Subgroups — #29837692

Order 15616.

528 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	#21458793	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#21448043	C₂	1	1	yes	maximal, characteristic, complemented
4	#21448041	C₂	1	2	yes	maximal, complemented
5	#21447966	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447965	C₂	1	2	yes	maximal, complemented
7	#21447859	C₂	1	1	yes	maximal, characteristic, complemented
8	#21447857	C₂	1	2	yes	maximal, complemented
9	C₁₆×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#21447668	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₂₂	—	2	1	no	—
13	#16840433	—	2	1	no	—
14	C₂×#14519720	C₂²	1	2	yes	—
15	#16840167	C₂²	1	1	yes	characteristic
16	#16840166	C₂²	1	2	yes	—
17	#16840139	C₂²	1	1	yes	characteristic
18	#16840137	C₂²	1	2	yes	—
19	#16840057	—	2	1	no	—
20	#16840055	—	2	2	no	—
21	#16839957	—	2	1	no	—
22	#16839956	—	2	2	no	—
23	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₆₁	C₂²	1	2	yes	—
25	#16839844	C₂²	1	2	yes	—
26	#16839589	C₂²	1	2	yes	—
27	C₄⋊#11758185	C₂²	1	1	yes	characteristic
28	#16839580	C₂²	1	2	yes	—
29	#16839564	C₂²	1	2	yes	—
30	#16839564	C₂²	1	2	yes	—
31	#16839563	C₂²	1	2	yes	—
32	#16839562	C₂²	1	2	yes	—
33	C₁₆×Dic₆₁	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₆₁	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#367	C₂²	1	1	yes	characteristic
38	C₆₁×#358	C₂²	1	1	yes	characteristic
39	C₆₁×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#11758222	—	4	1	no	—
42	C₂×D₄₈₈	—	2	1	no	—
43	C₂×D₄₈₈	C₂³	1	2	yes	—
44	C₂×#11758199	—	2	2	no	—
45	#14519777	—	2	1	no	—
46	#14519777	—	4	1	no	—
47	#14519776	—	2	2	no	—
48	#14519776	—	4	2	no	—
49	#14519774	—	2	1	no	—
50	#14519773	—	2	2	no	—
51	#14519751	—	4	1	no	—
52	#14519750	—	4	2	no	—
53	Q₈×Dic₆₁	—	2	2	no	—
54	C₄×Dic₁₂₂	—	4	1	no	—
55	C₄×Dic₁₂₂	D₄	1	1	yes	characteristic
56	#14519739	—	2	2	no	—
57	#14519720	—	2	4	no	—
58	C₂×#11758185	C₂³	1	2	yes	—
59	#14519711	—	2	1	no	—
60	#14519711	C₂³	1	2	yes	—
61	#14519710	—	2	2	no	—
62	C₄⋊#92418	D₄	1	1	yes	characteristic
63	#14519706	D₄	1	2	yes	—
64	#14519701	—	2	2	no	—
65	#14519697	—	2	2	no	—
66	C₈×Dic₆₁	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₆₁	—	2	1	no	—
70	C₄×D₄×C₆₁	—	2	1	no	—
71	SD₃₂×C₆₁	—	2	2	no	—
72	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
74	#107×C₆₁	C₂³	1	1	yes	characteristic
75	#107×C₆₁	C₂³	1	1	yes	characteristic
76	#103×C₆₁	—	2	1	no	—
77	#102×C₆₁	—	2	1	no	—
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₆₁	—	2	1	no	—
80	C₂×D₄×C₆₁	—	2	1	no	—
81	C₂²×C₄×C₆₁	—	4	1	no	—
82	C₂×C₂²⋊D₆₁	—	4	2	no	—
83	C₂×C₂²⋊D₆₁	—	4	2	no	—
84	#11758222	—	8	4	no	—
85	C₂×D₂₄₄	—	2	2	no	—
86	C₂×D₂₄₄	—	4	1	no	—
87	C₂×D₂₄₄	—	4	1	no	—
88	C₂×C₄×D₆₁	—	4	2	no	—
89	C₂×Dic₁₂₂	—	4	1	no	—
90	C₂×Dic₁₂₂	—	4	2	no	—
91	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
92	Q₁₆×C₆₁	—	2	1	no	—
93	Q₁₆×C₆₁	#41	1	2	yes	complemented
94	D₈×C₆₁	—	2	1	no	—
95	D₈×C₆₁	—	2	1	no	—
96	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₆₁	—	2	1	no	—
99	C₄⋊C₄×C₆₁	—	2	1	no	—
100	C₄⋊C₄×C₆₁	—	4	1	no	—
101	C₂²⋊C₄×C₆₁	—	4	1	no	—
102	C₄²×C₆₁	—	4	1	no	—
103	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
104	#11758199	—	8	2	no	—
105	#11758198	—	4	2	no	—
106	C₄⋊Dic₆₁	—	2	2	no	—
107	C₄⋊Dic₆₁	—	4	1	no	—
108	C₄⋊Dic₆₁	—	4	2	no	—
109	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
110	#11758196	—	4	2	no	—
111	#11758196	D₈	1	2	yes	—
112	C₄×Dic₆₁	—	4	2	no	—
113	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
114	C₂×#92418	C₂×D₄	1	2	yes	—
115	D₄₈₈	—	2	2	no	—
116	D₄₈₈	—	2	2	no	—
117	D₄₈₈	—	8	1	no	—
118	C₁₆×C₆₁	—	2	1	no	—
119	C₁₆×C₆₁	—	2	1	no	—
120	#11758185	—	2	2	no	—
121	C₂³×C₆₁	—	4	1	no	—
122	C₂²×D₆₁	—	4	2	no	—
123	Q₈×C₆₁	—	2	2	no	—
124	Q₈×C₆₁	—	4	1	no	—
125	D₄×C₆₁	—	4	1	no	—
126	D₄×C₆₁	—	4	1	no	—
127	C₂×C₄×C₆₁	—	4	1	no	—
128	C₂×C₄×C₆₁	—	4	1	no	—
129	C₂×C₄×C₆₁	—	4	1	no	—
130	C₂×C₄×C₆₁	—	8	2	no	—
131	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₆₁	—	8	4	no	—
135	C₂²⋊D₆₁	—	16	2	no	—
136	C₂×Dic₆₁	—	4	2	no	—
137	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
138	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
139	D₂₄₄	—	4	2	no	—
140	D₂₄₄	—	4	2	no	—
141	D₂₄₄	—	8	2	no	—
142	D₂₄₄	—	16	1	no	—
143	C₄×D₆₁	—	8	4	no	—
144	Dic₁₂₂	—	16	2	no	—
145	Dic₁₂₂	—	16	1	no	—
146	Dic₁₂₂	SD₃₂	1	4	yes	complemented
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	#92418	—	2	2	no	—
151	#33929	—	61	1	no	maximal, Sylow
152	C₂²×C₆₁	—	8	1	no	—
153	C₂²×C₆₁	—	8	1	no	—
154	C₂²×C₆₁	—	8	1	no	—
155	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₆₁	—	8	2	no	—
157	C₂×D₆₁	—	8	2	no	—
158	C₂×D₆₁	—	8	2	no	—
159	C₄×C₆₁	—	2	1	no	—
160	C₄×C₆₁	—	4	2	no	—
161	C₄×C₆₁	—	8	1	no	—
162	C₄×C₆₁	—	8	1	no	—
163	C₄×C₆₁	C₂×SD₃₂	1	2	yes	—
164	C₄×C₆₁	#548	1	1	yes	characteristic
165	C₄×C₆₁	#577	1	1	yes	characteristic
166	Dic₆₁	—	2	2	no	—
167	Dic₆₁	—	8	2	no	—
168	Dic₆₁	C₂×SD₃₂	1	4	yes	—
169	#3391	—	61	1	no	—
170	Q₈×Q₁₆	—	61	1	no	—
171	#2240	—	61	2	no	—
172	#2240	—	61	1	no	—
173	#2226	—	61	1	no	—
174	#2226	—	61	2	no	—
175	#2207	—	61	2	no	—
176	#2207	—	61	1	no	—
177	Q₈×C₁₆	—	61	1	no	—
178	C₄×SD₃₂	—	61	2	no	—
179	C₄×SD₃₂	D₆₁	1	1	yes	characteristic, complemented
180	C₂×C₆₁	—	16	1	no	—
181	C₂×C₆₁	#3358	1	1	yes	characteristic
182	C₂×C₆₁	C₂²×SD₃₂	1	1	yes	characteristic
183	C₂×C₆₁	#3415	1	1	yes	characteristic
184	D₆₁	—	16	2	no	—
185	Q₈²	—	122	1	no	—
186	#550	—	122	1	no	—
187	C₂×SD₃₂	—	61	2	no	—
188	C₂×SD₃₂	C₂×D₆₁	1	1	yes	characteristic
189	#500	—	61	2	no	—
190	#500	—	61	1	no	—
191	#493	—	61	2	no	—
192	#493	—	61	1	no	—
193	#477	—	122	2	no	—
194	#477	—	122	1	no	—
195	#459	—	122	2	no	—
196	#459	—	122	1	no	—
197	C₈×Q₈	—	61	1	no	—
198	C₄×Q₁₆	—	61	2	no	—
199	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
200	C₄×D₈	—	61	2	no	—
201	C₄×D₈	C₂×D₆₁	1	1	yes	characteristic
202	#367	—	61	2	no	—
203	#367	C₂×D₆₁	1	1	yes	characteristic
204	C₄⋊C₁₆	—	61	2	no	—
205	C₄⋊C₁₆	—	61	1	no	—
206	#358	C₂×D₆₁	1	1	yes	characteristic
207	#357	—	61	2	no	—
208	#357	—	61	2	no	—
209	#357	—	61	2	no	—
210	#357	—	61	2	no	—
211	#357	C₂×D₆₁	1	1	yes	characteristic
212	C₄×C₁₆	—	61	2	no	—
213	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
214	C₆₁	#33929	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	—	244	1	no	—
216	C₂×Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
217	C₂×D₈	—	61	2	no	—
218	C₂×D₈	—	122	2	no	—
219	C₂×D₈	—	122	1	no	—
220	C₂×D₈	C₂²×D₆₁	1	1	yes	characteristic
221	C₄⋊Q₈	—	122	2	no	—
222	C₄⋊Q₈	—	122	1	no	—
223	C₄⋊Q₈	—	244	1	no	—
224	C₄⋊Q₈	—	244	2	no	—
225	#127	—	122	2	no	—
226	#127	—	122	1	no	—
227	#121	—	244	2	no	—
228	#121	—	244	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	61	1	no	—
231	C₄×Q₈	—	122	2	no	—
232	C₄×Q₈	—	244	1	no	—
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	—	122	2	no	—
235	SD₃₂	—	2	2	no	—
236	SD₃₂	—	122	4	no	—
237	C₂×C₁₆	—	61	2	no	—
238	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
240	#107	—	61	2	no	—
241	#107	—	122	2	no	—
242	#107	—	122	1	no	—
243	#107	C₂²×D₆₁	1	1	yes	characteristic
244	#107	C₂²×D₆₁	1	1	yes	characteristic
245	C₄⋊C₈	—	61	2	no	—
246	C₄⋊C₈	—	61	1	no	—
247	#103	—	2	1	no	—
248	#103	—	122	2	no	—
249	#102	—	2	1	no	—
250	#102	—	122	2	no	—
251	C₄×C₈	—	61	2	no	—
252	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
253	#41	—	488	4	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	61	1	no	—
256	C₂×Q₈	—	244	2	no	—
257	C₂×Q₈	—	244	1	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	122	2	no	—
260	C₂×D₄	—	244	2	no	—
261	C₂×D₄	—	244	2	no	—
262	C₂×D₄	—	244	1	no	—
263	C₂×D₄	—	244	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	—	244	2	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	#11758222	1	2	yes	complemented
268	D₈	—	2	1	no	—
269	D₈	—	2	1	no	—
270	D₈	—	122	2	no	—
271	D₈	—	122	2	no	—
272	D₈	—	488	1	no	—
273	D₈	—	488	2	no	—
274	C₂×C₈	—	61	2	no	—
275	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	61	2	no	—
281	C₄⋊C₄	—	61	1	no	—
282	C₄⋊C₄	—	122	2	no	—
283	C₄⋊C₄	—	244	2	no	—
284	C₄⋊C₄	—	244	2	no	—
285	C₄⋊C₄	—	244	1	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	244	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	61	2	no	—
290	C₄²	—	244	2	no	—
291	C₄²	D₄×D₆₁	1	1	yes	characteristic
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₁₆	—	122	2	no	—
295	C₂³	—	4	1	no	—
296	C₂³	—	244	2	no	—
297	Q₈	—	2	2	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	61	4	no	—
300	Q₈	—	976	1	no	—
301	Q₈	—	976	2	no	—
302	D₄	—	4	1	no	—
303	D₄	—	4	1	no	—
304	D₄	—	976	1	no	—
305	D₄	—	976	2	no	—
306	D₄	—	488	4	no	—
307	D₄	—	488	2	no	—
308	D₄	—	244	2	no	—
309	D₄	—	244	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	61	2	no	—
315	C₂×C₄	—	61	2	no	—
316	C₂×C₄	—	488	4	no	—
317	C₂×C₄	—	244	2	no	—
318	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
319	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	122	2	no	—
323	C₈	C₂×#11758222	1	1	yes	characteristic
324	C₈	#14519824	1	1	yes	characteristic
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	488	2	no	—
329	C₂²	—	488	2	no	—
330	C₂²	—	488	2	no	—
331	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	61	4	no	—
337	C₄	—	122	2	no	—
338	C₄	—	488	2	no	—
339	C₄	SD₃₂×D₆₁	1	2	yes	—
340	C₄	#16840421	1	1	yes	characteristic
341	C₄	#16840569	1	1	yes	characteristic
342	C₂	—	16	1	no	—
343	C₂	—	976	2	no	—
344	C₂	#21458452	1	1	yes	characteristic
345	C₂	C₂×SD₃₂×D₆₁	1	1	yes	characteristic
346	C₂	#21458957	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₁
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₄×SD₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	#21458793	C₂
hypercenter	C₄×SD₃₂	D₆₁
hypocenter	C₆₁	#33929
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₂₂	C₂	semidirect
3. #21448043	C₂	semidirect
4. #21448041	C₂	semidirect
5. #21447966	C₂	semidirect
6. #21447965	C₂	semidirect
7. #21447859	C₂	semidirect
8. #21447857	C₂	semidirect
9. C₁₆×Dic₁₂₂	C₂	semidirect
10. #21447668	C₂	semidirect
11. C₄×SD₃₂×C₆₁	C₂	semidirect
93. Q₁₆×C₆₁	#41	semidirect
146. Dic₁₂₂	SD₃₂	semidirect
179. C₄×SD₃₂	D₆₁	semidirect
214. C₆₁	#33929	semidirect
267. Q₁₆	#11758222	semidirect

Maximal subgroups

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

subgroup	index
#21458793	2
Q₁₆×Dic₁₂₂	2
#21448043	2
#21448041	2
#21447966	2
#21447965	2
#21447859	2
#21447857	2
C₁₆×Dic₁₂₂	2
#21447668	2
C₄×SD₃₂×C₆₁	2
#33929	61