Subgroups — C₆₁×#24320 #29527753

Order 15616.

1092 conjugacy classes of subgroups.

The subgroup lattice has 632 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×#2542	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2060	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×#1859	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	C₂×C₂²⋊Q₈×C₆₁	—	2	2	no	—
26	C₆₁×#520	C₂²	1	2	yes	—
27	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#478	—	2	2	no	—
30	C₆₁×#476	—	2	1	no	—
31	C₆₁×#474	—	2	1	no	—
32	C₆₁×#442	—	2	1	no	—
33	C₆₁×#440	—	2	1	no	—
34	C₄×SD₁₆×C₆₁	—	2	1	no	—
35	C₄×D₈×C₆₁	—	2	1	no	—
36	C₆₁×#428	C₂²	1	2	yes	—
37	C₆₁×#428	C₂²	1	1	yes	characteristic
38	C₆₁×#423	C₂²	1	1	yes	characteristic
39	C₂×#104×C₆₁	C₂²	1	2	yes	—
40	C₆₁×#418	C₂²	1	1	yes	characteristic
41	C₆₁×#418	C₂²	1	2	yes	—
42	C₆₁×#417	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
44	C₆₁×#399	—	2	2	no	—
45	C₆₁×#390	C₂²	1	2	yes	—
46	C₆₁×#384	C₂²	1	2	yes	—
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	C₂²⋊Q₈×C₆₁	—	4	8	no	—
88	C₄×Q₈×C₆₁	—	2	2	no	—
89	C₄×Q₈×C₆₁	—	2	2	no	—
90	C₄×Q₈×C₆₁	—	2	2	no	—
91	C₄×Q₈×C₆₁	—	2	1	no	—
92	C₄×Q₈×C₆₁	—	2	1	no	—
93	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
95	C₄×D₄×C₆₁	—	2	2	no	—
96	C₄×D₄×C₆₁	—	2	1	no	—
97	C₄×D₄×C₆₁	—	2	4	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	—	2	1	no	—
103	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
104	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
105	#117×C₆₁	—	2	2	no	—
106	#117×C₆₁	—	2	1	no	—
107	#117×C₆₁	C₂³	1	2	yes	—
108	#117×C₆₁	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
111	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
112	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
113	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
117	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
118	C₂×C₄²×C₆₁	—	2	1	no	—
119	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
120	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
121	#107×C₆₁	—	2	1	no	—
122	#107×C₆₁	—	2	2	no	—
123	#106×C₆₁	—	2	1	no	—
124	#106×C₆₁	—	2	2	no	—
125	C₄⋊C₈×C₆₁	—	2	1	no	—
126	C₄⋊C₈×C₆₁	—	2	1	no	—
127	#104×C₆₁	—	2	4	no	—
128	#103×C₆₁	—	2	1	no	—
129	#102×C₆₁	—	2	1	no	—
130	#102×C₆₁	—	2	1	no	—
131	#102×C₆₁	—	2	1	no	—
132	#102×C₆₁	—	2	2	no	—
133	#102×C₆₁	—	2	2	no	—
134	#97×C₆₁	—	2	1	no	—
135	C₄×C₈×C₆₁	—	2	1	no	—
136	#95×C₆₁	—	2	1	no	—
137	C₂⁴×C₆₁	—	2	1	no	—
138	#41×C₆₁	—	2	1	no	—
139	#41×C₆₁	—	2	1	no	—
140	#41×C₆₁	C₂×Q₈	1	2	yes	—
141	#41×C₆₁	#41	1	2	yes	—
142	C₂×Q₈×C₆₁	—	2	4	no	—
143	C₂×Q₈×C₆₁	—	2	1	no	—
144	C₂×Q₈×C₆₁	—	2	2	no	—
145	C₂×Q₈×C₆₁	—	2	1	no	—
146	C₂×Q₈×C₆₁	—	2	2	no	—
147	C₂×Q₈×C₆₁	—	4	4	no	—
148	C₂×Q₈×C₆₁	—	8	4	no	—
149	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
156	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
157	C₂×D₄×C₆₁	C₂×Q₈	1	2	yes	complemented
158	C₂²×C₄×C₆₁	—	2	1	no	—
159	C₂²×C₄×C₆₁	—	2	2	no	—
160	C₂²×C₄×C₆₁	—	2	4	no	—
161	C₂²×C₄×C₆₁	—	2	1	no	—
162	C₂²×C₄×C₆₁	—	2	1	no	—
163	C₂²×C₄×C₆₁	—	2	1	no	—
164	C₂²×C₄×C₆₁	—	4	4	no	—
165	C₂²×C₄×C₆₁	—	4	1	no	—
166	C₂²×C₄×C₆₁	—	4	1	no	—
167	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
169	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
171	SD₁₆×C₆₁	—	2	4	no	—
172	D₈×C₆₁	—	2	4	no	—
173	M₁₆×C₆₁	—	2	1	no	—
174	M₁₆×C₆₁	—	2	2	no	—
175	M₁₆×C₆₁	—	2	1	no	—
176	M₁₆×C₆₁	—	2	1	no	—
177	M₁₆×C₆₁	—	2	2	no	—
178	M₁₆×C₆₁	C₂×Q₈	1	2	yes	complemented
179	C₂×C₈×C₆₁	—	2	1	no	—
180	C₂×C₈×C₆₁	—	2	1	no	—
181	C₂×C₈×C₆₁	—	2	2	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	4	no	—
186	C₄⋊C₄×C₆₁	—	2	4	no	—
187	C₄⋊C₄×C₆₁	—	2	1	no	—
188	C₄⋊C₄×C₆₁	—	2	2	no	—
189	C₄⋊C₄×C₆₁	—	2	2	no	—
190	C₄⋊C₄×C₆₁	—	2	1	no	—
191	C₄⋊C₄×C₆₁	—	2	1	no	—
192	C₄⋊C₄×C₆₁	—	2	1	no	—
193	C₄⋊C₄×C₆₁	—	2	2	no	—
194	C₄⋊C₄×C₆₁	—	2	2	no	—
195	C₄⋊C₄×C₆₁	—	2	2	no	—
196	C₄⋊C₄×C₆₁	—	2	2	no	—
197	C₄⋊C₄×C₆₁	—	2	1	no	—
198	C₄⋊C₄×C₆₁	—	2	1	no	—
199	C₄⋊C₄×C₆₁	—	4	8	no	—
200	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
201	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
202	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
203	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
204	C₂²⋊C₄×C₆₁	—	2	4	no	—
205	C₂²⋊C₄×C₆₁	—	2	2	no	—
206	C₂²⋊C₄×C₆₁	—	2	1	no	—
207	C₂²⋊C₄×C₆₁	—	2	1	no	—
208	C₂²⋊C₄×C₆₁	—	2	1	no	—
209	C₂²⋊C₄×C₆₁	—	4	8	no	—
210	C₂²⋊C₄×C₆₁	—	4	1	no	—
211	C₂²⋊C₄×C₆₁	—	4	1	no	—
212	C₄²×C₆₁	—	2	4	no	—
213	C₄²×C₆₁	—	2	1	no	—
214	C₄²×C₆₁	—	2	1	no	—
215	C₄²×C₆₁	—	2	1	no	—
216	C₄²×C₆₁	—	2	2	no	—
217	C₄²×C₆₁	—	2	2	no	—
218	C₄²×C₆₁	—	2	2	no	—
219	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
223	C₂³×C₆₁	—	2	1	no	—
224	C₂³×C₆₁	—	2	2	no	—
225	C₂³×C₆₁	—	4	2	no	—
226	C₂³×C₆₁	—	4	2	no	—
227	C₂³×C₆₁	—	4	1	no	—
228	C₂³×C₆₁	—	4	1	no	—
229	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
230	Q₈×C₆₁	—	2	1	no	—
231	Q₈×C₆₁	—	4	2	no	—
232	Q₈×C₆₁	—	4	4	no	—
233	Q₈×C₆₁	—	4	2	no	—
234	Q₈×C₆₁	—	8	2	no	—
235	Q₈×C₆₁	—	8	4	no	—
236	Q₈×C₆₁	C₂²⋊Q₈	1	2	yes	complemented
237	D₄×C₆₁	—	2	2	no	—
238	D₄×C₆₁	—	2	2	no	—
239	D₄×C₆₁	—	2	2	no	—
240	D₄×C₆₁	—	2	1	no	—
241	D₄×C₆₁	—	2	2	no	—
242	D₄×C₆₁	—	4	1	no	—
243	D₄×C₆₁	—	4	1	no	—
244	D₄×C₆₁	C₂²⋊Q₈	1	2	yes	complemented
245	C₂×C₄×C₆₁	—	2	1	no	—
246	C₂×C₄×C₆₁	—	2	2	no	—
247	C₂×C₄×C₆₁	—	2	2	no	—
248	C₂×C₄×C₆₁	—	2	2	no	—
249	C₂×C₄×C₆₁	—	2	4	no	—
250	C₂×C₄×C₆₁	—	2	2	no	—
251	C₂×C₄×C₆₁	—	2	1	no	—
252	C₂×C₄×C₆₁	—	2	1	no	—
253	C₂×C₄×C₆₁	—	2	4	no	—
254	C₂×C₄×C₆₁	—	2	2	no	—
255	C₂×C₄×C₆₁	—	4	1	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	2	no	—
260	C₂×C₄×C₆₁	—	4	4	no	—
261	C₂×C₄×C₆₁	—	4	2	no	—
262	C₂×C₄×C₆₁	—	4	1	no	—
263	C₂×C₄×C₆₁	—	8	2	no	—
264	C₂×C₄×C₆₁	—	8	4	no	—
265	C₂×C₄×C₆₁	—	8	2	no	—
266	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
267	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
268	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
269	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
270	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
273	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
275	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
276	C₈×C₆₁	—	2	2	no	—
277	C₈×C₆₁	—	4	1	no	—
278	C₈×C₆₁	—	4	2	no	—
279	#24320	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂²×C₆₁	—	2	1	no	—
281	C₂²×C₆₁	—	2	1	no	—
282	C₂²×C₆₁	—	2	2	no	—
283	C₂²×C₆₁	—	4	4	no	—
284	C₂²×C₆₁	—	4	1	no	—
285	C₂²×C₆₁	—	4	1	no	—
286	C₂²×C₆₁	—	4	2	no	—
287	C₂²×C₆₁	—	8	2	no	—
288	C₂²×C₆₁	—	8	1	no	—
289	C₂²×C₆₁	—	8	1	no	—
290	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
291	C₂²×C₆₁	#548	1	1	yes	characteristic
292	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
293	C₄×C₆₁	—	2	2	no	—
294	C₄×C₆₁	—	2	1	no	—
295	C₄×C₆₁	—	2	1	no	—
296	C₄×C₆₁	—	2	1	no	—
297	C₄×C₆₁	—	4	4	no	—
298	C₄×C₆₁	—	4	1	no	—
299	C₄×C₆₁	—	4	4	no	—
300	C₄×C₆₁	—	4	2	no	—
301	C₄×C₆₁	—	4	1	no	—
302	C₄×C₆₁	—	4	2	no	—
303	C₄×C₆₁	—	8	1	no	—
304	C₄×C₆₁	#453	1	2	yes	—
305	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
306	C₄×C₆₁	#536	1	1	yes	characteristic
307	C₄×C₆₁	#548	1	1	yes	characteristic
308	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
309	#3503	C₂×C₆₁	1	1	yes	characteristic, complemented
310	#3077	C₂×C₆₁	1	1	yes	characteristic, complemented
311	C₄×AGL(1,ℤ/8)	C₂×C₆₁	1	1	yes	characteristic
312	#2542	C₂×C₆₁	1	1	yes	characteristic, complemented
313	#2060	C₂×C₆₁	1	2	yes	complemented
314	#2027	C₂×C₆₁	1	2	yes	complemented
315	#1997	C₂×C₆₁	1	1	yes	characteristic, complemented
316	#1986	C₂×C₆₁	1	1	yes	characteristic, complemented
317	#1883	C₂×C₆₁	1	2	yes	—
318	#1859	C₂×C₆₁	1	2	yes	complemented
319	#1806	C₂×C₆₁	1	1	yes	characteristic, complemented
320	C₂×C₆₁	—	2	1	no	—
321	C₂×C₆₁	—	2	1	no	—
322	C₂×C₆₁	—	4	2	no	—
323	C₂×C₆₁	—	8	2	no	—
324	C₂×C₆₁	#2409	1	1	yes	characteristic
325	C₂×C₆₁	C₂×#453	1	1	yes	characteristic
326	C₂×C₆₁	#3020	1	1	yes	characteristic
327	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	1	yes	characteristic
328	Q₈²	—	2	1	no	—
329	#557	—	2	1	no	—
330	#554	—	2	1	no	—
331	#554	—	2	2	no	—
332	D₄×Q₈	—	2	2	no	—
333	D₄×Q₈	—	2	1	no	—
334	#544	—	2	2	no	—
335	#544	C₂²×C₆₁	1	2	yes	—
336	#533	—	2	1	no	—
337	#533	C₂²×C₆₁	1	1	yes	characteristic
338	C₂×C₄⋊Q₈	—	2	1	no	—
339	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
340	C₂×C₂²⋊Q₈	—	2	2	no	—
341	#520	C₂²×C₆₁	1	2	yes	—
342	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
343	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
344	#478	—	2	2	no	—
345	#476	—	2	1	no	—
346	#474	—	2	1	no	—
347	#442	—	2	1	no	—
348	#440	—	2	1	no	—
349	C₄×SD₁₆	—	2	1	no	—
350	C₄×D₈	—	2	1	no	—
351	#428	C₂²×C₆₁	1	2	yes	—
352	#428	C₂²×C₆₁	1	1	yes	characteristic
353	#423	C₂²×C₆₁	1	1	yes	characteristic
354	C₂×#104	C₂²×C₆₁	1	2	yes	—
355	#418	C₂²×C₆₁	1	2	yes	—
356	#418	C₂²×C₆₁	1	1	yes	characteristic
357	#417	C₂²×C₆₁	1	1	yes	characteristic
358	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
359	#399	—	2	2	no	—
360	#390	C₂²×C₆₁	1	2	yes	—
361	#384	C₂²×C₆₁	1	2	yes	—
362	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
363	#343	C₂²×C₆₁	1	2	yes	—
364	#343	C₂²×C₆₁	1	2	yes	—
365	#339	C₂²×C₆₁	1	2	yes	—
366	#339	C₂²×C₆₁	1	2	yes	—
367	#339	C₂²×C₆₁	1	2	yes	—
368	C₆₁	#24320	1	1	yes	characteristic, direct product factor, Sylow
369	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
370	C₂²×Q₈	—	2	2	no	—
371	C₂²×Q₈	D₄×C₆₁	1	2	yes	—
372	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
373	C₂³×C₄	—	2	1	no	—
374	AGL(1,ℤ/8)	—	2	2	no	—
375	AGL(1,ℤ/8)	Q₈×C₆₁	1	4	yes	complemented
376	C₂×SD₁₆	—	2	1	no	—
377	C₂×D₈	—	2	1	no	—
378	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
379	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
380	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
381	C₄⋊Q₈	—	2	2	no	—
382	C₄⋊Q₈	—	2	2	no	—
383	C₄⋊Q₈	—	2	4	no	—
384	C₄⋊Q₈	—	2	2	no	—
385	C₄⋊Q₈	—	2	2	no	—
386	C₄⋊Q₈	—	2	1	no	—
387	C₄⋊Q₈	—	2	1	no	—
388	C₄⋊Q₈	—	2	1	no	—
389	C₄⋊Q₈	—	2	1	no	—
390	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
391	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
392	#125	—	2	4	no	—
393	#125	—	2	2	no	—
394	#125	—	2	2	no	—
395	#125	D₄×C₆₁	1	2	yes	—
396	C₂²⋊Q₈	—	2	2	no	—
397	C₂²⋊Q₈	—	2	2	no	—
398	C₂²⋊Q₈	—	2	4	no	—
399	C₂²⋊Q₈	—	2	4	no	—
400	C₂²⋊Q₈	—	2	1	no	—
401	C₂²⋊Q₈	—	2	1	no	—
402	C₂²⋊Q₈	—	2	4	no	—
403	C₂²⋊Q₈	—	4	8	no	—
404	C₄×Q₈	—	2	2	no	—
405	C₄×Q₈	—	2	2	no	—
406	C₄×Q₈	—	2	2	no	—
407	C₄×Q₈	—	2	1	no	—
408	C₄×Q₈	—	2	1	no	—
409	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
410	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	4	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	1	no	—
418	C₄×D₄	—	2	1	no	—
419	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
420	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
421	#117	—	2	2	no	—
422	#117	—	2	1	no	—
423	#117	C₂³×C₆₁	1	1	yes	characteristic
424	#117	C₂³×C₆₁	1	2	yes	—
425	C₂×C₄⋊C₄	—	2	2	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	2	no	—
428	C₂×C₄⋊C₄	—	2	4	no	—
429	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
430	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₂²⋊C₄	—	2	4	no	—
434	C₂×C₄²	—	2	1	no	—
435	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
436	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
437	#107	—	2	1	no	—
438	#107	—	2	2	no	—
439	#106	—	2	1	no	—
440	#106	—	2	2	no	—
441	C₄⋊C₈	—	2	1	no	—
442	C₄⋊C₈	—	2	1	no	—
443	#104	—	2	4	no	—
444	#103	—	2	1	no	—
445	#102	—	2	1	no	—
446	#102	—	2	1	no	—
447	#102	—	2	1	no	—
448	#102	—	2	2	no	—
449	#102	—	2	2	no	—
450	#97	—	2	1	no	—
451	C₄×C₈	—	2	1	no	—
452	#95	—	2	1	no	—
453	C₂⁴	—	2	1	no	—
454	#41	—	2	1	no	—
455	#41	—	2	1	no	—
456	#41	C₂×Q₈×C₆₁	1	2	yes	—
457	#41	#41×C₆₁	1	2	yes	—
458	C₂×Q₈	—	2	4	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	—	2	2	no	—
461	C₂×Q₈	—	2	1	no	—
462	C₂×Q₈	—	2	2	no	—
463	C₂×Q₈	—	4	4	no	—
464	C₂×Q₈	—	8	4	no	—
465	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
473	C₂×D₄	C₂×Q₈×C₆₁	1	2	yes	complemented
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	4	4	no	—
481	C₂²×C₄	—	4	1	no	—
482	C₂²×C₄	—	4	1	no	—
483	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
485	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
487	SD₁₆	—	2	4	no	—
488	D₈	—	2	4	no	—
489	M₁₆	—	2	1	no	—
490	M₁₆	—	2	2	no	—
491	M₁₆	—	2	1	no	—
492	M₁₆	—	2	1	no	—
493	M₁₆	—	2	2	no	—
494	M₁₆	C₂×Q₈×C₆₁	1	2	yes	complemented
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	2	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	4	8	no	—
516	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
517	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
518	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
519	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	1	no	—
523	C₂²⋊C₄	—	2	1	no	—
524	C₂²⋊C₄	—	2	1	no	—
525	C₂²⋊C₄	—	4	8	no	—
526	C₂²⋊C₄	—	4	1	no	—
527	C₂²⋊C₄	—	4	1	no	—
528	C₄²	—	2	4	no	—
529	C₄²	—	2	1	no	—
530	C₄²	—	2	1	no	—
531	C₄²	—	2	1	no	—
532	C₄²	—	2	2	no	—
533	C₄²	—	2	2	no	—
534	C₄²	—	2	2	no	—
535	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
536	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
537	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
538	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
539	C₂³	—	2	1	no	—
540	C₂³	—	2	2	no	—
541	C₂³	—	4	2	no	—
542	C₂³	—	4	2	no	—
543	C₂³	—	4	1	no	—
544	C₂³	—	4	1	no	—
545	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
546	Q₈	—	2	1	no	—
547	Q₈	—	4	2	no	—
548	Q₈	—	4	4	no	—
549	Q₈	—	4	2	no	—
550	Q₈	—	8	2	no	—
551	Q₈	—	8	4	no	—
552	Q₈	C₂²⋊Q₈×C₆₁	1	2	yes	complemented
553	D₄	—	2	2	no	—
554	D₄	—	2	2	no	—
555	D₄	—	2	1	no	—
556	D₄	—	2	2	no	—
557	D₄	—	2	2	no	—
558	D₄	—	4	1	no	—
559	D₄	—	4	1	no	—
560	D₄	C₂²⋊Q₈×C₆₁	1	2	yes	complemented
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	2	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	8	2	no	—
580	C₂×C₄	—	8	4	no	—
581	C₂×C₄	—	8	2	no	—
582	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
583	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
584	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
585	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
586	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
587	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
588	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
589	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
590	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
591	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
592	C₈	—	2	2	no	—
593	C₈	—	4	1	no	—
594	C₈	—	4	2	no	—
595	C₂²	—	2	1	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	2	no	—
598	C₂²	—	4	4	no	—
599	C₂²	—	4	1	no	—
600	C₂²	—	4	1	no	—
601	C₂²	—	4	2	no	—
602	C₂²	—	8	2	no	—
603	C₂²	—	8	1	no	—
604	C₂²	—	8	1	no	—
605	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
606	C₂²	C₆₁×#548	1	1	yes	characteristic
607	C₂²	D₄×Q₈×C₆₁	1	1	yes	characteristic
608	C₄	—	2	2	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	1	no	—
611	C₄	—	2	1	no	—
612	C₄	—	4	4	no	—
613	C₄	—	4	1	no	—
614	C₄	—	4	4	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	1	no	—
617	C₄	—	4	2	no	—
618	C₄	—	8	1	no	—
619	C₄	C₆₁×#453	1	2	yes	—
620	C₄	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
621	C₄	C₆₁×#536	1	1	yes	characteristic
622	C₄	C₆₁×#548	1	1	yes	characteristic
623	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
624	C₂	—	2	1	no	—
625	C₂	—	2	1	no	—
626	C₂	—	4	2	no	—
627	C₂	—	8	2	no	—
628	C₂	C₆₁×#2409	1	1	yes	characteristic
629	C₂	C₂×C₆₁×#453	1	1	yes	characteristic
630	C₂	C₆₁×#3020	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×#2542	C₂	semidirect
5. C₆₁×#2060	C₂	semidirect
6. C₆₁×#2027	C₂	semidirect
7. C₆₁×#1997	C₂	semidirect
8. C₆₁×#1986	C₂	semidirect
10. C₆₁×#1859	C₂	semidirect
11. C₆₁×#1806	C₂	semidirect
59. AGL(1,ℤ/8)×C₆₁	Q₈	semidirect
157. C₂×D₄×C₆₁	C₂×Q₈	semidirect
178. M₁₆×C₆₁	C₂×Q₈	semidirect
236. Q₈×C₆₁	C₂²⋊Q₈	semidirect
244. D₄×C₆₁	C₂²⋊Q₈	semidirect
279. #24320	C₆₁	direct (N × H)
309. #3503	C₂×C₆₁	semidirect
310. #3077	C₂×C₆₁	semidirect
312. #2542	C₂×C₆₁	semidirect
313. #2060	C₂×C₆₁	semidirect
314. #2027	C₂×C₆₁	semidirect
315. #1997	C₂×C₆₁	semidirect
316. #1986	C₂×C₆₁	semidirect
318. #1859	C₂×C₆₁	semidirect
319. #1806	C₂×C₆₁	semidirect
368. C₆₁	#24320	direct (N × H)
375. AGL(1,ℤ/8)	Q₈×C₆₁	semidirect
473. C₂×D₄	C₂×Q₈×C₆₁	semidirect
494. M₁₆	C₂×Q₈×C₆₁	semidirect
552. Q₈	C₂²⋊Q₈×C₆₁	semidirect
560. D₄	C₂²⋊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×#2542	2
C₆₁×#2060	2
C₆₁×#2027	2
C₆₁×#1997	2
C₆₁×#1986	2
C₆₁×#1883	2
C₆₁×#1859	2
C₆₁×#1806	2
#24320	61

Subgroups — C61×#24320 #29527753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24320 #29527753