Subgroups — C₆₁×#61408 #29564841

Order 15616.

1858 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3507	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3402	C₂	1	4	yes	maximal, complemented
3	C₆₁×#3400	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3396	C₂	1	2	yes	maximal, complemented
5	C₆₁×#3395	C₂	1	2	yes	maximal, complemented
6	C₆₁×#3393	C₂	1	4	yes	maximal, complemented
7	C₆₁×#3156	C₂	1	2	yes	maximal, complemented
8	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#3046	C₂	1	4	yes	maximal, complemented
10	C₆₁×#3037	C₂	1	2	yes	maximal, complemented
11	C₆₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#2951	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#137×C₆₁	C₂²	1	4	yes	—
15	C₂×#135×C₆₁	C₂²	1	2	yes	—
16	C₂×#135×C₆₁	C₂²	1	2	yes	—
17	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#559	C₂²	1	4	yes	—
19	C₆₁×#557	C₂²	1	4	yes	—
20	C₆₁×#556	—	2	2	no	—
21	C₆₁×#556	C₂²	1	8	yes	—
22	C₆₁×#554	—	2	2	no	—
23	C₆₁×#552	—	2	4	no	—
24	C₆₁×#539	—	2	4	no	—
25	C₆₁×#536	C₂²	1	4	yes	—
26	C₆₁×#533	C₂²	1	1	yes	characteristic
27	C₆₁×#532	C₂²	1	2	yes	—
28	C₆₁×#529	—	2	4	no	—
29	C₆₁×#520	C₂²	1	4	yes	—
30	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
31	C₆₁×#497	C₂²	1	8	yes	—
32	C₆₁×#495	C₂²	1	2	yes	—
33	C₆₁×#495	C₂²	1	4	yes	—
34	C₆₁×#495	C₂²	1	2	yes	—
35	C₆₁×#494	C₂²	1	4	yes	—
36	C₆₁×#494	C₂²	1	1	yes	characteristic
37	C₆₁×#493	C₂²	1	1	yes	characteristic
38	C₆₁×#492	C₂²	1	2	yes	—
39	C₆₁×#464	C₂²	1	8	yes	—
40	C₆₁×#464	C₂²	1	8	yes	—
41	C₆₁×#463	C₂²	1	8	yes	—
42	C₆₁×#462	C₂²	1	4	yes	—
43	C₆₁×#462	C₂²	1	8	yes	—
44	C₆₁×#461	C₂²	1	8	yes	—
45	C₆₁×#460	C₂²	1	4	yes	—
46	C₆₁×#446	C₂²	1	2	yes	—
47	C₆₁×#446	C₂²	1	2	yes	—
48	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
49	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
50	C₆₁×#440	C₂²	1	8	yes	—
51	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
52	C₄×Q₁₆×C₆₁	C₂²	1	4	yes	—
53	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
54	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
55	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
56	C₄×D₈×C₆₁	C₂²	1	4	yes	—
57	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
58	C₆₁×#424	C₂²	1	2	yes	—
59	C₆₁×#423	C₂²	1	1	yes	characteristic
60	C₆₁×#417	C₂²	1	4	yes	—
61	C₂×#103×C₆₁	C₂²	1	2	yes	—
62	C₆₁×#405	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
64	C₂×#41×C₆₁	C₂³	1	4	yes	—
65	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
66	#137×C₆₁	—	4	8	no	—
67	#135×C₆₁	—	2	8	no	—
68	#135×C₆₁	—	4	4	no	—
69	C₂×Q₁₆×C₆₁	—	2	4	no	—
70	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
73	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
74	C₂×Q₁₆×C₆₁	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₆₁	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₆₁	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₆₁	C₂³	1	8	yes	—
78	C₂×D₈×C₆₁	C₂³	1	2	yes	—
79	C₂×D₈×C₆₁	C₂³	1	2	yes	—
80	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
81	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
84	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
85	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
86	C₄⋊Q₈×C₆₁	C₂³	1	8	yes	—
87	#127×C₆₁	C₂³	1	2	yes	—
88	#126×C₆₁	—	2	8	no	—
89	#126×C₆₁	—	2	8	no	—
90	#125×C₆₁	—	2	8	no	—
91	#125×C₆₁	—	2	4	no	—
92	#125×C₆₁	D₄	1	2	yes	—
93	#124×C₆₁	—	2	4	no	—
94	#124×C₆₁	C₂³	1	4	yes	—
95	#124×C₆₁	C₂³	1	8	yes	—
96	#123×C₆₁	—	2	8	no	—
97	#123×C₆₁	—	2	8	no	—
98	C₂²⋊Q₈×C₆₁	—	2	8	no	—
99	C₂²⋊Q₈×C₆₁	—	2	4	no	—
100	C₂²⋊Q₈×C₆₁	—	2	4	no	—
101	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
102	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
103	#121×C₆₁	—	2	2	no	—
104	#121×C₆₁	—	2	2	no	—
105	C₄×Q₈×C₆₁	—	2	4	no	—
106	C₄×Q₈×C₆₁	—	2	2	no	—
107	C₄×Q₈×C₆₁	—	2	2	no	—
108	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
109	C₄×Q₈×C₆₁	C₂³	1	8	yes	—
110	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
111	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
112	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
113	C₄×D₄×C₆₁	—	2	4	no	—
114	C₄×D₄×C₆₁	—	2	8	no	—
115	C₄×D₄×C₆₁	—	2	2	no	—
116	C₄×D₄×C₆₁	—	2	2	no	—
117	C₄×D₄×C₆₁	C₂³	1	8	yes	—
118	#117×C₆₁	—	2	4	no	—
119	#117×C₆₁	—	2	8	no	—
120	#117×C₆₁	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
122	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
123	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
124	C₂×C₄²×C₆₁	—	2	2	no	—
125	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
126	#107×C₆₁	C₂³	1	2	yes	—
127	#107×C₆₁	C₂³	1	2	yes	—
128	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
129	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
130	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
131	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
132	#103×C₆₁	C₂³	1	4	yes	—
133	#103×C₆₁	C₂³	1	4	yes	—
134	#103×C₆₁	C₂³	1	8	yes	—
135	#102×C₆₁	C₂³	1	8	yes	—
136	#97×C₆₁	C₂³	1	4	yes	—
137	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₆₁	C₂³	1	4	yes	—
142	#41×C₆₁	—	2	8	no	—
143	#41×C₆₁	—	2	8	no	—
144	C₂×Q₈×C₆₁	—	2	2	no	—
145	C₂×Q₈×C₆₁	—	2	4	no	—
146	C₂×Q₈×C₆₁	—	2	4	no	—
147	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
148	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
149	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
150	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
151	C₂×D₄×C₆₁	—	2	4	no	—
152	C₂×D₄×C₆₁	—	2	2	no	—
153	C₂×D₄×C₆₁	—	2	2	no	—
154	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₆₁	—	2	2	no	—
156	C₂²×C₄×C₆₁	—	2	2	no	—
157	C₂²×C₄×C₆₁	—	2	4	no	—
158	C₂²×C₄×C₆₁	—	2	8	no	—
159	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
162	Q₁₆×C₆₁	—	2	4	no	—
163	Q₁₆×C₆₁	—	2	4	no	—
164	Q₁₆×C₆₁	—	2	4	no	—
165	Q₁₆×C₆₁	—	4	2	no	—
166	Q₁₆×C₆₁	—	4	8	no	—
167	SD₁₆×C₆₁	—	2	8	no	—
168	SD₁₆×C₆₁	—	4	4	no	—
169	SD₁₆×C₆₁	—	4	8	no	—
170	D₈×C₆₁	—	2	4	no	—
171	D₈×C₆₁	—	4	2	no	—
172	M₁₆×C₆₁	—	2	4	no	—
173	C₂×C₈×C₆₁	—	2	2	no	—
174	C₂×C₈×C₆₁	#41	1	4	yes	—
175	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
176	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
178	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
179	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
180	C₄⋊C₄×C₆₁	—	2	2	no	—
181	C₄⋊C₄×C₆₁	—	2	8	no	—
182	C₄⋊C₄×C₆₁	—	2	4	no	—
183	C₄⋊C₄×C₆₁	—	2	4	no	—
184	C₄⋊C₄×C₆₁	—	2	8	no	—
185	C₄⋊C₄×C₆₁	—	2	2	no	—
186	C₄⋊C₄×C₆₁	—	2	2	no	—
187	C₄⋊C₄×C₆₁	—	2	8	no	—
188	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
189	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
190	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
191	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
192	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
195	C₂²⋊C₄×C₆₁	—	2	4	no	—
196	C₂²⋊C₄×C₆₁	—	2	8	no	—
197	C₂²⋊C₄×C₆₁	—	2	8	no	—
198	C₂²⋊C₄×C₆₁	—	2	4	no	—
199	C₂²⋊C₄×C₆₁	—	2	8	no	—
200	C₂²⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
201	C₄²×C₆₁	—	2	4	no	—
202	C₄²×C₆₁	—	2	4	no	—
203	C₄²×C₆₁	—	2	8	no	—
204	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
207	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
208	C₄²×C₆₁	C₂⁴	1	4	yes	—
209	C₂³×C₆₁	—	2	4	no	—
210	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
211	Q₈×C₆₁	—	2	4	no	—
212	Q₈×C₆₁	—	2	4	no	—
213	Q₈×C₆₁	—	2	4	no	—
214	Q₈×C₆₁	—	2	4	no	—
215	Q₈×C₆₁	—	2	4	no	—
216	Q₈×C₆₁	—	4	2	no	—
217	Q₈×C₆₁	—	4	4	no	—
218	D₄×C₆₁	—	2	4	no	—
219	D₄×C₆₁	—	2	4	no	—
220	D₄×C₆₁	—	4	8	no	—
221	D₄×C₆₁	—	4	4	no	—
222	D₄×C₆₁	—	8	2	no	—
223	C₂×C₄×C₆₁	—	2	2	no	—
224	C₂×C₄×C₆₁	—	2	4	no	—
225	C₂×C₄×C₆₁	—	2	4	no	—
226	C₂×C₄×C₆₁	—	2	4	no	—
227	C₂×C₄×C₆₁	—	4	1	no	—
228	C₂×C₄×C₆₁	—	4	4	no	—
229	C₂×C₄×C₆₁	—	4	8	no	—
230	C₂×C₄×C₆₁	—	8	8	no	—
231	C₂×C₄×C₆₁	—	8	2	no	—
232	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
234	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
237	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
240	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
241	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
242	C₈×C₆₁	—	2	1	no	—
243	C₈×C₆₁	—	2	1	no	—
244	C₈×C₆₁	—	2	4	no	—
245	C₈×C₆₁	C₂×#41	1	2	yes	—
246	C₈×C₆₁	C₂×#41	1	2	yes	—
247	#61408	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂²×C₆₁	—	2	1	no	—
249	C₂²×C₆₁	—	2	1	no	—
250	C₂²×C₆₁	—	4	4	no	—
251	C₂²×C₆₁	—	4	4	no	—
252	C₂²×C₆₁	—	4	4	no	—
253	C₂²×C₆₁	#550	1	2	yes	—
254	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
255	C₄×C₆₁	—	2	1	no	—
256	C₄×C₆₁	—	2	1	no	—
257	C₄×C₆₁	—	2	4	no	—
258	C₄×C₆₁	—	2	1	no	—
259	C₄×C₆₁	—	2	4	no	—
260	C₄×C₆₁	—	4	4	no	—
261	C₄×C₆₁	—	4	4	no	—
262	C₄×C₆₁	—	4	4	no	—
263	C₄×C₆₁	#550	1	2	yes	—
264	C₄×C₆₁	#578	1	2	yes	—
265	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
266	C₄×C₆₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
267	#3507	C₂×C₆₁	1	2	yes	complemented
268	#3402	C₂×C₆₁	1	4	yes	complemented
269	#3400	C₂×C₆₁	1	4	yes	complemented
270	#3396	C₂×C₆₁	1	2	yes	complemented
271	#3395	C₂×C₆₁	1	2	yes	complemented
272	#3393	C₂×C₆₁	1	4	yes	complemented
273	#3156	C₂×C₆₁	1	2	yes	complemented
274	#3147	C₂×C₆₁	1	1	yes	characteristic
275	#3046	C₂×C₆₁	1	4	yes	complemented
276	#3037	C₂×C₆₁	1	2	yes	complemented
277	#2960	C₂×C₆₁	1	1	yes	characteristic, complemented
278	#2951	C₂×C₆₁	1	2	yes	complemented
279	C₂×C₄×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
280	C₂×C₆₁	—	2	2	no	—
281	C₂×C₆₁	—	8	4	no	—
282	C₂×C₆₁	C₂×#550	1	1	yes	characteristic
283	C₂×C₆₁	C₂×#578	1	1	yes	characteristic
284	C₂×C₆₁	#3584	1	1	yes	characteristic
285	C₂×#137	C₂²×C₆₁	1	4	yes	—
286	C₂×#135	C₂²×C₆₁	1	2	yes	—
287	C₂×#135	C₂²×C₆₁	1	2	yes	—
288	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
289	#559	C₂²×C₆₁	1	4	yes	—
290	#557	C₂²×C₆₁	1	4	yes	—
291	#556	—	2	2	no	—
292	#556	C₂²×C₆₁	1	8	yes	—
293	#554	—	2	2	no	—
294	#552	—	2	4	no	—
295	#539	—	2	4	no	—
296	#536	C₂²×C₆₁	1	4	yes	—
297	#533	C₂²×C₆₁	1	1	yes	characteristic
298	#532	C₂²×C₆₁	1	2	yes	—
299	#529	—	2	4	no	—
300	#520	C₂²×C₆₁	1	4	yes	—
301	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
302	#497	C₂²×C₆₁	1	8	yes	—
303	#495	C₂²×C₆₁	1	2	yes	—
304	#495	C₂²×C₆₁	1	4	yes	—
305	#495	C₂²×C₆₁	1	2	yes	—
306	#494	C₂²×C₆₁	1	4	yes	—
307	#494	C₂²×C₆₁	1	1	yes	characteristic
308	#493	C₂²×C₆₁	1	1	yes	characteristic
309	#492	C₂²×C₆₁	1	2	yes	—
310	#464	C₂²×C₆₁	1	8	yes	—
311	#464	C₂²×C₆₁	1	8	yes	—
312	#463	C₂²×C₆₁	1	8	yes	—
313	#462	C₂²×C₆₁	1	4	yes	—
314	#462	C₂²×C₆₁	1	8	yes	—
315	#461	C₂²×C₆₁	1	8	yes	—
316	#460	C₂²×C₆₁	1	4	yes	—
317	#446	C₂²×C₆₁	1	2	yes	—
318	#446	C₂²×C₆₁	1	2	yes	—
319	C₈×Q₈	C₂²×C₆₁	1	2	yes	—
320	C₈×Q₈	C₂²×C₆₁	1	2	yes	—
321	#440	C₂²×C₆₁	1	8	yes	—
322	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
323	C₄×Q₁₆	C₂²×C₆₁	1	4	yes	—
324	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
325	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
326	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
327	C₄×D₈	C₂²×C₆₁	1	4	yes	—
328	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
329	#424	C₂²×C₆₁	1	2	yes	—
330	#423	C₂²×C₆₁	1	1	yes	characteristic
331	#417	C₂²×C₆₁	1	4	yes	—
332	C₂×#103	C₂²×C₆₁	1	2	yes	—
333	#405	C₂²×C₆₁	1	2	yes	—
334	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
335	C₆₁	#61408	1	1	yes	characteristic, direct product factor, Sylow
336	C₂×#41	C₂³×C₆₁	1	4	yes	—
337	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
338	#137	—	4	8	no	—
339	#135	—	2	8	no	—
340	#135	—	4	4	no	—
341	C₂×Q₁₆	—	2	4	no	—
342	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
343	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
344	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
345	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
346	C₂×Q₁₆	C₂³×C₆₁	1	8	yes	—
347	C₂×SD₁₆	C₂³×C₆₁	1	4	yes	—
348	C₂×SD₁₆	C₂³×C₆₁	1	4	yes	—
349	C₂×SD₁₆	C₂³×C₆₁	1	8	yes	—
350	C₂×D₈	C₂³×C₆₁	1	2	yes	—
351	C₂×D₈	C₂³×C₆₁	1	2	yes	—
352	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
353	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
354	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
355	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
356	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
357	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
358	C₄⋊Q₈	C₂³×C₆₁	1	8	yes	—
359	#127	C₂³×C₆₁	1	2	yes	—
360	#126	—	2	8	no	—
361	#126	—	2	8	no	—
362	#125	—	2	4	no	—
363	#125	—	2	8	no	—
364	#125	D₄×C₆₁	1	2	yes	—
365	#124	—	2	4	no	—
366	#124	C₂³×C₆₁	1	4	yes	—
367	#124	C₂³×C₆₁	1	8	yes	—
368	#123	—	2	8	no	—
369	#123	—	2	8	no	—
370	C₂²⋊Q₈	—	2	8	no	—
371	C₂²⋊Q₈	—	2	4	no	—
372	C₂²⋊Q₈	—	2	4	no	—
373	C₂²⋊Q₈	D₄×C₆₁	1	2	yes	—
374	C₂²⋊Q₈	D₄×C₆₁	1	2	yes	—
375	#121	—	2	2	no	—
376	#121	—	2	2	no	—
377	C₄×Q₈	—	2	4	no	—
378	C₄×Q₈	—	2	2	no	—
379	C₄×Q₈	—	2	2	no	—
380	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
381	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
382	C₄×Q₈	C₂³×C₆₁	1	8	yes	—
383	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
384	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
385	C₄×D₄	—	2	4	no	—
386	C₄×D₄	—	2	8	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	C₂³×C₆₁	1	8	yes	—
390	#117	—	2	4	no	—
391	#117	—	2	8	no	—
392	#117	C₂³×C₆₁	1	2	yes	—
393	C₂×C₄⋊C₄	—	2	2	no	—
394	C₂×C₄⋊C₄	—	2	4	no	—
395	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
396	C₂×C₄²	—	2	2	no	—
397	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
398	#107	C₂³×C₆₁	1	2	yes	—
399	#107	C₂³×C₆₁	1	2	yes	—
400	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
401	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
402	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
403	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
404	#103	C₂³×C₆₁	1	4	yes	—
405	#103	C₂³×C₆₁	1	4	yes	—
406	#103	C₂³×C₆₁	1	8	yes	—
407	#102	C₂³×C₆₁	1	8	yes	—
408	#97	C₂³×C₆₁	1	4	yes	—
409	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
410	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
411	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
412	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
413	C₄×C₈	C₂³×C₆₁	1	4	yes	—
414	#41	—	2	8	no	—
415	#41	—	2	8	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	—	2	4	no	—
419	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
420	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
421	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
422	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	8	no	—
431	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
433	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
434	Q₁₆	—	2	4	no	—
435	Q₁₆	—	2	4	no	—
436	Q₁₆	—	2	4	no	—
437	Q₁₆	—	4	2	no	—
438	Q₁₆	—	4	8	no	—
439	SD₁₆	—	2	8	no	—
440	SD₁₆	—	4	4	no	—
441	SD₁₆	—	4	8	no	—
442	D₈	—	2	4	no	—
443	D₈	—	4	2	no	—
444	M₁₆	—	2	4	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	#41×C₆₁	1	4	yes	—
447	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
448	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
449	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
450	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
451	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	8	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	8	no	—
460	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
461	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
462	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
463	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
464	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
465	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
466	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	8	no	—
472	C₂²⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
473	C₄²	—	2	4	no	—
474	C₄²	—	2	4	no	—
475	C₄²	—	2	8	no	—
476	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
477	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
478	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
479	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
480	C₄²	C₂⁴×C₆₁	1	4	yes	—
481	C₂³	—	2	4	no	—
482	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
483	Q₈	—	2	4	no	—
484	Q₈	—	2	4	no	—
485	Q₈	—	2	4	no	—
486	Q₈	—	2	4	no	—
487	Q₈	—	2	4	no	—
488	Q₈	—	4	2	no	—
489	Q₈	—	4	4	no	—
490	D₄	—	2	4	no	—
491	D₄	—	2	4	no	—
492	D₄	—	4	8	no	—
493	D₄	—	4	4	no	—
494	D₄	—	8	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	—	8	8	no	—
503	C₂×C₄	—	8	2	no	—
504	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
506	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
508	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
509	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
510	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
511	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
512	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
513	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
514	C₈	—	2	1	no	—
515	C₈	—	2	1	no	—
516	C₈	—	2	4	no	—
517	C₈	C₂×#41×C₆₁	1	2	yes	—
518	C₈	C₂×#41×C₆₁	1	2	yes	—
519	C₂²	—	2	1	no	—
520	C₂²	—	2	1	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	4	no	—
523	C₂²	—	4	4	no	—
524	C₂²	C₆₁×#550	1	2	yes	—
525	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
526	C₄	—	2	1	no	—
527	C₄	—	2	4	no	—
528	C₄	—	2	1	no	—
529	C₄	—	2	1	no	—
530	C₄	—	2	4	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	4	no	—
533	C₄	—	4	4	no	—
534	C₄	C₆₁×#550	1	2	yes	—
535	C₄	C₆₁×#578	1	2	yes	—
536	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
537	C₄	C₂×H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
538	C₂	—	2	2	no	—
539	C₂	—	8	4	no	—
540	C₂	C₂×C₆₁×#550	1	1	yes	characteristic
541	C₂	C₂×C₆₁×#578	1	1	yes	characteristic
542	C₂	C₆₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₆₁×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₆₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3507	C₂	semidirect
2. C₆₁×#3402	C₂	semidirect
3. C₆₁×#3400	C₂	semidirect
4. C₆₁×#3396	C₂	semidirect
5. C₆₁×#3395	C₂	semidirect
6. C₆₁×#3393	C₂	semidirect
7. C₆₁×#3156	C₂	semidirect
9. C₆₁×#3046	C₂	semidirect
10. C₆₁×#3037	C₂	semidirect
11. C₆₁×#2960	C₂	semidirect
12. C₆₁×#2951	C₂	semidirect
13. C₂×C₄×Q₁₆×C₆₁	C₂	semidirect
247. #61408	C₆₁	direct (N × H)
267. #3507	C₂×C₆₁	semidirect
268. #3402	C₂×C₆₁	semidirect
269. #3400	C₂×C₆₁	semidirect
270. #3396	C₂×C₆₁	semidirect
271. #3395	C₂×C₆₁	semidirect
272. #3393	C₂×C₆₁	semidirect
273. #3156	C₂×C₆₁	semidirect
275. #3046	C₂×C₆₁	semidirect
276. #3037	C₂×C₆₁	semidirect
277. #2960	C₂×C₆₁	semidirect
278. #2951	C₂×C₆₁	semidirect
279. C₂×C₄×Q₁₆	C₂×C₆₁	semidirect
335. C₆₁	#61408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#3507	2
C₆₁×#3402	2
C₆₁×#3400	2
C₆₁×#3396	2
C₆₁×#3395	2
C₆₁×#3393	2
C₆₁×#3156	2
C₆₁×#3147	2
C₆₁×#3046	2
C₆₁×#3037	2
C₆₁×#2960	2
C₆₁×#2951	2
C₂×C₄×Q₁₆×C₆₁	2
#61408	61

Subgroups — C61×#61408 #29564841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#61408 #29564841