Subgroups — C₆₁×#25989 #29529422

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2439	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2347	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2088	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2087	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2057	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1949	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1921	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1918	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1854	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1850	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#1825	C₂	1	1	yes	maximal, characteristic
14	C₆₁×#1808	C₂	1	1	yes	maximal, characteristic
15	C₆₁×#1735	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×#117×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×#107×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×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#396	—	2	1	no	—
28	C₆₁×#393	—	2	1	no	—
29	C₆₁×#391	—	2	1	no	—
30	C₆₁×#389	C₂²	1	1	yes	characteristic
31	C₆₁×#389	C₂²	1	1	yes	characteristic
32	C₆₁×#389	C₂²	1	1	yes	characteristic
33	C₆₁×#389	C₂²	1	1	yes	characteristic
34	C₆₁×#388	—	2	1	no	—
35	C₆₁×#388	—	2	1	no	—
36	C₆₁×#387	—	2	1	no	—
37	C₆₁×#383	—	2	1	no	—
38	C₆₁×#382	C₂²	1	1	yes	characteristic
39	C₆₁×#380	C₂²	1	1	yes	characteristic
40	C₆₁×#380	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
42	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
45	C₆₁×#340	C₂²	1	1	yes	characteristic
46	C₆₁×#340	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×#340	C₂²	1	1	yes	characteristic
56	C₆₁×#340	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	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₆₁	—	2	2	no	—
62	C₂²×C₈×C₆₁	—	2	2	no	—
63	C₂²×C₈×C₆₁	—	4	1	no	—
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	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₆₁	—	2	2	no	—
69	C₄⋊Q₈×C₆₁	—	4	1	no	—
70	#117×C₆₁	—	2	2	no	—
71	#117×C₆₁	—	2	2	no	—
72	#117×C₆₁	—	4	1	no	—
73	#117×C₆₁	Q₈	1	4	yes	complemented
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	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₁	C₂³	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×C₂²⋊C₄×C₆₁	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
93	C₂×C₄²×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	#107×C₆₁	—	2	2	no	—
100	#107×C₆₁	—	2	2	no	—
101	#107×C₆₁	—	4	1	no	—
102	#106×C₆₁	—	4	1	no	—
103	C₄⋊C₈×C₆₁	—	2	2	no	—
104	C₄⋊C₈×C₆₁	—	4	1	no	—
105	C₂²⋊C₈×C₆₁	—	2	2	no	—
106	#95×C₆₁	—	2	1	no	—
107	#95×C₆₁	—	2	1	no	—
108	#95×C₆₁	—	2	1	no	—
109	#95×C₆₁	—	2	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	D₄	1	1	yes	characteristic
117	#95×C₆₁	D₄	1	1	yes	characteristic
118	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Q₈×C₆₁	—	4	2	no	—
120	C₂×Q₈×C₆₁	—	4	2	no	—
121	C₂×Q₈×C₆₁	—	8	1	no	—
122	C₂²×C₄×C₆₁	—	2	1	no	—
123	C₂²×C₄×C₆₁	—	2	1	no	—
124	C₂²×C₄×C₆₁	—	2	1	no	—
125	C₂²×C₄×C₆₁	—	2	1	no	—
126	C₂²×C₄×C₆₁	—	2	1	no	—
127	C₂²×C₄×C₆₁	—	2	1	no	—
128	C₂²×C₄×C₆₁	—	2	1	no	—
129	C₂²×C₄×C₆₁	—	2	1	no	—
130	C₂²×C₄×C₆₁	—	2	1	no	—
131	C₂²×C₄×C₆₁	—	2	1	no	—
132	C₂²×C₄×C₆₁	—	2	1	no	—
133	C₂²×C₄×C₆₁	—	2	1	no	—
134	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
140	C₂²×C₄×C₆₁	#41	1	2	yes	—
141	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
142	C₂×C₈×C₆₁	—	2	1	no	—
143	C₂×C₈×C₆₁	—	2	1	no	—
144	C₂×C₈×C₆₁	—	2	1	no	—
145	C₂×C₈×C₆₁	—	2	1	no	—
146	C₂×C₈×C₆₁	—	2	1	no	—
147	C₂×C₈×C₆₁	—	2	1	no	—
148	C₂×C₈×C₆₁	—	2	1	no	—
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	—	4	4	no	—
154	C₂×C₈×C₆₁	#41	1	2	yes	—
155	C₄⋊C₄×C₆₁	—	2	1	no	—
156	C₄⋊C₄×C₆₁	—	2	1	no	—
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	—	2	1	no	—
162	C₄⋊C₄×C₆₁	—	2	1	no	—
163	C₄⋊C₄×C₆₁	—	2	1	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	—	4	2	no	—
169	C₄⋊C₄×C₆₁	—	4	2	no	—
170	C₄⋊C₄×C₆₁	—	4	2	no	—
171	C₄⋊C₄×C₆₁	—	4	2	no	—
172	C₄⋊C₄×C₆₁	—	8	1	no	—
173	C₄⋊C₄×C₆₁	C₂×Q₈	1	2	yes	—
174	C₄⋊C₄×C₆₁	C₂×Q₈	1	2	yes	—
175	C₄⋊C₄×C₆₁	#41	1	2	yes	—
176	C₂²⋊C₄×C₆₁	—	2	4	no	—
177	C₂²⋊C₄×C₆₁	—	2	2	no	—
178	C₂²⋊C₄×C₆₁	—	4	1	no	—
179	C₂²⋊C₄×C₆₁	—	4	2	no	—
180	C₂²⋊C₄×C₆₁	—	8	1	no	—
181	C₂²⋊C₄×C₆₁	—	8	1	no	—
182	C₄²×C₆₁	—	2	4	no	—
183	C₄²×C₆₁	—	2	1	no	—
184	C₄²×C₆₁	—	2	1	no	—
185	C₄²×C₆₁	—	2	1	no	—
186	C₄²×C₆₁	—	2	1	no	—
187	C₄²×C₆₁	—	4	2	no	—
188	C₄²×C₆₁	—	8	1	no	—
189	C₄²×C₆₁	—	8	1	no	—
190	C₂³×C₆₁	—	2	1	no	—
191	C₂³×C₆₁	—	2	1	no	—
192	C₂³×C₆₁	—	2	1	no	—
193	C₂³×C₆₁	—	2	1	no	—
194	C₂³×C₆₁	—	2	1	no	—
195	C₂³×C₆₁	—	2	1	no	—
196	C₂³×C₆₁	C₂²⋊Q₈	1	2	yes	—
197	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
198	Q₈×C₆₁	—	8	4	no	—
199	C₂×C₄×C₆₁	—	2	1	no	—
200	C₂×C₄×C₆₁	—	2	1	no	—
201	C₂×C₄×C₆₁	—	2	1	no	—
202	C₂×C₄×C₆₁	—	2	1	no	—
203	C₂×C₄×C₆₁	—	2	2	no	—
204	C₂×C₄×C₆₁	—	2	2	no	—
205	C₂×C₄×C₆₁	—	2	2	no	—
206	C₂×C₄×C₆₁	—	2	2	no	—
207	C₂×C₄×C₆₁	—	2	2	no	—
208	C₂×C₄×C₆₁	—	2	2	no	—
209	C₂×C₄×C₆₁	—	2	1	no	—
210	C₂×C₄×C₆₁	—	2	2	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	1	no	—
217	C₂×C₄×C₆₁	—	2	1	no	—
218	C₂×C₄×C₆₁	—	2	1	no	—
219	C₂×C₄×C₆₁	—	4	1	no	—
220	C₂×C₄×C₆₁	—	4	1	no	—
221	C₂×C₄×C₆₁	—	4	1	no	—
222	C₂×C₄×C₆₁	—	4	1	no	—
223	C₂×C₄×C₆₁	—	4	1	no	—
224	C₂×C₄×C₆₁	—	4	1	no	—
225	C₂×C₄×C₆₁	—	4	1	no	—
226	C₂×C₄×C₆₁	—	4	1	no	—
227	C₂×C₄×C₆₁	—	4	1	no	—
228	C₂×C₄×C₆₁	—	4	1	no	—
229	C₂×C₄×C₆₁	—	4	1	no	—
230	C₂×C₄×C₆₁	—	4	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	C₂²⋊Q₈	1	2	yes	—
241	C₂×C₄×C₆₁	#135	1	2	yes	—
242	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
243	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
244	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
245	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
246	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
247	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
248	C₈×C₆₁	—	2	2	no	—
249	C₈×C₆₁	—	4	1	no	—
250	C₈×C₆₁	—	4	1	no	—
251	C₈×C₆₁	—	4	1	no	—
252	#25989	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂²×C₆₁	—	2	2	no	—
254	C₂²×C₆₁	—	2	2	no	—
255	C₂²×C₆₁	—	2	2	no	—
256	C₂²×C₆₁	—	4	1	no	—
257	C₂²×C₆₁	—	4	1	no	—
258	C₂²×C₆₁	—	4	1	no	—
259	C₂²×C₆₁	—	4	1	no	—
260	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
261	C₂²×C₆₁	#546	1	1	yes	characteristic
262	C₂²×C₆₁	#547	1	1	yes	characteristic
263	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
264	C₂²×C₆₁	#575	1	1	yes	characteristic
265	C₂²×C₆₁	#576	1	1	yes	characteristic
266	C₂²×C₆₁	#578	1	1	yes	characteristic
267	C₄×C₆₁	—	2	2	no	—
268	C₄×C₆₁	—	2	2	no	—
269	C₄×C₆₁	—	4	2	no	—
270	C₄×C₆₁	—	4	2	no	—
271	C₄×C₆₁	—	4	1	no	—
272	C₄×C₆₁	—	4	1	no	—
273	C₄×C₆₁	—	4	2	no	—
274	C₄×C₆₁	—	4	1	no	—
275	C₄×C₆₁	—	8	1	no	—
276	C₄×C₆₁	—	8	1	no	—
277	C₄×C₆₁	—	8	1	no	—
278	C₄×C₆₁	—	8	1	no	—
279	C₄×C₆₁	—	8	1	no	—
280	C₄×C₆₁	#533	1	1	yes	characteristic
281	C₄×C₆₁	#544	1	1	yes	characteristic
282	C₄×C₆₁	#551	1	1	yes	characteristic
283	C₄×C₆₁	#552	1	1	yes	characteristic
284	#2439	C₂×C₆₁	1	1	yes	characteristic
285	#2347	C₂×C₆₁	1	1	yes	characteristic
286	#2101	C₂×C₆₁	1	1	yes	characteristic, complemented
287	#2088	C₂×C₆₁	1	1	yes	characteristic, complemented
288	#2087	C₂×C₆₁	1	1	yes	characteristic, complemented
289	#2057	C₂×C₆₁	1	1	yes	characteristic, complemented
290	#1988	C₂×C₆₁	1	1	yes	characteristic, complemented
291	#1949	C₂×C₆₁	1	1	yes	characteristic, complemented
292	#1921	C₂×C₆₁	1	1	yes	characteristic, complemented
293	#1918	C₂×C₆₁	1	1	yes	characteristic, complemented
294	#1854	C₂×C₆₁	1	1	yes	characteristic
295	#1850	C₂×C₆₁	1	1	yes	characteristic
296	#1825	C₂×C₆₁	1	1	yes	characteristic
297	#1808	C₂×C₆₁	1	1	yes	characteristic
298	#1735	C₂×C₆₁	1	1	yes	characteristic
299	C₂×C₆₁	—	4	2	no	—
300	C₂×C₆₁	#2438	1	1	yes	characteristic
301	C₂×C₆₁	#3076	1	1	yes	characteristic
302	C₂×C₆₁	#3083	1	1	yes	characteristic
303	C₂×C₆₁	#3297	1	1	yes	characteristic
304	C₂×C₆₁	#3313	1	1	yes	characteristic
305	C₂×C₆₁	#3342	1	1	yes	characteristic
306	C₂×C₆₁	#3347	1	1	yes	characteristic
307	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
308	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
309	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
310	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
311	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
312	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
313	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
314	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
315	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
316	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
317	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
318	#396	—	2	1	no	—
319	#393	—	2	1	no	—
320	#391	—	2	1	no	—
321	#389	C₂²×C₆₁	1	1	yes	characteristic
322	#389	C₂²×C₆₁	1	1	yes	characteristic
323	#389	C₂²×C₆₁	1	1	yes	characteristic
324	#389	C₂²×C₆₁	1	1	yes	characteristic
325	#388	—	2	1	no	—
326	#388	—	2	1	no	—
327	#387	—	2	1	no	—
328	#383	—	2	1	no	—
329	#382	C₂²×C₆₁	1	1	yes	characteristic
330	#380	C₂²×C₆₁	1	1	yes	characteristic
331	#380	C₂²×C₆₁	1	1	yes	characteristic
332	C₄×C₄⋊C₄	—	2	1	no	—
333	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
334	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
335	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
336	#340	C₂²×C₆₁	1	1	yes	characteristic
337	#340	C₂²×C₆₁	1	1	yes	characteristic
338	#340	C₂²×C₆₁	1	1	yes	characteristic
339	#340	C₂²×C₆₁	1	1	yes	characteristic
340	#340	C₂²×C₆₁	1	1	yes	characteristic
341	#340	C₂²×C₆₁	1	1	yes	characteristic
342	#340	C₂²×C₆₁	1	1	yes	characteristic
343	#340	C₂²×C₆₁	1	1	yes	characteristic
344	#340	C₂²×C₆₁	1	1	yes	characteristic
345	#340	C₂²×C₆₁	1	1	yes	characteristic
346	#340	C₂²×C₆₁	1	1	yes	characteristic
347	#340	C₂²×C₆₁	1	1	yes	characteristic
348	#336	C₂²×C₆₁	1	1	yes	characteristic
349	#336	C₂²×C₆₁	1	1	yes	characteristic
350	C₆₁	#25989	1	1	yes	characteristic, direct product factor, Sylow
351	C₂²×Q₈	—	2	1	no	—
352	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
353	C₂²×C₈	—	2	2	no	—
354	C₂²×C₈	—	2	2	no	—
355	C₂²×C₈	—	4	1	no	—
356	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
357	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
358	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
359	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
360	C₄⋊Q₈	—	2	2	no	—
361	C₄⋊Q₈	—	4	1	no	—
362	#117	—	2	2	no	—
363	#117	—	2	2	no	—
364	#117	—	4	1	no	—
365	#117	Q₈×C₆₁	1	4	yes	complemented
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
372	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	—	2	1	no	—
380	C₂×C₂²⋊C₄	—	2	1	no	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
384	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	—	2	1	no	—
388	C₂×C₄²	—	2	1	no	—
389	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
391	#107	—	2	2	no	—
392	#107	—	2	2	no	—
393	#107	—	4	1	no	—
394	#106	—	4	1	no	—
395	C₄⋊C₈	—	2	2	no	—
396	C₄⋊C₈	—	4	1	no	—
397	C₂²⋊C₈	—	2	2	no	—
398	#95	—	2	1	no	—
399	#95	—	2	1	no	—
400	#95	—	2	1	no	—
401	#95	—	2	1	no	—
402	#95	—	2	1	no	—
403	#95	—	2	1	no	—
404	#95	—	2	1	no	—
405	#95	—	2	1	no	—
406	#95	—	2	1	no	—
407	#95	—	2	1	no	—
408	#95	D₄×C₆₁	1	1	yes	characteristic
409	#95	D₄×C₆₁	1	1	yes	characteristic
410	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
411	C₂×Q₈	—	4	2	no	—
412	C₂×Q₈	—	4	2	no	—
413	C₂×Q₈	—	8	1	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
427	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
428	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
429	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
431	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
432	C₂²×C₄	#41×C₆₁	1	2	yes	—
433	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	4	4	no	—
446	C₂×C₈	#41×C₆₁	1	2	yes	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	4	2	no	—
461	C₄⋊C₄	—	4	2	no	—
462	C₄⋊C₄	—	4	2	no	—
463	C₄⋊C₄	—	4	2	no	—
464	C₄⋊C₄	—	8	1	no	—
465	C₄⋊C₄	C₂×Q₈×C₆₁	1	2	yes	—
466	C₄⋊C₄	C₂×Q₈×C₆₁	1	2	yes	—
467	C₄⋊C₄	#41×C₆₁	1	2	yes	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	2	no	—
470	C₂²⋊C₄	—	4	1	no	—
471	C₂²⋊C₄	—	4	2	no	—
472	C₂²⋊C₄	—	8	1	no	—
473	C₂²⋊C₄	—	8	1	no	—
474	C₄²	—	2	4	no	—
475	C₄²	—	2	1	no	—
476	C₄²	—	2	1	no	—
477	C₄²	—	2	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	4	2	no	—
480	C₄²	—	8	1	no	—
481	C₄²	—	8	1	no	—
482	C₂³	—	2	1	no	—
483	C₂³	—	2	1	no	—
484	C₂³	—	2	1	no	—
485	C₂³	—	2	1	no	—
486	C₂³	—	2	1	no	—
487	C₂³	—	2	1	no	—
488	C₂³	C₂²⋊Q₈×C₆₁	1	2	yes	—
489	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
490	Q₈	—	8	4	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
533	C₂×C₄	#135×C₆₁	1	2	yes	—
534	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
535	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
536	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
537	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
538	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
539	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
540	C₈	—	2	2	no	—
541	C₈	—	4	1	no	—
542	C₈	—	4	1	no	—
543	C₈	—	4	1	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	4	1	no	—
548	C₂²	—	4	1	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	1	no	—
551	C₂²	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
552	C₂²	C₆₁×#546	1	1	yes	characteristic
553	C₂²	C₆₁×#547	1	1	yes	characteristic
554	C₂²	C₂×#135×C₆₁	1	1	yes	characteristic
555	C₂²	C₆₁×#575	1	1	yes	characteristic
556	C₂²	C₆₁×#576	1	1	yes	characteristic
557	C₂²	C₆₁×#578	1	1	yes	characteristic
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	4	2	no	—
561	C₄	—	4	1	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	8	1	no	—
567	C₄	—	8	1	no	—
568	C₄	—	8	1	no	—
569	C₄	—	8	1	no	—
570	C₄	—	8	1	no	—
571	C₄	C₆₁×#533	1	1	yes	characteristic
572	C₄	C₆₁×#544	1	1	yes	characteristic
573	C₄	C₆₁×#551	1	1	yes	characteristic
574	C₄	C₆₁×#552	1	1	yes	characteristic
575	C₂	—	4	2	no	—
576	C₂	C₆₁×#2438	1	1	yes	characteristic
577	C₂	C₆₁×#3076	1	1	yes	characteristic
578	C₂	C₆₁×#3083	1	1	yes	characteristic
579	C₂	C₆₁×#3297	1	1	yes	characteristic
580	C₂	C₆₁×#3313	1	1	yes	characteristic
581	C₂	C₆₁×#3342	1	1	yes	characteristic
582	C₂	C₆₁×#3347	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₆₁×#2101	C₂	semidirect
4. C₆₁×#2088	C₂	semidirect
5. C₆₁×#2087	C₂	semidirect
6. C₆₁×#2057	C₂	semidirect
7. C₆₁×#1988	C₂	semidirect
8. C₆₁×#1949	C₂	semidirect
9. C₆₁×#1921	C₂	semidirect
10. C₆₁×#1918	C₂	semidirect
73. #117×C₆₁	Q₈	semidirect
252. #25989	C₆₁	direct (N × H)
286. #2101	C₂×C₆₁	semidirect
287. #2088	C₂×C₆₁	semidirect
288. #2087	C₂×C₆₁	semidirect
289. #2057	C₂×C₆₁	semidirect
290. #1988	C₂×C₆₁	semidirect
291. #1949	C₂×C₆₁	semidirect
292. #1921	C₂×C₆₁	semidirect
293. #1918	C₂×C₆₁	semidirect
350. C₆₁	#25989	direct (N × H)
365. #117	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×#2439	2
C₆₁×#2347	2
C₆₁×#2101	2
C₆₁×#2088	2
C₆₁×#2087	2
C₆₁×#2057	2
C₆₁×#1988	2
C₆₁×#1949	2
C₆₁×#1921	2
C₆₁×#1918	2
C₆₁×#1854	2
C₆₁×#1850	2
C₆₁×#1825	2
C₆₁×#1808	2
C₆₁×#1735	2
#25989	61

Subgroups — C61×#25989 #29529422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25989 #29529422