Subgroups — #29835520

Order 15616.

474 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456356	C₂	1	2	yes	maximal
2	#21448106	C₂	1	2	yes	maximal, complemented
3	#21448104	C₂	1	2	yes	maximal, complemented
4	#21448096	C₂	1	1	yes	maximal, characteristic, complemented
5	#21448094	C₂	1	1	yes	maximal, characteristic, complemented
6	#21448093	C₂	1	1	yes	maximal, characteristic, complemented
7	#21447606	C₂	1	2	yes	maximal
8	#21447600	C₂	1	1	yes	maximal, characteristic
9	#366×D₆₁	C₂	1	1	yes	maximal, characteristic
10	#21447414	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840392	—	2	2	no	—
13	C₂×#14519716	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
15	#16840172	—	2	2	no	—
16	#16840169	—	2	2	no	—
17	#16840167	C₂²	1	2	yes	—
18	#16840166	C₂²	1	2	yes	—
19	#16840163	C₂²	1	2	yes	—
20	#16839801	—	2	2	no	—
21	#16839799	C₂²	1	2	yes	—
22	#107×D₆₁	C₂²	1	2	yes	—
23	#16839716	C₂²	1	1	yes	characteristic
24	#16839589	C₂²	1	2	yes	—
25	#16839589	C₂²	1	2	yes	—
26	#16839588	C₂²	1	1	yes	characteristic
27	#16839588	C₂²	1	1	yes	characteristic
28	#16839588	C₂²	1	2	yes	—
29	#16839587	C₂²	1	2	yes	—
30	#16839587	C₂²	1	2	yes	—
31	#16839538	C₂²	1	1	yes	characteristic
32	#16839537	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
34	C₆₁×#499	C₂²	1	2	yes	—
35	C₆₁×#367	C₂²	1	2	yes	—
36	C₆₁×#366	C₂²	1	1	yes	characteristic
37	C₆₁×#366	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×#11758189	D₄	1	1	yes	characteristic
40	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
41	#14519777	—	2	2	no	—
42	#14519776	—	2	2	no	—
43	#14519776	—	4	2	no	—
44	#14519768	—	2	2	no	—
45	#14519767	—	4	2	no	—
46	#14519757	—	4	2	no	—
47	#14519755	—	4	2	no	—
48	#14519735	—	2	2	no	—
49	#14519734	D₄	1	1	yes	characteristic
50	C₄⋊C₄×D₆₁	—	2	2	no	—
51	#14519716	Q₈	1	4	yes	complemented
52	C₂×#11758185	C₂³	1	1	yes	characteristic
53	C₂×#11758185	C₂³	1	1	yes	characteristic
54	C₁₆×D₆₁	Q₈	1	4	yes	complemented
55	#14519711	—	2	2	no	—
56	#14519711	C₂³	1	2	yes	—
57	#14519711	C₂³	1	2	yes	—
58	#14519710	—	2	2	no	—
59	#14519710	—	2	2	no	—
60	#14519708	—	2	2	no	—
61	#14519708	—	2	2	no	—
62	#14519688	D₄	1	1	yes	characteristic
63	#14519687	D₄	1	1	yes	characteristic
64	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
65	#125×C₆₁	—	2	2	no	—
66	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
68	#107×C₆₁	C₂³	1	2	yes	—
69	#107×C₆₁	C₂³	1	2	yes	—
70	#106×C₆₁	—	2	2	no	—
71	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₆₁	—	4	2	no	—
73	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂×Dic₁₂₂	—	4	2	no	—
75	C₂×Dic₁₂₂	—	4	2	no	—
76	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₆₁	—	2	2	no	—
79	C₄⋊C₄×C₆₁	—	2	2	no	—
80	C₄⋊C₄×C₆₁	—	4	2	no	—
81	C₄⋊C₄×C₆₁	—	4	2	no	—
82	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
83	#11758198	—	4	2	no	—
84	#11758198	D₈	1	1	yes	characteristic
85	#11758198	D₈	1	1	yes	characteristic
86	C₄⋊Dic₆₁	—	2	2	no	—
87	C₄⋊Dic₆₁	—	2	2	no	—
88	C₄⋊Dic₆₁	—	4	2	no	—
89	C₄⋊Dic₆₁	—	4	2	no	—
90	#11758196	—	4	2	no	—
91	#11758196	—	4	2	no	—
92	#11758196	—	4	2	no	—
93	#11758196	D₈	1	1	yes	characteristic
94	#11758196	D₈	1	1	yes	characteristic
95	C₄×Dic₆₁	—	4	2	no	—
96	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×#92418	C₂×D₄	1	1	yes	characteristic
98	C₂×#92418	C₂×D₄	1	1	yes	characteristic
99	#11758189	Q₁₆	1	4	yes	—
100	C₈×D₆₁	C₂×Q₈	1	2	yes	—
101	C₈×D₆₁	C₂×Q₈	1	2	yes	—
102	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
103	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
104	#11758185	C₂×Q₈	1	2	yes	—
105	#11758185	C₂×Q₈	1	2	yes	—
106	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₆₁	—	4	2	no	—
113	C₂×Dic₆₁	—	4	2	no	—
114	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
115	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
116	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
117	C₄×D₆₁	—	16	2	no	—
118	C₄×D₆₁	C₄⋊Q₈	1	2	yes	—
119	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
120	Dic₁₂₂	—	16	2	no	—
121	Dic₁₂₂	—	16	2	no	—
122	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
123	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
125	#92418	C₄⋊Q₈	1	2	yes	—
126	#92418	C₂×Q₁₆	1	2	yes	—
127	#33729	—	61	1	no	maximal, Sylow
128	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
129	C₂×D₆₁	—	2	1	no	—
130	C₂×D₆₁	—	2	1	no	—
131	C₂×D₆₁	#500	1	2	yes	—
132	C₄×C₆₁	—	2	1	no	—
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	—	8	2	no	—
135	C₄×C₆₁	—	8	2	no	—
136	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
138	Dic₆₁	—	2	1	no	—
139	Dic₆₁	—	2	1	no	—
140	Dic₆₁	—	8	2	no	—
141	Dic₆₁	—	8	2	no	—
142	Dic₆₁	#500	1	2	yes	—
143	#3164	—	61	2	no	—
144	#2255	—	61	2	no	—
145	#2255	—	61	2	no	—
146	#2253	—	61	1	no	—
147	#2253	D₆₁	1	1	yes	characteristic, complemented
148	#2252	—	61	1	no	—
149	#2252	—	61	1	no	—
150	#2159	—	61	2	no	—
151	#2158	—	61	1	no	—
152	C₂×#366	—	61	1	no	—
153	#2108	—	61	1	no	—
154	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
155	C₂×C₆₁	#3415	1	1	yes	characteristic
156	C₂×C₆₁	#3417	1	1	yes	characteristic
157	D₆₁	—	2	2	no	—
158	#544	—	122	2	no	—
159	C₂×M₃₂	—	61	1	no	—
160	C₂²×C₁₆	—	61	1	no	—
161	#501	—	122	2	no	—
162	#501	—	122	2	no	—
163	#500	—	61	2	no	—
164	#500	—	61	2	no	—
165	#499	—	61	2	no	—
166	#499	C₂×D₆₁	1	2	yes	—
167	#428	—	122	2	no	—
168	#427	—	61	2	no	—
169	C₂×#107	—	61	2	no	—
170	#405	—	61	1	no	—
171	#367	—	61	2	no	—
172	#367	—	61	2	no	—
173	#367	C₂×D₆₁	1	2	yes	—
174	#366	—	61	1	no	—
175	#366	—	61	1	no	—
176	#366	—	61	2	no	—
177	#366	—	61	2	no	—
178	#366	—	61	2	no	—
179	#366	C₂×D₆₁	1	1	yes	characteristic
180	#366	C₂×D₆₁	1	1	yes	characteristic
181	#346	—	61	1	no	—
182	#346	—	61	1	no	—
183	C₄×C₁₆	—	61	1	no	—
184	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
185	C₆₁	#33729	1	1	yes	characteristic, complemented, Sylow
186	C₂×M₁₆	—	61	1	no	—
187	C₂²×C₈	—	61	1	no	—
188	C₄⋊Q₈	—	122	2	no	—
189	C₄⋊Q₈	—	122	2	no	—
190	C₄⋊Q₈	—	244	2	no	—
191	#125	—	2	2	no	—
192	#125	—	122	2	no	—
193	#125	—	244	2	no	—
194	C₂²⋊Q₈	—	244	2	no	—
195	C₂²⋊Q₈	—	244	2	no	—
196	#117	—	61	1	no	—
197	#117	—	122	2	no	—
198	C₂×C₄⋊C₄	—	122	2	no	—
199	M₃₂	—	61	4	no	—
200	C₂×C₁₆	—	61	1	no	—
201	C₂×C₁₆	—	61	1	no	—
202	C₂×C₁₆	—	61	4	no	—
203	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
204	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
205	#107	—	61	2	no	—
206	#107	—	61	2	no	—
207	#107	—	122	2	no	—
208	#107	—	122	2	no	—
209	#107	—	122	2	no	—
210	#107	C₂²×D₆₁	1	2	yes	—
211	#107	C₂²×D₆₁	1	2	yes	—
212	#106	—	2	2	no	—
213	#106	—	122	2	no	—
214	#106	—	122	2	no	—
215	#97	—	61	1	no	—
216	#97	—	61	1	no	—
217	C₄×C₈	—	61	1	no	—
218	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
219	C₂×Q₈	—	244	2	no	—
220	C₂×Q₈	—	244	2	no	—
221	C₂²×C₄	—	61	1	no	—
222	C₂²×C₄	—	244	2	no	—
223	M₁₆	—	61	4	no	—
224	C₂×C₈	—	61	1	no	—
225	C₂×C₈	—	61	1	no	—
226	C₂×C₈	—	61	2	no	—
227	C₂×C₈	—	61	2	no	—
228	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	61	1	no	—
235	C₄⋊C₄	—	61	1	no	—
236	C₄⋊C₄	—	122	2	no	—
237	C₄⋊C₄	—	122	2	no	—
238	C₄⋊C₄	—	244	2	no	—
239	C₄⋊C₄	—	244	2	no	—
240	C₄⋊C₄	—	244	2	no	—
241	C₄⋊C₄	—	244	2	no	—
242	C₄⋊C₄	—	244	2	no	—
243	C₂²⋊C₄	—	61	1	no	—
244	C₂²⋊C₄	—	61	1	no	—
245	C₂²⋊C₄	—	244	2	no	—
246	C₄²	—	61	1	no	—
247	C₄²	—	244	2	no	—
248	C₄²	D₄×D₆₁	1	1	yes	characteristic
249	C₁₆	—	61	2	no	—
250	C₁₆	—	61	2	no	—
251	C₁₆	Q₈×D₆₁	1	2	yes	—
252	C₁₆	Q₈×D₆₁	1	2	yes	—
253	C₂³	—	61	1	no	—
254	Q₈	—	976	2	no	—
255	Q₈	—	976	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	61	1	no	—
259	C₂×C₄	—	61	2	no	—
260	C₂×C₄	—	61	2	no	—
261	C₂×C₄	—	61	1	no	—
262	C₂×C₄	—	61	1	no	—
263	C₂×C₄	—	976	2	no	—
264	C₂×C₄	—	244	2	no	—
265	C₂×C₄	—	244	2	no	—
266	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
267	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
269	C₈	—	61	2	no	—
270	C₈	—	61	2	no	—
271	C₈	Q₁₆×D₆₁	1	2	yes	—
272	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
273	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
274	C₂²	—	61	2	no	—
275	C₂²	—	122	1	no	—
276	C₂²	—	122	1	no	—
277	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	61	2	no	—
283	C₄	—	122	1	no	—
284	C₄	—	122	1	no	—
285	C₄	—	488	2	no	—
286	C₄	—	488	2	no	—
287	C₄	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
288	C₄	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
289	C₂	—	122	2	no	—
290	C₂	#500×D₆₁	1	1	yes	characteristic
291	C₂	#21458957	1	1	yes	characteristic
292	C₂	#21458969	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×#2253	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂²×D₆₁	C₂×D₈
hypercenter	#2253	D₆₁
hypocenter	C₆₁	#33729
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #21448106	C₂	semidirect
3. #21448104	C₂	semidirect
4. #21448096	C₂	semidirect
5. #21448094	C₂	semidirect
6. #21448093	C₂	semidirect
11. C₆₁×#2253	C₂	semidirect
51. #14519716	Q₈	semidirect
54. C₁₆×D₆₁	Q₈	semidirect
147. #2253	D₆₁	semidirect
185. C₆₁	#33729	semidirect

Maximal subgroups

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

subgroup	index
#21456356	2
#21448106	2
#21448104	2
#21448096	2
#21448094	2
#21448093	2
#21447606	2
#21447600	2
#366×D₆₁	2
#21447414	2
C₆₁×#2253	2
#33729	61