Subgroups — C₆₁×#33516 #29536949

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#508	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3050	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2189	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2188	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2187	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2186	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2185	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#2184	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#2138	C₂	1	2	yes	maximal, complemented
11	C₂×C₆₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₁	—	2	1	no	—
13	C₂×#135×C₆₁	—	2	8	no	—
14	C₂×#135×C₆₁	—	2	1	no	—
15	C₂×#135×C₆₁	—	2	1	no	—
16	C₂×#135×C₆₁	C₂²	1	2	yes	—
17	C₂×#135×C₆₁	C₂²	1	2	yes	—
18	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic, complemented
19	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×SD₁₆×C₆₁	—	2	1	no	—
21	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#525	—	2	1	no	—
24	C₆₁×#508	—	2	8	no	—
25	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	complemented
26	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	complemented
27	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	complemented
28	C₂×D₁₆×C₆₁	C₂²	1	2	yes	complemented
29	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	complemented
30	C₆₁×#467	C₂²	1	1	yes	characteristic
31	C₆₁×#467	C₂²	1	1	yes	characteristic, complemented
32	C₆₁×#466	C₂²	1	1	yes	characteristic
33	C₆₁×#466	C₂²	1	1	yes	characteristic, complemented
34	C₆₁×#465	—	2	1	no	—
35	C₆₁×#465	—	2	1	no	—
36	C₆₁×#427	C₂²	1	1	yes	characteristic, complemented
37	C₆₁×#416	—	2	1	no	—
38	C₆₁×#358	C₂²	1	2	yes	complemented
39	C₆₁×#358	C₂²	1	2	yes	complemented
40	C₆₁×#357	C₂²	1	2	yes	complemented
41	C₆₁×#357	C₂²	1	2	yes	complemented
42	C₆₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
43	C₆₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
44	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
45	C₂×#41×C₆₁	—	2	8	no	—
46	C₂×#41×C₆₁	—	2	2	no	—
47	C₂×#41×C₆₁	—	2	2	no	—
48	C₂×#41×C₆₁	—	2	1	no	—
49	C₂×#41×C₆₁	—	4	4	no	—
50	C₂×#41×C₆₁	—	4	1	no	—
51	C₂×#41×C₆₁	—	4	1	no	—
52	C₂²×Q₈×C₆₁	—	2	1	no	—
53	C₂²×D₄×C₆₁	—	2	1	no	—
54	C₂²×D₄×C₆₁	—	4	1	no	—
55	C₂³×C₄×C₆₁	—	4	1	no	—
56	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
57	#135×C₆₁	—	2	8	no	—
58	#135×C₆₁	—	2	8	no	—
59	#135×C₆₁	—	2	4	no	—
60	#135×C₆₁	—	2	4	no	—
61	#135×C₆₁	—	2	4	no	—
62	#135×C₆₁	—	2	4	no	—
63	#135×C₆₁	D₄	1	8	yes	complemented
64	C₂×Q₁₆×C₆₁	—	2	4	no	—
65	C₂×Q₁₆×C₆₁	—	2	1	no	—
66	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
68	C₂×SD₁₆×C₆₁	—	2	8	no	—
69	C₂×SD₁₆×C₆₁	—	2	2	no	—
70	C₂×SD₁₆×C₆₁	—	2	2	no	—
71	C₂×SD₁₆×C₆₁	—	2	1	no	—
72	C₂×D₈×C₆₁	—	2	4	no	—
73	C₂×D₈×C₆₁	—	2	1	no	—
74	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₂×D₈×C₆₁	C₂³	1	2	yes	—
76	C₂²×C₈×C₆₁	—	2	4	no	—
77	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
83	#123×C₆₁	—	4	1	no	—
84	C₂²⋊Q₈×C₆₁	—	2	1	no	—
85	C₂²⋊Q₈×C₆₁	—	2	1	no	—
86	#121×C₆₁	—	2	1	no	—
87	#121×C₆₁	—	2	1	no	—
88	#120×C₆₁	—	4	1	no	—
89	C₄×D₄×C₆₁	—	4	1	no	—
90	C₄×D₄×C₆₁	—	4	1	no	—
91	#117×C₆₁	—	2	1	no	—
92	Q₃₂×C₆₁	—	2	4	no	—
93	SD₃₂×C₆₁	—	2	4	no	—
94	SD₃₂×C₆₁	—	2	4	no	—
95	D₁₆×C₆₁	—	2	4	no	—
96	C₂×C₁₆×C₆₁	—	2	4	no	—
97	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
98	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
99	#107×C₆₁	C₂³	1	1	yes	characteristic
100	#107×C₆₁	C₂³	1	1	yes	characteristic
101	#106×C₆₁	—	2	1	no	—
102	#103×C₆₁	—	2	1	no	—
103	#103×C₆₁	—	2	1	no	—
104	#102×C₆₁	—	2	1	no	—
105	#102×C₆₁	—	2	1	no	—
106	C₂⁴×C₆₁	—	4	1	no	—
107	C₂⁴×C₆₁	D₈	1	1	yes	characteristic
108	#41×C₆₁	—	2	8	no	—
109	#41×C₆₁	—	2	8	no	—
110	#41×C₆₁	—	4	8	no	—
111	#41×C₆₁	—	4	4	no	—
112	#41×C₆₁	—	4	8	no	—
113	#41×C₆₁	—	4	4	no	—
114	#41×C₆₁	—	4	2	no	—
115	#41×C₆₁	—	4	2	no	—
116	C₂×Q₈×C₆₁	—	2	4	no	—
117	C₂×Q₈×C₆₁	—	2	2	no	—
118	C₂×Q₈×C₆₁	—	2	1	no	—
119	C₂×Q₈×C₆₁	—	4	2	no	—
120	C₂×Q₈×C₆₁	—	4	1	no	—
121	C₂×D₄×C₆₁	—	2	4	no	—
122	C₂×D₄×C₆₁	—	2	2	no	—
123	C₂×D₄×C₆₁	—	2	1	no	—
124	C₂×D₄×C₆₁	—	4	2	no	—
125	C₂×D₄×C₆₁	—	4	8	no	—
126	C₂×D₄×C₆₁	—	4	2	no	—
127	C₂×D₄×C₆₁	—	4	2	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	—	4	1	no	—
132	C₂²×C₄×C₆₁	—	2	2	no	—
133	C₂²×C₄×C₆₁	—	2	2	no	—
134	C₂²×C₄×C₆₁	—	4	8	no	—
135	C₂²×C₄×C₆₁	—	4	1	no	—
136	C₂²×C₄×C₆₁	—	4	2	no	—
137	C₂²×C₄×C₆₁	—	4	2	no	—
138	C₂²×C₄×C₆₁	—	4	1	no	—
139	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
140	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
141	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
142	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
145	Q₁₆×C₆₁	—	2	4	no	—
146	Q₁₆×C₆₁	—	2	2	no	—
147	Q₁₆×C₆₁	—	2	2	no	—
148	Q₁₆×C₆₁	C₂×D₄	1	4	yes	—
149	SD₁₆×C₆₁	—	2	8	no	—
150	SD₁₆×C₆₁	—	2	8	no	—
151	SD₁₆×C₆₁	—	4	2	no	—
152	D₈×C₆₁	—	2	4	no	—
153	D₈×C₆₁	—	2	2	no	—
154	D₈×C₆₁	—	2	2	no	—
155	D₈×C₆₁	C₂×D₄	1	4	yes	—
156	C₂×C₈×C₆₁	—	2	2	no	—
157	C₂×C₈×C₆₁	—	2	2	no	—
158	C₂×C₈×C₆₁	—	2	2	no	—
159	C₂×C₈×C₆₁	—	2	2	no	—
160	C₂×C₈×C₆₁	—	2	2	no	—
161	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
165	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
166	C₄⋊C₄×C₆₁	—	2	1	no	—
167	C₄⋊C₄×C₆₁	—	2	1	no	—
168	C₄⋊C₄×C₆₁	—	4	1	no	—
169	C₄⋊C₄×C₆₁	—	4	1	no	—
170	C₂²⋊C₄×C₆₁	—	4	1	no	—
171	C₂²⋊C₄×C₆₁	—	4	1	no	—
172	C₂²⋊C₄×C₆₁	—	4	1	no	—
173	C₂²⋊C₄×C₆₁	—	4	1	no	—
174	C₂²⋊C₄×C₆₁	—	4	1	no	—
175	C₄²×C₆₁	—	4	1	no	—
176	C₁₆×C₆₁	—	2	2	no	—
177	C₁₆×C₆₁	—	2	2	no	—
178	C₂³×C₆₁	—	2	2	no	—
179	C₂³×C₆₁	—	2	2	no	—
180	C₂³×C₆₁	—	4	1	no	—
181	C₂³×C₆₁	—	4	4	no	—
182	C₂³×C₆₁	—	4	2	no	—
183	C₂³×C₆₁	—	4	1	no	—
184	C₂³×C₆₁	—	8	2	no	—
185	C₂³×C₆₁	—	8	2	no	—
186	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
187	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
188	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
189	Q₈×C₆₁	—	2	4	no	—
190	Q₈×C₆₁	—	4	4	no	—
191	Q₈×C₆₁	—	4	2	no	—
192	Q₈×C₆₁	—	4	1	no	—
193	Q₈×C₆₁	—	4	1	no	—
194	D₄×C₆₁	—	2	4	no	—
195	D₄×C₆₁	—	4	4	no	—
196	D₄×C₆₁	—	4	8	no	—
197	D₄×C₆₁	—	4	8	no	—
198	D₄×C₆₁	—	4	1	no	—
199	D₄×C₆₁	—	4	2	no	—
200	D₄×C₆₁	—	4	1	no	—
201	D₄×C₆₁	—	8	2	no	—
202	D₄×C₆₁	—	8	2	no	—
203	D₄×C₆₁	—	8	2	no	—
204	C₂×C₄×C₆₁	—	2	2	no	—
205	C₂×C₄×C₆₁	—	2	1	no	—
206	C₂×C₄×C₆₁	—	2	1	no	—
207	C₂×C₄×C₆₁	—	2	1	no	—
208	C₂×C₄×C₆₁	—	2	1	no	—
209	C₂×C₄×C₆₁	—	2	2	no	—
210	C₂×C₄×C₆₁	—	2	2	no	—
211	C₂×C₄×C₆₁	—	4	8	no	—
212	C₂×C₄×C₆₁	—	4	4	no	—
213	C₂×C₄×C₆₁	—	4	2	no	—
214	C₂×C₄×C₆₁	—	4	8	no	—
215	C₂×C₄×C₆₁	—	4	1	no	—
216	C₂×C₄×C₆₁	—	4	1	no	—
217	C₂×C₄×C₆₁	—	4	1	no	—
218	C₂×C₄×C₆₁	—	8	2	no	—
219	C₂×C₄×C₆₁	#120	1	2	yes	—
220	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
221	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
222	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
223	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
224	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
225	C₈×C₆₁	—	2	1	no	—
226	C₈×C₆₁	—	2	1	no	—
227	C₈×C₆₁	#120	1	2	yes	—
228	C₈×C₆₁	C₂²×D₄	1	2	yes	—
229	#33516	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂²×C₆₁	—	2	1	no	—
231	C₂²×C₆₁	—	2	1	no	—
232	C₂²×C₆₁	—	2	1	no	—
233	C₂²×C₆₁	—	2	1	no	—
234	C₂²×C₆₁	—	2	1	no	—
235	C₂²×C₆₁	—	2	1	no	—
236	C₂²×C₆₁	—	2	1	no	—
237	C₂²×C₆₁	—	4	4	no	—
238	C₂²×C₆₁	—	4	4	no	—
239	C₂²×C₆₁	—	8	8	no	—
240	C₂²×C₆₁	—	8	2	no	—
241	C₂²×C₆₁	—	8	2	no	—
242	C₂²×C₆₁	—	8	1	no	—
243	C₂²×C₆₁	—	8	1	no	—
244	C₂²×C₆₁	—	8	1	no	—
245	C₂²×C₆₁	#447	1	2	yes	—
246	C₂²×C₆₁	#508	1	2	yes	complemented
247	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
248	C₄×C₆₁	—	2	1	no	—
249	C₄×C₆₁	—	2	1	no	—
250	C₄×C₆₁	—	4	4	no	—
251	C₄×C₆₁	—	8	1	no	—
252	C₄×C₆₁	—	8	1	no	—
253	C₄×C₆₁	—	8	1	no	—
254	C₄×C₆₁	#447	1	2	yes	—
255	C₄×C₆₁	C₂×#120	1	1	yes	characteristic
256	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₂²×#135	C₂×C₆₁	1	1	yes	characteristic, complemented
258	C₂×#508	C₂×C₆₁	1	2	yes	complemented
259	#3050	C₂×C₆₁	1	1	yes	characteristic, complemented
260	#2189	C₂×C₆₁	1	2	yes	complemented
261	#2188	C₂×C₆₁	1	1	yes	characteristic, complemented
262	#2187	C₂×C₆₁	1	1	yes	characteristic, complemented
263	#2186	C₂×C₆₁	1	2	yes	complemented
264	#2185	C₂×C₆₁	1	1	yes	characteristic, complemented
265	#2184	C₂×C₆₁	1	1	yes	characteristic, complemented
266	#2138	C₂×C₆₁	1	2	yes	complemented
267	C₂×C₂²⋊C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
268	C₂×C₆₁	—	2	2	no	—
269	C₂×C₆₁	—	2	2	no	—
270	C₂×C₆₁	—	4	1	no	—
271	C₂×C₆₁	—	8	4	no	—
272	C₂×C₆₁	—	16	1	no	—
273	C₂×C₆₁	C₂×#447	1	1	yes	characteristic
274	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
275	C₂×C₆₁	#3414	1	1	yes	characteristic
276	C₂²×#41	—	2	1	no	—
277	C₂×#135	—	2	8	no	—
278	C₂×#135	—	2	1	no	—
279	C₂×#135	—	2	1	no	—
280	C₂×#135	C₂²×C₆₁	1	2	yes	—
281	C₂×#135	C₂²×C₆₁	1	2	yes	—
282	C₂×#135	C₂²×C₆₁	1	1	yes	characteristic, complemented
283	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
284	C₂²×SD₁₆	—	2	1	no	—
285	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
286	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
287	#525	—	2	1	no	—
288	#508	—	2	8	no	—
289	C₂×Q₃₂	C₂²×C₆₁	1	2	yes	complemented
290	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	complemented
291	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	complemented
292	C₂×D₁₆	C₂²×C₆₁	1	2	yes	complemented
293	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	complemented
294	#467	C₂²×C₆₁	1	1	yes	characteristic
295	#467	C₂²×C₆₁	1	1	yes	characteristic, complemented
296	#466	C₂²×C₆₁	1	1	yes	characteristic
297	#466	C₂²×C₆₁	1	1	yes	characteristic, complemented
298	#465	—	2	1	no	—
299	#465	—	2	1	no	—
300	#427	C₂²×C₆₁	1	1	yes	characteristic, complemented
301	#416	—	2	1	no	—
302	#358	C₂²×C₆₁	1	2	yes	complemented
303	#358	C₂²×C₆₁	1	2	yes	complemented
304	#357	C₂²×C₆₁	1	2	yes	complemented
305	#357	C₂²×C₆₁	1	2	yes	complemented
306	C₂²⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
307	C₂²⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
308	C₂²⋊C₁₆	C₂²×C₆₁	1	2	yes	complemented
309	C₆₁	#33516	1	1	yes	characteristic, direct product factor, Sylow
310	C₂×#41	—	2	8	no	—
311	C₂×#41	—	2	2	no	—
312	C₂×#41	—	2	2	no	—
313	C₂×#41	—	2	1	no	—
314	C₂×#41	—	4	4	no	—
315	C₂×#41	—	4	1	no	—
316	C₂×#41	—	4	1	no	—
317	C₂²×Q₈	—	2	1	no	—
318	C₂²×D₄	—	2	1	no	—
319	C₂²×D₄	—	4	1	no	—
320	C₂³×C₄	—	4	1	no	—
321	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
322	#135	—	2	8	no	—
323	#135	—	2	8	no	—
324	#135	—	2	4	no	—
325	#135	—	2	4	no	—
326	#135	—	2	4	no	—
327	#135	—	2	4	no	—
328	#135	D₄×C₆₁	1	8	yes	complemented
329	C₂×Q₁₆	—	2	4	no	—
330	C₂×Q₁₆	—	2	1	no	—
331	C₂×Q₁₆	C₂³×C₆₁	1	1	yes	characteristic
332	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
333	C₂×SD₁₆	—	2	8	no	—
334	C₂×SD₁₆	—	2	2	no	—
335	C₂×SD₁₆	—	2	2	no	—
336	C₂×SD₁₆	—	2	1	no	—
337	C₂×D₈	—	2	4	no	—
338	C₂×D₈	—	2	1	no	—
339	C₂×D₈	C₂³×C₆₁	1	1	yes	characteristic
340	C₂×D₈	C₂³×C₆₁	1	2	yes	—
341	C₂²×C₈	—	2	4	no	—
342	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
343	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
344	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
346	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
347	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
348	#123	—	4	1	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	#121	—	2	1	no	—
352	#121	—	2	1	no	—
353	#120	—	4	1	no	—
354	C₄×D₄	—	4	1	no	—
355	C₄×D₄	—	4	1	no	—
356	#117	—	2	1	no	—
357	Q₃₂	—	2	4	no	—
358	SD₃₂	—	2	4	no	—
359	SD₃₂	—	2	4	no	—
360	D₁₆	—	2	4	no	—
361	C₂×C₁₆	—	2	4	no	—
362	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
363	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
364	#107	C₂³×C₆₁	1	1	yes	characteristic
365	#107	C₂³×C₆₁	1	1	yes	characteristic
366	#106	—	2	1	no	—
367	#103	—	2	1	no	—
368	#103	—	2	1	no	—
369	#102	—	2	1	no	—
370	#102	—	2	1	no	—
371	C₂⁴	—	4	1	no	—
372	C₂⁴	D₈×C₆₁	1	1	yes	characteristic
373	#41	—	2	8	no	—
374	#41	—	2	8	no	—
375	#41	—	4	8	no	—
376	#41	—	4	4	no	—
377	#41	—	4	8	no	—
378	#41	—	4	4	no	—
379	#41	—	4	2	no	—
380	#41	—	4	2	no	—
381	C₂×Q₈	—	2	4	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	1	no	—
384	C₂×Q₈	—	4	2	no	—
385	C₂×Q₈	—	4	1	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	4	2	no	—
390	C₂×D₄	—	4	8	no	—
391	C₂×D₄	—	4	2	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂×D₄	—	4	1	no	—
394	C₂×D₄	—	4	1	no	—
395	C₂×D₄	—	4	1	no	—
396	C₂×D₄	—	4	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	4	8	no	—
400	C₂²×C₄	—	4	1	no	—
401	C₂²×C₄	—	4	2	no	—
402	C₂²×C₄	—	4	2	no	—
403	C₂²×C₄	—	4	1	no	—
404	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
405	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
406	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
409	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
410	Q₁₆	—	2	4	no	—
411	Q₁₆	—	2	2	no	—
412	Q₁₆	—	2	2	no	—
413	Q₁₆	C₂×D₄×C₆₁	1	4	yes	—
414	SD₁₆	—	2	8	no	—
415	SD₁₆	—	2	8	no	—
416	SD₁₆	—	4	2	no	—
417	D₈	—	2	4	no	—
418	D₈	—	2	2	no	—
419	D₈	—	2	2	no	—
420	D₈	C₂×D₄×C₆₁	1	4	yes	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
427	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
428	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
429	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
430	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	4	1	no	—
434	C₄⋊C₄	—	4	1	no	—
435	C₂²⋊C₄	—	4	1	no	—
436	C₂²⋊C₄	—	4	1	no	—
437	C₂²⋊C₄	—	4	1	no	—
438	C₂²⋊C₄	—	4	1	no	—
439	C₂²⋊C₄	—	4	1	no	—
440	C₄²	—	4	1	no	—
441	C₁₆	—	2	2	no	—
442	C₁₆	—	2	2	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	4	1	no	—
446	C₂³	—	4	4	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	1	no	—
449	C₂³	—	8	2	no	—
450	C₂³	—	8	2	no	—
451	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
452	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
453	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
454	Q₈	—	2	4	no	—
455	Q₈	—	4	4	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	4	1	no	—
458	Q₈	—	4	1	no	—
459	D₄	—	2	4	no	—
460	D₄	—	4	4	no	—
461	D₄	—	4	8	no	—
462	D₄	—	4	8	no	—
463	D₄	—	4	2	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	1	no	—
466	D₄	—	8	2	no	—
467	D₄	—	8	2	no	—
468	D₄	—	8	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	8	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	8	2	no	—
484	C₂×C₄	#120×C₆₁	1	2	yes	—
485	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
486	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
487	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
488	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
489	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
490	C₈	—	2	1	no	—
491	C₈	—	2	1	no	—
492	C₈	#120×C₆₁	1	2	yes	—
493	C₈	C₂²×D₄×C₆₁	1	2	yes	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	1	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	8	8	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	—	8	1	no	—
507	C₂²	—	8	1	no	—
508	C₂²	—	8	1	no	—
509	C₂²	C₆₁×#447	1	2	yes	—
510	C₂²	C₆₁×#508	1	2	yes	complemented
511	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	4	4	no	—
515	C₄	—	8	1	no	—
516	C₄	—	8	1	no	—
517	C₄	—	8	1	no	—
518	C₄	C₆₁×#447	1	2	yes	—
519	C₄	C₂×#120×C₆₁	1	1	yes	characteristic
520	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	2	2	no	—
523	C₂	—	4	1	no	—
524	C₂	—	8	4	no	—
525	C₂	—	16	1	no	—
526	C₂	C₂×C₆₁×#447	1	1	yes	characteristic
527	C₂	C₂×C₆₁×#508	1	1	yes	characteristic
528	C₂	C₆₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#135×C₆₁	C₂	semidirect
2. C₂×C₆₁×#508	C₂	semidirect
3. C₆₁×#3050	C₂	semidirect
4. C₆₁×#2189	C₂	semidirect
5. C₆₁×#2188	C₂	semidirect
6. C₆₁×#2187	C₂	semidirect
7. C₆₁×#2186	C₂	semidirect
8. C₆₁×#2185	C₂	semidirect
9. C₆₁×#2184	C₂	semidirect
10. C₆₁×#2138	C₂	semidirect
11. C₂×C₆₁×C₂²⋊C₁₆	C₂	semidirect
18. C₂×#135×C₆₁	C₂²	semidirect
25. C₂×Q₃₂×C₆₁	C₂²	semidirect
26. C₂×SD₃₂×C₆₁	C₂²	semidirect
27. C₂×SD₃₂×C₆₁	C₂²	semidirect
28. C₂×D₁₆×C₆₁	C₂²	semidirect
29. C₂²×C₁₆×C₆₁	C₂²	semidirect
31. C₆₁×#467	C₂²	semidirect
33. C₆₁×#466	C₂²	semidirect
36. C₆₁×#427	C₂²	semidirect
38. C₆₁×#358	C₂²	semidirect
39. C₆₁×#358	C₂²	semidirect
40. C₆₁×#357	C₂²	semidirect
41. C₆₁×#357	C₂²	semidirect
44. C₆₁×C₂²⋊C₁₆	C₂²	semidirect
63. #135×C₆₁	D₄	semidirect
229. #33516	C₆₁	direct (N × H)
246. C₂²×C₆₁	#508	semidirect
257. C₂²×#135	C₂×C₆₁	semidirect
258. C₂×#508	C₂×C₆₁	semidirect
259. #3050	C₂×C₆₁	semidirect
260. #2189	C₂×C₆₁	semidirect
261. #2188	C₂×C₆₁	semidirect
262. #2187	C₂×C₆₁	semidirect
263. #2186	C₂×C₆₁	semidirect
264. #2185	C₂×C₆₁	semidirect
265. #2184	C₂×C₆₁	semidirect
266. #2138	C₂×C₆₁	semidirect
267. C₂×C₂²⋊C₁₆	C₂×C₆₁	semidirect
282. C₂×#135	C₂²×C₆₁	semidirect
289. C₂×Q₃₂	C₂²×C₆₁	semidirect
290. C₂×SD₃₂	C₂²×C₆₁	semidirect
291. C₂×SD₃₂	C₂²×C₆₁	semidirect
292. C₂×D₁₆	C₂²×C₆₁	semidirect
293. C₂²×C₁₆	C₂²×C₆₁	semidirect
295. #467	C₂²×C₆₁	semidirect
297. #466	C₂²×C₆₁	semidirect
300. #427	C₂²×C₆₁	semidirect
302. #358	C₂²×C₆₁	semidirect
303. #358	C₂²×C₆₁	semidirect
304. #357	C₂²×C₆₁	semidirect
305. #357	C₂²×C₆₁	semidirect
308. C₂²⋊C₁₆	C₂²×C₆₁	semidirect
309. C₆₁	#33516	direct (N × H)
328. #135	D₄×C₆₁	semidirect
510. C₂²	C₆₁×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₁	2
C₂×C₆₁×#508	2
C₆₁×#3050	2
C₆₁×#2189	2
C₆₁×#2188	2
C₆₁×#2187	2
C₆₁×#2186	2
C₆₁×#2185	2
C₆₁×#2184	2
C₆₁×#2138	2
C₂×C₆₁×C₂²⋊C₁₆	2
#33516	61

Subgroups — C61×#33516 #29536949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33516 #29536949