Subgroups — #29662650

Order 15616.

586 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21455854	C₂	1	2	yes	maximal
2	C₂×#16839892	C₂	1	1	yes	maximal, characteristic
3	#21445344	C₂	1	2	yes	maximal, complemented
4	#21445334	C₂	1	2	yes	maximal, complemented
5	#21445322	C₂	1	4	yes	maximal, complemented
6	#21444567	C₂	1	1	yes	maximal, characteristic
7	#21444476	C₂	1	2	yes	maximal
8	C₆₁×#3160	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
10	#16840095	—	2	2	no	—
11	#16840054	C₂²	1	4	yes	—
12	#16839969	C₂²	1	4	yes	—
13	#16839917	—	2	2	no	—
14	#16839892	—	2	1	no	—
15	#16839892	—	2	1	no	—
16	#16839892	C₂²	1	2	yes	—
17	#16839892	C₂²	1	2	yes	—
18	C₄×#11758202	—	2	4	no	—
19	#16839815	C₂²	1	2	yes	—
20	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
21	C₂×#14519706	—	2	1	no	—
22	C₂×#14519687	—	2	1	no	—
23	#16839515	C₂²	1	2	yes	—
24	#16839513	C₂²	1	2	yes	—
25	#16839511	C₂²	1	4	yes	—
26	#16839508	C₂²	1	4	yes	—
27	C₆₁×#533	C₂²	1	2	yes	—
28	C₆₁×#500	C₂²	1	1	yes	characteristic
29	C₆₁×#500	C₂²	1	1	yes	characteristic
30	C₆₁×#499	—	2	1	no	—
31	C₆₁×#499	—	2	1	no	—
32	C₆₁×#498	C₂²	1	2	yes	—
33	C₆₁×#427	—	2	1	no	—
34	C₆₁×#427	—	2	1	no	—
35	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₂₂	D₄	1	1	yes	characteristic
37	C₂×#11758202	—	2	4	no	—
38	C₂²×#92418	—	2	1	no	—
39	C₄×Dic₁₂₂	—	2	1	no	—
40	C₄×Dic₁₂₂	—	2	1	no	—
41	C₄×Dic₁₂₂	D₄	1	2	yes	—
42	C₄×Dic₁₂₂	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
44	C₂×#11758196	—	2	1	no	—
45	C₂×C₄×Dic₆₁	—	2	1	no	—
46	#14519710	—	2	4	no	—
47	C₄⋊#92418	—	2	1	no	—
48	C₄⋊#92418	C₂³	1	4	yes	—
49	C₄⋊#92418	C₂³	1	2	yes	—
50	#14519706	—	2	2	no	—
51	#14519706	—	2	2	no	—
52	#14519701	—	2	4	no	—
53	#14519700	—	2	4	no	—
54	C₂²⋊#92418	—	2	2	no	—
55	#14519687	—	2	2	no	—
56	#14519687	—	2	2	no	—
57	C₄×#92418	—	2	2	no	—
58	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
62	#125×C₆₁	—	2	2	no	—
63	C₂²⋊Q₈×C₆₁	—	2	2	no	—
64	C₂²⋊Q₈×C₆₁	—	2	2	no	—
65	C₄×Q₈×C₆₁	—	2	2	no	—
66	C₄×Q₈×C₆₁	—	2	2	no	—
67	#117×C₆₁	—	2	2	no	—
68	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
69	#107×C₆₁	—	2	1	no	—
70	#107×C₆₁	—	2	1	no	—
71	#107×C₆₁	—	2	1	no	—
72	#107×C₆₁	—	2	1	no	—
73	#106×C₆₁	—	2	2	no	—
74	#106×C₆₁	—	2	2	no	—
75	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
76	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₆₁	—	2	2	no	—
80	C₂×Q₈×C₆₁	—	2	2	no	—
81	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
84	C₂²×Dic₆₁	—	2	1	no	—
85	C₂²×Dic₆₁	—	2	1	no	—
86	C₂×Dic₁₂₂	—	2	1	no	—
87	C₂×Dic₁₂₂	—	4	2	no	—
88	C₂×Dic₁₂₂	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₁	—	2	2	no	—
90	C₂×C₈×C₆₁	Q₁₆	1	4	yes	complemented
91	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₄⋊C₄×C₆₁	—	4	2	no	—
100	C₄⋊C₄×C₆₁	—	4	2	no	—
101	C₂²⋊C₄×C₆₁	—	4	2	no	—
102	C₄²×C₆₁	—	4	2	no	—
103	C₄²×C₆₁	C₂×D₄	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	#11758202	—	4	8	no	—
108	C₄⋊Dic₆₁	—	2	1	no	—
109	C₄⋊Dic₆₁	C₂×D₄	1	2	yes	—
110	#11758196	—	2	2	no	—
111	#11758196	—	2	2	no	—
112	C₄×Dic₆₁	—	2	2	no	—
113	C₄×Dic₆₁	—	2	2	no	—
114	C₂×#92418	—	2	4	no	—
115	C₂×#92418	—	2	2	no	—
116	C₂×#92418	—	4	2	no	—
117	C₂³×C₆₁	#121	1	1	yes	characteristic
118	Q₈×C₆₁	—	4	4	no	—
119	Q₈×C₆₁	—	8	2	no	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	—	4	1	no	—
122	C₂×C₄×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	—	4	2	no	—
124	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
125	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
126	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
128	C₂×C₄×C₆₁	#137	1	2	yes	—
129	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₆₁	—	2	2	no	—
133	C₂×Dic₆₁	—	2	2	no	—
134	C₂×Dic₆₁	—	4	2	no	—
135	C₂×Dic₆₁	—	4	2	no	—
136	Dic₁₂₂	—	2	4	no	—
137	Dic₁₂₂	—	4	2	no	—
138	C₈×C₆₁	—	2	1	no	—
139	C₈×C₆₁	—	2	1	no	—
140	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
141	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
142	#92418	—	4	2	no	—
143	#92418	—	4	4	no	—
144	#19150	—	61	1	no	maximal, Sylow
145	C₂²×C₆₁	—	2	1	no	—
146	C₂²×C₆₁	—	2	1	no	—
147	C₂²×C₆₁	#462	1	2	yes	—
148	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	—	2	1	no	—
151	C₄×C₆₁	—	2	1	no	—
152	C₄×C₆₁	—	8	2	no	—
153	C₄×C₆₁	—	8	2	no	—
154	C₄×C₆₁	#462	1	2	yes	—
155	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
156	C₄×C₆₁	C₂×#137	1	1	yes	characteristic
157	C₄×C₆₁	#575	1	2	yes	—
158	Dic₆₁	—	4	2	no	—
159	Dic₆₁	—	4	2	no	—
160	#3160	D₆₁	1	1	yes	characteristic
161	#3104	—	61	2	no	—
162	C₂×#446	—	61	1	no	—
163	#1696	—	61	2	no	—
164	#1694	—	61	2	no	—
165	#1692	—	61	4	no	—
166	#1564	—	61	1	no	—
167	#1542	—	61	2	no	—
168	C₂×C₆₁	—	2	2	no	—
169	C₂×C₆₁	C₂×#462	1	1	yes	characteristic
170	C₂×C₆₁	#3054	1	1	yes	characteristic
171	C₂×C₆₁	#3067	1	1	yes	characteristic
172	#533	C₂×D₆₁	1	2	yes	—
173	C₂×C₄×Q₈	—	61	1	no	—
174	#500	C₂×D₆₁	1	1	yes	characteristic
175	#500	C₂×D₆₁	1	1	yes	characteristic
176	#499	—	2	1	no	—
177	#499	—	2	1	no	—
178	#498	C₂×D₆₁	1	2	yes	—
179	#484	—	122	2	no	—
180	#477	—	61	4	no	—
181	#462	—	61	4	no	—
182	#451	—	122	2	no	—
183	#446	—	61	2	no	—
184	#446	—	61	2	no	—
185	#446	—	122	1	no	—
186	#446	—	122	1	no	—
187	C₄×Q₁₆	—	122	4	no	—
188	#431	—	61	2	no	—
189	#427	—	2	1	no	—
190	#427	—	2	1	no	—
191	C₂×C₄⋊C₈	—	61	1	no	—
192	C₂×C₄⋊C₈	—	122	1	no	—
193	C₂×#97	—	122	1	no	—
194	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
195	#335	—	61	2	no	—
196	#334	—	61	2	no	—
197	#333	—	61	4	no	—
198	#332	—	61	4	no	—
199	C₆₁	#19150	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	61	1	no	—
201	C₂×Q₁₆	—	122	4	no	—
202	C₂²×C₈	—	122	1	no	—
203	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
204	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
205	C₄⋊Q₈	C₂²×D₆₁	1	2	yes	—
206	C₄⋊Q₈	C₂²×D₆₁	1	2	yes	—
207	#125	—	2	2	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	61	2	no	—
213	C₄×Q₈	—	61	2	no	—
214	C₄×Q₈	—	122	1	no	—
215	C₄×Q₈	—	122	1	no	—
216	#117	—	2	2	no	—
217	C₂×C₄⋊C₄	—	61	1	no	—
218	C₂×C₄⋊C₄	—	122	1	no	—
219	C₂×C₄²	—	122	1	no	—
220	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#107	—	2	1	no	—
225	#107	—	122	4	no	—
226	#106	—	2	2	no	—
227	#106	—	2	2	no	—
228	C₄⋊C₈	—	61	2	no	—
229	C₄⋊C₈	—	61	4	no	—
230	C₄⋊C₈	—	122	2	no	—
231	C₄⋊C₈	—	122	2	no	—
232	C₄⋊C₈	—	122	1	no	—
233	#103	—	122	4	no	—
234	#103	—	122	4	no	—
235	C₂²⋊C₈	—	122	2	no	—
236	#97	—	122	2	no	—
237	#97	—	122	2	no	—
238	C₄×C₈	—	122	2	no	—
239	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
240	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	61	2	no	—
246	C₂×Q₈	—	122	1	no	—
247	C₂×Q₈	—	244	2	no	—
248	C₂²×C₄	—	122	1	no	—
249	C₂²×C₄	—	122	1	no	—
250	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
251	C₂²×C₄	#11758224	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
253	Q₁₆	—	244	8	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	122	4	no	—
256	C₂×C₈	—	122	2	no	—
257	C₂×C₈	—	244	2	no	—
258	C₂×C₈	#11758202	1	4	yes	complemented
259	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	61	2	no	—
270	C₄⋊C₄	—	122	2	no	—
271	C₄⋊C₄	—	122	1	no	—
272	C₄⋊C₄	—	122	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	—	122	2	no	—
276	C₄²	—	122	2	no	—
277	C₄²	D₄×D₆₁	1	1	yes	characteristic
278	C₄²	#11758224	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
280	C₄²	C₂³×D₆₁	1	1	yes	characteristic
281	C₂³	#14519748	1	1	yes	characteristic
282	Q₈	—	4	4	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	122	4	no	—
285	Q₈	—	244	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	122	2	no	—
291	C₂×C₄	—	122	2	no	—
292	C₂×C₄	—	244	2	no	—
293	C₂×C₄	—	244	2	no	—
294	C₂×C₄	#14519748	1	1	yes	characteristic
295	C₂×C₄	#14519748	1	1	yes	characteristic
296	C₂×C₄	#14519748	1	1	yes	characteristic
297	C₂×C₄	C₂×#11758202	1	2	yes	—
298	C₂×C₄	#14519808	1	2	yes	—
299	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
301	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	244	2	no	—
305	C₈	—	244	4	no	—
306	C₈	C₂×#11758202	1	2	yes	—
307	C₈	C₂×#11758202	1	2	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	#16839969	1	2	yes	—
311	C₂²	C₂×#14519748	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	244	2	no	—
318	C₄	—	244	2	no	—
319	C₄	#16839969	1	2	yes	—
320	C₄	C₂²×#11758202	1	1	yes	characteristic
321	C₄	C₂×#14519808	1	1	yes	characteristic
322	C₄	#16840559	1	2	yes	—
323	C₂	—	2	2	no	—
324	C₂	C₂×#16839969	1	1	yes	characteristic
325	C₂	#21455424	1	1	yes	characteristic
326	C₂	#21455535	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519748
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#3160	C₂
socle	C₂²×C₆₁	C₂×#121
prime radical	C₂³×C₆₁	#121
hypercenter	#3160	D₆₁
hypocenter	C₆₁	#19150
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #21445344	C₂	semidirect
4. #21445334	C₂	semidirect
5. #21445322	C₂	semidirect
90. C₂×C₈×C₆₁	Q₁₆	semidirect
199. C₆₁	#19150	semidirect
258. C₂×C₈	#11758202	semidirect

Maximal subgroups

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

subgroup	index
#21455854	2
C₂×#16839892	2
#21445344	2
#21445334	2
#21445322	2
#21444567	2
#21444476	2
C₆₁×#3160	2
#19150	61