Subgroups — #29836901

Order 15616.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458213	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₂₄₄	C₂	1	1	yes	maximal, characteristic
3	#21448047	C₂	1	1	yes	maximal, characteristic, complemented
4	#21448023	C₂	1	2	yes	maximal, complemented
5	#21447971	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447909	C₂	1	2	yes	maximal, complemented
7	#21447869	C₂	1	1	yes	maximal, characteristic, complemented
8	#21447751	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₂₄₄	C₂	1	1	yes	maximal, characteristic
10	#358×D₆₁	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₁	C₂²	1	2	yes	—
13	#16840472	—	2	1	no	—
14	Q₈×D₂₄₄	—	2	1	no	—
15	C₂×#14519723	C₂²	1	2	yes	—
16	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
17	#16840167	C₂²	1	1	yes	characteristic
18	#16840142	C₂²	1	1	yes	characteristic
19	#16840096	—	2	2	no	—
20	#16840057	—	2	1	no	—
21	#16840004	C₂²	1	2	yes	—
22	#16840000	C₂²	1	2	yes	—
23	#16839972	—	2	1	no	—
24	#16839919	—	2	2	no	—
25	C₈×D₂₄₄	C₂²	1	1	yes	characteristic
26	#107×D₆₁	C₂²	1	2	yes	—
27	#16839587	C₂²	1	2	yes	—
28	C₄⋊#11758185	C₂²	1	1	yes	characteristic
29	#16839567	C₂²	1	2	yes	—
30	#16839567	C₂²	1	2	yes	—
31	#16839566	C₂²	1	2	yes	—
32	#16839565	C₂²	1	2	yes	—
33	#16839541	C₂²	1	2	yes	—
34	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#366	C₂²	1	1	yes	characteristic
38	C₆₁×#358	C₂²	1	1	yes	characteristic
39	C₆₁×#358	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₆₁	—	2	2	no	—
42	C₂×Dic₂₄₄	—	2	1	no	—
43	C₂×Dic₂₄₄	C₂³	1	2	yes	—
44	C₂×#11758202	—	2	2	no	—
45	Q₁₆×D₆₁	—	2	4	no	—
46	Q₁₆×D₆₁	—	4	2	no	—
47	C₂×C₈×D₆₁	C₂³	1	2	yes	—
48	#14519777	—	2	1	no	—
49	#14519777	—	2	1	no	—
50	#14519777	—	4	1	no	—
51	#14519769	—	4	1	no	—
52	#14519757	—	2	2	no	—
53	#14519757	—	4	2	no	—
54	#14519755	—	4	2	no	—
55	#14519753	—	2	2	no	—
56	C₄×D₂₄₄	—	4	1	no	—
57	C₄×D₂₄₄	—	4	1	no	—
58	C₄×D₂₄₄	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₆₁	—	2	2	no	—
60	#14519723	—	4	2	no	—
61	C₂×#11758185	C₂³	1	2	yes	—
62	C₁₆×D₆₁	—	4	2	no	—
63	#14519711	—	2	1	no	—
64	#14519711	C₂³	1	2	yes	—
65	#14519710	—	2	2	no	—
66	C₄⋊#92418	D₄	1	1	yes	characteristic
67	#14519701	—	2	2	no	—
68	#14519700	—	2	2	no	—
69	#14519689	D₄	1	2	yes	—
70	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₆₁	—	2	1	no	—
73	C₄×Q₈×C₆₁	—	2	1	no	—
74	Q₃₂×C₆₁	—	2	2	no	—
75	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
77	#107×C₆₁	C₂³	1	1	yes	characteristic
78	#107×C₆₁	C₂³	1	1	yes	characteristic
79	#103×C₆₁	—	2	1	no	—
80	#103×C₆₁	—	2	1	no	—
81	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₆₁	—	2	1	no	—
83	C₂×Q₈×C₆₁	—	2	1	no	—
84	Q₈×D₆₁	—	4	4	no	—
85	Q₈×D₆₁	—	8	2	no	—
86	C₂×D₂₄₄	D₈	1	1	yes	characteristic
87	C₂×C₄×D₆₁	—	4	2	no	—
88	C₂×C₄×D₆₁	—	4	2	no	—
89	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
90	C₂×Dic₁₂₂	—	2	2	no	—
91	C₂×Dic₁₂₂	—	4	2	no	—
92	C₂×Dic₁₂₂	—	4	1	no	—
93	Q₁₆×C₆₁	—	2	1	no	—
94	Q₁₆×C₆₁	—	2	1	no	—
95	Q₁₆×C₆₁	—	2	1	no	—
96	Q₁₆×C₆₁	#41	1	2	yes	—
97	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₆₁	—	2	1	no	—
100	C₄⋊C₄×C₆₁	—	2	1	no	—
101	C₄⋊C₄×C₆₁	—	4	1	no	—
102	C₄⋊C₄×C₆₁	—	4	1	no	—
103	C₄²×C₆₁	—	4	1	no	—
104	C₄²×C₆₁	—	4	1	no	—
105	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
106	#11758202	—	4	4	no	—
107	#11758198	—	4	2	no	—
108	#11758198	—	4	2	no	—
109	#11758198	D₈	1	2	yes	—
110	C₄⋊Dic₆₁	—	2	2	no	—
111	C₄⋊Dic₆₁	—	4	1	no	—
112	C₄⋊Dic₆₁	—	4	1	no	—
113	C₄⋊Dic₆₁	—	4	1	no	—
114	C₄⋊Dic₆₁	—	4	2	no	—
115	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
116	#11758196	—	4	2	no	—
117	#11758196	—	4	2	no	—
118	C₂×#92418	C₂×D₄	1	2	yes	—
119	Dic₂₄₄	—	2	2	no	—
120	Dic₂₄₄	—	2	2	no	—
121	Dic₂₄₄	—	8	1	no	—
122	C₈×D₆₁	—	2	2	no	—
123	C₈×D₆₁	—	2	2	no	—
124	C₁₆×C₆₁	—	2	1	no	—
125	C₁₆×C₆₁	—	2	1	no	—
126	#11758185	—	2	2	no	—
127	C₂²×D₆₁	C₂×D₈	1	2	yes	—
128	Q₈×C₆₁	—	2	2	no	—
129	Q₈×C₆₁	—	4	1	no	—
130	Q₈×C₆₁	—	4	1	no	—
131	Q₈×C₆₁	—	4	1	no	—
132	C₂×C₄×C₆₁	—	4	1	no	—
133	C₂×C₄×C₆₁	—	4	1	no	—
134	C₂×C₄×C₆₁	—	4	1	no	—
135	C₂×C₄×C₆₁	—	4	1	no	—
136	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₆₁	—	4	2	no	—
140	C₂×Dic₆₁	—	4	2	no	—
141	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
142	D₂₄₄	Q₃₂	1	4	yes	complemented
143	C₄×D₆₁	—	2	2	no	—
144	C₄×D₆₁	—	2	2	no	—
145	C₄×D₆₁	—	8	4	no	—
146	C₄×D₆₁	—	16	2	no	—
147	Dic₁₂₂	—	4	2	no	—
148	Dic₁₂₂	—	4	2	no	—
149	Dic₁₂₂	—	8	4	no	—
150	Dic₁₂₂	—	16	1	no	—
151	C₈×C₆₁	—	2	1	no	—
152	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
153	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
154	#92418	—	2	2	no	—
155	#33866	—	61	1	no	maximal, Sylow
156	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₆₁	—	2	2	no	—
158	C₂×D₆₁	—	2	2	no	—
159	C₂×D₆₁	C₂×Q₃₂	1	4	yes	—
160	C₄×C₆₁	—	2	1	no	—
161	C₄×C₆₁	—	4	2	no	—
162	C₄×C₆₁	—	8	1	no	—
163	C₄×C₆₁	—	8	1	no	—
164	C₄×C₆₁	—	8	1	no	—
165	C₄×C₆₁	C₂×Q₃₂	1	2	yes	—
166	C₄×C₆₁	#546	1	1	yes	characteristic
167	C₄×C₆₁	#577	1	1	yes	characteristic
168	Dic₆₁	—	2	2	no	—
169	Dic₆₁	—	8	2	no	—
170	Dic₆₁	—	8	2	no	—
171	#3338	—	61	1	no	—
172	D₄×Q₁₆	—	61	1	no	—
173	#2241	—	61	1	no	—
174	#2236	—	61	2	no	—
175	#2227	—	61	1	no	—
176	#2216	—	61	2	no	—
177	#2209	—	61	1	no	—
178	#2188	—	61	2	no	—
179	C₄×Q₃₂	D₆₁	1	1	yes	characteristic, complemented
180	D₄×C₁₆	—	61	1	no	—
181	C₂×#358	—	61	2	no	—
182	C₂×C₆₁	#3294	1	1	yes	characteristic
183	C₂×C₆₁	C₂²×Q₃₂	1	1	yes	characteristic
184	C₂×C₆₁	#3416	1	1	yes	characteristic
185	D₆₁	—	2	4	no	—
186	C₂²×Q₁₆	—	61	2	no	—
187	#554	—	122	1	no	—
188	D₄×Q₈	—	122	1	no	—
189	C₂×Q₃₂	—	61	2	no	—
190	C₂×Q₃₂	C₂×D₆₁	1	1	yes	characteristic
191	C₂²×C₁₆	—	61	2	no	—
192	#500	—	61	1	no	—
193	#494	—	61	1	no	—
194	#484	—	122	2	no	—
195	#477	—	122	1	no	—
196	#467	—	61	2	no	—
197	#467	—	61	2	no	—
198	#462	—	122	1	no	—
199	#451	—	122	2	no	—
200	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
201	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
202	C₈×D₄	—	61	1	no	—
203	C₂×#107	—	61	2	no	—
204	#366	—	61	2	no	—
205	#366	C₂×D₆₁	1	1	yes	characteristic
206	C₄⋊C₁₆	—	61	1	no	—
207	#358	—	61	2	no	—
208	#358	—	61	2	no	—
209	#358	—	61	2	no	—
210	#358	—	61	2	no	—
211	#358	C₂×D₆₁	1	1	yes	characteristic
212	#358	C₂×D₆₁	1	1	yes	characteristic
213	C₂²⋊C₁₆	—	61	2	no	—
214	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
215	C₆₁	#33866	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	—	122	2	no	—
217	C₂×Q₁₆	—	61	2	no	—
218	C₂×Q₁₆	—	122	2	no	—
219	C₂×Q₁₆	—	122	4	no	—
220	C₂×Q₁₆	—	122	1	no	—
221	C₂×Q₁₆	—	244	2	no	—
222	C₂×Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
223	C₂×Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
224	C₂²×C₈	—	61	2	no	—
225	C₄⋊Q₈	—	122	1	no	—
226	C₄⋊Q₈	—	122	1	no	—
227	C₄⋊Q₈	—	244	1	no	—
228	#125	—	244	1	no	—
229	C₂²⋊Q₈	—	122	2	no	—
230	C₂²⋊Q₈	—	122	2	no	—
231	C₂²⋊Q₈	—	244	2	no	—
232	C₂²⋊Q₈	—	244	2	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×D₄	—	61	1	no	—
236	C₄×D₄	—	244	1	no	—
237	C₄×D₄	—	244	1	no	—
238	C₂×C₄⋊C₄	—	122	2	no	—
239	Q₃₂	—	2	2	no	—
240	Q₃₂	—	244	2	no	—
241	C₂×C₁₆	—	61	2	no	—
242	C₂×C₁₆	—	244	2	no	—
243	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
244	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
245	#107	—	61	2	no	—
246	#107	—	122	2	no	—
247	#107	—	122	1	no	—
248	#107	C₂²×D₆₁	1	1	yes	characteristic
249	#107	C₂²×D₆₁	1	1	yes	characteristic
250	C₄⋊C₈	—	61	1	no	—
251	#103	—	2	1	no	—
252	#103	—	2	1	no	—
253	#103	—	122	2	no	—
254	#103	—	122	2	no	—
255	C₂²⋊C₈	—	61	2	no	—
256	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	122	2	no	—
260	C₂×Q₈	—	488	2	no	—
261	C₂×Q₈	—	244	2	no	—
262	C₂×Q₈	—	244	4	no	—
263	C₂×Q₈	—	244	1	no	—
264	C₂×D₄	—	61	1	no	—
265	C₂²×C₄	—	61	2	no	—
266	C₂²×C₄	—	244	2	no	—
267	C₂²×C₄	—	244	2	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	122	2	no	—
272	Q₁₆	—	122	2	no	—
273	Q₁₆	—	488	1	no	—
274	Q₁₆	—	244	4	no	—
275	Q₁₆	#11758222	1	2	yes	—
276	C₂×C₈	—	61	2	no	—
277	C₂×C₈	—	122	2	no	—
278	C₂×C₈	—	122	2	no	—
279	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
280	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	61	1	no	—
286	C₄⋊C₄	—	122	2	no	—
287	C₄⋊C₄	—	244	2	no	—
288	C₄⋊C₄	—	244	2	no	—
289	C₄⋊C₄	—	244	2	no	—
290	C₄⋊C₄	—	244	1	no	—
291	C₄⋊C₄	—	244	1	no	—
292	C₄⋊C₄	—	244	1	no	—
293	C₂²⋊C₄	—	61	2	no	—
294	C₂²⋊C₄	—	244	2	no	—
295	C₂²⋊C₄	—	244	2	no	—
296	C₄²	—	4	1	no	—
297	C₄²	—	4	1	no	—
298	C₄²	D₄×D₆₁	1	1	yes	characteristic
299	C₁₆	—	2	1	no	—
300	C₁₆	—	2	1	no	—
301	C₁₆	—	122	2	no	—
302	C₂³	—	61	2	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	976	1	no	—
308	Q₈	—	488	4	no	—
309	Q₈	—	244	2	no	—
310	Q₈	—	244	2	no	—
311	D₄	—	61	4	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	61	2	no	—
317	C₂×C₄	—	122	2	no	—
318	C₂×C₄	—	122	2	no	—
319	C₂×C₄	—	976	2	no	—
320	C₂×C₄	—	488	4	no	—
321	C₂×C₄	—	244	2	no	—
322	C₂×C₄	—	244	2	no	—
323	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
324	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	122	2	no	—
328	C₈	C₂×#11758222	1	1	yes	characteristic
329	C₈	#14519823	1	1	yes	characteristic
330	C₂²	—	61	4	no	—
331	C₂²	—	122	2	no	—
332	C₂²	—	122	2	no	—
333	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	—	122	2	no	—
340	C₄	—	488	2	no	—
341	C₄	—	488	2	no	—
342	C₄	Q₃₂×D₆₁	1	2	yes	—
343	C₄	#16840408	1	1	yes	characteristic
344	C₄	#16840569	1	1	yes	characteristic
345	C₂	—	122	4	no	—
346	C₂	#21457634	1	1	yes	characteristic
347	C₂	C₂×Q₃₂×D₆₁	1	1	yes	characteristic
348	C₂	#21458962	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×Q₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂×D₂₄₄	D₈
hypercenter	C₄×Q₃₂	D₆₁
hypocenter	C₆₁	#33866
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #21448047	C₂	semidirect
4. #21448023	C₂	semidirect
5. #21447971	C₂	semidirect
6. #21447909	C₂	semidirect
7. #21447869	C₂	semidirect
8. #21447751	C₂	semidirect
10. #358×D₆₁	C₂	semidirect
11. C₄×Q₃₂×C₆₁	C₂	semidirect
142. D₂₄₄	Q₃₂	semidirect
179. C₄×Q₃₂	D₆₁	semidirect
215. C₆₁	#33866	semidirect

Maximal subgroups

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

subgroup	index
#21458213	2
Q₁₆×D₂₄₄	2
#21448047	2
#21448023	2
#21447971	2
#21447909	2
#21447869	2
#21447751	2
C₁₆×D₂₄₄	2
#358×D₆₁	2
C₄×Q₃₂×C₆₁	2
#33866	61