Subgroups — #29837752

Order 15616.

518 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458794	C₂	1	1	yes	maximal, characteristic
2	#21448062	C₂	1	2	yes	maximal, complemented
3	#21448044	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447973	C₂	1	2	yes	maximal, complemented
5	#21447953	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447859	C₂	1	2	yes	maximal, complemented
7	#21447859	C₂	1	1	yes	maximal, characteristic, complemented
8	#21447728	C₂	1	1	yes	maximal, characteristic, complemented
9	#21447681	C₂	1	2	yes	maximal, complemented
10	C₄×#14519720	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#3383	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840434	—	2	1	no	—
13	C₂×#14519720	C₂²	1	2	yes	—
14	C₂×#14519720	C₂²	1	1	yes	characteristic
15	#16840139	C₂²	1	1	yes	characteristic
16	#16840139	C₂²	1	2	yes	—
17	#16839958	—	2	1	no	—
18	#16839958	—	2	2	no	—
19	C₄×D₄₈₈	C₂²	1	2	yes	—
20	C₄×D₄₈₈	C₂²	1	1	yes	characteristic
21	#16839589	C₂²	1	1	yes	characteristic
22	#16839587	C₂²	1	2	yes	—
23	C₄⋊#11758185	C₂²	1	1	yes	characteristic
24	C₄⋊#11758185	C₂²	1	2	yes	—
25	#16839566	C₂²	1	1	yes	characteristic
26	#16839562	C₂²	1	2	yes	—
27	#16839562	C₂²	1	2	yes	—
28	#16839562	C₂²	1	2	yes	—
29	#16839562	C₂²	1	2	yes	—
30	#16839562	C₂²	1	1	yes	characteristic
31	#16839538	C₂²	1	2	yes	—
32	C₄×#11758185	C₂²	1	1	yes	characteristic
33	C₆₁×#557	—	2	1	no	—
34	C₆₁×#500	C₂²	1	1	yes	characteristic
35	C₆₁×#499	C₂²	1	2	yes	—
36	C₆₁×#479	—	2	2	no	—
37	C₆₁×#475	—	2	1	no	—
38	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
40	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#11758226	—	4	1	no	—
42	C₂×D₄₈₈	—	2	2	no	—
43	C₂×D₄₈₈	—	2	1	no	—
44	C₂×D₄₈₈	C₂³	1	2	yes	—
45	C₂×D₄₈₈	C₂³	1	1	yes	characteristic
46	#14519774	—	2	2	no	—
47	#14519774	—	2	1	no	—
48	#14519752	—	4	1	no	—
49	#14519752	—	4	2	no	—
50	C₄×D₂₄₄	—	2	2	no	—
51	C₄×D₂₄₄	—	2	1	no	—
52	#14519720	—	2	4	no	—
53	#14519720	—	2	2	no	—
54	C₂×#11758185	C₂³	1	1	yes	characteristic
55	C₂×#11758185	C₂³	1	1	yes	characteristic
56	C₂×#11758185	C₂³	1	2	yes	—
57	#14519711	C₂³	1	1	yes	characteristic
58	#14519699	—	2	2	no	—
59	#14519699	—	2	1	no	—
60	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₆₁	—	2	1	no	—
62	C₄⋊Q₈×C₆₁	—	4	1	no	—
63	#125×C₆₁	—	2	2	no	—
64	#125×C₆₁	—	4	2	no	—
65	C₄×Q₈×C₆₁	—	2	1	no	—
66	C₄×Q₈×C₆₁	—	2	2	no	—
67	C₄×Q₈×C₆₁	—	4	1	no	—
68	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
69	#107×C₆₁	—	2	1	no	—
70	#107×C₆₁	C₂³	1	2	yes	—
71	#107×C₆₁	C₂³	1	1	yes	characteristic
72	#106×C₆₁	—	2	2	no	—
73	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
75	#103×C₆₁	—	2	2	no	—
76	#103×C₆₁	—	2	1	no	—
77	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂³	1	2	yes	—
79	C₂×Q₈×C₆₁	—	2	1	no	—
80	C₂×Q₈×C₆₁	—	4	1	no	—
81	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
82	#11758226	—	8	4	no	—
83	C₂×D₂₄₄	—	2	2	no	—
84	C₂×D₂₄₄	—	2	1	no	—
85	C₂×D₂₄₄	—	4	2	no	—
86	C₂×D₂₄₄	—	4	1	no	—
87	C₂×D₂₄₄	—	4	2	no	—
88	C₂×D₂₄₄	—	4	1	no	—
89	C₂×C₄×D₆₁	—	4	2	no	—
90	C₂×C₄×D₆₁	—	4	1	no	—
91	Q₁₆×C₆₁	—	2	1	no	—
92	Q₁₆×C₆₁	#41	1	2	yes	complemented
93	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₆₁	—	2	1	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	—	4	1	no	—
99	C₄⋊C₄×C₆₁	—	4	2	no	—
100	C₄⋊C₄×C₆₁	—	4	2	no	—
101	C₄⋊C₄×C₆₁	—	4	2	no	—
102	C₄⋊C₄×C₆₁	—	4	1	no	—
103	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
104	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
105	C₄²×C₆₁	—	4	2	no	—
106	C₄²×C₆₁	—	4	1	no	—
107	C₄²×C₆₁	C₂×D₄	1	2	yes	—
108	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
109	#11758198	—	4	2	no	—
110	#11758198	—	4	1	no	—
111	C₄⋊Dic₆₁	—	2	1	no	—
112	D₄₈₈	—	2	2	no	—
113	D₄₈₈	—	2	1	no	—
114	D₄₈₈	—	2	2	no	—
115	D₄₈₈	—	2	1	no	—
116	D₄₈₈	—	8	2	no	—
117	D₄₈₈	—	8	1	no	—
118	#11758185	—	2	1	no	—
119	#11758185	—	2	2	no	—
120	#11758185	—	2	1	no	—
121	C₂²×D₆₁	—	4	2	no	—
122	C₂²×D₆₁	—	4	1	no	—
123	Q₈×C₆₁	—	2	2	no	—
124	Q₈×C₆₁	—	2	2	no	—
125	Q₈×C₆₁	—	4	1	no	—
126	Q₈×C₆₁	—	8	2	no	—
127	C₂×C₄×C₆₁	—	4	1	no	—
128	C₂×C₄×C₆₁	—	4	1	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₆₁	—	4	1	no	—
136	D₂₄₄	—	4	2	no	—
137	D₂₄₄	—	4	1	no	—
138	D₂₄₄	—	4	1	no	—
139	D₂₄₄	—	4	2	no	—
140	D₂₄₄	—	8	4	no	—
141	D₂₄₄	—	8	2	no	—
142	D₂₄₄	—	16	2	no	—
143	D₂₄₄	—	16	1	no	—
144	C₄×D₆₁	—	8	4	no	—
145	C₄×D₆₁	—	8	2	no	—
146	C₈×C₆₁	—	2	2	no	—
147	C₈×C₆₁	—	2	1	no	—
148	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
149	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
150	#33935	—	61	1	no	maximal, Sylow
151	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₆₁	—	8	2	no	—
153	C₂×D₆₁	—	8	1	no	—
154	C₂×D₆₁	—	8	2	no	—
155	C₂×D₆₁	—	8	1	no	—
156	C₂×D₆₁	—	8	2	no	—
157	C₂×D₆₁	—	8	1	no	—
158	C₄×C₆₁	—	2	2	no	—
159	C₄×C₆₁	—	2	2	no	—
160	C₄×C₆₁	—	2	1	no	—
161	C₄×C₆₁	—	4	2	no	—
162	C₄×C₆₁	—	8	2	no	—
163	C₄×C₆₁	—	8	1	no	—
164	C₄×C₆₁	#509	1	2	yes	—
165	C₄×C₆₁	#548	1	1	yes	characteristic
166	C₄×C₆₁	#577	1	1	yes	characteristic
167	Dic₆₁	—	8	1	no	—
168	#3391	—	61	1	no	—
169	#3383	D₆₁	1	1	yes	characteristic, complemented
170	#2244	—	61	2	no	—
171	#2240	—	61	1	no	—
172	#2228	—	61	2	no	—
173	#2224	—	61	1	no	—
174	#2207	—	61	2	no	—
175	#2207	—	61	1	no	—
176	#2183	—	61	1	no	—
177	#2175	—	61	2	no	—
178	C₄×SD₃₂	—	61	1	no	—
179	C₂×C₆₁	#3358	1	1	yes	characteristic
180	C₂×C₆₁	C₂×#509	1	1	yes	characteristic
181	C₂×C₆₁	#3416	1	1	yes	characteristic
182	D₆₁	—	16	2	no	—
183	D₆₁	—	16	1	no	—
184	#557	—	2	1	no	—
185	#550	—	122	1	no	—
186	C₂×SD₃₂	—	61	1	no	—
187	C₂×SD₃₂	—	61	2	no	—
188	#500	C₂×D₆₁	1	1	yes	characteristic
189	#499	C₂×D₆₁	1	2	yes	—
190	#493	—	61	1	no	—
191	#493	—	61	2	no	—
192	#479	—	2	2	no	—
193	#475	—	2	1	no	—
194	#459	—	122	1	no	—
195	#459	—	122	2	no	—
196	C₈×Q₈	C₂×D₆₁	1	1	yes	characteristic
197	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
198	C₄×Q₁₆	C₂×D₆₁	1	2	yes	—
199	C₄×D₈	—	61	1	no	—
200	C₄×D₈	—	61	2	no	—
201	#367	—	61	1	no	—
202	#366	—	61	2	no	—
203	C₄⋊C₁₆	—	61	1	no	—
204	C₄⋊C₁₆	—	61	2	no	—
205	#358	—	61	1	no	—
206	#357	—	61	1	no	—
207	#357	—	61	2	no	—
208	#357	—	61	2	no	—
209	#357	—	61	2	no	—
210	#357	—	61	2	no	—
211	#346	—	61	2	no	—
212	C₄×C₁₆	—	61	1	no	—
213	C₆₁	#33935	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	244	1	no	—
215	C₂×Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
216	C₂×D₈	—	61	1	no	—
217	C₂×D₈	—	61	2	no	—
218	C₂×D₈	—	122	2	no	—
219	C₂×D₈	—	122	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	4	1	no	—
222	#127	—	122	2	no	—
223	#127	—	122	1	no	—
224	#125	—	2	2	no	—
225	#125	—	4	2	no	—
226	#121	—	244	1	no	—
227	#121	—	244	2	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×Q₈	—	4	1	no	—
231	C₄×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
232	C₄×D₄	—	122	2	no	—
233	C₄×D₄	—	122	1	no	—
234	SD₃₂	—	122	2	no	—
235	SD₃₂	—	122	4	no	—
236	C₂×C₁₆	—	61	1	no	—
237	C₂×C₁₆	—	61	1	no	—
238	C₂×C₁₆	—	61	2	no	—
239	#107	—	2	1	no	—
240	#107	—	61	1	no	—
241	#107	C₂²×D₆₁	1	2	yes	—
242	#107	C₂²×D₆₁	1	1	yes	characteristic
243	#106	—	2	2	no	—
244	C₄⋊C₈	C₂²⋊D₆₁	1	1	yes	characteristic
245	C₄⋊C₈	C₂²⋊D₆₁	1	2	yes	—
246	#103	—	2	2	no	—
247	#103	—	2	1	no	—
248	#102	—	122	2	no	—
249	#102	—	122	1	no	—
250	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₆₁	1	2	yes	—
252	#41	—	488	4	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	#11758199	1	1	yes	characteristic
256	C₂×D₄	—	122	2	no	—
257	C₂×D₄	—	122	1	no	—
258	C₂×D₄	—	244	2	no	—
259	C₂×D₄	—	244	1	no	—
260	C₂×D₄	—	244	2	no	—
261	C₂×D₄	—	244	1	no	—
262	C₂²×C₄	—	244	2	no	—
263	C₂²×C₄	—	244	1	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	#11758226	1	2	yes	complemented
266	D₈	—	122	1	no	—
267	D₈	—	122	2	no	—
268	D₈	—	122	1	no	—
269	D₈	—	122	2	no	—
270	D₈	—	488	2	no	—
271	D₈	—	488	1	no	—
272	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	122	1	no	—
283	C₄⋊C₄	#11758199	1	1	yes	characteristic
284	C₄⋊C₄	#11758199	1	2	yes	—
285	C₂²⋊C₄	—	244	2	no	—
286	C₂²⋊C₄	—	244	1	no	—
287	C₄²	—	4	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
290	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
291	C₁₆	—	122	1	no	—
292	C₁₆	—	122	2	no	—
293	C₁₆	—	122	1	no	—
294	C₂³	—	244	2	no	—
295	C₂³	—	244	1	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	2	2	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	8	2	no	—
300	D₄	—	976	2	no	—
301	D₄	—	976	1	no	—
302	D₄	—	488	4	no	—
303	D₄	—	488	2	no	—
304	D₄	—	244	1	no	—
305	D₄	—	244	2	no	—
306	D₄	—	244	1	no	—
307	D₄	—	244	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	488	4	no	—
312	C₂×C₄	—	488	2	no	—
313	C₂×C₄	—	244	1	no	—
314	C₂×C₄	C₂×#11758199	1	2	yes	—
315	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
316	C₂×C₄	C₂×#11758199	1	2	yes	—
317	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	C₂×#11758226	1	1	yes	characteristic
322	C₈	#14519825	1	1	yes	characteristic
323	C₂²	—	488	2	no	—
324	C₂²	—	488	1	no	—
325	C₂²	—	488	2	no	—
326	C₂²	—	488	1	no	—
327	C₂²	—	488	2	no	—
328	C₂²	—	488	1	no	—
329	C₂²	C₂²×#11758199	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	488	1	no	—
337	C₄	#16840197	1	2	yes	—
338	C₄	#16840423	1	1	yes	characteristic
339	C₄	#16840570	1	1	yes	characteristic
340	C₂	—	976	2	no	—
341	C₂	—	976	1	no	—
342	C₂	#21458454	1	1	yes	characteristic
343	C₂	C₂×#16840197	1	1	yes	characteristic
344	C₂	#21458963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11758199
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#3383	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	#21458794	C₂
hypercenter	#3383	D₆₁
hypocenter	C₆₁	#33935
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

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

N	H ≅ G/N	type
2. #21448062	C₂	semidirect
3. #21448044	C₂	semidirect
4. #21447973	C₂	semidirect
5. #21447953	C₂	semidirect
6. #21447859	C₂	semidirect
7. #21447859	C₂	semidirect
8. #21447728	C₂	semidirect
9. #21447681	C₂	semidirect
10. C₄×#14519720	C₂	semidirect
11. C₆₁×#3383	C₂	semidirect
92. Q₁₆×C₆₁	#41	semidirect
169. #3383	D₆₁	semidirect
213. C₆₁	#33935	semidirect
265. Q₁₆	#11758226	semidirect

Maximal subgroups

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

subgroup	index
#21458794	2
#21448062	2
#21448044	2
#21447973	2
#21447953	2
#21447859	2
#21447728	2
#21447681	2
C₄×#14519720	2
C₆₁×#3383	2
#33935	61