Subgroups — C₆₁×#24278 #29527711

Order 15616.

1436 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SD₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#471	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3004	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3003	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2929	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2046	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2019	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1977	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1810	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1784	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1783	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#137×C₆₁	C₂²	1	2	yes	complemented
15	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₆₁	—	2	8	no	—
17	C₂²×SD₁₆×C₆₁	—	2	2	no	—
18	C₂²×SD₁₆×C₆₁	—	2	2	no	—
19	C₂²×SD₁₆×C₆₁	C₂²	1	2	yes	—
20	C₂²×SD₁₆×C₆₁	C₂²	1	2	yes	—
21	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic, complemented
22	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#525	C₂²	1	1	yes	characteristic
24	C₆₁×#471	—	2	8	no	—
25	C₆₁×#451	—	2	2	no	—
26	C₆₁×#450	—	2	2	no	—
27	C₆₁×#448	—	2	2	no	—
28	C₆₁×#447	—	2	2	no	—
29	C₆₁×#435	—	2	2	no	—
30	C₈×D₄×C₆₁	—	2	2	no	—
31	C₂×#108×C₆₁	C₂²	1	2	yes	complemented
32	C₆₁×#424	C₂²	1	1	yes	characteristic, complemented
33	C₆₁×#419	C₂²	1	2	yes	complemented
34	C₆₁×#418	C₂²	1	2	yes	—
35	C₂×#101×C₆₁	C₂²	1	2	yes	complemented
36	C₂×#101×C₆₁	C₂²	1	2	yes	complemented
37	C₂×#100×C₆₁	C₂²	1	2	yes	—
38	C₂×#100×C₆₁	C₂²	1	2	yes	—
39	C₆₁×#408	C₂²	1	1	yes	characteristic, complemented
40	C₆₁×#407	C₂²	1	1	yes	characteristic
41	C₆₁×#407	C₂²	1	1	yes	characteristic
42	C₆₁×#407	C₂²	1	1	yes	characteristic
43	C₆₁×#405	C₂²	1	1	yes	characteristic, complemented
44	C₆₁×#341	C₂²	1	2	yes	complemented
45	C₂⁵×C₆₁	—	2	1	no	—
46	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
47	C₂²×Q₈×C₆₁	—	2	1	no	—
48	C₂²×Q₈×C₆₁	—	2	4	no	—
49	C₂²×Q₈×C₆₁	—	2	2	no	—
50	C₂²×Q₈×C₆₁	—	2	2	no	—
51	C₂²×Q₈×C₆₁	—	4	2	no	—
52	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
53	C₂²×D₄×C₆₁	—	2	4	no	—
54	C₂²×D₄×C₆₁	—	2	2	no	—
55	C₂²×D₄×C₆₁	—	2	2	no	—
56	C₂²×D₄×C₆₁	—	2	1	no	—
57	C₂²×D₄×C₆₁	—	4	2	no	—
58	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₁	—	2	1	no	—
60	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
61	#137×C₆₁	—	2	4	no	—
62	#137×C₆₁	—	4	2	no	—
63	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
64	AGL(1,ℤ/8)×C₆₁	—	4	2	no	—
65	C₂×Q₁₆×C₆₁	—	2	2	no	—
66	C₂×SD₁₆×C₆₁	—	2	16	no	—
67	C₂×SD₁₆×C₆₁	—	2	16	no	—
68	C₂×SD₁₆×C₆₁	—	2	4	no	—
69	C₂×SD₁₆×C₆₁	—	2	4	no	—
70	C₂×SD₁₆×C₆₁	—	2	2	no	—
71	C₂×SD₁₆×C₆₁	—	2	2	no	—
72	C₂×SD₁₆×C₆₁	—	2	4	no	—
73	C₂×SD₁₆×C₆₁	—	2	2	no	—
74	C₂×SD₁₆×C₆₁	—	2	2	no	—
75	C₂×SD₁₆×C₆₁	—	2	4	no	—
76	C₂×SD₁₆×C₆₁	D₄	1	8	yes	complemented
77	C₂×D₈×C₆₁	—	2	2	no	—
78	#131×C₆₁	—	2	4	no	—
79	C₂×M₁₆×C₆₁	—	2	1	no	—
80	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
82	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
83	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₆₁	—	2	4	no	—
85	C₂²×C₈×C₆₁	—	2	1	no	—
86	C₂²×C₈×C₆₁	D₄	1	2	yes	—
87	C₂²×C₈×C₆₁	D₄	1	2	yes	—
88	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
90	#123×C₆₁	—	2	1	no	—
91	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
92	#121×C₆₁	D₄	1	2	yes	—
93	#120×C₆₁	—	2	1	no	—
94	C₄×D₄×C₆₁	—	2	2	no	—
95	#117×C₆₁	C₂³	1	1	yes	characteristic
96	#108×C₆₁	—	2	4	no	—
97	C₄⋊C₈×C₆₁	—	2	2	no	—
98	#103×C₆₁	—	2	4	no	—
99	#102×C₆₁	—	2	4	no	—
100	#101×C₆₁	—	2	4	no	—
101	#101×C₆₁	—	4	2	no	—
102	#100×C₆₁	—	2	4	no	—
103	#100×C₆₁	—	4	2	no	—
104	C₂²⋊C₈×C₆₁	—	2	2	no	—
105	C₂²⋊C₈×C₆₁	—	2	2	no	—
106	C₂²⋊C₈×C₆₁	—	2	2	no	—
107	C₂²⋊C₈×C₆₁	—	2	2	no	—
108	#97×C₆₁	—	2	1	no	—
109	C₄×C₈×C₆₁	—	2	1	no	—
110	C₂⁴×C₆₁	—	2	2	no	—
111	C₂⁴×C₆₁	—	4	4	no	—
112	C₂⁴×C₆₁	—	4	2	no	—
113	C₂⁴×C₆₁	—	8	4	no	—
114	C₂⁴×C₆₁	#41	1	1	yes	characteristic
115	#41×C₆₁	—	2	2	no	—
116	#41×C₆₁	—	2	2	no	—
117	C₂×Q₈×C₆₁	—	2	2	no	—
118	C₂×Q₈×C₆₁	—	2	8	no	—
119	C₂×Q₈×C₆₁	—	2	2	no	—
120	C₂×Q₈×C₆₁	—	2	2	no	—
121	C₂×Q₈×C₆₁	—	2	2	no	—
122	C₂×Q₈×C₆₁	—	2	2	no	—
123	C₂×Q₈×C₆₁	—	4	4	no	—
124	C₂×Q₈×C₆₁	—	4	8	no	—
125	C₂×Q₈×C₆₁	—	4	4	no	—
126	C₂×Q₈×C₆₁	—	4	2	no	—
127	C₂×Q₈×C₆₁	C₂×D₄	1	4	yes	—
128	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
129	C₂×D₄×C₆₁	—	2	8	no	—
130	C₂×D₄×C₆₁	—	2	2	no	—
131	C₂×D₄×C₆₁	—	2	1	no	—
132	C₂×D₄×C₆₁	—	2	2	no	—
133	C₂×D₄×C₆₁	—	2	2	no	—
134	C₂×D₄×C₆₁	—	2	2	no	—
135	C₂×D₄×C₆₁	—	2	2	no	—
136	C₂×D₄×C₆₁	—	2	2	no	—
137	C₂×D₄×C₆₁	—	4	8	no	—
138	C₂×D₄×C₆₁	—	4	4	no	—
139	C₂×D₄×C₆₁	—	4	4	no	—
140	C₂×D₄×C₆₁	—	4	2	no	—
141	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
142	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₁	—	2	2	no	—
144	C₂²×C₄×C₆₁	—	2	2	no	—
145	C₂²×C₄×C₆₁	—	2	2	no	—
146	C₂²×C₄×C₆₁	—	2	1	no	—
147	C₂²×C₄×C₆₁	—	4	4	no	—
148	C₂²×C₄×C₆₁	—	4	2	no	—
149	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
152	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
153	Q₁₆×C₆₁	—	4	4	no	—
154	SD₁₆×C₆₁	—	2	16	no	—
155	SD₁₆×C₆₁	—	2	8	no	—
156	SD₁₆×C₆₁	—	2	8	no	—
157	SD₁₆×C₆₁	—	4	4	no	—
158	SD₁₆×C₆₁	—	4	4	no	—
159	D₈×C₆₁	—	4	4	no	—
160	M₁₆×C₆₁	—	2	2	no	—
161	M₁₆×C₆₁	—	2	2	no	—
162	M₁₆×C₆₁	—	2	2	no	—
163	M₁₆×C₆₁	—	2	2	no	—
164	M₁₆×C₆₁	—	2	2	no	—
165	M₁₆×C₆₁	—	2	2	no	—
166	M₁₆×C₆₁	—	4	2	no	—
167	C₂×C₈×C₆₁	—	2	2	no	—
168	C₂×C₈×C₆₁	—	2	4	no	—
169	C₂×C₈×C₆₁	—	2	4	no	—
170	C₂×C₈×C₆₁	—	2	1	no	—
171	C₂×C₈×C₆₁	—	2	2	no	—
172	C₂×C₈×C₆₁	—	2	2	no	—
173	C₂×C₈×C₆₁	—	2	1	no	—
174	C₂×C₈×C₆₁	—	4	2	no	—
175	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
176	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
179	C₄⋊C₄×C₆₁	—	2	2	no	—
180	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
181	C₂²⋊C₄×C₆₁	—	2	2	no	—
182	C₂²⋊C₄×C₆₁	—	2	1	no	—
183	C₂²⋊C₄×C₆₁	—	2	2	no	—
184	C₄²×C₆₁	—	2	1	no	—
185	C₂³×C₆₁	—	2	2	no	—
186	C₂³×C₆₁	—	2	4	no	—
187	C₂³×C₆₁	—	2	1	no	—
188	C₂³×C₆₁	—	2	4	no	—
189	C₂³×C₆₁	—	4	8	no	—
190	C₂³×C₆₁	—	4	2	no	—
191	C₂³×C₆₁	—	4	2	no	—
192	C₂³×C₆₁	—	8	16	no	—
193	C₂³×C₆₁	—	8	4	no	—
194	C₂³×C₆₁	—	8	4	no	—
195	C₂³×C₆₁	—	8	4	no	—
196	C₂³×C₆₁	#121	1	2	yes	—
197	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
198	Q₈×C₆₁	—	2	4	no	—
199	Q₈×C₆₁	—	2	2	no	—
200	Q₈×C₆₁	—	2	4	no	—
201	Q₈×C₆₁	—	4	8	no	—
202	Q₈×C₆₁	—	4	2	no	—
203	Q₈×C₆₁	—	4	2	no	—
204	D₄×C₆₁	—	2	4	no	—
205	D₄×C₆₁	—	2	2	no	—
206	D₄×C₆₁	—	2	4	no	—
207	D₄×C₆₁	—	4	8	no	—
208	D₄×C₆₁	—	4	2	no	—
209	D₄×C₆₁	—	4	4	no	—
210	D₄×C₆₁	—	4	2	no	—
211	D₄×C₆₁	—	4	2	no	—
212	C₂×C₄×C₆₁	—	2	2	no	—
213	C₂×C₄×C₆₁	—	2	2	no	—
214	C₂×C₄×C₆₁	—	2	2	no	—
215	C₂×C₄×C₆₁	—	2	4	no	—
216	C₂×C₄×C₆₁	—	2	2	no	—
217	C₂×C₄×C₆₁	—	2	1	no	—
218	C₂×C₄×C₆₁	—	2	4	no	—
219	C₂×C₄×C₆₁	—	4	8	no	—
220	C₂×C₄×C₆₁	—	4	2	no	—
221	C₂×C₄×C₆₁	—	4	2	no	—
222	C₂×C₄×C₆₁	—	4	1	no	—
223	C₂×C₄×C₆₁	—	4	1	no	—
224	C₂×C₄×C₆₁	#121	1	2	yes	—
225	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
229	C₈×C₆₁	—	2	2	no	—
230	C₈×C₆₁	—	4	1	no	—
231	C₈×C₆₁	—	4	1	no	—
232	C₈×C₆₁	—	4	2	no	—
233	C₈×C₆₁	—	4	2	no	—
234	C₈×C₆₁	#120	1	2	yes	—
235	C₈×C₆₁	#120	1	2	yes	—
236	#24278	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂²×C₆₁	—	2	8	no	—
238	C₂²×C₆₁	—	2	2	no	—
239	C₂²×C₆₁	—	2	2	no	—
240	C₂²×C₆₁	—	2	2	no	—
241	C₂²×C₆₁	—	4	1	no	—
242	C₂²×C₆₁	—	4	4	no	—
243	C₂²×C₆₁	—	4	2	no	—
244	C₂²×C₆₁	—	8	4	no	—
245	C₂²×C₆₁	—	8	4	no	—
246	C₂²×C₆₁	—	8	16	no	—
247	C₂²×C₆₁	—	8	4	no	—
248	C₂²×C₆₁	#471	1	4	yes	complemented
249	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
250	C₂²×C₆₁	#546	1	2	yes	—
251	C₄×C₆₁	—	2	2	no	—
252	C₄×C₆₁	—	4	4	no	—
253	C₄×C₆₁	—	4	1	no	—
254	C₄×C₆₁	—	4	2	no	—
255	C₄×C₆₁	C₂×#120	1	1	yes	characteristic
256	C₄×C₆₁	#535	1	1	yes	characteristic
257	C₄×C₆₁	D₄²	1	2	yes	—
258	C₂³×SD₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
259	C₂×#471	C₂×C₆₁	1	2	yes	complemented
260	#3004	C₂×C₆₁	1	1	yes	characteristic, complemented
261	#3003	C₂×C₆₁	1	1	yes	characteristic
262	#2929	C₂×C₆₁	1	1	yes	characteristic, complemented
263	#2046	C₂×C₆₁	1	2	yes	complemented
264	#2019	C₂×C₆₁	1	2	yes	complemented
265	#1977	C₂×C₆₁	1	2	yes	complemented
266	#1810	C₂×C₆₁	1	1	yes	characteristic, complemented
267	#1784	C₂×C₆₁	1	1	yes	characteristic, complemented
268	#1783	C₂×C₆₁	1	1	yes	characteristic, complemented
269	C₂×C₆₁	—	2	2	no	—
270	C₂×C₆₁	—	2	4	no	—
271	C₂×C₆₁	—	8	1	no	—
272	C₂×C₆₁	—	8	4	no	—
273	C₂×C₆₁	#2408	1	1	yes	characteristic
274	C₂×C₆₁	C₂×#471	1	2	yes	—
275	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic
276	C₂³×D₄	C₂²×C₆₁	1	1	yes	characteristic
277	C₂×#137	C₂²×C₆₁	1	2	yes	complemented
278	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	—
279	C₂²×SD₁₆	—	2	8	no	—
280	C₂²×SD₁₆	—	2	2	no	—
281	C₂²×SD₁₆	—	2	2	no	—
282	C₂²×SD₁₆	C₂²×C₆₁	1	2	yes	—
283	C₂²×SD₁₆	C₂²×C₆₁	1	2	yes	—
284	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic, complemented
285	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
286	#525	C₂²×C₆₁	1	1	yes	characteristic
287	#471	—	2	8	no	—
288	#451	—	2	2	no	—
289	#450	—	2	2	no	—
290	#448	—	2	2	no	—
291	#447	—	2	2	no	—
292	#435	—	2	2	no	—
293	C₈×D₄	—	2	2	no	—
294	C₂×#108	C₂²×C₆₁	1	2	yes	complemented
295	#424	C₂²×C₆₁	1	1	yes	characteristic, complemented
296	#419	C₂²×C₆₁	1	2	yes	complemented
297	#418	C₂²×C₆₁	1	2	yes	—
298	C₂×#101	C₂²×C₆₁	1	2	yes	complemented
299	C₂×#101	C₂²×C₆₁	1	2	yes	complemented
300	C₂×#100	C₂²×C₆₁	1	2	yes	—
301	C₂×#100	C₂²×C₆₁	1	2	yes	—
302	#408	C₂²×C₆₁	1	1	yes	characteristic, complemented
303	#407	C₂²×C₆₁	1	1	yes	characteristic
304	#407	C₂²×C₆₁	1	1	yes	characteristic
305	#407	C₂²×C₆₁	1	1	yes	characteristic
306	#405	C₂²×C₆₁	1	1	yes	characteristic, complemented
307	#341	C₂²×C₆₁	1	2	yes	complemented
308	C₆₁	#24278	1	1	yes	characteristic, direct product factor, Sylow
309	C₂⁵	—	2	1	no	—
310	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
311	C₂²×Q₈	—	2	1	no	—
312	C₂²×Q₈	—	2	4	no	—
313	C₂²×Q₈	—	2	2	no	—
314	C₂²×Q₈	—	2	2	no	—
315	C₂²×Q₈	—	4	2	no	—
316	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
317	C₂²×D₄	—	2	4	no	—
318	C₂²×D₄	—	2	2	no	—
319	C₂²×D₄	—	2	2	no	—
320	C₂²×D₄	—	2	1	no	—
321	C₂²×D₄	—	4	2	no	—
322	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
323	C₂³×C₄	—	2	1	no	—
324	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
325	#137	—	2	4	no	—
326	#137	—	4	2	no	—
327	AGL(1,ℤ/8)	—	2	4	no	—
328	AGL(1,ℤ/8)	—	4	2	no	—
329	C₂×Q₁₆	—	2	2	no	—
330	C₂×SD₁₆	—	2	16	no	—
331	C₂×SD₁₆	—	2	16	no	—
332	C₂×SD₁₆	—	2	4	no	—
333	C₂×SD₁₆	—	2	4	no	—
334	C₂×SD₁₆	—	2	2	no	—
335	C₂×SD₁₆	—	2	2	no	—
336	C₂×SD₁₆	—	2	4	no	—
337	C₂×SD₁₆	—	2	2	no	—
338	C₂×SD₁₆	—	2	2	no	—
339	C₂×SD₁₆	—	2	4	no	—
340	C₂×SD₁₆	D₄×C₆₁	1	8	yes	complemented
341	C₂×D₈	—	2	2	no	—
342	#131	—	2	4	no	—
343	C₂×M₁₆	—	2	1	no	—
344	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
345	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
346	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
347	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
348	C₂²×C₈	—	2	4	no	—
349	C₂²×C₈	—	2	1	no	—
350	C₂²×C₈	D₄×C₆₁	1	2	yes	—
351	C₂²×C₈	D₄×C₆₁	1	2	yes	—
352	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
354	#123	—	2	1	no	—
355	C₂²⋊Q₈	D₄×C₆₁	1	2	yes	—
356	#121	D₄×C₆₁	1	2	yes	—
357	#120	—	2	1	no	—
358	C₄×D₄	—	2	2	no	—
359	#117	C₂³×C₆₁	1	1	yes	characteristic
360	#108	—	2	4	no	—
361	C₄⋊C₈	—	2	2	no	—
362	#103	—	2	4	no	—
363	#102	—	2	4	no	—
364	#101	—	2	4	no	—
365	#101	—	4	2	no	—
366	#100	—	2	4	no	—
367	#100	—	4	2	no	—
368	C₂²⋊C₈	—	2	2	no	—
369	C₂²⋊C₈	—	2	2	no	—
370	C₂²⋊C₈	—	2	2	no	—
371	C₂²⋊C₈	—	2	2	no	—
372	#97	—	2	1	no	—
373	C₄×C₈	—	2	1	no	—
374	C₂⁴	—	2	2	no	—
375	C₂⁴	—	4	4	no	—
376	C₂⁴	—	4	2	no	—
377	C₂⁴	—	8	4	no	—
378	C₂⁴	#41×C₆₁	1	1	yes	characteristic
379	#41	—	2	2	no	—
380	#41	—	2	2	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	8	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	4	4	no	—
388	C₂×Q₈	—	4	8	no	—
389	C₂×Q₈	—	4	4	no	—
390	C₂×Q₈	—	4	2	no	—
391	C₂×Q₈	C₂×D₄×C₆₁	1	4	yes	—
392	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
393	C₂×D₄	—	2	8	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	1	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	4	8	no	—
402	C₂×D₄	—	4	4	no	—
403	C₂×D₄	—	4	4	no	—
404	C₂×D₄	—	4	2	no	—
405	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
406	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	4	4	no	—
412	C₂²×C₄	—	4	2	no	—
413	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
415	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
417	Q₁₆	—	4	4	no	—
418	SD₁₆	—	2	16	no	—
419	SD₁₆	—	2	8	no	—
420	SD₁₆	—	2	8	no	—
421	SD₁₆	—	4	4	no	—
422	SD₁₆	—	4	4	no	—
423	D₈	—	4	4	no	—
424	M₁₆	—	2	2	no	—
425	M₁₆	—	2	2	no	—
426	M₁₆	—	2	2	no	—
427	M₁₆	—	2	2	no	—
428	M₁₆	—	2	2	no	—
429	M₁₆	—	2	2	no	—
430	M₁₆	—	4	2	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	2	4	no	—
433	C₂×C₈	—	2	4	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	4	2	no	—
439	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
440	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
441	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
442	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
445	C₂²⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	—	2	1	no	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₄²	—	2	1	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	4	no	—
451	C₂³	—	2	1	no	—
452	C₂³	—	2	4	no	—
453	C₂³	—	4	8	no	—
454	C₂³	—	4	2	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	8	4	no	—
457	C₂³	—	8	16	no	—
458	C₂³	—	8	4	no	—
459	C₂³	—	8	4	no	—
460	C₂³	#121×C₆₁	1	2	yes	—
461	C₂³	C₂×#41×C₆₁	1	1	yes	characteristic
462	Q₈	—	2	2	no	—
463	Q₈	—	2	4	no	—
464	Q₈	—	2	4	no	—
465	Q₈	—	4	8	no	—
466	Q₈	—	4	2	no	—
467	Q₈	—	4	2	no	—
468	D₄	—	2	2	no	—
469	D₄	—	2	4	no	—
470	D₄	—	2	4	no	—
471	D₄	—	4	8	no	—
472	D₄	—	4	2	no	—
473	D₄	—	4	4	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	4	8	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	#121×C₆₁	1	2	yes	—
489	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
490	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
492	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
493	C₈	—	2	2	no	—
494	C₈	—	4	1	no	—
495	C₈	—	4	2	no	—
496	C₈	—	4	2	no	—
497	C₈	—	4	1	no	—
498	C₈	#120×C₆₁	1	2	yes	—
499	C₈	#120×C₆₁	1	2	yes	—
500	C₂²	—	2	8	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	4	4	no	—
507	C₂²	—	8	4	no	—
508	C₂²	—	8	4	no	—
509	C₂²	—	8	16	no	—
510	C₂²	—	8	4	no	—
511	C₂²	C₆₁×#471	1	4	yes	complemented
512	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
513	C₂²	C₆₁×#546	1	2	yes	—
514	C₄	—	2	2	no	—
515	C₄	—	4	2	no	—
516	C₄	—	4	4	no	—
517	C₄	—	4	1	no	—
518	C₄	C₂×#120×C₆₁	1	1	yes	characteristic
519	C₄	C₆₁×#535	1	1	yes	characteristic
520	C₄	D₄²×C₆₁	1	2	yes	—
521	C₂	—	2	2	no	—
522	C₂	—	2	4	no	—
523	C₂	—	8	1	no	—
524	C₂	—	8	4	no	—
525	C₂	C₆₁×#2408	1	1	yes	characteristic
526	C₂	C₂×C₆₁×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₆₁×#3003	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×SD₁₆×C₆₁	C₂	semidirect
2. C₂×C₆₁×#471	C₂	semidirect
3. C₆₁×#3004	C₂	semidirect
5. C₆₁×#2929	C₂	semidirect
6. C₆₁×#2046	C₂	semidirect
7. C₆₁×#2019	C₂	semidirect
8. C₆₁×#1977	C₂	semidirect
9. C₆₁×#1810	C₂	semidirect
10. C₆₁×#1784	C₂	semidirect
11. C₆₁×#1783	C₂	semidirect
14. C₂×#137×C₆₁	C₂²	semidirect
21. C₂×#131×C₆₁	C₂²	semidirect
31. C₂×#108×C₆₁	C₂²	semidirect
32. C₆₁×#424	C₂²	semidirect
33. C₆₁×#419	C₂²	semidirect
35. C₂×#101×C₆₁	C₂²	semidirect
36. C₂×#101×C₆₁	C₂²	semidirect
39. C₆₁×#408	C₂²	semidirect
43. C₆₁×#405	C₂²	semidirect
44. C₆₁×#341	C₂²	semidirect
76. C₂×SD₁₆×C₆₁	D₄	semidirect
236. #24278	C₆₁	direct (N × H)
248. C₂²×C₆₁	#471	semidirect
258. C₂³×SD₁₆	C₂×C₆₁	semidirect
259. C₂×#471	C₂×C₆₁	semidirect
260. #3004	C₂×C₆₁	semidirect
262. #2929	C₂×C₆₁	semidirect
263. #2046	C₂×C₆₁	semidirect
264. #2019	C₂×C₆₁	semidirect
265. #1977	C₂×C₆₁	semidirect
266. #1810	C₂×C₆₁	semidirect
267. #1784	C₂×C₆₁	semidirect
268. #1783	C₂×C₆₁	semidirect
277. C₂×#137	C₂²×C₆₁	semidirect
284. C₂×#131	C₂²×C₆₁	semidirect
294. C₂×#108	C₂²×C₆₁	semidirect
295. #424	C₂²×C₆₁	semidirect
296. #419	C₂²×C₆₁	semidirect
298. C₂×#101	C₂²×C₆₁	semidirect
299. C₂×#101	C₂²×C₆₁	semidirect
302. #408	C₂²×C₆₁	semidirect
306. #405	C₂²×C₆₁	semidirect
307. #341	C₂²×C₆₁	semidirect
308. C₆₁	#24278	direct (N × H)
340. C₂×SD₁₆	D₄×C₆₁	semidirect
511. C₂²	C₆₁×#471	semidirect

Maximal subgroups

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

subgroup	index
C₂³×SD₁₆×C₆₁	2
C₂×C₆₁×#471	2
C₆₁×#3004	2
C₆₁×#3003	2
C₆₁×#2929	2
C₆₁×#2046	2
C₆₁×#2019	2
C₆₁×#1977	2
C₆₁×#1810	2
C₆₁×#1784	2
C₆₁×#1783	2
#24278	61

Subgroups — C61×#24278 #29527711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24278 #29527711