Subgroups — C₆₁×#19312 #29522745

Order 15616.

768 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3115	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3038	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2970	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1688	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1686	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1667	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1665	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1569	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1557	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1555	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₆₁	—	2	1	no	—
13	C₂×AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
14	C₆₁×#532	C₂²	1	1	yes	characteristic
15	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
16	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#496	—	2	1	no	—
18	C₆₁×#496	—	2	1	no	—
19	C₆₁×#493	C₂²	1	2	yes	—
20	C₆₁×#492	C₂²	1	2	yes	—
21	C₆₁×#482	—	2	1	no	—
22	C₆₁×#475	C₂²	1	1	yes	characteristic, complemented
23	C₆₁×#474	C₂²	1	1	yes	characteristic, complemented
24	C₆₁×#460	—	2	1	no	—
25	C₆₁×#460	C₂²	1	2	yes	—
26	C₆₁×#460	C₂²	1	1	yes	characteristic, complemented
27	C₆₁×#459	—	2	1	no	—
28	C₆₁×#459	C₂²	1	2	yes	—
29	C₆₁×#459	C₂²	1	1	yes	characteristic, complemented
30	C₆₁×#449	—	2	1	no	—
31	C₆₁×#446	C₂²	1	2	yes	complemented
32	C₆₁×#442	—	2	1	no	—
33	C₆₁×#440	—	2	1	no	—
34	C₆₁×#435	—	2	2	no	—
35	C₈×D₄×C₆₁	C₂²	1	2	yes	complemented
36	C₆₁×#432	—	2	1	no	—
37	C₆₁×#432	C₂²	1	1	yes	characteristic
38	C₆₁×#432	C₂²	1	1	yes	characteristic
39	C₆₁×#423	C₂²	1	1	yes	characteristic
40	C₆₁×#418	—	2	1	no	—
41	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
42	C₂×#97×C₆₁	—	2	1	no	—
43	C₆₁×#335	C₂²	1	2	yes	complemented
44	C₆₁×#334	C₂²	1	2	yes	complemented
45	C₆₁×#332	C₂²	1	2	yes	complemented
46	C₆₁×#331	C₂²	1	2	yes	—
47	C₆₁×#331	C₂²	1	2	yes	—
48	C₆₁×#331	C₂²	1	2	yes	complemented
49	C₂×#41×C₆₁	—	2	1	no	—
50	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
51	C₂²×D₄×C₆₁	—	2	1	no	—
52	C₂²×D₄×C₆₁	—	2	1	no	—
53	C₂²×D₄×C₆₁	—	2	1	no	—
54	AGL(1,ℤ/8)×C₆₁	—	4	8	no	—
55	AGL(1,ℤ/8)×C₆₁	—	4	4	no	—
56	C₂×SD₁₆×C₆₁	—	2	2	no	—
57	C₂×SD₁₆×C₆₁	—	2	1	no	—
58	C₂×SD₁₆×C₆₁	—	2	2	no	—
59	C₂×SD₁₆×C₆₁	—	2	2	no	—
60	C₂×D₈×C₆₁	—	2	2	no	—
61	C₂×D₈×C₆₁	—	2	1	no	—
62	C₂×D₈×C₆₁	—	2	2	no	—
63	C₂×D₈×C₆₁	—	2	2	no	—
64	C₂×M₁₆×C₆₁	—	2	1	no	—
65	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
66	C₂²×C₈×C₆₁	—	2	2	no	—
67	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
68	#127×C₆₁	—	2	1	no	—
69	#127×C₆₁	—	2	1	no	—
70	#127×C₆₁	—	2	1	no	—
71	#127×C₆₁	C₂³	1	2	yes	—
72	#127×C₆₁	C₂³	1	1	yes	characteristic
73	#124×C₆₁	—	2	1	no	—
74	#121×C₆₁	—	2	1	no	—
75	#121×C₆₁	—	2	1	no	—
76	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
77	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₄×D₄×C₆₁	—	2	1	no	—
79	C₄×D₄×C₆₁	—	2	1	no	—
80	C₄×D₄×C₆₁	—	2	1	no	—
81	C₄×D₄×C₆₁	—	2	1	no	—
82	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
83	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
84	#117×C₆₁	—	2	1	no	—
85	#117×C₆₁	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₆₁	—	2	1	no	—
87	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
88	#107×C₆₁	—	2	1	no	—
89	#106×C₆₁	—	2	1	no	—
90	C₄⋊C₈×C₆₁	—	2	2	no	—
91	C₄⋊C₈×C₆₁	—	2	1	no	—
92	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
93	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
96	#103×C₆₁	—	2	1	no	—
97	#102×C₆₁	—	2	1	no	—
98	#102×C₆₁	—	2	2	no	—
99	#102×C₆₁	—	2	2	no	—
100	#102×C₆₁	—	2	1	no	—
101	#102×C₆₁	—	2	1	no	—
102	C₂²⋊C₈×C₆₁	—	2	1	no	—
103	C₂²⋊C₈×C₆₁	—	2	1	no	—
104	C₂²⋊C₈×C₆₁	—	2	2	no	—
105	#97×C₆₁	—	2	2	no	—
106	#97×C₆₁	—	2	1	no	—
107	#97×C₆₁	—	2	1	no	—
108	#97×C₆₁	D₄	1	1	yes	characteristic
109	#97×C₆₁	D₄	1	1	yes	characteristic
110	C₄×C₈×C₆₁	C₂³	1	2	yes	—
111	C₂⁴×C₆₁	—	4	1	no	—
112	#41×C₆₁	—	2	2	no	—
113	#41×C₆₁	—	4	1	no	—
114	#41×C₆₁	—	4	4	no	—
115	C₂×Q₈×C₆₁	—	2	1	no	—
116	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
117	C₂×D₄×C₆₁	—	2	1	no	—
118	C₂×D₄×C₆₁	—	2	1	no	—
119	C₂×D₄×C₆₁	—	2	2	no	—
120	C₂×D₄×C₆₁	—	2	1	no	—
121	C₂×D₄×C₆₁	—	2	2	no	—
122	C₂×D₄×C₆₁	—	2	1	no	—
123	C₂×D₄×C₆₁	—	2	2	no	—
124	C₂×D₄×C₆₁	—	4	4	no	—
125	C₂×D₄×C₆₁	—	4	2	no	—
126	C₂×D₄×C₆₁	—	4	2	no	—
127	C₂×D₄×C₆₁	—	4	1	no	—
128	C₂×D₄×C₆₁	—	4	1	no	—
129	C₂×D₄×C₆₁	—	4	1	no	—
130	C₂×D₄×C₆₁	—	4	1	no	—
131	C₂×D₄×C₆₁	—	8	2	no	—
132	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₁	—	2	1	no	—
134	C₂²×C₄×C₆₁	—	2	1	no	—
135	C₂²×C₄×C₆₁	—	2	1	no	—
136	C₂²×C₄×C₆₁	—	4	1	no	—
137	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
140	SD₁₆×C₆₁	—	4	4	no	—
141	SD₁₆×C₆₁	—	4	2	no	—
142	SD₁₆×C₆₁	—	4	4	no	—
143	SD₁₆×C₆₁	—	4	4	no	—
144	D₈×C₆₁	—	4	4	no	—
145	D₈×C₆₁	—	4	2	no	—
146	D₈×C₆₁	—	4	4	no	—
147	D₈×C₆₁	—	4	4	no	—
148	M₁₆×C₆₁	—	2	4	no	—
149	M₁₆×C₆₁	—	4	2	no	—
150	C₂×C₈×C₆₁	—	2	2	no	—
151	C₂×C₈×C₆₁	—	2	1	no	—
152	C₂×C₈×C₆₁	—	2	1	no	—
153	C₂×C₈×C₆₁	—	2	2	no	—
154	C₂×C₈×C₆₁	—	4	4	no	—
155	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
156	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₆₁	—	2	1	no	—
158	C₄⋊C₄×C₆₁	—	2	1	no	—
159	C₄⋊C₄×C₆₁	—	2	1	no	—
160	C₄⋊C₄×C₆₁	—	2	1	no	—
161	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
162	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
163	C₂²⋊C₄×C₆₁	—	2	1	no	—
164	C₂²⋊C₄×C₆₁	—	2	1	no	—
165	C₂²⋊C₄×C₆₁	—	2	1	no	—
166	C₂²⋊C₄×C₆₁	—	4	1	no	—
167	C₂²⋊C₄×C₆₁	—	4	1	no	—
168	C₄²×C₆₁	—	2	1	no	—
169	C₄²×C₆₁	—	2	1	no	—
170	C₄²×C₆₁	—	2	1	no	—
171	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₆₁	#41	1	1	yes	characteristic
174	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
175	C₂³×C₆₁	—	2	1	no	—
176	C₂³×C₆₁	—	4	1	no	—
177	C₂³×C₆₁	—	4	2	no	—
178	C₂³×C₆₁	—	8	2	no	—
179	C₂³×C₆₁	—	8	2	no	—
180	C₂³×C₆₁	—	8	2	no	—
181	C₂³×C₆₁	#121	1	1	yes	characteristic
182	Q₈×C₆₁	—	2	1	no	—
183	Q₈×C₆₁	—	2	1	no	—
184	Q₈×C₆₁	—	4	2	no	—
185	D₄×C₆₁	—	2	1	no	—
186	D₄×C₆₁	—	4	2	no	—
187	D₄×C₆₁	—	4	2	no	—
188	D₄×C₆₁	—	4	2	no	—
189	D₄×C₆₁	—	4	2	no	—
190	D₄×C₆₁	—	4	2	no	—
191	D₄×C₆₁	—	4	1	no	—
192	D₄×C₆₁	—	4	1	no	—
193	D₄×C₆₁	—	4	2	no	—
194	D₄×C₆₁	—	8	1	no	—
195	D₄×C₆₁	—	8	2	no	—
196	D₄×C₆₁	—	8	2	no	—
197	D₄×C₆₁	—	8	2	no	—
198	D₄×C₆₁	—	8	2	no	—
199	D₄×C₆₁	—	8	2	no	—
200	D₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	complemented
201	C₂×C₄×C₆₁	—	2	2	no	—
202	C₂×C₄×C₆₁	—	2	1	no	—
203	C₂×C₄×C₆₁	—	2	1	no	—
204	C₂×C₄×C₆₁	—	2	1	no	—
205	C₂×C₄×C₆₁	—	2	1	no	—
206	C₂×C₄×C₆₁	—	2	1	no	—
207	C₂×C₄×C₆₁	—	4	1	no	—
208	C₂×C₄×C₆₁	—	4	2	no	—
209	C₂×C₄×C₆₁	—	4	2	no	—
210	C₂×C₄×C₆₁	—	4	1	no	—
211	C₂×C₄×C₆₁	—	8	1	no	—
212	C₂×C₄×C₆₁	—	8	2	no	—
213	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
214	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
215	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
216	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
219	C₈×C₆₁	—	2	2	no	—
220	C₈×C₆₁	—	4	2	no	—
221	C₈×C₆₁	—	4	2	no	—
222	C₈×C₆₁	—	4	1	no	—
223	C₈×C₆₁	—	4	1	no	—
224	C₈×C₆₁	AGL(1,ℤ/8)	1	4	yes	complemented
225	#19312	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂²×C₆₁	—	2	1	no	—
227	C₂²×C₆₁	—	2	1	no	—
228	C₂²×C₆₁	—	2	2	no	—
229	C₂²×C₆₁	—	2	1	no	—
230	C₂²×C₆₁	—	4	1	no	—
231	C₂²×C₆₁	—	4	1	no	—
232	C₂²×C₆₁	—	8	1	no	—
233	C₂²×C₆₁	—	8	2	no	—
234	C₂²×C₆₁	—	8	2	no	—
235	C₂²×C₆₁	—	8	1	no	—
236	C₂²×C₆₁	—	8	2	no	—
237	C₂²×C₆₁	—	8	1	no	—
238	C₂²×C₆₁	—	16	4	no	—
239	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
240	C₄×C₆₁	—	2	1	no	—
241	C₄×C₆₁	—	2	1	no	—
242	C₄×C₆₁	—	2	1	no	—
243	C₄×C₆₁	—	2	1	no	—
244	C₄×C₆₁	—	4	1	no	—
245	C₄×C₆₁	—	4	1	no	—
246	C₄×C₆₁	—	4	1	no	—
247	C₄×C₆₁	—	8	1	no	—
248	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
249	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₄×C₆₁	#575	1	1	yes	characteristic
251	#3151	C₂×C₆₁	1	1	yes	characteristic, complemented
252	#3115	C₂×C₆₁	1	1	yes	characteristic, complemented
253	#3038	C₂×C₆₁	1	1	yes	characteristic, complemented
254	#2970	C₂×C₆₁	1	1	yes	characteristic, complemented
255	#1688	C₂×C₆₁	1	2	yes	complemented
256	#1686	C₂×C₆₁	1	2	yes	complemented
257	#1667	C₂×C₆₁	1	2	yes	complemented
258	#1665	C₂×C₆₁	1	2	yes	complemented
259	#1569	C₂×C₆₁	1	1	yes	characteristic, complemented
260	#1557	C₂×C₆₁	1	1	yes	characteristic, complemented
261	#1555	C₂×C₆₁	1	1	yes	characteristic, complemented
262	C₂×C₆₁	—	4	1	no	—
263	C₂×C₆₁	—	4	2	no	—
264	C₂×C₆₁	—	16	1	no	—
265	C₂×C₆₁	—	16	2	no	—
266	C₂×C₆₁	#3038	1	1	yes	characteristic
267	C₂×C₆₁	#3055	1	1	yes	characteristic
268	C₂×C₆₁	#3068	1	1	yes	characteristic
269	C₂×AGL(1,ℤ/8)	—	2	1	no	—
270	C₂×AGL(1,ℤ/8)	—	2	2	no	—
271	#532	C₂²×C₆₁	1	1	yes	characteristic
272	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
273	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
274	#496	—	2	1	no	—
275	#496	—	2	1	no	—
276	#493	C₂²×C₆₁	1	2	yes	—
277	#492	C₂²×C₆₁	1	2	yes	—
278	#482	—	2	1	no	—
279	#475	C₂²×C₆₁	1	1	yes	characteristic, complemented
280	#474	C₂²×C₆₁	1	1	yes	characteristic, complemented
281	#460	—	2	1	no	—
282	#460	C₂²×C₆₁	1	2	yes	—
283	#460	C₂²×C₆₁	1	1	yes	characteristic, complemented
284	#459	—	2	1	no	—
285	#459	C₂²×C₆₁	1	2	yes	—
286	#459	C₂²×C₆₁	1	1	yes	characteristic, complemented
287	#449	—	2	1	no	—
288	#446	C₂²×C₆₁	1	2	yes	complemented
289	#442	—	2	1	no	—
290	#440	—	2	1	no	—
291	#435	—	2	2	no	—
292	C₈×D₄	C₂²×C₆₁	1	2	yes	complemented
293	#432	—	2	1	no	—
294	#432	C₂²×C₆₁	1	1	yes	characteristic
295	#432	C₂²×C₆₁	1	1	yes	characteristic
296	#423	C₂²×C₆₁	1	1	yes	characteristic
297	#418	—	2	1	no	—
298	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
299	C₂×#97	—	2	1	no	—
300	#335	C₂²×C₆₁	1	2	yes	complemented
301	#334	C₂²×C₆₁	1	2	yes	complemented
302	#332	C₂²×C₆₁	1	2	yes	complemented
303	#331	C₂²×C₆₁	1	2	yes	—
304	#331	C₂²×C₆₁	1	2	yes	—
305	#331	C₂²×C₆₁	1	2	yes	complemented
306	C₆₁	#19312	1	1	yes	characteristic, direct product factor, Sylow
307	C₂×#41	—	2	1	no	—
308	C₂×#41	D₄×C₆₁	1	1	yes	characteristic
309	C₂²×D₄	—	2	1	no	—
310	C₂²×D₄	—	2	1	no	—
311	C₂²×D₄	—	2	1	no	—
312	AGL(1,ℤ/8)	—	4	8	no	—
313	AGL(1,ℤ/8)	—	4	4	no	—
314	C₂×SD₁₆	—	2	2	no	—
315	C₂×SD₁₆	—	2	1	no	—
316	C₂×SD₁₆	—	2	2	no	—
317	C₂×SD₁₆	—	2	2	no	—
318	C₂×D₈	—	2	2	no	—
319	C₂×D₈	—	2	1	no	—
320	C₂×D₈	—	2	2	no	—
321	C₂×D₈	—	2	2	no	—
322	C₂×M₁₆	—	2	1	no	—
323	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
324	C₂²×C₈	—	2	2	no	—
325	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
326	#127	—	2	1	no	—
327	#127	—	2	1	no	—
328	#127	—	2	1	no	—
329	#127	C₂³×C₆₁	1	2	yes	—
330	#127	C₂³×C₆₁	1	1	yes	characteristic
331	#124	—	2	1	no	—
332	#121	—	2	1	no	—
333	#121	—	2	1	no	—
334	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
335	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	—	2	1	no	—
339	C₄×D₄	—	2	1	no	—
340	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
341	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
342	#117	—	2	1	no	—
343	#117	D₄×C₆₁	1	1	yes	characteristic
344	C₂×C₄²	—	2	1	no	—
345	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
346	#107	—	2	1	no	—
347	#106	—	2	1	no	—
348	C₄⋊C₈	—	2	2	no	—
349	C₄⋊C₈	—	2	1	no	—
350	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
351	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
352	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
353	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
354	#103	—	2	1	no	—
355	#102	—	2	1	no	—
356	#102	—	2	2	no	—
357	#102	—	2	2	no	—
358	#102	—	2	1	no	—
359	#102	—	2	1	no	—
360	C₂²⋊C₈	—	2	1	no	—
361	C₂²⋊C₈	—	2	1	no	—
362	C₂²⋊C₈	—	2	2	no	—
363	#97	—	2	2	no	—
364	#97	—	2	1	no	—
365	#97	—	2	1	no	—
366	#97	D₄×C₆₁	1	1	yes	characteristic
367	#97	D₄×C₆₁	1	1	yes	characteristic
368	C₄×C₈	C₂³×C₆₁	1	2	yes	—
369	C₂⁴	—	4	1	no	—
370	#41	—	2	2	no	—
371	#41	—	4	1	no	—
372	#41	—	4	4	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	4	4	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	—	4	1	no	—
387	C₂×D₄	—	4	1	no	—
388	C₂×D₄	—	4	1	no	—
389	C₂×D₄	—	8	2	no	—
390	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	4	1	no	—
395	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
397	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
398	SD₁₆	—	4	4	no	—
399	SD₁₆	—	4	2	no	—
400	SD₁₆	—	4	4	no	—
401	SD₁₆	—	4	4	no	—
402	D₈	—	4	4	no	—
403	D₈	—	4	2	no	—
404	D₈	—	4	4	no	—
405	D₈	—	4	4	no	—
406	M₁₆	—	2	4	no	—
407	M₁₆	—	4	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	4	4	no	—
413	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
414	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
420	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
421	C₂²⋊C₄	—	2	1	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₂²⋊C₄	—	4	1	no	—
425	C₂²⋊C₄	—	4	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
430	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
431	C₄²	#41×C₆₁	1	1	yes	characteristic
432	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
433	C₂³	—	2	1	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	8	2	no	—
437	C₂³	—	8	2	no	—
438	C₂³	—	8	2	no	—
439	C₂³	#121×C₆₁	1	1	yes	characteristic
440	Q₈	—	2	1	no	—
441	Q₈	—	2	1	no	—
442	Q₈	—	4	2	no	—
443	D₄	—	2	1	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	2	no	—
446	D₄	—	4	2	no	—
447	D₄	—	4	2	no	—
448	D₄	—	4	2	no	—
449	D₄	—	4	1	no	—
450	D₄	—	4	1	no	—
451	D₄	—	4	2	no	—
452	D₄	—	8	1	no	—
453	D₄	—	8	2	no	—
454	D₄	—	8	2	no	—
455	D₄	—	8	2	no	—
456	D₄	—	8	2	no	—
457	D₄	—	8	2	no	—
458	D₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	complemented
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	8	1	no	—
470	C₂×C₄	—	8	2	no	—
471	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
472	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
473	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
474	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
476	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
477	C₈	—	2	2	no	—
478	C₈	—	4	2	no	—
479	C₈	—	4	2	no	—
480	C₈	—	4	1	no	—
481	C₈	—	4	1	no	—
482	C₈	AGL(1,ℤ/8)×C₆₁	1	4	yes	complemented
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	4	1	no	—
488	C₂²	—	4	1	no	—
489	C₂²	—	8	1	no	—
490	C₂²	—	8	2	no	—
491	C₂²	—	8	2	no	—
492	C₂²	—	8	1	no	—
493	C₂²	—	8	2	no	—
494	C₂²	—	8	1	no	—
495	C₂²	—	16	4	no	—
496	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
497	C₄	—	2	1	no	—
498	C₄	—	2	1	no	—
499	C₄	—	2	1	no	—
500	C₄	—	2	1	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	1	no	—
503	C₄	—	4	1	no	—
504	C₄	—	8	1	no	—
505	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	2	yes	—
506	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
507	C₄	C₆₁×#575	1	1	yes	characteristic
508	C₂	—	4	1	no	—
509	C₂	—	4	2	no	—
510	C₂	—	16	2	no	—
511	C₂	—	16	1	no	—
512	C₂	C₆₁×#3038	1	1	yes	characteristic
513	C₂	C₆₁×#3055	1	1	yes	characteristic
514	C₂	C₆₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
LCS	G ▷ C₄² ▷ C₂² ▷ 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×#3151	C₂	semidirect
2. C₆₁×#3115	C₂	semidirect
3. C₆₁×#3038	C₂	semidirect
4. C₆₁×#2970	C₂	semidirect
5. C₆₁×#1688	C₂	semidirect
6. C₆₁×#1686	C₂	semidirect
7. C₆₁×#1667	C₂	semidirect
8. C₆₁×#1665	C₂	semidirect
9. C₆₁×#1569	C₂	semidirect
10. C₆₁×#1557	C₂	semidirect
11. C₆₁×#1555	C₂	semidirect
22. C₆₁×#475	C₂²	semidirect
23. C₆₁×#474	C₂²	semidirect
26. C₆₁×#460	C₂²	semidirect
29. C₆₁×#459	C₂²	semidirect
31. C₆₁×#446	C₂²	semidirect
35. C₈×D₄×C₆₁	C₂²	semidirect
43. C₆₁×#335	C₂²	semidirect
44. C₆₁×#334	C₂²	semidirect
45. C₆₁×#332	C₂²	semidirect
48. C₆₁×#331	C₂²	semidirect
200. D₄×C₆₁	AGL(1,ℤ/8)	semidirect
224. C₈×C₆₁	AGL(1,ℤ/8)	semidirect
225. #19312	C₆₁	direct (N × H)
251. #3151	C₂×C₆₁	semidirect
252. #3115	C₂×C₆₁	semidirect
253. #3038	C₂×C₆₁	semidirect
254. #2970	C₂×C₆₁	semidirect
255. #1688	C₂×C₆₁	semidirect
256. #1686	C₂×C₆₁	semidirect
257. #1667	C₂×C₆₁	semidirect
258. #1665	C₂×C₆₁	semidirect
259. #1569	C₂×C₆₁	semidirect
260. #1557	C₂×C₆₁	semidirect
261. #1555	C₂×C₆₁	semidirect
279. #475	C₂²×C₆₁	semidirect
280. #474	C₂²×C₆₁	semidirect
283. #460	C₂²×C₆₁	semidirect
286. #459	C₂²×C₆₁	semidirect
288. #446	C₂²×C₆₁	semidirect
292. C₈×D₄	C₂²×C₆₁	semidirect
300. #335	C₂²×C₆₁	semidirect
301. #334	C₂²×C₆₁	semidirect
302. #332	C₂²×C₆₁	semidirect
305. #331	C₂²×C₆₁	semidirect
306. C₆₁	#19312	direct (N × H)
458. D₄	AGL(1,ℤ/8)×C₆₁	semidirect
482. C₈	AGL(1,ℤ/8)×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3151	2
C₆₁×#3115	2
C₆₁×#3038	2
C₆₁×#2970	2
C₆₁×#1688	2
C₆₁×#1686	2
C₆₁×#1667	2
C₆₁×#1665	2
C₆₁×#1569	2
C₆₁×#1557	2
C₆₁×#1555	2
#19312	61

Subgroups — C61×#19312 #29522745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19312 #29522745