#92418	—	2	1	no	—
149	#92418	—	2	1	no	—
150	#92418	C₂²×C₈	1	4	yes	—
151	#17788	—	61	1	no	maximal, Sylow
152	C₂²×C₆₁	—	2	1	no	—
153	C₂²×C₆₁	—	2	1	no	—
154	C₂²×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
155	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂×D₆₁	—	2	2	no	—
157	C₂×D₆₁	—	2	2	no	—
158	C₂×D₆₁	C₂×C₄⋊C₈	1	4	yes	—
159	C₄×C₆₁	—	2	2	no	—
160	C₄×C₆₁	—	2	2	no	—
161	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
162	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
163	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
164	Dic₆₁	—	2	1	no	—
165	Dic₆₁	—	2	1	no	—
166	Dic₆₁	—	2	1	no	—
167	Dic₆₁	C₂×C₄⋊C₈	1	4	yes	—
168	Dic₆₁	#577	1	2	yes	—
169	#2988	—	61	2	no	—
170	C₄×#131	—	61	1	no	—
171	#1565	—	61	4	no	—
172	#1565	—	61	2	no	—
173	#1564	—	61	1	no	—
174	#1564	—	61	2	no	—
175	C₂×#334	—	61	1	no	—
176	C₂×#334	—	61	2	no	—
177	C₂×C₆₁	—	2	2	no	—
178	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
179	C₂×C₆₁	#2912	1	1	yes	characteristic
180	C₂×C₆₁	#2916	1	1	yes	characteristic
181	D₆₁	—	2	4	no	—
182	C₂×#131	—	61	1	no	—
183	C₂×#131	—	61	1	no	—
184	C₄×#41	—	61	1	no	—
185	C₈×Q₈	—	122	2	no	—
186	C₈×D₄	—	122	4	no	—
187	C₈×D₄	—	122	2	no	—
188	#431	—	61	2	no	—
189	#431	—	61	4	no	—
190	C₂×C₄⋊C₈	—	61	2	no	—
191	C₂×C₄⋊C₈	—	61	4	no	—
192	#405	—	61	1	no	—
193	#405	—	61	2	no	—
194	#405	—	61	2	no	—
195	#405	#6923	1	1	yes	characteristic
196	C₄×M₁₆	—	61	1	no	—
197	C₄×M₁₆	—	61	2	no	—
198	C₂×C₄×C₈	—	61	1	no	—
199	C₂×C₄×C₈	—	61	2	no	—
200	#335	—	61	2	no	—
201	#335	—	61	4	no	—
202	#334	—	61	2	no	—
203	#334	—	61	1	no	—
204	#334	—	61	2	no	—
205	#334	—	61	4	no	—
206	#334	—	61	1	no	—
207	C₆₁	#17788	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	61	1	no	—
209	#131	—	61	8	no	—
210	#131	—	122	4	no	—
211	C₂×M₁₆	—	61	1	no	—
212	C₂×M₁₆	—	61	2	no	—
213	C₂×M₁₆	—	61	2	no	—
214	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
215	C₂²×C₈	—	61	2	no	—
216	C₂²×C₈	—	61	1	no	—
217	C₂²×C₈	—	61	2	no	—
218	C₂²×C₈	—	122	2	no	—
219	C₂²×C₈	—	122	4	no	—
220	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
221	C₄×Q₈	—	61	1	no	—
222	C₄×Q₈	—	61	1	no	—
223	C₄×D₄	—	61	1	no	—
224	C₄×D₄	—	61	1	no	—
225	C₄×D₄	—	61	2	no	—
226	C₄×D₄	—	61	2	no	—
227	#117	—	61	2	no	—
228	#117	C₂×#6923	1	1	yes	characteristic
229	C₂×C₄²	—	61	1	no	—
230	C₂×C₄²	—	61	2	no	—
231	C₄⋊C₈	—	61	4	no	—
232	C₄⋊C₈	—	61	2	no	—
233	C₄⋊C₈	—	61	2	no	—
234	C₄⋊C₈	—	122	2	no	—
235	C₄⋊C₈	—	122	4	no	—
236	C₂²⋊C₈	—	122	2	no	—
237	C₂²⋊C₈	—	122	4	no	—
238	#97	—	61	2	no	—
239	#97	—	61	2	no	—
240	#97	C₂×#6923	1	2	yes	—
241	C₄×C₈	—	61	1	no	—
242	C₄×C₈	—	61	2	no	—
243	C₄×C₈	—	61	1	no	—
244	C₄×C₈	—	61	1	no	—
245	C₄×C₈	—	61	1	no	—
246	C₄×C₈	—	61	2	no	—
247	C₄×C₈	—	122	2	no	—
248	C₄×C₈	—	122	4	no	—
249	C₄×C₈	C₂×#6923	1	2	yes	—
250	#41	—	61	4	no	—
251	#41	—	61	4	no	—
252	C₂×Q₈	—	61	1	no	—
253	C₂×D₄	—	61	2	no	—
254	C₂×D₄	—	61	1	no	—
255	C₂²×C₄	—	61	2	no	—
256	C₂²×C₄	—	61	1	no	—
257	C₂²×C₄	—	61	1	no	—
258	C₂²×C₄	—	61	2	no	—
259	C₂²×C₄	—	61	2	no	—
260	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
261	M₁₆	—	2	2	no	—
262	M₁₆	—	61	4	no	—
263	M₁₆	—	61	8	no	—
264	M₁₆	—	122	4	no	—
265	C₂×C₈	—	61	2	no	—
266	C₂×C₈	—	61	8	no	—
267	C₂×C₈	—	61	1	no	—
268	C₂×C₈	—	61	1	no	—
269	C₂×C₈	—	122	4	no	—
270	C₂×C₈	—	122	2	no	—
271	C₂×C₈	—	122	2	no	—
272	C₂×C₈	—	122	2	no	—
273	C₂×C₈	—	122	4	no	—
274	C₂×C₈	—	488	4	no	—
275	C₂×C₈	—	488	2	no	—
276	C₂×C₈	D₆₁⋊C₈	1	4	yes	complemented
277	C₂×C₈	C₂²×#6923	1	2	yes	—
278	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
279	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
280	C₄⋊C₄	—	61	2	no	—
281	C₄⋊C₄	—	61	2	no	—
282	C₄⋊C₄	C₄⋊#6923	1	1	yes	characteristic
283	C₄⋊C₄	C₄⋊#6923	1	1	yes	characteristic
284	C₂²⋊C₄	—	61	2	no	—
285	C₂²⋊C₄	—	61	2	no	—
286	C₂²⋊C₄	C₄⋊#6923	1	2	yes	—
287	C₄²	—	61	2	no	—
288	C₄²	—	61	1	no	—
289	C₄²	—	61	2	no	—
290	C₄²	—	61	1	no	—
291	C₄²	—	61	1	no	—
292	C₄²	—	61	1	no	—
293	C₄²	C₂²×#6923	1	2	yes	—
294	C₂³	—	61	2	no	—
295	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
296	Q₈	—	61	4	no	—
297	D₄	—	61	8	no	—
298	D₄	—	61	4	no	—
299	C₂×C₄	—	61	2	no	—
300	C₂×C₄	—	61	1	no	—
301	C₂×C₄	—	61	4	no	—
302	C₂×C₄	—	61	4	no	—
303	C₂×C₄	—	61	1	no	—
304	C₂×C₄	—	61	1	no	—
305	C₂×C₄	—	61	1	no	—
306	C₂×C₄	—	122	4	no	—
307	C₂×C₄	—	122	2	no	—
308	C₂×C₄	—	244	2	no	—
309	C₂×C₄	—	244	1	no	—
310	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
313	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
314	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
315	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	61	4	no	—
318	C₈	—	122	1	no	—
319	C₈	—	122	1	no	—
320	C₈	—	244	2	no	—
321	C₈	—	244	2	no	—
322	C₈	—	244	4	no	—
323	C₈	C₂×D₆₁⋊C₈	1	2	yes	—
324	C₈	C₂×D₆₁⋊C₈	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	61	4	no	—
328	C₂²	—	122	2	no	—
329	C₂²	—	122	2	no	—
330	C₂²	D₆₁⋊(C₄⋊C₈)	1	2	yes	—
331	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	61	4	no	—
335	C₄	—	61	2	no	—
336	C₄	—	122	1	no	—
337	C₄	—	122	1	no	—
338	C₄	—	122	1	no	—
339	C₄	D₆₁⋊(C₄⋊C₈)	1	2	yes	—
340	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
341	C₄	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	122	4	no	—
344	C₂	C₂×D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
345	C₂	#21462473	1	1	yes	characteristic
346	C₂	#21462480	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6923
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#405	C₄
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#11758222	C₂×C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	#405	#6923
hypocenter	C₆₁	#17788
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #21461372	C₂	semidirect
3. #21461371	C₂	semidirect
4. #21461367	C₂	semidirect
5. #21461366	C₂	semidirect
6. D₆₁⋊#334	C₂	semidirect
7. C₂×#16840623	C₂	semidirect
47. #14519783	C₈	semidirect
93. C₂×C₈×C₆₁	C₂×C₈	semidirect
111. #11758194	C₂×C₈	semidirect
117. #11758189	C₂×C₈	semidirect
119. C₈×D₆₁	C₂×C₈	semidirect
207. C₆₁	#17788	semidirect
276. C₂×C₈	D₆₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#21462736	2
#21461372	2
#21461371	2
#21461367	2
#21461366	2
D₆₁⋊#334	2
C₂×#16840623	2
#21454241	2
#17788	61