Subgroups — C₆₁×#25749 #29529182

Order 15616.

1652 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3040	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₆₁×#455	C₂	1	2	yes	maximal, complemented
5	C₂×C₆₁×#453	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2657	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2004	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2002	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1963	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1759	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₁	C₂²	1	2	yes	—
13	C₂×#137×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#137×C₆₁	C₂²	1	2	yes	complemented
15	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	complemented
17	C₆₁×#550	—	2	2	no	—
18	D₄²×C₆₁	—	2	4	no	—
19	D₄²×C₆₁	—	2	2	no	—
20	C₆₁×#535	—	2	2	no	—
21	C₆₁×#535	C₂²	1	2	yes	—
22	C₆₁×#532	—	2	1	no	—
23	C₆₁×#532	C₂²	1	1	yes	characteristic
24	C₂×#127×C₆₁	—	2	1	no	—
25	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×#127×C₆₁	C₂²	1	2	yes	complemented
27	C₂×#124×C₆₁	C₂²	1	2	yes	complemented
28	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic, complemented
29	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#497	—	2	1	no	—
31	C₆₁×#496	—	2	1	no	—
32	C₆₁×#495	—	2	2	no	—
33	C₆₁×#464	—	2	2	no	—
34	C₆₁×#463	—	2	2	no	—
35	C₆₁×#455	—	2	8	no	—
36	C₆₁×#453	—	2	8	no	—
37	C₆₁×#423	C₂²	1	1	yes	characteristic, complemented
38	C₂×#104×C₆₁	C₂²	1	2	yes	—
39	C₂×#104×C₆₁	C₂²	1	2	yes	complemented
40	C₂×#104×C₆₁	C₂²	1	2	yes	complemented
41	C₆₁×#419	C₂²	1	1	yes	characteristic, complemented
42	C₆₁×#418	C₂²	1	1	yes	characteristic, complemented
43	C₂×#100×C₆₁	C₂²	1	2	yes	complemented
44	C₂×#100×C₆₁	C₂²	1	2	yes	complemented
45	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
46	C₆₁×#392	—	2	4	no	—
47	C₆₁×#376	C₂²	1	1	yes	characteristic, complemented
48	C₆₁×#339	C₂²	1	2	yes	complemented
49	H(5,𝔽₂)×C₆₁	—	2	4	no	—
50	H(5,𝔽₂)×C₆₁	—	2	4	no	—
51	H(5,𝔽₂)×C₆₁	—	4	2	no	—
52	H(5,𝔽₂)×C₆₁	—	4	2	no	—
53	C₂×#41×C₆₁	—	2	2	no	—
54	C₂×#41×C₆₁	—	2	2	no	—
55	C₂×#41×C₆₁	C₂³	1	2	yes	—
56	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
57	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₆₁	—	2	4	no	—
59	C₂²×D₄×C₆₁	—	2	4	no	—
60	C₂²×D₄×C₆₁	—	2	2	no	—
61	C₂²×D₄×C₆₁	—	2	2	no	—
62	C₂²×D₄×C₆₁	—	2	4	no	—
63	C₂²×D₄×C₆₁	D₄	1	4	yes	—
64	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
66	#137×C₆₁	—	2	4	no	—
67	#137×C₆₁	—	2	4	no	—
68	#137×C₆₁	—	4	2	no	—
69	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
70	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
71	AGL(1,ℤ/8)×C₆₁	—	4	2	no	—
72	C₂×Q₁₆×C₆₁	—	2	1	no	—
73	C₂×Q₁₆×C₆₁	—	2	2	no	—
74	C₂×SD₁₆×C₆₁	—	2	1	no	—
75	C₂×SD₁₆×C₆₁	—	2	2	no	—
76	C₂×SD₁₆×C₆₁	—	2	1	no	—
77	C₂×SD₁₆×C₆₁	—	2	2	no	—
78	C₂×D₈×C₆₁	—	2	1	no	—
79	C₂×D₈×C₆₁	—	2	2	no	—
80	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
82	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
83	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic, complemented
84	#127×C₆₁	—	2	4	no	—
85	#127×C₆₁	—	2	2	no	—
86	#127×C₆₁	—	2	1	no	—
87	#127×C₆₁	—	2	1	no	—
88	#127×C₆₁	—	2	2	no	—
89	#127×C₆₁	—	2	4	no	—
90	#127×C₆₁	—	4	2	no	—
91	#127×C₆₁	D₄	1	1	yes	characteristic, complemented
92	#124×C₆₁	—	2	2	no	—
93	#124×C₆₁	—	2	2	no	—
94	#124×C₆₁	—	2	1	no	—
95	#124×C₆₁	—	2	1	no	—
96	#124×C₆₁	—	2	4	no	—
97	#124×C₆₁	—	4	2	no	—
98	#124×C₆₁	D₄	1	2	yes	complemented
99	#121×C₆₁	—	2	4	no	—
100	#121×C₆₁	—	2	8	no	—
101	#121×C₆₁	—	2	2	no	—
102	#121×C₆₁	—	2	4	no	—
103	#120×C₆₁	—	2	8	no	—
104	#120×C₆₁	—	2	4	no	—
105	C₄×Q₈×C₆₁	D₄	1	2	yes	complemented
106	C₄×D₄×C₆₁	—	2	4	no	—
107	C₄×D₄×C₆₁	—	2	2	no	—
108	C₄×D₄×C₆₁	—	2	2	no	—
109	C₄×D₄×C₆₁	D₄	1	2	yes	complemented
110	#117×C₆₁	—	2	1	no	—
111	#117×C₆₁	C₂³	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
113	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
114	C₂×C₄²×C₆₁	—	2	1	no	—
115	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
116	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
117	C₄⋊C₈×C₆₁	—	2	2	no	—
118	#104×C₆₁	—	2	4	no	—
119	#104×C₆₁	—	2	4	no	—
120	#104×C₆₁	D₄	1	8	yes	complemented
121	#103×C₆₁	—	2	2	no	—
122	#102×C₆₁	—	2	2	no	—
123	#100×C₆₁	—	2	4	no	—
124	#100×C₆₁	—	2	4	no	—
125	#97×C₆₁	—	2	1	no	—
126	C₄×C₈×C₆₁	—	2	1	no	—
127	#95×C₆₁	—	2	2	no	—
128	C₂⁴×C₆₁	—	2	4	no	—
129	C₂⁴×C₆₁	—	2	1	no	—
130	C₂⁴×C₆₁	C₂×D₄	1	4	yes	—
131	#41×C₆₁	—	2	2	no	—
132	#41×C₆₁	—	2	2	no	—
133	#41×C₆₁	—	2	2	no	—
134	#41×C₆₁	—	2	2	no	—
135	#41×C₆₁	—	2	2	no	—
136	#41×C₆₁	—	4	4	no	—
137	#41×C₆₁	—	4	4	no	—
138	#41×C₆₁	—	4	4	no	—
139	#41×C₆₁	—	4	4	no	—
140	#41×C₆₁	C₂×D₄	1	4	yes	—
141	C₂×Q₈×C₆₁	—	2	1	no	—
142	C₂×Q₈×C₆₁	—	2	1	no	—
143	C₂×Q₈×C₆₁	—	2	2	no	—
144	C₂×Q₈×C₆₁	—	2	2	no	—
145	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
146	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
147	C₂×D₄×C₆₁	—	2	1	no	—
148	C₂×D₄×C₆₁	—	2	4	no	—
149	C₂×D₄×C₆₁	—	2	4	no	—
150	C₂×D₄×C₆₁	—	2	2	no	—
151	C₂×D₄×C₆₁	—	2	8	no	—
152	C₂×D₄×C₆₁	—	2	4	no	—
153	C₂×D₄×C₆₁	—	2	2	no	—
154	C₂×D₄×C₆₁	—	2	4	no	—
155	C₂×D₄×C₆₁	—	2	2	no	—
156	C₂×D₄×C₆₁	—	2	2	no	—
157	C₂×D₄×C₆₁	—	2	2	no	—
158	C₂×D₄×C₆₁	—	2	2	no	—
159	C₂×D₄×C₆₁	—	2	1	no	—
160	C₂×D₄×C₆₁	—	2	1	no	—
161	C₂×D₄×C₆₁	—	2	2	no	—
162	C₂×D₄×C₆₁	—	2	2	no	—
163	C₂×D₄×C₆₁	—	2	2	no	—
164	C₂×D₄×C₆₁	—	2	2	no	—
165	C₂×D₄×C₆₁	—	4	4	no	—
166	C₂×D₄×C₆₁	—	4	4	no	—
167	C₂×D₄×C₆₁	—	4	8	no	—
168	C₂×D₄×C₆₁	—	4	8	no	—
169	C₂×D₄×C₆₁	—	4	8	no	—
170	C₂×D₄×C₆₁	—	4	4	no	—
171	C₂×D₄×C₆₁	—	4	4	no	—
172	C₂×D₄×C₆₁	—	8	8	no	—
173	C₂×D₄×C₆₁	—	8	4	no	—
174	C₂×D₄×C₆₁	—	8	8	no	—
175	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₁	—	2	1	no	—
178	C₂²×C₄×C₆₁	—	2	4	no	—
179	C₂²×C₄×C₆₁	—	2	2	no	—
180	C₂²×C₄×C₆₁	—	2	2	no	—
181	C₂²×C₄×C₆₁	—	2	2	no	—
182	C₂²×C₄×C₆₁	—	2	1	no	—
183	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
185	Q₁₆×C₆₁	—	4	4	no	—
186	Q₁₆×C₆₁	—	4	2	no	—
187	SD₁₆×C₆₁	—	4	4	no	—
188	SD₁₆×C₆₁	—	4	4	no	—
189	SD₁₆×C₆₁	—	4	2	no	—
190	SD₁₆×C₆₁	—	4	2	no	—
191	D₈×C₆₁	—	4	4	no	—
192	D₈×C₆₁	—	4	2	no	—
193	M₁₆×C₆₁	—	2	2	no	—
194	M₁₆×C₆₁	—	2	2	no	—
195	M₁₆×C₆₁	—	2	2	no	—
196	M₁₆×C₆₁	C₂×D₄	1	4	yes	complemented
197	C₂×C₈×C₆₁	—	2	2	no	—
198	C₂×C₈×C₆₁	—	2	1	no	—
199	C₂×C₈×C₆₁	—	2	1	no	—
200	C₄⋊C₄×C₆₁	—	2	2	no	—
201	C₄⋊C₄×C₆₁	—	2	2	no	—
202	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
203	C₂²⋊C₄×C₆₁	—	2	8	no	—
204	C₂²⋊C₄×C₆₁	—	2	1	no	—
205	C₂²⋊C₄×C₆₁	—	2	2	no	—
206	C₂²⋊C₄×C₆₁	—	2	4	no	—
207	C₂²⋊C₄×C₆₁	—	4	8	no	—
208	C₂²⋊C₄×C₆₁	—	4	2	no	—
209	C₂²⋊C₄×C₆₁	—	4	2	no	—
210	C₄²×C₆₁	—	2	2	no	—
211	C₄²×C₆₁	—	2	1	no	—
212	C₄²×C₆₁	—	2	2	no	—
213	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₆₁	C₂×D₄	1	2	yes	—
216	C₄²×C₆₁	C₂×D₄	1	4	yes	complemented
217	C₂³×C₆₁	—	2	1	no	—
218	C₂³×C₆₁	—	2	4	no	—
219	C₂³×C₆₁	—	2	2	no	—
220	C₂³×C₆₁	—	2	4	no	—
221	C₂³×C₆₁	—	2	4	no	—
222	C₂³×C₆₁	—	4	8	no	—
223	C₂³×C₆₁	—	4	8	no	—
224	C₂³×C₆₁	—	4	4	no	—
225	C₂³×C₆₁	—	4	1	no	—
226	C₂³×C₆₁	—	4	2	no	—
227	C₂³×C₆₁	—	8	8	no	—
228	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
229	Q₈×C₆₁	—	2	1	no	—
230	Q₈×C₆₁	—	2	2	no	—
231	Q₈×C₆₁	—	2	2	no	—
232	Q₈×C₆₁	—	4	1	no	—
233	Q₈×C₆₁	—	4	1	no	—
234	Q₈×C₆₁	—	4	2	no	—
235	Q₈×C₆₁	#127	1	2	yes	—
236	D₄×C₆₁	—	2	4	no	—
237	D₄×C₆₁	—	2	1	no	—
238	D₄×C₆₁	—	2	2	no	—
239	D₄×C₆₁	—	2	2	no	—
240	D₄×C₆₁	—	4	8	no	—
241	D₄×C₆₁	—	4	8	no	—
242	D₄×C₆₁	—	4	4	no	—
243	D₄×C₆₁	—	4	8	no	—
244	D₄×C₆₁	—	4	2	no	—
245	D₄×C₆₁	—	4	2	no	—
246	D₄×C₆₁	—	4	2	no	—
247	D₄×C₆₁	—	4	1	no	—
248	D₄×C₆₁	—	4	2	no	—
249	D₄×C₆₁	—	4	1	no	—
250	D₄×C₆₁	—	4	2	no	—
251	D₄×C₆₁	—	4	2	no	—
252	D₄×C₆₁	—	4	2	no	—
253	D₄×C₆₁	—	8	4	no	—
254	D₄×C₆₁	—	8	8	no	—
255	D₄×C₆₁	—	8	16	no	—
256	D₄×C₆₁	#127	1	2	yes	—
257	C₂×C₄×C₆₁	—	2	1	no	—
258	C₂×C₄×C₆₁	—	2	4	no	—
259	C₂×C₄×C₆₁	—	2	2	no	—
260	C₂×C₄×C₆₁	—	2	2	no	—
261	C₂×C₄×C₆₁	—	2	4	no	—
262	C₂×C₄×C₆₁	—	4	2	no	—
263	C₂×C₄×C₆₁	—	4	8	no	—
264	C₂×C₄×C₆₁	—	4	4	no	—
265	C₂×C₄×C₆₁	—	4	4	no	—
266	C₂×C₄×C₆₁	—	4	2	no	—
267	C₂×C₄×C₆₁	—	4	2	no	—
268	C₂×C₄×C₆₁	—	4	1	no	—
269	C₂×C₄×C₆₁	—	4	2	no	—
270	C₂×C₄×C₆₁	#120	1	2	yes	—
271	C₂×C₄×C₆₁	#120	1	2	yes	—
272	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
276	C₈×C₆₁	—	2	4	no	—
277	C₈×C₆₁	—	4	1	no	—
278	C₈×C₆₁	—	4	1	no	—
279	#25749	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂²×C₆₁	—	2	2	no	—
281	C₂²×C₆₁	—	2	2	no	—
282	C₂²×C₆₁	—	4	2	no	—
283	C₂²×C₆₁	—	4	2	no	—
284	C₂²×C₆₁	—	4	4	no	—
285	C₂²×C₆₁	—	4	8	no	—
286	C₂²×C₆₁	—	4	4	no	—
287	C₂²×C₆₁	—	8	4	no	—
288	C₂²×C₆₁	—	8	4	no	—
289	C₂²×C₆₁	—	8	4	no	—
290	C₂²×C₆₁	—	8	1	no	—
291	C₂²×C₆₁	—	8	2	no	—
292	C₂²×C₆₁	—	16	8	no	—
293	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
294	C₂²×C₆₁	D₄²	1	2	yes	—
295	C₄×C₆₁	—	2	2	no	—
296	C₄×C₆₁	—	2	2	no	—
297	C₄×C₆₁	—	2	2	no	—
298	C₄×C₆₁	—	4	4	no	—
299	C₄×C₆₁	—	4	2	no	—
300	C₄×C₆₁	—	4	2	no	—
301	C₄×C₆₁	—	8	1	no	—
302	C₄×C₆₁	C₂×#127	1	1	yes	characteristic
303	C₄×C₆₁	#535	1	1	yes	characteristic
304	C₄×C₆₁	D₄²	1	2	yes	—
305	#3498	C₂×C₆₁	1	1	yes	characteristic, complemented
306	#3153	C₂×C₆₁	1	1	yes	characteristic, complemented
307	#3040	C₂×C₆₁	1	1	yes	characteristic, complemented
308	C₂×#455	C₂×C₆₁	1	2	yes	complemented
309	C₂×#453	C₂×C₆₁	1	2	yes	complemented
310	#2657	C₂×C₆₁	1	1	yes	characteristic, complemented
311	#2004	C₂×C₆₁	1	2	yes	complemented
312	#2002	C₂×C₆₁	1	2	yes	complemented
313	#1965	C₂×C₆₁	1	1	yes	characteristic, complemented
314	#1963	C₂×C₆₁	1	1	yes	characteristic, complemented
315	#1759	C₂×C₆₁	1	1	yes	characteristic, complemented
316	C₂×C₆₁	—	2	2	no	—
317	C₂×C₆₁	—	4	2	no	—
318	C₂×C₆₁	—	8	4	no	—
319	C₂×C₆₁	—	8	2	no	—
320	C₂×C₆₁	—	16	1	no	—
321	C₂×C₆₁	#2433	1	1	yes	characteristic
322	C₂×C₆₁	#3019	1	2	yes	—
323	C₂×H(5,𝔽₂)	C₂²×C₆₁	1	2	yes	—
324	C₂×#137	C₂²×C₆₁	1	1	yes	characteristic
325	C₂×#137	C₂²×C₆₁	1	2	yes	complemented
326	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	1	yes	characteristic
327	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	complemented
328	#550	—	2	2	no	—
329	D₄²	—	2	4	no	—
330	D₄²	—	2	2	no	—
331	#535	—	2	2	no	—
332	#535	C₂²×C₆₁	1	2	yes	—
333	#532	—	2	1	no	—
334	#532	C₂²×C₆₁	1	1	yes	characteristic
335	C₂×#127	—	2	1	no	—
336	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
337	C₂×#127	C₂²×C₆₁	1	2	yes	complemented
338	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic, complemented
339	C₂×#124	C₂²×C₆₁	1	2	yes	complemented
340	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
341	#497	—	2	1	no	—
342	#496	—	2	1	no	—
343	#495	—	2	2	no	—
344	#464	—	2	2	no	—
345	#463	—	2	2	no	—
346	#455	—	2	8	no	—
347	#453	—	2	8	no	—
348	#423	C₂²×C₆₁	1	1	yes	characteristic, complemented
349	C₂×#104	C₂²×C₆₁	1	2	yes	—
350	C₂×#104	C₂²×C₆₁	1	2	yes	complemented
351	C₂×#104	C₂²×C₆₁	1	2	yes	complemented
352	#419	C₂²×C₆₁	1	1	yes	characteristic, complemented
353	#418	C₂²×C₆₁	1	1	yes	characteristic, complemented
354	C₂×#100	C₂²×C₆₁	1	2	yes	complemented
355	C₂×#100	C₂²×C₆₁	1	2	yes	complemented
356	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic, complemented
357	#392	—	2	4	no	—
358	#376	C₂²×C₆₁	1	1	yes	characteristic, complemented
359	#339	C₂²×C₆₁	1	2	yes	complemented
360	C₆₁	#25749	1	1	yes	characteristic, direct product factor, Sylow
361	H(5,𝔽₂)	—	2	4	no	—
362	H(5,𝔽₂)	—	2	4	no	—
363	H(5,𝔽₂)	—	4	2	no	—
364	H(5,𝔽₂)	—	4	2	no	—
365	C₂×#41	—	2	2	no	—
366	C₂×#41	—	2	2	no	—
367	C₂×#41	C₂³×C₆₁	1	2	yes	—
368	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
369	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
370	C₂²×D₄	—	2	4	no	—
371	C₂²×D₄	—	2	4	no	—
372	C₂²×D₄	—	2	2	no	—
373	C₂²×D₄	—	2	2	no	—
374	C₂²×D₄	—	2	4	no	—
375	C₂²×D₄	D₄×C₆₁	1	4	yes	—
376	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
377	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
378	#137	—	2	4	no	—
379	#137	—	2	4	no	—
380	#137	—	4	2	no	—
381	AGL(1,ℤ/8)	—	2	4	no	—
382	AGL(1,ℤ/8)	—	2	4	no	—
383	AGL(1,ℤ/8)	—	4	2	no	—
384	C₂×Q₁₆	—	2	1	no	—
385	C₂×Q₁₆	—	2	2	no	—
386	C₂×SD₁₆	—	2	1	no	—
387	C₂×SD₁₆	—	2	2	no	—
388	C₂×SD₁₆	—	2	1	no	—
389	C₂×SD₁₆	—	2	2	no	—
390	C₂×D₈	—	2	1	no	—
391	C₂×D₈	—	2	2	no	—
392	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
393	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
394	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
395	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic, complemented
396	#127	—	2	4	no	—
397	#127	—	2	2	no	—
398	#127	—	2	1	no	—
399	#127	—	2	1	no	—
400	#127	—	2	2	no	—
401	#127	—	2	4	no	—
402	#127	—	4	2	no	—
403	#127	D₄×C₆₁	1	1	yes	characteristic, complemented
404	#124	—	2	2	no	—
405	#124	—	2	2	no	—
406	#124	—	2	1	no	—
407	#124	—	2	1	no	—
408	#124	—	2	4	no	—
409	#124	—	4	2	no	—
410	#124	D₄×C₆₁	1	2	yes	complemented
411	#121	—	2	4	no	—
412	#121	—	2	8	no	—
413	#121	—	2	2	no	—
414	#121	—	2	4	no	—
415	#120	—	2	8	no	—
416	#120	—	2	4	no	—
417	C₄×Q₈	D₄×C₆₁	1	2	yes	complemented
418	C₄×D₄	—	2	4	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	—	2	2	no	—
421	C₄×D₄	D₄×C₆₁	1	2	yes	complemented
422	#117	—	2	1	no	—
423	#117	C₂³×C₆₁	1	1	yes	characteristic
424	C₂×C₂²⋊C₄	—	2	4	no	—
425	C₂×C₂²⋊C₄	—	2	2	no	—
426	C₂×C₄²	—	2	1	no	—
427	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
428	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
429	C₄⋊C₈	—	2	2	no	—
430	#104	—	2	4	no	—
431	#104	—	2	4	no	—
432	#104	D₄×C₆₁	1	8	yes	complemented
433	#103	—	2	2	no	—
434	#102	—	2	2	no	—
435	#100	—	2	4	no	—
436	#100	—	2	4	no	—
437	#97	—	2	1	no	—
438	C₄×C₈	—	2	1	no	—
439	#95	—	2	2	no	—
440	C₂⁴	—	2	4	no	—
441	C₂⁴	—	2	1	no	—
442	C₂⁴	C₂×D₄×C₆₁	1	4	yes	—
443	#41	—	2	2	no	—
444	#41	—	2	2	no	—
445	#41	—	2	2	no	—
446	#41	—	2	2	no	—
447	#41	—	2	2	no	—
448	#41	—	4	4	no	—
449	#41	—	4	4	no	—
450	#41	—	4	4	no	—
451	#41	—	4	4	no	—
452	#41	C₂×D₄×C₆₁	1	4	yes	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
458	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	2	8	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	2	no	—
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	—	2	2	no	—
476	C₂×D₄	—	2	2	no	—
477	C₂×D₄	—	4	4	no	—
478	C₂×D₄	—	4	4	no	—
479	C₂×D₄	—	4	8	no	—
480	C₂×D₄	—	4	8	no	—
481	C₂×D₄	—	4	8	no	—
482	C₂×D₄	—	4	4	no	—
483	C₂×D₄	—	4	4	no	—
484	C₂×D₄	—	8	8	no	—
485	C₂×D₄	—	8	4	no	—
486	C₂×D₄	—	8	8	no	—
487	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
488	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	—	2	2	no	—
493	C₂²×C₄	—	2	2	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
496	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
497	Q₁₆	—	4	4	no	—
498	Q₁₆	—	4	2	no	—
499	SD₁₆	—	4	4	no	—
500	SD₁₆	—	4	4	no	—
501	SD₁₆	—	4	2	no	—
502	SD₁₆	—	4	2	no	—
503	D₈	—	4	4	no	—
504	D₈	—	4	2	no	—
505	M₁₆	—	2	2	no	—
506	M₁₆	—	2	2	no	—
507	M₁₆	—	2	2	no	—
508	M₁₆	C₂×D₄×C₆₁	1	4	yes	complemented
509	C₂×C₈	—	2	2	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
515	C₂²⋊C₄	—	2	8	no	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	4	8	no	—
520	C₂²⋊C₄	—	4	2	no	—
521	C₂²⋊C₄	—	4	2	no	—
522	C₄²	—	2	2	no	—
523	C₄²	—	2	1	no	—
524	C₄²	—	2	2	no	—
525	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
526	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
527	C₄²	C₂×D₄×C₆₁	1	2	yes	—
528	C₄²	C₂×D₄×C₆₁	1	4	yes	complemented
529	C₂³	—	2	1	no	—
530	C₂³	—	2	4	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	4	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	4	8	no	—
535	C₂³	—	4	8	no	—
536	C₂³	—	4	4	no	—
537	C₂³	—	4	2	no	—
538	C₂³	—	4	1	no	—
539	C₂³	—	8	8	no	—
540	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
541	Q₈	—	2	1	no	—
542	Q₈	—	2	2	no	—
543	Q₈	—	2	2	no	—
544	Q₈	—	4	1	no	—
545	Q₈	—	4	1	no	—
546	Q₈	—	4	2	no	—
547	Q₈	#127×C₆₁	1	2	yes	—
548	D₄	—	2	4	no	—
549	D₄	—	2	1	no	—
550	D₄	—	2	2	no	—
551	D₄	—	2	2	no	—
552	D₄	—	4	8	no	—
553	D₄	—	4	8	no	—
554	D₄	—	4	4	no	—
555	D₄	—	4	8	no	—
556	D₄	—	4	2	no	—
557	D₄	—	4	2	no	—
558	D₄	—	4	2	no	—
559	D₄	—	4	1	no	—
560	D₄	—	4	2	no	—
561	D₄	—	4	1	no	—
562	D₄	—	4	2	no	—
563	D₄	—	4	2	no	—
564	D₄	—	4	2	no	—
565	D₄	—	8	4	no	—
566	D₄	—	8	8	no	—
567	D₄	—	8	16	no	—
568	D₄	#127×C₆₁	1	2	yes	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	8	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	4	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	4	2	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	#120×C₆₁	1	2	yes	—
583	C₂×C₄	#120×C₆₁	1	2	yes	—
584	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
586	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
587	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
588	C₈	—	2	4	no	—
589	C₈	—	4	1	no	—
590	C₈	—	4	1	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	2	2	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	4	no	—
596	C₂²	—	4	8	no	—
597	C₂²	—	4	4	no	—
598	C₂²	—	8	4	no	—
599	C₂²	—	8	4	no	—
600	C₂²	—	8	4	no	—
601	C₂²	—	8	1	no	—
602	C₂²	—	8	2	no	—
603	C₂²	—	16	8	no	—
604	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
605	C₂²	D₄²×C₆₁	1	2	yes	—
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	2	no	—
612	C₄	—	8	1	no	—
613	C₄	C₂×#127×C₆₁	1	1	yes	characteristic
614	C₄	C₆₁×#535	1	1	yes	characteristic
615	C₄	D₄²×C₆₁	1	2	yes	—
616	C₂	—	2	2	no	—
617	C₂	—	4	2	no	—
618	C₂	—	8	4	no	—
619	C₂	—	8	2	no	—
620	C₂	—	16	1	no	—
621	C₂	C₆₁×#2433	1	1	yes	characteristic
622	C₂	C₆₁×#3019	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3498	C₂	semidirect
2. C₆₁×#3153	C₂	semidirect
3. C₆₁×#3040	C₂	semidirect
4. C₂×C₆₁×#455	C₂	semidirect
5. C₂×C₆₁×#453	C₂	semidirect
6. C₆₁×#2657	C₂	semidirect
7. C₆₁×#2004	C₂	semidirect
8. C₆₁×#2002	C₂	semidirect
9. C₆₁×#1965	C₂	semidirect
10. C₆₁×#1963	C₂	semidirect
11. C₆₁×#1759	C₂	semidirect
14. C₂×#137×C₆₁	C₂²	semidirect
16. C₂×AGL(1,ℤ/8)×C₆₁	C₂²	semidirect
26. C₂×#127×C₆₁	C₂²	semidirect
27. C₂×#124×C₆₁	C₂²	semidirect
28. C₂×#124×C₆₁	C₂²	semidirect
37. C₆₁×#423	C₂²	semidirect
39. C₂×#104×C₆₁	C₂²	semidirect
40. C₂×#104×C₆₁	C₂²	semidirect
41. C₆₁×#419	C₂²	semidirect
42. C₆₁×#418	C₂²	semidirect
43. C₂×#100×C₆₁	C₂²	semidirect
44. C₂×#100×C₆₁	C₂²	semidirect
45. C₄×M₁₆×C₆₁	C₂²	semidirect
47. C₆₁×#376	C₂²	semidirect
48. C₆₁×#339	C₂²	semidirect
83. C₄⋊Q₈×C₆₁	D₄	semidirect
91. #127×C₆₁	D₄	semidirect
98. #124×C₆₁	D₄	semidirect
105. C₄×Q₈×C₆₁	D₄	semidirect
109. C₄×D₄×C₆₁	D₄	semidirect
120. #104×C₆₁	D₄	semidirect
196. M₁₆×C₆₁	C₂×D₄	semidirect
216. C₄²×C₆₁	C₂×D₄	semidirect
279. #25749	C₆₁	direct (N × H)
305. #3498	C₂×C₆₁	semidirect
306. #3153	C₂×C₆₁	semidirect
307. #3040	C₂×C₆₁	semidirect
308. C₂×#455	C₂×C₆₁	semidirect
309. C₂×#453	C₂×C₆₁	semidirect
310. #2657	C₂×C₆₁	semidirect
311. #2004	C₂×C₆₁	semidirect
312. #2002	C₂×C₆₁	semidirect
313. #1965	C₂×C₆₁	semidirect
314. #1963	C₂×C₆₁	semidirect
315. #1759	C₂×C₆₁	semidirect
325. C₂×#137	C₂²×C₆₁	semidirect
327. C₂×AGL(1,ℤ/8)	C₂²×C₆₁	semidirect
337. C₂×#127	C₂²×C₆₁	semidirect
338. C₂×#124	C₂²×C₆₁	semidirect
339. C₂×#124	C₂²×C₆₁	semidirect
348. #423	C₂²×C₆₁	semidirect
350. C₂×#104	C₂²×C₆₁	semidirect
351. C₂×#104	C₂²×C₆₁	semidirect
352. #419	C₂²×C₆₁	semidirect
353. #418	C₂²×C₆₁	semidirect
354. C₂×#100	C₂²×C₆₁	semidirect
355. C₂×#100	C₂²×C₆₁	semidirect
356. C₄×M₁₆	C₂²×C₆₁	semidirect
358. #376	C₂²×C₆₁	semidirect
359. #339	C₂²×C₆₁	semidirect
360. C₆₁	#25749	direct (N × H)
395. C₄⋊Q₈	D₄×C₆₁	semidirect
403. #127	D₄×C₆₁	semidirect
410. #124	D₄×C₆₁	semidirect
417. C₄×Q₈	D₄×C₆₁	semidirect
421. C₄×D₄	D₄×C₆₁	semidirect
432. #104	D₄×C₆₁	semidirect
508. M₁₆	C₂×D₄×C₆₁	semidirect
528. C₄²	C₂×D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3498	2
C₆₁×#3153	2
C₆₁×#3040	2
C₂×C₆₁×#455	2
C₂×C₆₁×#453	2
C₆₁×#2657	2
C₆₁×#2004	2
C₆₁×#2002	2
C₆₁×#1965	2
C₆₁×#1963	2
C₆₁×#1759	2
#25749	61

Subgroups — C61×#25749 #29529182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25749 #29529182