Subgroups — C₆₁×#24228 #29527661

Order 15616.

1430 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3496	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3037	C₂	1	2	yes	maximal, complemented
3	C₂×C₆₁×#455	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2326	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2039	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2007	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1963	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1792	C₂	1	2	yes	maximal, complemented
10	C₂×H(5,𝔽₂)×C₆₁	C₂²	1	2	yes	—
11	C₂×#137×C₆₁	C₂²	1	4	yes	complemented
12	C₆₁×#547	—	2	2	no	—
13	C₆₁×#546	—	2	4	no	—
14	D₄²×C₆₁	—	2	2	no	—
15	C₆₁×#537	C₂²	1	2	yes	—
16	C₆₁×#534	—	2	2	no	—
17	C₆₁×#532	C₂²	1	2	yes	—
18	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×#121×C₆₁	—	2	2	no	—
20	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#464	—	2	4	no	—
23	C₆₁×#462	—	2	2	no	—
24	C₆₁×#460	—	2	2	no	—
25	C₆₁×#455	—	2	8	no	—
26	C₆₁×#423	C₂²	1	2	yes	complemented
27	C₂×#104×C₆₁	C₂²	1	4	yes	complemented
28	C₂×#104×C₆₁	C₂²	1	4	yes	complemented
29	C₆₁×#417	C₂²	1	4	yes	complemented
30	C₂×#100×C₆₁	C₂²	1	2	yes	—
31	C₆₁×#391	C₂²	1	1	yes	characteristic, complemented
32	C₆₁×#390	C₂²	1	1	yes	characteristic
33	C₆₁×#388	—	2	2	no	—
34	C₆₁×#385	—	2	2	no	—
35	C₆₁×#384	C₂²	1	1	yes	characteristic, complemented
36	C₆₁×#376	C₂²	1	1	yes	characteristic, complemented
37	C₆₁×#343	C₂²	1	2	yes	complemented
38	H(5,𝔽₂)×C₆₁	—	2	4	no	—
39	H(5,𝔽₂)×C₆₁	—	2	4	no	—
40	H(5,𝔽₂)×C₆₁	—	4	4	no	—
41	C₂×#41×C₆₁	—	2	4	no	—
42	C₂×#41×C₆₁	C₂³	1	4	yes	—
43	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₆₁	—	2	4	no	—
45	C₂²×D₄×C₆₁	—	2	2	no	—
46	C₂²×D₄×C₆₁	—	2	4	no	—
47	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₆₁	D₄	1	2	yes	—
49	#137×C₆₁	—	2	8	no	—
50	#137×C₆₁	—	4	4	no	—
51	C₂×Q₁₆×C₆₁	—	2	4	no	—
52	C₂×SD₁₆×C₆₁	—	2	4	no	—
53	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
54	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
55	#127×C₆₁	D₄	1	2	yes	—
56	#124×C₆₁	—	2	2	no	—
57	#124×C₆₁	D₄	1	4	yes	—
58	#124×C₆₁	D₄	1	4	yes	complemented
59	#123×C₆₁	—	2	8	no	—
60	C₂²⋊Q₈×C₆₁	—	2	4	no	—
61	#121×C₆₁	—	2	4	no	—
62	#121×C₆₁	—	2	4	no	—
63	#121×C₆₁	—	2	4	no	—
64	#121×C₆₁	—	2	2	no	—
65	#121×C₆₁	—	2	2	no	—
66	#121×C₆₁	—	2	4	no	—
67	#120×C₆₁	—	2	8	no	—
68	C₄×Q₈×C₆₁	—	2	2	no	—
69	C₄×Q₈×C₆₁	—	2	2	no	—
70	C₄×D₄×C₆₁	—	2	2	no	—
71	C₄×D₄×C₆₁	—	2	2	no	—
72	C₄×D₄×C₆₁	—	2	2	no	—
73	C₄×D₄×C₆₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
80	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
83	C₂×C₄²×C₆₁	—	2	1	no	—
84	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₆₁	—	2	2	no	—
88	C₄⋊C₈×C₆₁	—	2	2	no	—
89	#104×C₆₁	—	2	8	no	—
90	#104×C₆₁	—	4	4	no	—
91	#103×C₆₁	—	2	4	no	—
92	#102×C₆₁	—	2	4	no	—
93	#100×C₆₁	—	2	4	no	—
94	#95×C₆₁	—	2	1	no	—
95	#95×C₆₁	—	2	1	no	—
96	#95×C₆₁	—	2	2	no	—
97	C₂⁴×C₆₁	—	2	4	no	—
98	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
99	#41×C₆₁	—	2	4	no	—
100	#41×C₆₁	—	2	4	no	—
101	#41×C₆₁	—	2	4	no	—
102	#41×C₆₁	—	2	4	no	—
103	#41×C₆₁	—	4	8	no	—
104	#41×C₆₁	—	4	8	no	—
105	C₂×Q₈×C₆₁	—	2	1	no	—
106	C₂×Q₈×C₆₁	—	2	2	no	—
107	C₂×Q₈×C₆₁	—	2	2	no	—
108	C₂×Q₈×C₆₁	C₂×D₄	1	4	yes	—
109	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	complemented
110	C₂×D₄×C₆₁	—	2	8	no	—
111	C₂×D₄×C₆₁	—	2	4	no	—
112	C₂×D₄×C₆₁	—	2	4	no	—
113	C₂×D₄×C₆₁	—	2	2	no	—
114	C₂×D₄×C₆₁	—	2	2	no	—
115	C₂×D₄×C₆₁	—	2	2	no	—
116	C₂×D₄×C₆₁	—	2	2	no	—
117	C₂×D₄×C₆₁	—	4	8	no	—
118	C₂×D₄×C₆₁	—	4	8	no	—
119	C₂×D₄×C₆₁	—	4	8	no	—
120	C₂×D₄×C₆₁	—	4	8	no	—
121	C₂×D₄×C₆₁	—	8	4	no	—
122	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
123	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
124	C₂×D₄×C₆₁	#41	1	2	yes	—
125	C₂²×C₄×C₆₁	—	2	4	no	—
126	C₂²×C₄×C₆₁	—	2	1	no	—
127	C₂²×C₄×C₆₁	—	2	1	no	—
128	C₂²×C₄×C₆₁	—	2	2	no	—
129	C₂²×C₄×C₆₁	—	2	4	no	—
130	C₂²×C₄×C₆₁	—	2	2	no	—
131	C₂²×C₄×C₆₁	—	2	1	no	—
132	C₂²×C₄×C₆₁	—	2	1	no	—
133	C₂²×C₄×C₆₁	—	8	2	no	—
134	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
138	Q₁₆×C₆₁	—	4	8	no	—
139	SD₁₆×C₆₁	—	4	8	no	—
140	M₁₆×C₆₁	—	2	4	no	—
141	M₁₆×C₆₁	—	2	4	no	—
142	C₂×C₈×C₆₁	—	2	4	no	—
143	C₄⋊C₄×C₆₁	—	2	4	no	—
144	C₄⋊C₄×C₆₁	—	2	2	no	—
145	C₄⋊C₄×C₆₁	—	2	2	no	—
146	C₄⋊C₄×C₆₁	—	2	2	no	—
147	C₄⋊C₄×C₆₁	—	2	2	no	—
148	C₄⋊C₄×C₆₁	—	4	1	no	—
149	C₄⋊C₄×C₆₁	—	4	1	no	—
150	C₄⋊C₄×C₆₁	—	8	2	no	—
151	C₂²⋊C₄×C₆₁	—	2	8	no	—
152	C₂²⋊C₄×C₆₁	—	2	8	no	—
153	C₂²⋊C₄×C₆₁	—	2	4	no	—
154	C₂²⋊C₄×C₆₁	—	2	4	no	—
155	C₂²⋊C₄×C₆₁	—	2	4	no	—
156	C₂²⋊C₄×C₆₁	—	8	2	no	—
157	C₄²×C₆₁	—	2	1	no	—
158	C₄²×C₆₁	—	2	2	no	—
159	C₄²×C₆₁	—	4	2	no	—
160	C₄²×C₆₁	C₂×D₄	1	2	yes	—
161	C₄²×C₆₁	C₂×D₄	1	2	yes	—
162	C₄²×C₆₁	C₂×D₄	1	2	yes	complemented
163	C₂³×C₆₁	—	2	2	no	—
164	C₂³×C₆₁	—	2	2	no	—
165	C₂³×C₆₁	—	2	4	no	—
166	C₂³×C₆₁	—	4	2	no	—
167	C₂³×C₆₁	—	4	2	no	—
168	C₂³×C₆₁	—	4	8	no	—
169	C₂³×C₆₁	—	8	4	no	—
170	C₂³×C₆₁	—	8	4	no	—
171	C₂³×C₆₁	#121	1	4	yes	—
172	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
173	Q₈×C₆₁	—	2	4	no	—
174	Q₈×C₆₁	—	2	4	no	—
175	Q₈×C₆₁	—	2	4	no	—
176	Q₈×C₆₁	—	4	2	no	—
177	D₄×C₆₁	—	2	4	no	—
178	D₄×C₆₁	—	2	4	no	—
179	D₄×C₆₁	—	2	4	no	—
180	D₄×C₆₁	—	2	4	no	—
181	D₄×C₆₁	—	4	4	no	—
182	D₄×C₆₁	—	4	16	no	—
183	D₄×C₆₁	—	4	4	no	—
184	D₄×C₆₁	—	4	4	no	—
185	D₄×C₆₁	—	8	2	no	—
186	D₄×C₆₁	—	8	4	no	—
187	C₂×C₄×C₆₁	—	2	2	no	—
188	C₂×C₄×C₆₁	—	2	4	no	—
189	C₂×C₄×C₆₁	—	2	2	no	—
190	C₂×C₄×C₆₁	—	2	2	no	—
191	C₂×C₄×C₆₁	—	4	1	no	—
192	C₂×C₄×C₆₁	—	4	1	no	—
193	C₂×C₄×C₆₁	—	4	4	no	—
194	C₂×C₄×C₆₁	—	4	1	no	—
195	C₂×C₄×C₆₁	—	4	1	no	—
196	C₂×C₄×C₆₁	—	4	8	no	—
197	C₂×C₄×C₆₁	—	4	1	no	—
198	C₂×C₄×C₆₁	—	4	1	no	—
199	C₂×C₄×C₆₁	—	4	4	no	—
200	C₂×C₄×C₆₁	—	4	4	no	—
201	C₂×C₄×C₆₁	—	4	2	no	—
202	C₂×C₄×C₆₁	—	4	1	no	—
203	C₂×C₄×C₆₁	—	4	2	no	—
204	C₂×C₄×C₆₁	—	4	1	no	—
205	C₂×C₄×C₆₁	—	4	1	no	—
206	C₂×C₄×C₆₁	—	4	1	no	—
207	C₂×C₄×C₆₁	—	8	2	no	—
208	C₂×C₄×C₆₁	#120	1	2	yes	—
209	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
210	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
211	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
215	C₈×C₆₁	—	4	4	no	—
216	#24228	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂²×C₆₁	—	2	1	no	—
218	C₂²×C₆₁	—	2	1	no	—
219	C₂²×C₆₁	—	2	4	no	—
220	C₂²×C₆₁	—	4	4	no	—
221	C₂²×C₆₁	—	4	2	no	—
222	C₂²×C₆₁	—	4	2	no	—
223	C₂²×C₆₁	—	4	4	no	—
224	C₂²×C₆₁	—	4	4	no	—
225	C₂²×C₆₁	—	4	4	no	—
226	C₂²×C₆₁	—	8	2	no	—
227	C₂²×C₆₁	—	16	8	no	—
228	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
229	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
230	C₂²×C₆₁	#535	1	1	yes	characteristic
231	C₄×C₆₁	—	2	1	no	—
232	C₄×C₆₁	—	2	1	no	—
233	C₄×C₆₁	—	2	2	no	—
234	C₄×C₆₁	—	4	2	no	—
235	C₄×C₆₁	—	4	4	no	—
236	C₄×C₆₁	—	4	2	no	—
237	C₄×C₆₁	—	4	2	no	—
238	C₄×C₆₁	—	8	1	no	—
239	C₄×C₆₁	—	8	1	no	—
240	C₄×C₆₁	D₄²	1	2	yes	—
241	C₄×C₆₁	#546	1	2	yes	—
242	#3496	C₂×C₆₁	1	1	yes	characteristic
243	#3037	C₂×C₆₁	1	2	yes	complemented
244	C₂×#455	C₂×C₆₁	1	2	yes	complemented
245	#2326	C₂×C₆₁	1	1	yes	characteristic, complemented
246	#2039	C₂×C₆₁	1	2	yes	complemented
247	#2007	C₂×C₆₁	1	2	yes	complemented
248	#1963	C₂×C₆₁	1	2	yes	complemented
249	#1800	C₂×C₆₁	1	1	yes	characteristic, complemented
250	#1792	C₂×C₆₁	1	2	yes	complemented
251	C₂×C₆₁	—	2	1	no	—
252	C₂×C₆₁	—	2	1	no	—
253	C₂×C₆₁	—	4	4	no	—
254	C₂×C₆₁	—	8	4	no	—
255	C₂×C₆₁	#2408	1	1	yes	characteristic
256	C₂×C₆₁	#3019	1	1	yes	characteristic
257	C₂×C₆₁	#3022	1	1	yes	characteristic
258	C₂×H(5,𝔽₂)	C₂²×C₆₁	1	2	yes	—
259	C₂×#137	C₂²×C₆₁	1	4	yes	complemented
260	#547	—	2	2	no	—
261	#546	—	2	4	no	—
262	D₄²	—	2	2	no	—
263	#537	C₂²×C₆₁	1	2	yes	—
264	#534	—	2	2	no	—
265	#532	C₂²×C₆₁	1	2	yes	—
266	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
267	C₂×#121	—	2	2	no	—
268	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
269	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
270	#464	—	2	4	no	—
271	#462	—	2	2	no	—
272	#460	—	2	2	no	—
273	#455	—	2	8	no	—
274	#423	C₂²×C₆₁	1	2	yes	complemented
275	C₂×#104	C₂²×C₆₁	1	4	yes	complemented
276	C₂×#104	C₂²×C₆₁	1	4	yes	complemented
277	#417	C₂²×C₆₁	1	4	yes	complemented
278	C₂×#100	C₂²×C₆₁	1	2	yes	—
279	#391	C₂²×C₆₁	1	1	yes	characteristic, complemented
280	#390	C₂²×C₆₁	1	1	yes	characteristic
281	#388	—	2	2	no	—
282	#385	—	2	2	no	—
283	#384	C₂²×C₆₁	1	1	yes	characteristic, complemented
284	#376	C₂²×C₆₁	1	1	yes	characteristic, complemented
285	#343	C₂²×C₆₁	1	2	yes	complemented
286	C₆₁	#24228	1	1	yes	characteristic, direct product factor, Sylow
287	H(5,𝔽₂)	—	2	4	no	—
288	H(5,𝔽₂)	—	2	4	no	—
289	H(5,𝔽₂)	—	4	4	no	—
290	C₂×#41	—	2	4	no	—
291	C₂×#41	C₂³×C₆₁	1	4	yes	—
292	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
293	C₂²×D₄	—	2	4	no	—
294	C₂²×D₄	—	2	2	no	—
295	C₂²×D₄	—	2	4	no	—
296	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
297	C₂³×C₄	D₄×C₆₁	1	2	yes	—
298	#137	—	2	8	no	—
299	#137	—	4	4	no	—
300	C₂×Q₁₆	—	2	4	no	—
301	C₂×SD₁₆	—	2	4	no	—
302	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
303	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
304	#127	D₄×C₆₁	1	2	yes	—
305	#124	—	2	2	no	—
306	#124	D₄×C₆₁	1	4	yes	—
307	#124	D₄×C₆₁	1	4	yes	complemented
308	#123	—	2	8	no	—
309	C₂²⋊Q₈	—	2	4	no	—
310	#121	—	2	4	no	—
311	#121	—	2	4	no	—
312	#121	—	2	4	no	—
313	#121	—	2	2	no	—
314	#121	—	2	4	no	—
315	#121	—	2	2	no	—
316	#120	—	2	8	no	—
317	C₄×Q₈	—	2	2	no	—
318	C₄×Q₈	—	2	2	no	—
319	C₄×D₄	—	2	2	no	—
320	C₄×D₄	—	2	2	no	—
321	C₄×D₄	—	2	2	no	—
322	C₄×D₄	—	2	2	no	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	—	2	4	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
336	C₄⋊C₈	—	2	2	no	—
337	C₄⋊C₈	—	2	2	no	—
338	#104	—	2	8	no	—
339	#104	—	4	4	no	—
340	#103	—	2	4	no	—
341	#102	—	2	4	no	—
342	#100	—	2	4	no	—
343	#95	—	2	1	no	—
344	#95	—	2	1	no	—
345	#95	—	2	2	no	—
346	C₂⁴	—	2	4	no	—
347	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
348	#41	—	2	4	no	—
349	#41	—	2	4	no	—
350	#41	—	2	4	no	—
351	#41	—	2	4	no	—
352	#41	—	4	8	no	—
353	#41	—	4	8	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	C₂×D₄×C₆₁	1	4	yes	—
358	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	complemented
359	C₂×D₄	—	2	8	no	—
360	C₂×D₄	—	2	4	no	—
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	4	8	no	—
367	C₂×D₄	—	4	8	no	—
368	C₂×D₄	—	4	8	no	—
369	C₂×D₄	—	4	8	no	—
370	C₂×D₄	—	8	4	no	—
371	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
372	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
373	C₂×D₄	#41×C₆₁	1	2	yes	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	8	2	no	—
383	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
384	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
386	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
387	Q₁₆	—	4	8	no	—
388	SD₁₆	—	4	8	no	—
389	M₁₆	—	2	4	no	—
390	M₁₆	—	2	4	no	—
391	C₂×C₈	—	2	4	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	4	1	no	—
398	C₄⋊C₄	—	4	1	no	—
399	C₄⋊C₄	—	8	2	no	—
400	C₂²⋊C₄	—	2	8	no	—
401	C₂²⋊C₄	—	2	8	no	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₂²⋊C₄	—	8	2	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	4	2	no	—
409	C₄²	C₂×D₄×C₆₁	1	2	yes	—
410	C₄²	C₂×D₄×C₆₁	1	2	yes	—
411	C₄²	C₂×D₄×C₆₁	1	2	yes	complemented
412	C₂³	—	2	2	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	4	8	no	—
418	C₂³	—	8	4	no	—
419	C₂³	—	8	4	no	—
420	C₂³	#121×C₆₁	1	4	yes	—
421	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
422	Q₈	—	2	4	no	—
423	Q₈	—	2	4	no	—
424	Q₈	—	2	4	no	—
425	Q₈	—	4	2	no	—
426	D₄	—	2	4	no	—
427	D₄	—	2	4	no	—
428	D₄	—	2	4	no	—
429	D₄	—	2	4	no	—
430	D₄	—	4	4	no	—
431	D₄	—	4	16	no	—
432	D₄	—	4	4	no	—
433	D₄	—	4	4	no	—
434	D₄	—	8	2	no	—
435	D₄	—	8	4	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	4	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	8	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	4	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	8	2	no	—
457	C₂×C₄	#120×C₆₁	1	2	yes	—
458	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
459	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
461	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
462	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
463	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
464	C₈	—	4	4	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	16	8	no	—
476	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
477	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
478	C₂²	C₆₁×#535	1	1	yes	characteristic
479	C₄	—	2	1	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	2	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	4	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	2	no	—
486	C₄	—	8	1	no	—
487	C₄	—	8	1	no	—
488	C₄	D₄²×C₆₁	1	2	yes	—
489	C₄	C₆₁×#546	1	2	yes	—
490	C₂	—	2	1	no	—
491	C₂	—	2	1	no	—
492	C₂	—	4	4	no	—
493	C₂	—	8	4	no	—
494	C₂	C₆₁×#2408	1	1	yes	characteristic
495	C₂	C₆₁×#3019	1	1	yes	characteristic
496	C₂	C₆₁×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₆₁×#3496	C₂
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
2. C₆₁×#3037	C₂	semidirect
3. C₂×C₆₁×#455	C₂	semidirect
4. C₆₁×#2326	C₂	semidirect
5. C₆₁×#2039	C₂	semidirect
6. C₆₁×#2007	C₂	semidirect
7. C₆₁×#1963	C₂	semidirect
8. C₆₁×#1800	C₂	semidirect
9. C₆₁×#1792	C₂	semidirect
11. C₂×#137×C₆₁	C₂²	semidirect
20. C₂×C₄×Q₈×C₆₁	C₂²	semidirect
26. C₆₁×#423	C₂²	semidirect
27. C₂×#104×C₆₁	C₂²	semidirect
28. C₂×#104×C₆₁	C₂²	semidirect
29. C₆₁×#417	C₂²	semidirect
31. C₆₁×#391	C₂²	semidirect
35. C₆₁×#384	C₂²	semidirect
36. C₆₁×#376	C₂²	semidirect
37. C₆₁×#343	C₂²	semidirect
58. #124×C₆₁	D₄	semidirect
109. C₂×Q₈×C₆₁	C₂×D₄	semidirect
162. C₄²×C₆₁	C₂×D₄	semidirect
216. #24228	C₆₁	direct (N × H)
243. #3037	C₂×C₆₁	semidirect
244. C₂×#455	C₂×C₆₁	semidirect
245. #2326	C₂×C₆₁	semidirect
246. #2039	C₂×C₆₁	semidirect
247. #2007	C₂×C₆₁	semidirect
248. #1963	C₂×C₆₁	semidirect
249. #1800	C₂×C₆₁	semidirect
250. #1792	C₂×C₆₁	semidirect
259. C₂×#137	C₂²×C₆₁	semidirect
268. C₂×C₄×Q₈	C₂²×C₆₁	semidirect
274. #423	C₂²×C₆₁	semidirect
275. C₂×#104	C₂²×C₆₁	semidirect
276. C₂×#104	C₂²×C₆₁	semidirect
277. #417	C₂²×C₆₁	semidirect
279. #391	C₂²×C₆₁	semidirect
283. #384	C₂²×C₆₁	semidirect
284. #376	C₂²×C₆₁	semidirect
285. #343	C₂²×C₆₁	semidirect
286. C₆₁	#24228	direct (N × H)
307. #124	D₄×C₆₁	semidirect
358. C₂×Q₈	C₂×D₄×C₆₁	semidirect
411. 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×#3496	2
C₆₁×#3037	2
C₂×C₆₁×#455	2
C₆₁×#2326	2
C₆₁×#2039	2
C₆₁×#2007	2
C₆₁×#1963	2
C₆₁×#1800	2
C₆₁×#1792	2
#24228	61

Subgroups — C61×#24228 #29527661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24228 #29527661