Subgroups — C₆₁×#27019 #29530452

Order 15616.

742 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2452	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2065	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2058	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2057	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2023	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1944	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1926	C₂	1	1	yes	maximal, characteristic
10	C₆₁×#1925	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1920	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#1918	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₁×#1917	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#1862	C₂	1	1	yes	maximal, characteristic
15	C₆₁×#1762	C₂	1	1	yes	maximal, characteristic
16	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
19	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#396	—	2	1	no	—
31	C₆₁×#393	—	2	1	no	—
32	C₆₁×#391	—	2	1	no	—
33	C₆₁×#391	—	2	1	no	—
34	C₆₁×#390	C₂²	1	1	yes	characteristic
35	C₆₁×#390	C₂²	1	1	yes	characteristic
36	C₆₁×#389	—	2	1	no	—
37	C₆₁×#389	C₂²	1	1	yes	characteristic
38	C₆₁×#389	C₂²	1	1	yes	characteristic
39	C₆₁×#389	C₂²	1	1	yes	characteristic
40	C₆₁×#389	C₂²	1	1	yes	characteristic
41	C₆₁×#385	—	2	1	no	—
42	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
43	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
47	C₆₁×#340	C₂²	1	1	yes	characteristic
48	C₆₁×#340	C₂²	1	1	yes	characteristic
49	C₆₁×#340	C₂²	1	1	yes	characteristic
50	C₆₁×#340	C₂²	1	1	yes	characteristic
51	C₆₁×#340	C₂²	1	1	yes	characteristic
52	C₆₁×#340	C₂²	1	1	yes	characteristic
53	C₆₁×#340	C₂²	1	1	yes	characteristic
54	C₆₁×#340	C₂²	1	1	yes	characteristic
55	C₆₁×#336	C₂²	1	1	yes	characteristic
56	C₆₁×#336	C₂²	1	1	yes	characteristic
57	C₆₁×#336	C₂²	1	1	yes	characteristic
58	C₆₁×#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₆₁	—	2	1	no	—
60	C₂²×Q₈×C₆₁	—	2	1	no	—
61	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₁	—	2	1	no	—
63	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₆₁	—	2	2	no	—
68	C₄⋊Q₈×C₆₁	—	2	2	no	—
69	C₄⋊Q₈×C₆₁	—	2	2	no	—
70	C₄⋊Q₈×C₆₁	—	4	1	no	—
71	C₂²⋊Q₈×C₆₁	—	4	4	no	—
72	C₄×D₄×C₆₁	—	2	2	no	—
73	C₄×D₄×C₆₁	—	4	1	no	—
74	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
94	C₂×C₄²×C₆₁	—	2	1	no	—
95	C₂×C₄²×C₆₁	—	2	1	no	—
96	C₂×C₄²×C₆₁	—	2	1	no	—
97	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
100	#107×C₆₁	—	2	2	no	—
101	#107×C₆₁	—	4	1	no	—
102	#106×C₆₁	—	2	2	no	—
103	#106×C₆₁	—	2	2	no	—
104	C₄⋊C₈×C₆₁	—	4	1	no	—
105	#102×C₆₁	—	2	2	no	—
106	#102×C₆₁	—	2	2	no	—
107	#102×C₆₁	—	4	1	no	—
108	#102×C₆₁	Q₈	1	4	yes	complemented
109	#97×C₆₁	—	4	1	no	—
110	#95×C₆₁	—	2	1	no	—
111	#95×C₆₁	—	2	1	no	—
112	#95×C₆₁	—	2	1	no	—
113	#95×C₆₁	—	2	1	no	—
114	#95×C₆₁	—	2	1	no	—
115	#95×C₆₁	—	2	1	no	—
116	#95×C₆₁	—	2	1	no	—
117	#95×C₆₁	D₄	1	1	yes	characteristic
118	#95×C₆₁	D₄	1	1	yes	characteristic
119	C₂⁴×C₆₁	—	2	1	no	—
120	C₂×Q₈×C₆₁	—	2	4	no	—
121	C₂×Q₈×C₆₁	—	4	2	no	—
122	C₂×Q₈×C₆₁	—	4	2	no	—
123	C₂×Q₈×C₆₁	—	8	1	no	—
124	C₂×Q₈×C₆₁	—	8	1	no	—
125	C₂×Q₈×C₆₁	—	8	1	no	—
126	C₂×D₄×C₆₁	—	2	1	no	—
127	C₂×D₄×C₆₁	—	2	1	no	—
128	C₂×D₄×C₆₁	—	2	1	no	—
129	C₂×D₄×C₆₁	—	2	1	no	—
130	C₂×D₄×C₆₁	—	2	1	no	—
131	C₂×D₄×C₆₁	C₂×Q₈	1	2	yes	—
132	C₂²×C₄×C₆₁	—	2	1	no	—
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	—	2	1	no	—
137	C₂²×C₄×C₆₁	—	2	1	no	—
138	C₂²×C₄×C₆₁	—	2	1	no	—
139	C₂²×C₄×C₆₁	—	4	2	no	—
140	C₂²×C₄×C₆₁	—	4	2	no	—
141	C₂²×C₄×C₆₁	—	8	1	no	—
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	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
149	C₂×C₈×C₆₁	—	2	1	no	—
150	C₂×C₈×C₆₁	—	2	1	no	—
151	C₂×C₈×C₆₁	—	2	1	no	—
152	C₂×C₈×C₆₁	—	2	1	no	—
153	C₂×C₈×C₆₁	—	2	1	no	—
154	C₂×C₈×C₆₁	—	2	1	no	—
155	C₂×C₈×C₆₁	—	2	1	no	—
156	C₂×C₈×C₆₁	—	2	1	no	—
157	C₂×C₈×C₆₁	—	2	1	no	—
158	C₂×C₈×C₆₁	—	2	1	no	—
159	C₂×C₈×C₆₁	—	2	1	no	—
160	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
161	C₄⋊C₄×C₆₁	—	2	4	no	—
162	C₄⋊C₄×C₆₁	—	2	2	no	—
163	C₄⋊C₄×C₆₁	—	2	2	no	—
164	C₄⋊C₄×C₆₁	—	2	1	no	—
165	C₄⋊C₄×C₆₁	—	2	1	no	—
166	C₄⋊C₄×C₆₁	—	2	1	no	—
167	C₄⋊C₄×C₆₁	—	2	1	no	—
168	C₄⋊C₄×C₆₁	—	2	1	no	—
169	C₄⋊C₄×C₆₁	—	2	1	no	—
170	C₄⋊C₄×C₆₁	—	2	1	no	—
171	C₄⋊C₄×C₆₁	—	2	1	no	—
172	C₄⋊C₄×C₆₁	—	2	1	no	—
173	C₄⋊C₄×C₆₁	—	2	1	no	—
174	C₄⋊C₄×C₆₁	—	4	2	no	—
175	C₄⋊C₄×C₆₁	—	4	2	no	—
176	C₄⋊C₄×C₆₁	—	4	2	no	—
177	C₄⋊C₄×C₆₁	—	8	1	no	—
178	C₄⋊C₄×C₆₁	—	8	1	no	—
179	C₄⋊C₄×C₆₁	C₂×Q₈	1	2	yes	—
180	C₄⋊C₄×C₆₁	#41	1	2	yes	—
181	C₄⋊C₄×C₆₁	#41	1	2	yes	—
182	C₄⋊C₄×C₆₁	#41	1	2	yes	—
183	C₂²⋊C₄×C₆₁	—	4	2	no	—
184	C₂²⋊C₄×C₆₁	—	4	2	no	—
185	C₂²⋊C₄×C₆₁	—	4	2	no	—
186	C₂²⋊C₄×C₆₁	—	4	2	no	—
187	C₂²⋊C₄×C₆₁	—	8	1	no	—
188	C₄²×C₆₁	—	2	1	no	—
189	C₄²×C₆₁	—	2	1	no	—
190	C₄²×C₆₁	—	2	1	no	—
191	C₄²×C₆₁	—	2	1	no	—
192	C₄²×C₆₁	—	2	1	no	—
193	C₄²×C₆₁	—	2	1	no	—
194	C₄²×C₆₁	—	4	2	no	—
195	C₄²×C₆₁	—	8	1	no	—
196	C₄²×C₆₁	—	8	1	no	—
197	C₂³×C₆₁	—	2	2	no	—
198	C₂³×C₆₁	—	4	1	no	—
199	C₂³×C₆₁	—	4	1	no	—
200	C₂³×C₆₁	—	4	1	no	—
201	C₂³×C₆₁	—	4	1	no	—
202	C₂³×C₆₁	—	4	1	no	—
203	C₂³×C₆₁	—	4	1	no	—
204	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₆₁	—	8	4	no	—
206	Q₈×C₆₁	—	8	4	no	—
207	D₄×C₆₁	—	2	2	no	—
208	D₄×C₆₁	—	2	2	no	—
209	D₄×C₆₁	—	4	1	no	—
210	D₄×C₆₁	—	4	1	no	—
211	C₂×C₄×C₆₁	—	2	1	no	—
212	C₂×C₄×C₆₁	—	2	1	no	—
213	C₂×C₄×C₆₁	—	2	1	no	—
214	C₂×C₄×C₆₁	—	2	1	no	—
215	C₂×C₄×C₆₁	—	2	1	no	—
216	C₂×C₄×C₆₁	—	2	2	no	—
217	C₂×C₄×C₆₁	—	2	2	no	—
218	C₂×C₄×C₆₁	—	2	1	no	—
219	C₂×C₄×C₆₁	—	2	1	no	—
220	C₂×C₄×C₆₁	—	2	1	no	—
221	C₂×C₄×C₆₁	—	2	2	no	—
222	C₂×C₄×C₆₁	—	2	2	no	—
223	C₂×C₄×C₆₁	—	2	1	no	—
224	C₂×C₄×C₆₁	—	2	1	no	—
225	C₂×C₄×C₆₁	—	2	1	no	—
226	C₂×C₄×C₆₁	—	2	1	no	—
227	C₂×C₄×C₆₁	—	2	1	no	—
228	C₂×C₄×C₆₁	—	2	1	no	—
229	C₂×C₄×C₆₁	—	2	1	no	—
230	C₂×C₄×C₆₁	—	2	1	no	—
231	C₂×C₄×C₆₁	—	4	1	no	—
232	C₂×C₄×C₆₁	—	4	1	no	—
233	C₂×C₄×C₆₁	—	4	1	no	—
234	C₂×C₄×C₆₁	—	4	1	no	—
235	C₂×C₄×C₆₁	—	4	1	no	—
236	C₂×C₄×C₆₁	—	4	1	no	—
237	C₂×C₄×C₆₁	—	4	1	no	—
238	C₂×C₄×C₆₁	—	4	1	no	—
239	C₂×C₄×C₆₁	—	4	1	no	—
240	C₂×C₄×C₆₁	—	4	1	no	—
241	C₂×C₄×C₆₁	—	4	1	no	—
242	C₂×C₄×C₆₁	—	4	1	no	—
243	C₂×C₄×C₆₁	—	4	1	no	—
244	C₂×C₄×C₆₁	—	4	1	no	—
245	C₂×C₄×C₆₁	—	4	1	no	—
246	C₂×C₄×C₆₁	—	4	1	no	—
247	C₂×C₄×C₆₁	—	4	1	no	—
248	C₂×C₄×C₆₁	—	8	4	no	—
249	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
250	C₂×C₄×C₆₁	#137	1	2	yes	—
251	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
252	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
255	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
257	C₈×C₆₁	—	2	2	no	—
258	C₈×C₆₁	—	4	1	no	—
259	C₈×C₆₁	—	4	1	no	—
260	C₈×C₆₁	—	4	1	no	—
261	#27019	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₂²×C₆₁	—	4	2	no	—
263	C₂²×C₆₁	—	4	2	no	—
264	C₂²×C₆₁	—	4	2	no	—
265	C₂²×C₆₁	—	8	1	no	—
266	C₂²×C₆₁	—	8	1	no	—
267	C₂²×C₆₁	—	8	1	no	—
268	C₂²×C₆₁	—	8	1	no	—
269	C₂²×C₆₁	#525	1	1	yes	characteristic
270	C₂²×C₆₁	#546	1	1	yes	characteristic
271	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
272	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₂²×C₆₁	C₂×#137	1	1	yes	characteristic
274	C₂²×C₆₁	#576	1	1	yes	characteristic
275	C₂²×C₆₁	#577	1	1	yes	characteristic
276	C₄×C₆₁	—	2	2	no	—
277	C₄×C₆₁	—	2	2	no	—
278	C₄×C₆₁	—	4	1	no	—
279	C₄×C₆₁	—	4	2	no	—
280	C₄×C₆₁	—	4	2	no	—
281	C₄×C₆₁	—	4	2	no	—
282	C₄×C₆₁	—	4	1	no	—
283	C₄×C₆₁	—	4	1	no	—
284	C₄×C₆₁	—	4	1	no	—
285	C₄×C₆₁	—	4	2	no	—
286	C₄×C₆₁	—	8	1	no	—
287	C₄×C₆₁	—	8	1	no	—
288	C₄×C₆₁	—	8	1	no	—
289	C₄×C₆₁	#533	1	1	yes	characteristic
290	C₄×C₆₁	#542	1	1	yes	characteristic
291	C₄×C₆₁	#554	1	1	yes	characteristic
292	C₄×C₆₁	#556	1	1	yes	characteristic
293	#2452	C₂×C₆₁	1	1	yes	characteristic
294	#2369	C₂×C₆₁	1	1	yes	characteristic, complemented
295	#2065	C₂×C₆₁	1	1	yes	characteristic
296	#2058	C₂×C₆₁	1	1	yes	characteristic, complemented
297	#2057	C₂×C₆₁	1	1	yes	characteristic, complemented
298	#2023	C₂×C₆₁	1	1	yes	characteristic
299	#1987	C₂×C₆₁	1	1	yes	characteristic, complemented
300	#1944	C₂×C₆₁	1	1	yes	characteristic, complemented
301	#1926	C₂×C₆₁	1	1	yes	characteristic
302	#1925	C₂×C₆₁	1	1	yes	characteristic
303	#1920	C₂×C₆₁	1	1	yes	characteristic, complemented
304	#1918	C₂×C₆₁	1	1	yes	characteristic, complemented
305	#1917	C₂×C₆₁	1	1	yes	characteristic, complemented
306	#1862	C₂×C₆₁	1	1	yes	characteristic
307	#1762	C₂×C₆₁	1	1	yes	characteristic
308	C₂×C₆₁	—	8	2	no	—
309	C₂×C₆₁	#2465	1	1	yes	characteristic
310	C₂×C₆₁	#3112	1	1	yes	characteristic
311	C₂×C₆₁	#3118	1	1	yes	characteristic
312	C₂×C₆₁	#3316	1	1	yes	characteristic
313	C₂×C₆₁	#3325	1	1	yes	characteristic
314	C₂×C₆₁	#3384	1	1	yes	characteristic
315	C₂×C₆₁	#3385	1	1	yes	characteristic
316	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
317	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
318	C₂×C₂²⋊Q₈	—	2	1	no	—
319	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
320	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
321	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
322	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
323	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
324	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
325	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
326	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
327	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
328	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
329	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
330	#396	—	2	1	no	—
331	#393	—	2	1	no	—
332	#391	—	2	1	no	—
333	#391	—	2	1	no	—
334	#390	C₂²×C₆₁	1	1	yes	characteristic
335	#390	C₂²×C₆₁	1	1	yes	characteristic
336	#389	—	2	1	no	—
337	#389	C₂²×C₆₁	1	1	yes	characteristic
338	#389	C₂²×C₆₁	1	1	yes	characteristic
339	#389	C₂²×C₆₁	1	1	yes	characteristic
340	#389	C₂²×C₆₁	1	1	yes	characteristic
341	#385	—	2	1	no	—
342	C₄×C₄⋊C₄	—	2	1	no	—
343	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
344	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
345	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
346	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
347	#340	C₂²×C₆₁	1	1	yes	characteristic
348	#340	C₂²×C₆₁	1	1	yes	characteristic
349	#340	C₂²×C₆₁	1	1	yes	characteristic
350	#340	C₂²×C₆₁	1	1	yes	characteristic
351	#340	C₂²×C₆₁	1	1	yes	characteristic
352	#340	C₂²×C₆₁	1	1	yes	characteristic
353	#340	C₂²×C₆₁	1	1	yes	characteristic
354	#340	C₂²×C₆₁	1	1	yes	characteristic
355	#336	C₂²×C₆₁	1	1	yes	characteristic
356	#336	C₂²×C₆₁	1	1	yes	characteristic
357	#336	C₂²×C₆₁	1	1	yes	characteristic
358	#336	C₂²×C₆₁	1	1	yes	characteristic
359	C₆₁	#27019	1	1	yes	characteristic, direct product factor, Sylow
360	C₂²×Q₈	—	2	1	no	—
361	C₂²×Q₈	—	2	1	no	—
362	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
363	C₂³×C₄	—	2	1	no	—
364	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
365	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
368	C₄⋊Q₈	—	2	2	no	—
369	C₄⋊Q₈	—	2	2	no	—
370	C₄⋊Q₈	—	2	2	no	—
371	C₄⋊Q₈	—	4	1	no	—
372	C₂²⋊Q₈	—	4	4	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	—	4	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
382	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
385	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
390	C₂×C₂²⋊C₄	—	2	1	no	—
391	C₂×C₂²⋊C₄	—	2	1	no	—
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₄²	—	2	1	no	—
396	C₂×C₄²	—	2	1	no	—
397	C₂×C₄²	—	2	1	no	—
398	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
401	#107	—	2	2	no	—
402	#107	—	4	1	no	—
403	#106	—	2	2	no	—
404	#106	—	2	2	no	—
405	C₄⋊C₈	—	4	1	no	—
406	#102	—	2	2	no	—
407	#102	—	2	2	no	—
408	#102	—	4	1	no	—
409	#102	Q₈×C₆₁	1	4	yes	complemented
410	#97	—	4	1	no	—
411	#95	—	2	1	no	—
412	#95	—	2	1	no	—
413	#95	—	2	1	no	—
414	#95	—	2	1	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	#95	—	2	1	no	—
418	#95	D₄×C₆₁	1	1	yes	characteristic
419	#95	D₄×C₆₁	1	1	yes	characteristic
420	C₂⁴	—	2	1	no	—
421	C₂×Q₈	—	2	4	no	—
422	C₂×Q₈	—	4	2	no	—
423	C₂×Q₈	—	4	2	no	—
424	C₂×Q₈	—	8	1	no	—
425	C₂×Q₈	—	8	1	no	—
426	C₂×Q₈	—	8	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	C₂×Q₈×C₆₁	1	2	yes	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	4	2	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	8	1	no	—
443	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
444	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	4	2	no	—
476	C₄⋊C₄	—	4	2	no	—
477	C₄⋊C₄	—	4	2	no	—
478	C₄⋊C₄	—	8	1	no	—
479	C₄⋊C₄	—	8	1	no	—
480	C₄⋊C₄	C₂×Q₈×C₆₁	1	2	yes	—
481	C₄⋊C₄	#41×C₆₁	1	2	yes	—
482	C₄⋊C₄	#41×C₆₁	1	2	yes	—
483	C₄⋊C₄	#41×C₆₁	1	2	yes	—
484	C₂²⋊C₄	—	4	2	no	—
485	C₂²⋊C₄	—	4	2	no	—
486	C₂²⋊C₄	—	4	2	no	—
487	C₂²⋊C₄	—	4	2	no	—
488	C₂²⋊C₄	—	8	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	4	2	no	—
496	C₄²	—	8	1	no	—
497	C₄²	—	8	1	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	4	1	no	—
500	C₂³	—	4	1	no	—
501	C₂³	—	4	1	no	—
502	C₂³	—	4	1	no	—
503	C₂³	—	4	1	no	—
504	C₂³	—	4	1	no	—
505	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
506	Q₈	—	8	4	no	—
507	Q₈	—	8	4	no	—
508	D₄	—	2	2	no	—
509	D₄	—	2	2	no	—
510	D₄	—	4	1	no	—
511	D₄	—	4	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	8	4	no	—
550	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
551	C₂×C₄	#137×C₆₁	1	2	yes	—
552	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
553	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
554	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
555	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
556	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
557	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
558	C₈	—	2	2	no	—
559	C₈	—	4	1	no	—
560	C₈	—	4	1	no	—
561	C₈	—	4	1	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	8	1	no	—
566	C₂²	—	8	1	no	—
567	C₂²	—	8	1	no	—
568	C₂²	—	8	1	no	—
569	C₂²	C₆₁×#525	1	1	yes	characteristic
570	C₂²	C₆₁×#546	1	1	yes	characteristic
571	C₂²	D₄×Q₈×C₆₁	1	1	yes	characteristic
572	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
573	C₂²	C₂×#137×C₆₁	1	1	yes	characteristic
574	C₂²	C₆₁×#576	1	1	yes	characteristic
575	C₂²	C₆₁×#577	1	1	yes	characteristic
576	C₄	—	2	2	no	—
577	C₄	—	2	2	no	—
578	C₄	—	4	1	no	—
579	C₄	—	4	1	no	—
580	C₄	—	4	1	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	2	no	—
586	C₄	—	8	1	no	—
587	C₄	—	8	1	no	—
588	C₄	—	8	1	no	—
589	C₄	C₆₁×#533	1	1	yes	characteristic
590	C₄	C₆₁×#542	1	1	yes	characteristic
591	C₄	C₆₁×#554	1	1	yes	characteristic
592	C₄	C₆₁×#556	1	1	yes	characteristic
593	C₂	—	8	2	no	—
594	C₂	C₆₁×#2465	1	1	yes	characteristic
595	C₂	C₆₁×#3112	1	1	yes	characteristic
596	C₂	C₆₁×#3118	1	1	yes	characteristic
597	C₂	C₆₁×#3316	1	1	yes	characteristic
598	C₂	C₆₁×#3325	1	1	yes	characteristic
599	C₂	C₆₁×#3384	1	1	yes	characteristic
600	C₂	C₆₁×#3385	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂²×D₄×C₆₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄×C₄⋊C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2369	C₂	semidirect
4. C₆₁×#2058	C₂	semidirect
5. C₆₁×#2057	C₂	semidirect
7. C₆₁×#1987	C₂	semidirect
8. C₆₁×#1944	C₂	semidirect
11. C₆₁×#1920	C₂	semidirect
12. C₆₁×#1918	C₂	semidirect
13. C₆₁×#1917	C₂	semidirect
108. #102×C₆₁	Q₈	semidirect
261. #27019	C₆₁	direct (N × H)
294. #2369	C₂×C₆₁	semidirect
296. #2058	C₂×C₆₁	semidirect
297. #2057	C₂×C₆₁	semidirect
299. #1987	C₂×C₆₁	semidirect
300. #1944	C₂×C₆₁	semidirect
303. #1920	C₂×C₆₁	semidirect
304. #1918	C₂×C₆₁	semidirect
305. #1917	C₂×C₆₁	semidirect
359. C₆₁	#27019	direct (N × H)
409. #102	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×#2452	2
C₆₁×#2369	2
C₆₁×#2065	2
C₆₁×#2058	2
C₆₁×#2057	2
C₆₁×#2023	2
C₆₁×#1987	2
C₆₁×#1944	2
C₆₁×#1926	2
C₆₁×#1925	2
C₆₁×#1920	2
C₆₁×#1918	2
C₆₁×#1917	2
C₆₁×#1862	2
C₆₁×#1762	2
#27019	61

Subgroups — C61×#27019 #29530452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#27019 #29530452