Subgroups — C₆₁×#24321 #29527754

Order 15616.

1036 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3077	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×AGL(1,ℤ/8)×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2563	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2061	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2027	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1986	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1883	C₂	1	2	yes	maximal
10	C₆₁×#1860	C₂	1	2	yes	maximal, complemented
11	C₆₁×#1806	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	1	yes	characteristic
13	Q₈²×C₆₁	—	2	1	no	—
14	C₆₁×#557	—	2	1	no	—
15	C₆₁×#554	—	2	1	no	—
16	C₆₁×#554	—	2	2	no	—
17	D₄×Q₈×C₆₁	—	2	2	no	—
18	D₄×Q₈×C₆₁	—	2	1	no	—
19	C₆₁×#544	—	2	2	no	—
20	C₆₁×#544	C₂²	1	2	yes	—
21	C₆₁×#533	—	2	1	no	—
22	C₆₁×#533	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
24	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×#520	C₂²	1	2	yes	—
26	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#478	—	2	2	no	—
29	C₆₁×#476	—	2	1	no	—
30	C₆₁×#474	—	2	1	no	—
31	C₆₁×#442	—	2	1	no	—
32	C₆₁×#440	—	2	1	no	—
33	C₄×SD₁₆×C₆₁	—	2	1	no	—
34	C₄×D₈×C₆₁	—	2	1	no	—
35	C₆₁×#428	C₂²	1	2	yes	—
36	C₆₁×#428	C₂²	1	1	yes	characteristic
37	C₆₁×#423	C₂²	1	1	yes	characteristic
38	C₂×#104×C₆₁	C₂²	1	2	yes	—
39	C₆₁×#418	C₂²	1	1	yes	characteristic
40	C₆₁×#418	C₂²	1	2	yes	—
41	C₆₁×#417	C₂²	1	1	yes	characteristic
42	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
43	C₆₁×#393	—	2	2	no	—
44	C₆₁×#390	C₂²	1	2	yes	—
45	C₆₁×#382	C₂²	1	2	yes	—
46	C₆₁×#380	—	2	2	no	—
47	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
48	C₆₁×#343	C₂²	1	2	yes	—
49	C₆₁×#343	C₂²	1	2	yes	—
50	C₆₁×#339	C₂²	1	2	yes	—
51	C₆₁×#339	C₂²	1	2	yes	—
52	C₆₁×#339	C₂²	1	2	yes	—
53	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₆₁	—	2	2	no	—
55	C₂²×Q₈×C₆₁	D₄	1	2	yes	—
56	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₁	—	2	1	no	—
58	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
59	AGL(1,ℤ/8)×C₆₁	Q₈	1	4	yes	complemented
60	C₂×SD₁₆×C₆₁	—	2	1	no	—
61	C₂×D₈×C₆₁	—	2	1	no	—
62	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
64	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₁	—	2	2	no	—
66	C₄⋊Q₈×C₆₁	—	2	2	no	—
67	C₄⋊Q₈×C₆₁	—	2	4	no	—
68	C₄⋊Q₈×C₆₁	—	2	2	no	—
69	C₄⋊Q₈×C₆₁	—	2	2	no	—
70	C₄⋊Q₈×C₆₁	—	2	1	no	—
71	C₄⋊Q₈×C₆₁	—	2	1	no	—
72	C₄⋊Q₈×C₆₁	—	2	1	no	—
73	C₄⋊Q₈×C₆₁	—	2	1	no	—
74	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
75	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
76	#125×C₆₁	—	2	4	no	—
77	#125×C₆₁	—	2	2	no	—
78	#125×C₆₁	—	2	2	no	—
79	#125×C₆₁	D₄	1	2	yes	—
80	C₂²⋊Q₈×C₆₁	—	2	2	no	—
81	C₂²⋊Q₈×C₆₁	—	2	2	no	—
82	C₂²⋊Q₈×C₆₁	—	2	4	no	—
83	C₂²⋊Q₈×C₆₁	—	2	4	no	—
84	C₂²⋊Q₈×C₆₁	—	2	1	no	—
85	C₂²⋊Q₈×C₆₁	—	2	1	no	—
86	C₂²⋊Q₈×C₆₁	—	2	4	no	—
87	C₄×Q₈×C₆₁	—	2	2	no	—
88	C₄×Q₈×C₆₁	—	2	2	no	—
89	C₄×Q₈×C₆₁	—	2	2	no	—
90	C₄×Q₈×C₆₁	—	2	1	no	—
91	C₄×Q₈×C₆₁	—	2	1	no	—
92	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
93	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
94	C₄×D₄×C₆₁	—	2	2	no	—
95	C₄×D₄×C₆₁	—	2	1	no	—
96	C₄×D₄×C₆₁	—	2	4	no	—
97	C₄×D₄×C₆₁	—	2	1	no	—
98	C₄×D₄×C₆₁	—	2	1	no	—
99	C₄×D₄×C₆₁	—	2	1	no	—
100	C₄×D₄×C₆₁	—	2	1	no	—
101	C₄×D₄×C₆₁	—	2	1	no	—
102	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
103	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
104	#117×C₆₁	—	2	2	no	—
105	#117×C₆₁	—	2	1	no	—
106	#117×C₆₁	C₂³	1	2	yes	—
107	#117×C₆₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
109	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
111	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
112	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
116	C₂×C₄²×C₆₁	—	2	1	no	—
117	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
118	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
119	#107×C₆₁	—	2	1	no	—
120	#107×C₆₁	—	2	2	no	—
121	#106×C₆₁	—	2	1	no	—
122	#106×C₆₁	—	2	2	no	—
123	C₄⋊C₈×C₆₁	—	2	1	no	—
124	C₄⋊C₈×C₆₁	—	2	1	no	—
125	#104×C₆₁	—	2	4	no	—
126	#103×C₆₁	—	2	1	no	—
127	#102×C₆₁	—	2	1	no	—
128	#102×C₆₁	—	2	1	no	—
129	#102×C₆₁	—	2	1	no	—
130	#102×C₆₁	—	2	2	no	—
131	#102×C₆₁	—	2	2	no	—
132	#97×C₆₁	—	2	1	no	—
133	C₄×C₈×C₆₁	—	2	1	no	—
134	#95×C₆₁	—	2	1	no	—
135	#95×C₆₁	—	2	4	no	—
136	C₂⁴×C₆₁	—	2	1	no	—
137	#41×C₆₁	—	2	1	no	—
138	#41×C₆₁	—	2	1	no	—
139	#41×C₆₁	C₂×Q₈	1	2	yes	—
140	#41×C₆₁	#41	1	2	yes	—
141	C₂×Q₈×C₆₁	—	2	4	no	—
142	C₂×Q₈×C₆₁	—	2	1	no	—
143	C₂×Q₈×C₆₁	—	2	2	no	—
144	C₂×Q₈×C₆₁	—	2	1	no	—
145	C₂×Q₈×C₆₁	—	2	2	no	—
146	C₂×Q₈×C₆₁	—	8	2	no	—
147	C₂×Q₈×C₆₁	—	8	4	no	—
148	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
149	C₂×D₄×C₆₁	—	2	1	no	—
150	C₂×D₄×C₆₁	—	2	1	no	—
151	C₂×D₄×C₆₁	—	2	1	no	—
152	C₂×D₄×C₆₁	—	2	1	no	—
153	C₂×D₄×C₆₁	—	2	1	no	—
154	C₂×D₄×C₆₁	—	2	1	no	—
155	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
156	C₂×D₄×C₆₁	C₂×Q₈	1	2	yes	complemented
157	C₂²×C₄×C₆₁	—	2	1	no	—
158	C₂²×C₄×C₆₁	—	2	4	no	—
159	C₂²×C₄×C₆₁	—	2	1	no	—
160	C₂²×C₄×C₆₁	—	2	1	no	—
161	C₂²×C₄×C₆₁	—	2	1	no	—
162	C₂²×C₄×C₆₁	—	2	2	no	—
163	C₂²×C₄×C₆₁	—	4	4	no	—
164	C₂²×C₄×C₆₁	—	4	1	no	—
165	C₂²×C₄×C₆₁	—	4	1	no	—
166	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
170	SD₁₆×C₆₁	—	2	4	no	—
171	D₈×C₆₁	—	2	4	no	—
172	M₁₆×C₆₁	—	2	1	no	—
173	M₁₆×C₆₁	—	2	2	no	—
174	M₁₆×C₆₁	—	2	1	no	—
175	M₁₆×C₆₁	—	2	1	no	—
176	M₁₆×C₆₁	—	2	2	no	—
177	M₁₆×C₆₁	C₂×Q₈	1	2	yes	complemented
178	C₂×C₈×C₆₁	—	2	1	no	—
179	C₂×C₈×C₆₁	—	2	1	no	—
180	C₂×C₈×C₆₁	—	2	2	no	—
181	C₄⋊C₄×C₆₁	—	2	4	no	—
182	C₄⋊C₄×C₆₁	—	2	4	no	—
183	C₄⋊C₄×C₆₁	—	2	4	no	—
184	C₄⋊C₄×C₆₁	—	2	4	no	—
185	C₄⋊C₄×C₆₁	—	2	1	no	—
186	C₄⋊C₄×C₆₁	—	2	2	no	—
187	C₄⋊C₄×C₆₁	—	2	2	no	—
188	C₄⋊C₄×C₆₁	—	2	1	no	—
189	C₄⋊C₄×C₆₁	—	2	1	no	—
190	C₄⋊C₄×C₆₁	—	2	1	no	—
191	C₄⋊C₄×C₆₁	—	2	2	no	—
192	C₄⋊C₄×C₆₁	—	2	2	no	—
193	C₄⋊C₄×C₆₁	—	2	2	no	—
194	C₄⋊C₄×C₆₁	—	2	2	no	—
195	C₄⋊C₄×C₆₁	—	2	1	no	—
196	C₄⋊C₄×C₆₁	—	2	1	no	—
197	C₄⋊C₄×C₆₁	—	4	2	no	—
198	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
199	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
201	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
202	C₂²⋊C₄×C₆₁	—	2	4	no	—
203	C₂²⋊C₄×C₆₁	—	2	2	no	—
204	C₂²⋊C₄×C₆₁	—	2	1	no	—
205	C₂²⋊C₄×C₆₁	—	2	1	no	—
206	C₂²⋊C₄×C₆₁	—	2	1	no	—
207	C₂²⋊C₄×C₆₁	—	4	1	no	—
208	C₂²⋊C₄×C₆₁	—	4	1	no	—
209	C₂²⋊C₄×C₆₁	—	8	4	no	—
210	C₄²×C₆₁	—	2	4	no	—
211	C₄²×C₆₁	—	2	1	no	—
212	C₄²×C₆₁	—	2	1	no	—
213	C₄²×C₆₁	—	2	1	no	—
214	C₄²×C₆₁	—	2	2	no	—
215	C₄²×C₆₁	—	2	2	no	—
216	C₄²×C₆₁	—	2	2	no	—
217	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
221	C₂³×C₆₁	—	2	1	no	—
222	C₂³×C₆₁	—	2	2	no	—
223	C₂³×C₆₁	—	4	2	no	—
224	C₂³×C₆₁	—	4	2	no	—
225	C₂³×C₆₁	—	4	1	no	—
226	C₂³×C₆₁	—	4	1	no	—
227	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
228	Q₈×C₆₁	—	2	1	no	—
229	Q₈×C₆₁	—	4	4	no	—
230	Q₈×C₆₁	—	4	2	no	—
231	Q₈×C₆₁	—	4	2	no	—
232	Q₈×C₆₁	—	8	2	no	—
233	Q₈×C₆₁	—	8	4	no	—
234	Q₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
235	D₄×C₆₁	—	2	2	no	—
236	D₄×C₆₁	—	2	2	no	—
237	D₄×C₆₁	—	2	2	no	—
238	D₄×C₆₁	—	2	1	no	—
239	D₄×C₆₁	—	2	2	no	—
240	D₄×C₆₁	—	4	1	no	—
241	D₄×C₆₁	—	4	1	no	—
242	D₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
243	C₂×C₄×C₆₁	—	2	1	no	—
244	C₂×C₄×C₆₁	—	2	2	no	—
245	C₂×C₄×C₆₁	—	2	2	no	—
246	C₂×C₄×C₆₁	—	2	2	no	—
247	C₂×C₄×C₆₁	—	2	2	no	—
248	C₂×C₄×C₆₁	—	2	1	no	—
249	C₂×C₄×C₆₁	—	2	1	no	—
250	C₂×C₄×C₆₁	—	2	4	no	—
251	C₂×C₄×C₆₁	—	2	2	no	—
252	C₂×C₄×C₆₁	—	4	1	no	—
253	C₂×C₄×C₆₁	—	4	2	no	—
254	C₂×C₄×C₆₁	—	4	2	no	—
255	C₂×C₄×C₆₁	—	4	4	no	—
256	C₂×C₄×C₆₁	—	4	2	no	—
257	C₂×C₄×C₆₁	—	4	2	no	—
258	C₂×C₄×C₆₁	—	4	2	no	—
259	C₂×C₄×C₆₁	—	4	1	no	—
260	C₂×C₄×C₆₁	—	4	2	no	—
261	C₂×C₄×C₆₁	—	8	2	no	—
262	C₂×C₄×C₆₁	—	8	4	no	—
263	C₂×C₄×C₆₁	—	8	2	no	—
264	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
265	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
266	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
267	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
268	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
271	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
272	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
273	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
274	C₈×C₆₁	—	2	2	no	—
275	C₈×C₆₁	—	4	1	no	—
276	C₈×C₆₁	—	4	2	no	—
277	#24321	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂²×C₆₁	—	2	1	no	—
279	C₂²×C₆₁	—	2	1	no	—
280	C₂²×C₆₁	—	2	2	no	—
281	C₂²×C₆₁	—	4	4	no	—
282	C₂²×C₆₁	—	4	1	no	—
283	C₂²×C₆₁	—	4	1	no	—
284	C₂²×C₆₁	—	4	2	no	—
285	C₂²×C₆₁	—	8	2	no	—
286	C₂²×C₆₁	—	8	1	no	—
287	C₂²×C₆₁	—	8	1	no	—
288	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
289	C₂²×C₆₁	#548	1	1	yes	characteristic
290	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
291	C₄×C₆₁	—	2	2	no	—
292	C₄×C₆₁	—	2	1	no	—
293	C₄×C₆₁	—	2	1	no	—
294	C₄×C₆₁	—	2	1	no	—
295	C₄×C₆₁	—	4	1	no	—
296	C₄×C₆₁	—	4	2	no	—
297	C₄×C₆₁	—	4	1	no	—
298	C₄×C₆₁	—	4	4	no	—
299	C₄×C₆₁	—	4	2	no	—
300	C₄×C₆₁	—	8	2	no	—
301	C₄×C₆₁	—	8	1	no	—
302	C₄×C₆₁	#454	1	2	yes	—
303	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
304	C₄×C₆₁	#536	1	1	yes	characteristic
305	C₄×C₆₁	#548	1	1	yes	characteristic
306	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
307	#3503	C₂×C₆₁	1	1	yes	characteristic, complemented
308	#3077	C₂×C₆₁	1	1	yes	characteristic, complemented
309	C₄×AGL(1,ℤ/8)	C₂×C₆₁	1	1	yes	characteristic
310	#2563	C₂×C₆₁	1	1	yes	characteristic, complemented
311	#2061	C₂×C₆₁	1	2	yes	complemented
312	#2027	C₂×C₆₁	1	2	yes	complemented
313	#1997	C₂×C₆₁	1	1	yes	characteristic, complemented
314	#1986	C₂×C₆₁	1	1	yes	characteristic, complemented
315	#1883	C₂×C₆₁	1	2	yes	—
316	#1860	C₂×C₆₁	1	2	yes	complemented
317	#1806	C₂×C₆₁	1	1	yes	characteristic, complemented
318	C₂×C₆₁	—	2	1	no	—
319	C₂×C₆₁	—	2	1	no	—
320	C₂×C₆₁	—	4	2	no	—
321	C₂×C₆₁	—	8	2	no	—
322	C₂×C₆₁	#2409	1	1	yes	characteristic
323	C₂×C₆₁	C₂×#454	1	1	yes	characteristic
324	C₂×C₆₁	#3019	1	1	yes	characteristic
325	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	1	yes	characteristic
326	Q₈²	—	2	1	no	—
327	#557	—	2	1	no	—
328	#554	—	2	1	no	—
329	#554	—	2	2	no	—
330	D₄×Q₈	—	2	2	no	—
331	D₄×Q₈	—	2	1	no	—
332	#544	—	2	2	no	—
333	#544	C₂²×C₆₁	1	2	yes	—
334	#533	—	2	1	no	—
335	#533	C₂²×C₆₁	1	1	yes	characteristic
336	C₂×C₄⋊Q₈	—	2	1	no	—
337	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
338	#520	C₂²×C₆₁	1	2	yes	—
339	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
340	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
341	#478	—	2	2	no	—
342	#476	—	2	1	no	—
343	#474	—	2	1	no	—
344	#442	—	2	1	no	—
345	#440	—	2	1	no	—
346	C₄×SD₁₆	—	2	1	no	—
347	C₄×D₈	—	2	1	no	—
348	#428	C₂²×C₆₁	1	2	yes	—
349	#428	C₂²×C₆₁	1	1	yes	characteristic
350	#423	C₂²×C₆₁	1	1	yes	characteristic
351	C₂×#104	C₂²×C₆₁	1	2	yes	—
352	#418	C₂²×C₆₁	1	2	yes	—
353	#418	C₂²×C₆₁	1	1	yes	characteristic
354	#417	C₂²×C₆₁	1	1	yes	characteristic
355	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
356	#393	—	2	2	no	—
357	#390	C₂²×C₆₁	1	2	yes	—
358	#382	C₂²×C₆₁	1	2	yes	—
359	#380	—	2	2	no	—
360	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
361	#343	C₂²×C₆₁	1	2	yes	—
362	#343	C₂²×C₆₁	1	2	yes	—
363	#339	C₂²×C₆₁	1	2	yes	—
364	#339	C₂²×C₆₁	1	2	yes	—
365	#339	C₂²×C₆₁	1	2	yes	—
366	C₆₁	#24321	1	1	yes	characteristic, direct product factor, Sylow
367	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
368	C₂²×Q₈	—	2	2	no	—
369	C₂²×Q₈	D₄×C₆₁	1	2	yes	—
370	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
371	C₂³×C₄	—	2	1	no	—
372	AGL(1,ℤ/8)	—	2	2	no	—
373	AGL(1,ℤ/8)	Q₈×C₆₁	1	4	yes	complemented
374	C₂×SD₁₆	—	2	1	no	—
375	C₂×D₈	—	2	1	no	—
376	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
377	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
378	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
379	C₄⋊Q₈	—	2	2	no	—
380	C₄⋊Q₈	—	2	2	no	—
381	C₄⋊Q₈	—	2	4	no	—
382	C₄⋊Q₈	—	2	2	no	—
383	C₄⋊Q₈	—	2	2	no	—
384	C₄⋊Q₈	—	2	1	no	—
385	C₄⋊Q₈	—	2	1	no	—
386	C₄⋊Q₈	—	2	1	no	—
387	C₄⋊Q₈	—	2	1	no	—
388	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
389	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
390	#125	—	2	4	no	—
391	#125	—	2	2	no	—
392	#125	—	2	2	no	—
393	#125	D₄×C₆₁	1	2	yes	—
394	C₂²⋊Q₈	—	2	2	no	—
395	C₂²⋊Q₈	—	2	2	no	—
396	C₂²⋊Q₈	—	2	4	no	—
397	C₂²⋊Q₈	—	2	4	no	—
398	C₂²⋊Q₈	—	2	1	no	—
399	C₂²⋊Q₈	—	2	1	no	—
400	C₂²⋊Q₈	—	2	4	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₄×Q₈	—	2	2	no	—
404	C₄×Q₈	—	2	1	no	—
405	C₄×Q₈	—	2	1	no	—
406	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
407	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	1	no	—
411	C₄×D₄	—	2	4	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
417	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
418	#117	—	2	2	no	—
419	#117	—	2	1	no	—
420	#117	C₂³×C₆₁	1	1	yes	characteristic
421	#117	C₂³×C₆₁	1	2	yes	—
422	C₂×C₄⋊C₄	—	2	2	no	—
423	C₂×C₄⋊C₄	—	2	2	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	4	no	—
430	C₂×C₄²	—	2	1	no	—
431	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
432	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
433	#107	—	2	1	no	—
434	#107	—	2	2	no	—
435	#106	—	2	1	no	—
436	#106	—	2	2	no	—
437	C₄⋊C₈	—	2	1	no	—
438	C₄⋊C₈	—	2	1	no	—
439	#104	—	2	4	no	—
440	#103	—	2	1	no	—
441	#102	—	2	1	no	—
442	#102	—	2	1	no	—
443	#102	—	2	1	no	—
444	#102	—	2	2	no	—
445	#102	—	2	2	no	—
446	#97	—	2	1	no	—
447	C₄×C₈	—	2	1	no	—
448	#95	—	2	1	no	—
449	#95	—	2	4	no	—
450	C₂⁴	—	2	1	no	—
451	#41	—	2	1	no	—
452	#41	—	2	1	no	—
453	#41	C₂×Q₈×C₆₁	1	2	yes	—
454	#41	#41×C₆₁	1	2	yes	—
455	C₂×Q₈	—	2	4	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	—	8	2	no	—
461	C₂×Q₈	—	8	4	no	—
462	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
470	C₂×D₄	C₂×Q₈×C₆₁	1	2	yes	complemented
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	4	4	no	—
478	C₂²×C₄	—	4	1	no	—
479	C₂²×C₄	—	4	1	no	—
480	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
482	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
484	SD₁₆	—	2	4	no	—
485	D₈	—	2	4	no	—
486	M₁₆	—	2	1	no	—
487	M₁₆	—	2	2	no	—
488	M₁₆	—	2	1	no	—
489	M₁₆	—	2	1	no	—
490	M₁₆	—	2	2	no	—
491	M₁₆	C₂×Q₈×C₆₁	1	2	yes	complemented
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	2	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	4	2	no	—
512	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
513	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
514	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
515	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	1	no	—
519	C₂²⋊C₄	—	2	1	no	—
520	C₂²⋊C₄	—	2	1	no	—
521	C₂²⋊C₄	—	4	1	no	—
522	C₂²⋊C₄	—	4	1	no	—
523	C₂²⋊C₄	—	8	4	no	—
524	C₄²	—	2	4	no	—
525	C₄²	—	2	1	no	—
526	C₄²	—	2	1	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	2	2	no	—
529	C₄²	—	2	2	no	—
530	C₄²	—	2	2	no	—
531	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
532	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
533	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
534	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
535	C₂³	—	2	1	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	4	2	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	4	1	no	—
540	C₂³	—	4	1	no	—
541	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
542	Q₈	—	2	1	no	—
543	Q₈	—	4	4	no	—
544	Q₈	—	4	2	no	—
545	Q₈	—	4	2	no	—
546	Q₈	—	8	2	no	—
547	Q₈	—	8	4	no	—
548	Q₈	C₂²⋊Q₈×C₆₁	1	2	yes	—
549	D₄	—	2	2	no	—
550	D₄	—	2	2	no	—
551	D₄	—	2	1	no	—
552	D₄	—	2	2	no	—
553	D₄	—	2	2	no	—
554	D₄	—	4	1	no	—
555	D₄	—	4	1	no	—
556	D₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	4	2	no	—
569	C₂×C₄	—	4	4	no	—
570	C₂×C₄	—	4	2	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	2	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	8	2	no	—
576	C₂×C₄	—	8	4	no	—
577	C₂×C₄	—	8	2	no	—
578	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
579	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
580	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
581	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
582	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
583	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
584	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
585	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
586	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
587	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
588	C₈	—	2	2	no	—
589	C₈	—	4	1	no	—
590	C₈	—	4	2	no	—
591	C₂²	—	2	1	no	—
592	C₂²	—	2	1	no	—
593	C₂²	—	2	2	no	—
594	C₂²	—	4	4	no	—
595	C₂²	—	4	1	no	—
596	C₂²	—	4	1	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	8	2	no	—
599	C₂²	—	8	1	no	—
600	C₂²	—	8	1	no	—
601	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
602	C₂²	C₆₁×#548	1	1	yes	characteristic
603	C₂²	D₄×Q₈×C₆₁	1	1	yes	characteristic
604	C₄	—	2	2	no	—
605	C₄	—	2	1	no	—
606	C₄	—	2	1	no	—
607	C₄	—	2	1	no	—
608	C₄	—	4	1	no	—
609	C₄	—	4	2	no	—
610	C₄	—	4	1	no	—
611	C₄	—	4	4	no	—
612	C₄	—	4	2	no	—
613	C₄	—	8	2	no	—
614	C₄	—	8	1	no	—
615	C₄	C₆₁×#454	1	2	yes	—
616	C₄	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
617	C₄	C₆₁×#536	1	1	yes	characteristic
618	C₄	C₆₁×#548	1	1	yes	characteristic
619	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
620	C₂	—	2	1	no	—
621	C₂	—	2	1	no	—
622	C₂	—	4	2	no	—
623	C₂	—	8	2	no	—
624	C₂	C₆₁×#2409	1	1	yes	characteristic
625	C₂	C₂×C₆₁×#454	1	1	yes	characteristic
626	C₂	C₆₁×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×AGL(1,ℤ/8)×C₆₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3503	C₂	semidirect
2. C₆₁×#3077	C₂	semidirect
4. C₆₁×#2563	C₂	semidirect
5. C₆₁×#2061	C₂	semidirect
6. C₆₁×#2027	C₂	semidirect
7. C₆₁×#1997	C₂	semidirect
8. C₆₁×#1986	C₂	semidirect
10. C₆₁×#1860	C₂	semidirect
11. C₆₁×#1806	C₂	semidirect
59. AGL(1,ℤ/8)×C₆₁	Q₈	semidirect
156. C₂×D₄×C₆₁	C₂×Q₈	semidirect
177. M₁₆×C₆₁	C₂×Q₈	semidirect
277. #24321	C₆₁	direct (N × H)
307. #3503	C₂×C₆₁	semidirect
308. #3077	C₂×C₆₁	semidirect
310. #2563	C₂×C₆₁	semidirect
311. #2061	C₂×C₆₁	semidirect
312. #2027	C₂×C₆₁	semidirect
313. #1997	C₂×C₆₁	semidirect
314. #1986	C₂×C₆₁	semidirect
316. #1860	C₂×C₆₁	semidirect
317. #1806	C₂×C₆₁	semidirect
366. C₆₁	#24321	direct (N × H)
373. AGL(1,ℤ/8)	Q₈×C₆₁	semidirect
470. C₂×D₄	C₂×Q₈×C₆₁	semidirect
491. M₁₆	C₂×Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3503	2
C₆₁×#3077	2
C₄×AGL(1,ℤ/8)×C₆₁	2
C₆₁×#2563	2
C₆₁×#2061	2
C₆₁×#2027	2
C₆₁×#1997	2
C₆₁×#1986	2
C₆₁×#1883	2
C₆₁×#1860	2
C₆₁×#1806	2
#24321	61

Subgroups — C61×#24321 #29527754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24321 #29527754