Subgroups — #29837718

Order 15616.

484 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458829	C₂	1	1	yes	maximal, characteristic
2	#21458712	C₂	1	1	yes	maximal, characteristic, complemented
3	#21448052	C₂	1	2	yes	maximal, complemented
4	#21448044	C₂	1	1	yes	maximal, characteristic
5	#21447975	C₂	1	2	yes	maximal, complemented
6	#21447884	C₂	1	2	yes	maximal, complemented
7	#21447860	C₂	1	1	yes	maximal, characteristic
8	#21447721	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14519723	C₂	1	1	yes	maximal, characteristic, complemented
10	#21447670	C₂	1	2	yes	maximal, complemented
11	C₆₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840497	—	2	1	no	—
13	#16840487	—	2	1	no	—
14	C₂×#14519723	C₂²	1	1	yes	characteristic
15	C₂×#14519722	C₂²	1	2	yes	—
16	#16840163	C₂²	1	2	yes	—
17	#16840147	C₂²	1	2	yes	—
18	#16840139	C₂²	1	1	yes	characteristic
19	#16840068	—	2	2	no	—
20	#16840044	—	2	1	no	—
21	#16839980	—	2	2	no	—
22	#16839962	—	2	1	no	—
23	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
25	#16839856	C₂²	1	2	yes	—
26	#16839844	C₂²	1	2	yes	—
27	#16839587	C₂²	1	1	yes	characteristic
28	#16839580	C₂²	1	2	yes	—
29	#16839566	C₂²	1	1	yes	characteristic
30	#16839565	C₂²	1	1	yes	characteristic
31	#16839564	C₂²	1	2	yes	—
32	#16839562	C₂²	1	2	yes	—
33	#16839562	C₂²	1	2	yes	—
34	C₄×#11758185	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₆₁	C₂²	1	2	yes	—
36	C₆₁×#500	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
38	C₆₁×#367	C₂²	1	2	yes	—
39	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₆₁×#358	C₂²	1	2	yes	—
41	C₂×Dic₂₄₄	C₂³	1	1	yes	characteristic
42	C₂×#11758201	—	2	2	no	—
43	C₂×D₄₈₈	—	2	1	no	—
44	C₂×D₄₈₈	C₂³	1	2	yes	—
45	#14519775	—	4	1	no	—
46	#14519774	—	2	1	no	—
47	#14519768	—	2	2	no	—
48	#14519767	—	4	2	no	—
49	#14519766	—	2	2	no	—
50	#14519765	—	4	1	no	—
51	#14519762	—	4	2	no	—
52	Q₈×Dic₆₁	—	4	1	no	—
53	C₄×Dic₁₂₂	—	2	1	no	—
54	C₄×Dic₁₂₂	D₄	1	1	yes	characteristic
55	#14519742	—	2	2	no	—
56	C₄×D₂₄₄	—	4	1	no	—
57	#14519739	—	2	2	no	—
58	#14519723	—	2	2	no	—
59	#14519722	—	2	4	no	—
60	C₂×#11758185	C₂³	1	1	yes	characteristic
61	C₂×#11758185	C₂³	1	1	yes	characteristic
62	#14519711	C₂³	1	1	yes	characteristic
63	#14519708	—	2	2	no	—
64	C₄⋊#92418	D₄	1	1	yes	characteristic
65	#14519706	D₄	1	2	yes	—
66	#14519702	—	2	1	no	—
67	#14519700	—	2	2	no	—
68	#14519697	—	2	2	no	—
69	C₈×Dic₆₁	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₆₁	—	2	1	no	—
72	C₄×Q₈×C₆₁	—	2	1	no	—
73	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
74	#107×C₆₁	—	2	1	no	—
75	#107×C₆₁	C₂³	1	2	yes	—
76	#107×C₆₁	C₂³	1	1	yes	characteristic
77	#103×C₆₁	—	2	1	no	—
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₆₁	—	2	1	no	—
80	C₂×Q₈×C₆₁	—	4	1	no	—
81	C₂×D₂₄₄	—	2	2	no	—
82	C₂×D₂₄₄	—	4	1	no	—
83	C₂×D₂₄₄	—	4	1	no	—
84	C₂×C₄×D₆₁	—	4	2	no	—
85	C₂×Dic₁₂₂	—	2	1	no	—
86	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
87	Q₁₆×C₆₁	—	2	1	no	—
88	Q₁₆×C₆₁	—	2	1	no	—
89	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₆₁	—	2	1	no	—
92	C₄⋊C₄×C₆₁	—	2	2	no	—
93	C₄⋊C₄×C₆₁	—	4	1	no	—
94	C₄⋊C₄×C₆₁	—	4	1	no	—
95	C₄²×C₆₁	—	4	1	no	—
96	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
97	#11758201	—	8	2	no	—
98	#11758198	—	4	2	no	—
99	#11758198	—	4	2	no	—
100	#11758198	—	4	2	no	—
101	C₄⋊Dic₆₁	—	2	1	no	—
102	C₄⋊Dic₆₁	—	4	2	no	—
103	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
104	#11758196	—	4	1	no	—
105	#11758196	—	4	2	no	—
106	#11758196	—	4	2	no	—
107	#11758196	D₈	1	2	yes	—
108	C₄×Dic₆₁	—	4	1	no	—
109	C₄×Dic₆₁	—	4	2	no	—
110	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
111	C₂×#92418	C₂×D₄	1	2	yes	—
112	Dic₂₄₄	—	2	1	no	—
113	Dic₂₄₄	#41	1	2	yes	—
114	D₄₈₈	—	2	2	no	—
115	D₄₈₈	—	2	2	no	—
116	D₄₈₈	—	8	1	no	—
117	C₁₆×C₆₁	—	2	2	no	—
118	#11758185	—	2	1	no	—
119	#11758185	—	2	1	no	—
120	C₂²×D₆₁	—	4	2	no	—
121	Q₈×C₆₁	—	4	1	no	—
122	Q₈×C₆₁	—	4	1	no	—
123	Q₈×C₆₁	—	8	2	no	—
124	C₂×C₄×C₆₁	—	4	2	no	—
125	C₂×C₄×C₆₁	—	4	1	no	—
126	C₂×C₄×C₆₁	—	4	1	no	—
127	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₆₁	—	4	1	no	—
131	C₂×Dic₆₁	—	4	1	no	—
132	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
133	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
134	D₂₄₄	—	4	2	no	—
135	D₂₄₄	—	4	2	no	—
136	D₂₄₄	—	8	2	no	—
137	D₂₄₄	—	16	1	no	—
138	C₄×D₆₁	—	16	2	no	—
139	Dic₁₂₂	—	2	2	no	—
140	Dic₁₂₂	—	2	2	no	—
141	Dic₁₂₂	—	4	1	no	—
142	C₈×C₆₁	—	2	1	no	—
143	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
144	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
145	#92418	—	2	2	no	—
146	#33932	—	61	1	no	maximal, Sylow
147	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₆₁	—	8	2	no	—
149	C₂×D₆₁	—	8	2	no	—
150	C₂×D₆₁	—	8	2	no	—
151	C₄×C₆₁	—	2	1	no	—
152	C₄×C₆₁	—	8	1	no	—
153	C₄×C₆₁	—	8	2	no	—
154	C₄×C₆₁	—	8	1	no	—
155	C₄×C₆₁	#508	1	2	yes	—
156	C₄×C₆₁	#548	1	1	yes	characteristic
157	C₄×C₆₁	#577	1	1	yes	characteristic
158	Dic₆₁	—	2	2	no	—
159	Dic₆₁	—	2	2	no	—
160	Dic₆₁	—	4	2	no	—
161	Dic₆₁	—	8	1	no	—
162	#3396	—	61	1	no	—
163	#3383	—	61	1	no	—
164	#2242	—	61	2	no	—
165	#2240	—	61	1	no	—
166	#2228	—	61	2	no	—
167	#2225	D₆₁	1	1	yes	characteristic, complemented
168	#2211	—	61	2	no	—
169	#2207	—	61	1	no	—
170	Q₈×C₁₆	—	61	1	no	—
171	C₄×Q₃₂	—	61	1	no	—
172	C₄×SD₃₂	—	61	2	no	—
173	C₂×C₆₁	#3358	1	1	yes	characteristic
174	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
175	C₂×C₆₁	#3416	1	1	yes	characteristic
176	D₆₁	—	16	2	no	—
177	#559	—	122	1	no	—
178	#557	—	122	1	no	—
179	C₂×Q₃₂	—	61	1	no	—
180	C₂×SD₃₂	—	61	2	no	—
181	#500	C₂×D₆₁	1	1	yes	characteristic
182	#499	—	61	2	no	—
183	#495	—	61	2	no	—
184	#493	—	61	1	no	—
185	#479	—	122	2	no	—
186	#475	—	122	1	no	—
187	#463	—	122	2	no	—
188	#460	—	122	1	no	—
189	C₈×Q₈	—	61	1	no	—
190	C₄×Q₁₆	—	61	2	no	—
191	C₄×Q₁₆	—	61	1	no	—
192	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
193	C₄×D₈	—	61	2	no	—
194	#367	C₂×D₆₁	1	2	yes	—
195	#366	—	61	1	no	—
196	C₄⋊C₁₆	—	61	2	no	—
197	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
198	#358	—	61	1	no	—
199	#358	—	61	1	no	—
200	#358	C₂×D₆₁	1	2	yes	—
201	#357	—	61	2	no	—
202	#357	—	61	2	no	—
203	#357	—	61	2	no	—
204	C₄×C₁₆	—	61	2	no	—
205	C₄×C₁₆	—	61	1	no	—
206	C₆₁	#33932	1	1	yes	characteristic, complemented, Sylow
207	C₂×Q₁₆	—	61	1	no	—
208	C₂×Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
209	C₂×SD₁₆	—	122	2	no	—
210	C₂×D₈	—	61	2	no	—
211	C₂×D₈	—	122	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	244	1	no	—
214	#127	—	122	1	no	—
215	#125	—	122	2	no	—
216	#125	—	244	2	no	—
217	#124	—	122	2	no	—
218	#124	—	244	1	no	—
219	#123	—	244	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	61	1	no	—
222	C₄×Q₈	—	122	2	no	—
223	C₄×Q₈	—	122	1	no	—
224	C₄×Q₈	—	244	1	no	—
225	C₄×D₄	—	122	2	no	—
226	C₄×D₄	—	244	1	no	—
227	Q₃₂	—	122	2	no	—
228	SD₃₂	—	122	4	no	—
229	C₂×C₁₆	—	61	1	no	—
230	C₂×C₁₆	—	61	1	no	—
231	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
232	#107	—	2	1	no	—
233	#107	—	61	1	no	—
234	#107	C₂²×D₆₁	1	2	yes	—
235	#107	C₂²×D₆₁	1	1	yes	characteristic
236	#106	—	122	2	no	—
237	C₄⋊C₈	—	61	2	no	—
238	C₄⋊C₈	—	61	1	no	—
239	#103	—	2	1	no	—
240	#103	—	122	2	no	—
241	#103	—	122	1	no	—
242	#102	—	122	2	no	—
243	C₄×C₈	—	61	2	no	—
244	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	61	1	no	—
248	C₂×Q₈	—	122	1	no	—
249	C₂×D₄	—	122	2	no	—
250	C₂×D₄	—	244	1	no	—
251	C₂×D₄	—	244	1	no	—
252	C₂²×C₄	—	244	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	61	2	no	—
256	Q₁₆	—	122	1	no	—
257	SD₁₆	—	488	2	no	—
258	D₈	—	122	2	no	—
259	D₈	—	122	2	no	—
260	D₈	—	488	1	no	—
261	C₂×C₈	—	61	2	no	—
262	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	61	2	no	—
269	C₄⋊C₄	—	61	1	no	—
270	C₄⋊C₄	—	122	1	no	—
271	C₄⋊C₄	—	244	2	no	—
272	C₄⋊C₄	—	244	2	no	—
273	C₄⋊C₄	—	244	2	no	—
274	C₄⋊C₄	—	244	1	no	—
275	C₂²⋊C₄	—	244	2	no	—
276	C₂²⋊C₄	—	244	2	no	—
277	C₂²⋊C₄	—	244	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	61	2	no	—
280	C₄²	—	244	2	no	—
281	C₄²	—	244	1	no	—
282	C₄²	D₄×D₆₁	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	122	1	no	—
285	C₁₆	—	122	1	no	—
286	C₂³	—	244	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	122	2	no	—
291	Q₈	—	122	2	no	—
292	Q₈	—	244	1	no	—
293	D₄	—	976	1	no	—
294	D₄	—	488	2	no	—
295	D₄	—	244	2	no	—
296	D₄	—	244	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	61	2	no	—
301	C₂×C₄	—	61	2	no	—
302	C₂×C₄	—	976	2	no	—
303	C₂×C₄	—	244	1	no	—
304	C₂×C₄	—	244	1	no	—
305	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
306	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	122	2	no	—
310	C₈	#41×D₆₁	1	1	yes	characteristic
311	C₈	#14519825	1	1	yes	characteristic
312	C₂²	—	488	2	no	—
313	C₂²	—	488	2	no	—
314	C₂²	—	488	2	no	—
315	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	122	2	no	—
321	C₄	—	122	2	no	—
322	C₄	—	488	1	no	—
323	C₄	—	244	2	no	—
324	C₄	#16840193	1	2	yes	—
325	C₄	#16840422	1	1	yes	characteristic
326	C₄	#16840571	1	1	yes	characteristic
327	C₂	—	976	2	no	—
328	C₂	#21458453	1	1	yes	characteristic
329	C₂	C₂×#16840193	1	1	yes	characteristic
330	C₂	#21458964	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₁
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2225	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	#16840139	C₂²
hypercenter	#2225	D₆₁
hypocenter	C₆₁	#33932
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
2. #21458712	C₂	semidirect
3. #21448052	C₂	semidirect
5. #21447975	C₂	semidirect
6. #21447884	C₂	semidirect
8. #21447721	C₂	semidirect
9. C₄×#14519723	C₂	semidirect
10. #21447670	C₂	semidirect
11. C₆₁×#2225	C₂	semidirect
167. #2225	D₆₁	semidirect
206. C₆₁	#33932	semidirect

Maximal subgroups

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

subgroup	index
#21458829	2
#21458712	2
#21448052	2
#21448044	2
#21447975	2
#21447884	2
#21447860	2
#21447721	2
C₄×#14519723	2
#21447670	2
C₆₁×#2225	2
#33932	61