Subgroups — C₆₁×#60432 #29563865

Order 15616.

2646 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#578	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×2₋¹⁺⁴×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3046	C₂	1	6	yes	maximal, complemented
5	C₆₁×#3044	C₂	1	6	yes	maximal, complemented
6	C₆₁×#3039	C₂	1	3	yes	maximal, complemented
7	C₆₁×#3037	C₂	1	3	yes	maximal, complemented
8	C₆₁×#3035	C₂	1	3	yes	maximal, complemented
9	C₂×C₆₁×#462	C₂	1	3	yes	maximal, complemented
10	C₆₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2898	C₂	1	2	yes	maximal, complemented
12	C₂×2₋¹⁺⁴×C₆₁	C₂²	1	2	yes	—
13	C₂×2₋¹⁺⁴×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#578	—	2	16	no	—
15	C₂×#137×C₆₁	C₂²	1	6	yes	—
16	C₂×#137×C₆₁	C₂²	1	6	yes	—
17	C₂×#135×C₆₁	C₂²	1	6	yes	—
18	C₂²×Q₁₆×C₆₁	C₂²	1	6	yes	—
19	C₂×#131×C₆₁	C₂²	1	2	yes	—
20	D₄×Q₈×C₆₁	—	2	1	no	—
21	D₄×Q₈×C₆₁	—	2	1	no	—
22	C₆₁×#547	—	2	3	no	—
23	C₆₁×#547	—	2	3	no	—
24	C₆₁×#536	C₂²	1	6	yes	—
25	C₆₁×#532	C₂²	1	3	yes	—
26	C₂×C₄⋊Q₈×C₆₁	C₂²	1	3	yes	—
27	C₆₁×#520	—	2	3	no	—
28	C₆₁×#520	C₂²	1	3	yes	—
29	C₆₁×#520	C₂²	1	1	yes	characteristic
30	C₆₁×#519	—	2	1	no	—
31	C₆₁×#519	C₂²	1	3	yes	—
32	C₄×#41×C₆₁	—	2	3	no	—
33	C₄×#41×C₆₁	C₂²	1	3	yes	—
34	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
36	C₂×C₄×Q₈×C₆₁	C₂²	1	3	yes	—
37	C₆₁×#464	C₂²	1	6	yes	—
38	C₆₁×#464	C₂²	1	12	yes	—
39	C₆₁×#463	C₂²	1	6	yes	—
40	C₆₁×#462	C₂²	1	3	yes	—
41	C₆₁×#462	C₂²	1	6	yes	—
42	C₆₁×#462	C₂²	1	12	yes	—
43	C₆₁×#462	C₂²	1	6	yes	—
44	C₆₁×#461	C₂²	1	3	yes	—
45	C₆₁×#461	C₂²	1	6	yes	—
46	C₆₁×#460	C₂²	1	3	yes	—
47	C₆₁×#459	C₂²	1	1	yes	characteristic
48	C₆₁×#424	C₂²	1	6	yes	—
49	C₆₁×#423	C₂²	1	3	yes	—
50	C₂×C₄⋊C₈×C₆₁	C₂²	1	3	yes	—
51	C₆₁×#419	C₂²	1	6	yes	—
52	C₆₁×#417	C₂²	1	6	yes	—
53	C₆₁×#416	C₂²	1	6	yes	—
54	C₂×#103×C₆₁	C₂²	1	6	yes	—
55	2₋¹⁺⁴×C₆₁	—	2	4	no	—
56	2₋¹⁺⁴×C₆₁	—	2	8	no	—
57	2₋¹⁺⁴×C₆₁	—	2	8	no	—
58	2₋¹⁺⁴×C₆₁	D₄	1	8	yes	complemented
59	C₂×#41×C₆₁	—	2	3	no	—
60	C₂×#41×C₆₁	—	2	6	no	—
61	C₂×#41×C₆₁	C₂³	1	6	yes	—
62	C₂×#41×C₆₁	C₂³	1	3	yes	—
63	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₆₁	—	2	1	no	—
65	C₂²×Q₈×C₆₁	—	2	2	no	—
66	C₂²×Q₈×C₆₁	C₂³	1	6	yes	—
67	C₂²×Q₈×C₆₁	C₂³	1	3	yes	—
68	#137×C₆₁	—	2	24	no	—
69	#137×C₆₁	—	2	24	no	—
70	#135×C₆₁	—	2	24	no	—
71	C₂×Q₁₆×C₆₁	—	2	24	no	—
72	C₂×Q₁₆×C₆₁	C₂³	1	6	yes	—
73	C₂×Q₁₆×C₆₁	C₂³	1	12	yes	—
74	C₂×SD₁₆×C₆₁	C₂³	1	6	yes	—
75	C₂×SD₁₆×C₆₁	C₂³	1	6	yes	—
76	C₂×D₈×C₆₁	C₂³	1	2	yes	—
77	#131×C₆₁	—	2	8	no	—
78	C₂×M₁₆×C₆₁	C₂³	1	6	yes	—
79	C₂²×C₈×C₆₁	C₂³	1	6	yes	—
80	C₄⋊Q₈×C₆₁	—	2	3	no	—
81	C₄⋊Q₈×C₆₁	—	2	3	no	—
82	C₄⋊Q₈×C₆₁	C₂³	1	3	yes	—
83	C₄⋊Q₈×C₆₁	C₂³	1	6	yes	—
84	#127×C₆₁	C₂³	1	1	yes	characteristic
85	#124×C₆₁	C₂³	1	6	yes	—
86	#123×C₆₁	—	2	12	no	—
87	C₂²⋊Q₈×C₆₁	—	2	6	no	—
88	C₂²⋊Q₈×C₆₁	—	2	6	no	—
89	#121×C₆₁	—	2	3	no	—
90	#121×C₆₁	—	2	3	no	—
91	C₄×Q₈×C₆₁	—	2	1	no	—
92	C₄×Q₈×C₆₁	—	2	3	no	—
93	C₄×Q₈×C₆₁	—	2	3	no	—
94	C₄×Q₈×C₆₁	—	2	3	no	—
95	C₄×Q₈×C₆₁	—	2	3	no	—
96	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
97	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
98	C₄×Q₈×C₆₁	C₂³	1	3	yes	—
99	C₄×Q₈×C₆₁	C₂³	1	6	yes	—
100	C₄×Q₈×C₆₁	C₂³	1	3	yes	—
101	C₄×D₄×C₆₁	—	2	3	no	—
102	C₄×D₄×C₆₁	—	2	3	no	—
103	C₄×D₄×C₆₁	—	2	3	no	—
104	C₄×D₄×C₆₁	—	2	6	no	—
105	C₄×D₄×C₆₁	—	2	3	no	—
106	C₄×D₄×C₆₁	—	2	3	no	—
107	C₄×D₄×C₆₁	D₄	1	3	yes	—
108	C₄×D₄×C₆₁	D₄	1	3	yes	—
109	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
110	C₄×D₄×C₆₁	C₂³	1	3	yes	—
111	#117×C₆₁	—	2	3	no	—
112	#117×C₆₁	—	2	6	no	—
113	#117×C₆₁	—	2	3	no	—
114	#117×C₆₁	C₂³	1	3	yes	—
115	#117×C₆₁	C₂³	1	3	yes	—
116	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
117	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
118	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
119	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
120	C₂×C₄⋊C₄×C₆₁	C₂³	1	3	yes	—
121	C₂×C₄²×C₆₁	—	2	3	no	—
122	C₂×C₄²×C₆₁	—	2	3	no	—
123	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
124	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
125	C₄⋊C₈×C₆₁	C₂³	1	3	yes	—
126	C₄⋊C₈×C₆₁	C₂³	1	6	yes	—
127	C₄⋊C₈×C₆₁	C₂³	1	6	yes	—
128	#103×C₆₁	C₂³	1	6	yes	—
129	#103×C₆₁	C₂³	1	12	yes	—
130	#103×C₆₁	C₂³	1	6	yes	—
131	#102×C₆₁	C₂³	1	2	yes	—
132	#102×C₆₁	C₂³	1	6	yes	—
133	#41×C₆₁	—	2	24	no	—
134	#41×C₆₁	—	2	6	no	—
135	#41×C₆₁	—	2	12	no	—
136	#41×C₆₁	—	2	12	no	—
137	#41×C₆₁	—	4	24	no	—
138	#41×C₆₁	C₂×D₄	1	12	yes	—
139	#41×C₆₁	C₂×D₄	1	4	yes	—
140	#41×C₆₁	#41	1	4	yes	—
141	C₂×Q₈×C₆₁	—	2	8	no	—
142	C₂×Q₈×C₆₁	—	2	3	no	—
143	C₂×Q₈×C₆₁	—	2	6	no	—
144	C₂×Q₈×C₆₁	—	2	6	no	—
145	C₂×Q₈×C₆₁	—	2	12	no	—
146	C₂×Q₈×C₆₁	—	2	12	no	—
147	C₂×Q₈×C₆₁	—	4	8	no	—
148	C₂×Q₈×C₆₁	C₂×D₄	1	12	yes	—
149	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
150	C₂×Q₈×C₆₁	C₂⁴	1	6	yes	—
151	C₂×Q₈×C₆₁	C₂⁴	1	3	yes	—
152	C₂×D₄×C₆₁	—	2	3	no	—
153	C₂×D₄×C₆₁	—	2	6	no	—
154	C₂×D₄×C₆₁	—	2	1	no	—
155	C₂×D₄×C₆₁	—	2	1	no	—
156	C₂×D₄×C₆₁	C₂×D₄	1	3	yes	—
157	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
158	C₂×D₄×C₆₁	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₆₁	—	2	3	no	—
160	C₂²×C₄×C₆₁	—	2	3	no	—
161	C₂²×C₄×C₆₁	—	2	6	no	—
162	C₂²×C₄×C₆₁	—	2	3	no	—
163	C₂²×C₄×C₆₁	—	2	1	no	—
164	C₂²×C₄×C₆₁	—	2	6	no	—
165	C₂²×C₄×C₆₁	—	2	3	no	—
166	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
167	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
168	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
169	Q₁₆×C₆₁	—	2	12	no	—
170	Q₁₆×C₆₁	—	2	24	no	—
171	SD₁₆×C₆₁	—	2	12	no	—
172	SD₁₆×C₆₁	—	2	12	no	—
173	D₈×C₆₁	—	2	4	no	—
174	M₁₆×C₆₁	—	2	12	no	—
175	C₂×C₈×C₆₁	—	2	12	no	—
176	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
177	C₂×C₈×C₆₁	C₂⁴	1	6	yes	—
178	C₄⋊C₄×C₆₁	—	2	6	no	—
179	C₄⋊C₄×C₆₁	—	2	3	no	—
180	C₄⋊C₄×C₆₁	—	2	3	no	—
181	C₄⋊C₄×C₆₁	—	2	3	no	—
182	C₄⋊C₄×C₆₁	—	2	3	no	—
183	C₄⋊C₄×C₆₁	—	2	6	no	—
184	C₄⋊C₄×C₆₁	—	2	6	no	—
185	C₄⋊C₄×C₆₁	—	2	3	no	—
186	C₄⋊C₄×C₆₁	C₂×D₄	1	3	yes	—
187	C₄⋊C₄×C₆₁	C₂×D₄	1	3	yes	—
188	C₄⋊C₄×C₆₁	C₂⁴	1	3	yes	—
189	C₄⋊C₄×C₆₁	C₂⁴	1	1	yes	characteristic
190	C₂²⋊C₄×C₆₁	—	2	3	no	—
191	C₂²⋊C₄×C₆₁	—	2	1	no	—
192	C₂²⋊C₄×C₆₁	—	2	3	no	—
193	C₂²⋊C₄×C₆₁	—	2	6	no	—
194	C₂²⋊C₄×C₆₁	—	2	6	no	—
195	C₂²⋊C₄×C₆₁	C₂×D₄	1	6	yes	—
196	C₄²×C₆₁	—	2	3	no	—
197	C₄²×C₆₁	—	2	3	no	—
198	C₄²×C₆₁	—	2	6	no	—
199	C₄²×C₆₁	—	2	3	no	—
200	C₄²×C₆₁	C₂×D₄	1	3	yes	—
201	C₄²×C₆₁	C₂×D₄	1	3	yes	—
202	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
203	C₄²×C₆₁	C₂⁴	1	3	yes	—
204	C₂³×C₆₁	—	2	1	no	—
205	C₂³×C₆₁	—	2	2	no	—
206	C₂³×C₆₁	C₂²×D₄	1	3	yes	—
207	Q₈×C₆₁	—	2	12	no	—
208	Q₈×C₆₁	—	2	3	no	—
209	Q₈×C₆₁	—	2	6	no	—
210	Q₈×C₆₁	—	2	6	no	—
211	Q₈×C₆₁	—	4	12	no	—
212	Q₈×C₆₁	#121	1	4	yes	—
213	Q₈×C₆₁	C₂²×D₄	1	6	yes	—
214	D₄×C₆₁	—	2	12	no	—
215	D₄×C₆₁	—	2	2	no	—
216	D₄×C₆₁	—	2	1	no	—
217	D₄×C₆₁	—	2	2	no	—
218	D₄×C₆₁	—	4	2	no	—
219	D₄×C₆₁	—	4	12	no	—
220	D₄×C₆₁	—	4	2	no	—
221	D₄×C₆₁	#121	1	12	yes	—
222	D₄×C₆₁	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₆₁	—	2	12	no	—
224	C₂×C₄×C₆₁	—	2	3	no	—
225	C₂×C₄×C₆₁	—	2	12	no	—
226	C₂×C₄×C₆₁	—	2	6	no	—
227	C₂×C₄×C₆₁	—	2	1	no	—
228	C₂×C₄×C₆₁	—	2	3	no	—
229	C₂×C₄×C₆₁	—	2	6	no	—
230	C₂×C₄×C₆₁	—	4	3	no	—
231	C₂×C₄×C₆₁	—	4	12	no	—
232	C₂×C₄×C₆₁	—	4	12	no	—
233	C₂×C₄×C₆₁	—	4	2	no	—
234	C₂×C₄×C₆₁	—	8	3	no	—
235	C₂×C₄×C₆₁	—	8	3	no	—
236	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
237	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
238	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
240	C₂×C₄×C₆₁	C₂²×D₄	1	6	yes	—
241	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₆₁	C₂×#41	1	6	yes	—
243	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
244	C₈×C₆₁	—	2	2	no	—
245	C₈×C₆₁	—	2	6	no	—
246	#60432	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂²×C₆₁	—	2	3	no	—
248	C₂²×C₆₁	—	2	3	no	—
249	C₂²×C₆₁	—	2	2	no	—
250	C₂²×C₆₁	—	4	1	no	—
251	C₂²×C₆₁	—	4	1	no	—
252	C₂²×C₆₁	—	4	2	no	—
253	C₂²×C₆₁	—	4	2	no	—
254	C₂²×C₆₁	—	4	2	no	—
255	C₂²×C₆₁	C₂×#121	1	6	yes	—
256	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
257	C₄×C₆₁	—	2	1	no	—
258	C₄×C₆₁	—	2	3	no	—
259	C₄×C₆₁	—	2	6	no	—
260	C₄×C₆₁	—	2	3	no	—
261	C₄×C₆₁	—	4	3	no	—
262	C₄×C₆₁	—	4	6	no	—
263	C₄×C₆₁	—	4	1	no	—
264	C₄×C₆₁	C₂×#121	1	6	yes	—
265	C₄×C₆₁	#578	1	2	yes	complemented
266	C₄×C₆₁	C₂³×D₄	1	1	yes	characteristic
267	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
268	C₂×#578	C₂×C₆₁	1	2	yes	complemented
269	#3499	C₂×C₆₁	1	1	yes	characteristic, complemented
270	C₄×2₋¹⁺⁴	C₂×C₆₁	1	1	yes	characteristic, complemented
271	#3046	C₂×C₆₁	1	6	yes	complemented
272	#3044	C₂×C₆₁	1	6	yes	complemented
273	#3039	C₂×C₆₁	1	3	yes	complemented
274	#3037	C₂×C₆₁	1	3	yes	complemented
275	#3035	C₂×C₆₁	1	3	yes	complemented
276	C₂×#462	C₂×C₆₁	1	3	yes	complemented
277	#2906	C₂×C₆₁	1	1	yes	characteristic, complemented
278	#2898	C₂×C₆₁	1	2	yes	complemented
279	C₂×C₆₁	—	2	6	no	—
280	C₂×C₆₁	—	4	2	no	—
281	C₂×C₆₁	—	8	2	no	—
282	C₂×C₆₁	C₂²×#121	1	1	yes	characteristic
283	C₂×C₆₁	C₂×#578	1	1	yes	characteristic
284	C₂×C₆₁	#3586	1	1	yes	characteristic
285	C₂×2₋¹⁺⁴	C₂²×C₆₁	1	2	yes	—
286	C₂×2₋¹⁺⁴	C₂²×C₆₁	1	1	yes	characteristic
287	#578	—	2	16	no	—
288	C₂×#137	C₂²×C₆₁	1	6	yes	—
289	C₂×#137	C₂²×C₆₁	1	6	yes	—
290	C₂×#135	C₂²×C₆₁	1	6	yes	—
291	C₂²×Q₁₆	C₂²×C₆₁	1	6	yes	—
292	C₂×#131	C₂²×C₆₁	1	2	yes	—
293	D₄×Q₈	—	2	1	no	—
294	D₄×Q₈	—	2	1	no	—
295	#547	—	2	3	no	—
296	#547	—	2	3	no	—
297	#536	C₂²×C₆₁	1	6	yes	—
298	#532	C₂²×C₆₁	1	3	yes	—
299	C₂×C₄⋊Q₈	C₂²×C₆₁	1	3	yes	—
300	#520	—	2	3	no	—
301	#520	C₂²×C₆₁	1	3	yes	—
302	#520	C₂²×C₆₁	1	1	yes	characteristic
303	#519	—	2	1	no	—
304	#519	C₂²×C₆₁	1	3	yes	—
305	C₄×#41	—	2	3	no	—
306	C₄×#41	C₂²×C₆₁	1	3	yes	—
307	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
308	C₂×C₄×Q₈	—	2	1	no	—
309	C₂×C₄×Q₈	C₂²×C₆₁	1	3	yes	—
310	#464	C₂²×C₆₁	1	6	yes	—
311	#464	C₂²×C₆₁	1	12	yes	—
312	#463	C₂²×C₆₁	1	6	yes	—
313	#462	C₂²×C₆₁	1	3	yes	—
314	#462	C₂²×C₆₁	1	6	yes	—
315	#462	C₂²×C₆₁	1	12	yes	—
316	#462	C₂²×C₆₁	1	6	yes	—
317	#461	C₂²×C₆₁	1	3	yes	—
318	#461	C₂²×C₆₁	1	6	yes	—
319	#460	C₂²×C₆₁	1	3	yes	—
320	#459	C₂²×C₆₁	1	1	yes	characteristic
321	#424	C₂²×C₆₁	1	6	yes	—
322	#423	C₂²×C₆₁	1	3	yes	—
323	C₂×C₄⋊C₈	C₂²×C₆₁	1	3	yes	—
324	#419	C₂²×C₆₁	1	6	yes	—
325	#417	C₂²×C₆₁	1	6	yes	—
326	#416	C₂²×C₆₁	1	6	yes	—
327	C₂×#103	C₂²×C₆₁	1	6	yes	—
328	C₆₁	#60432	1	1	yes	characteristic, direct product factor, Sylow
329	2₋¹⁺⁴	—	2	4	no	—
330	2₋¹⁺⁴	—	2	8	no	—
331	2₋¹⁺⁴	—	2	8	no	—
332	2₋¹⁺⁴	D₄×C₆₁	1	8	yes	complemented
333	C₂×#41	—	2	3	no	—
334	C₂×#41	—	2	6	no	—
335	C₂×#41	C₂³×C₆₁	1	6	yes	—
336	C₂×#41	C₂³×C₆₁	1	3	yes	—
337	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
338	C₂²×Q₈	—	2	1	no	—
339	C₂²×Q₈	—	2	2	no	—
340	C₂²×Q₈	C₂³×C₆₁	1	6	yes	—
341	C₂²×Q₈	C₂³×C₆₁	1	3	yes	—
342	#137	—	2	24	no	—
343	#137	—	2	24	no	—
344	#135	—	2	24	no	—
345	C₂×Q₁₆	—	2	24	no	—
346	C₂×Q₁₆	C₂³×C₆₁	1	6	yes	—
347	C₂×Q₁₆	C₂³×C₆₁	1	12	yes	—
348	C₂×SD₁₆	C₂³×C₆₁	1	6	yes	—
349	C₂×SD₁₆	C₂³×C₆₁	1	6	yes	—
350	C₂×D₈	C₂³×C₆₁	1	2	yes	—
351	#131	—	2	8	no	—
352	C₂×M₁₆	C₂³×C₆₁	1	6	yes	—
353	C₂²×C₈	C₂³×C₆₁	1	6	yes	—
354	C₄⋊Q₈	—	2	3	no	—
355	C₄⋊Q₈	—	2	3	no	—
356	C₄⋊Q₈	C₂³×C₆₁	1	3	yes	—
357	C₄⋊Q₈	C₂³×C₆₁	1	6	yes	—
358	#127	C₂³×C₆₁	1	1	yes	characteristic
359	#124	C₂³×C₆₁	1	6	yes	—
360	#123	—	2	12	no	—
361	C₂²⋊Q₈	—	2	6	no	—
362	C₂²⋊Q₈	—	2	6	no	—
363	#121	—	2	3	no	—
364	#121	—	2	3	no	—
365	C₄×Q₈	—	2	1	no	—
366	C₄×Q₈	—	2	3	no	—
367	C₄×Q₈	—	2	3	no	—
368	C₄×Q₈	—	2	3	no	—
369	C₄×Q₈	—	2	3	no	—
370	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
371	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
372	C₄×Q₈	C₂³×C₆₁	1	3	yes	—
373	C₄×Q₈	C₂³×C₆₁	1	6	yes	—
374	C₄×Q₈	C₂³×C₆₁	1	3	yes	—
375	C₄×D₄	—	2	3	no	—
376	C₄×D₄	—	2	3	no	—
377	C₄×D₄	—	2	3	no	—
378	C₄×D₄	—	2	6	no	—
379	C₄×D₄	—	2	3	no	—
380	C₄×D₄	—	2	3	no	—
381	C₄×D₄	D₄×C₆₁	1	3	yes	—
382	C₄×D₄	D₄×C₆₁	1	3	yes	—
383	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
384	C₄×D₄	C₂³×C₆₁	1	3	yes	—
385	#117	—	2	3	no	—
386	#117	—	2	6	no	—
387	#117	—	2	3	no	—
388	#117	C₂³×C₆₁	1	3	yes	—
389	#117	C₂³×C₆₁	1	3	yes	—
390	C₂×C₄⋊C₄	—	2	3	no	—
391	C₂×C₄⋊C₄	—	2	3	no	—
392	C₂×C₄⋊C₄	—	2	3	no	—
393	C₂×C₄⋊C₄	—	2	3	no	—
394	C₂×C₄⋊C₄	C₂³×C₆₁	1	3	yes	—
395	C₂×C₄²	—	2	3	no	—
396	C₂×C₄²	—	2	3	no	—
397	C₂×C₄²	C₂³×C₆₁	1	3	yes	—
398	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
399	C₄⋊C₈	C₂³×C₆₁	1	3	yes	—
400	C₄⋊C₈	C₂³×C₆₁	1	6	yes	—
401	C₄⋊C₈	C₂³×C₆₁	1	6	yes	—
402	#103	C₂³×C₆₁	1	6	yes	—
403	#103	C₂³×C₆₁	1	12	yes	—
404	#103	C₂³×C₆₁	1	6	yes	—
405	#102	C₂³×C₆₁	1	2	yes	—
406	#102	C₂³×C₆₁	1	6	yes	—
407	#41	—	2	24	no	—
408	#41	—	2	6	no	—
409	#41	—	2	12	no	—
410	#41	—	2	12	no	—
411	#41	—	4	24	no	—
412	#41	C₂×D₄×C₆₁	1	12	yes	—
413	#41	C₂×D₄×C₆₁	1	4	yes	—
414	#41	#41×C₆₁	1	4	yes	—
415	C₂×Q₈	—	2	8	no	—
416	C₂×Q₈	—	2	3	no	—
417	C₂×Q₈	—	2	6	no	—
418	C₂×Q₈	—	2	6	no	—
419	C₂×Q₈	—	2	12	no	—
420	C₂×Q₈	—	2	12	no	—
421	C₂×Q₈	—	4	8	no	—
422	C₂×Q₈	C₂×D₄×C₆₁	1	12	yes	—
423	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
424	C₂×Q₈	C₂⁴×C₆₁	1	6	yes	—
425	C₂×Q₈	C₂⁴×C₆₁	1	3	yes	—
426	C₂×D₄	—	2	3	no	—
427	C₂×D₄	—	2	6	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	C₂×D₄×C₆₁	1	3	yes	—
431	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
432	C₂×D₄	C₂⁴×C₆₁	1	1	yes	characteristic
433	C₂²×C₄	—	2	3	no	—
434	C₂²×C₄	—	2	3	no	—
435	C₂²×C₄	—	2	6	no	—
436	C₂²×C₄	—	2	3	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	6	no	—
439	C₂²×C₄	—	2	3	no	—
440	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
441	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
442	C₂²×C₄	C₂⁴×C₆₁	1	3	yes	—
443	Q₁₆	—	2	12	no	—
444	Q₁₆	—	2	24	no	—
445	SD₁₆	—	2	12	no	—
446	SD₁₆	—	2	12	no	—
447	D₈	—	2	4	no	—
448	M₁₆	—	2	12	no	—
449	C₂×C₈	—	2	12	no	—
450	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
451	C₂×C₈	C₂⁴×C₆₁	1	6	yes	—
452	C₄⋊C₄	—	2	6	no	—
453	C₄⋊C₄	—	2	3	no	—
454	C₄⋊C₄	—	2	3	no	—
455	C₄⋊C₄	—	2	3	no	—
456	C₄⋊C₄	—	2	3	no	—
457	C₄⋊C₄	—	2	6	no	—
458	C₄⋊C₄	—	2	6	no	—
459	C₄⋊C₄	—	2	3	no	—
460	C₄⋊C₄	C₂×D₄×C₆₁	1	3	yes	—
461	C₄⋊C₄	C₂×D₄×C₆₁	1	3	yes	—
462	C₄⋊C₄	C₂⁴×C₆₁	1	3	yes	—
463	C₄⋊C₄	C₂⁴×C₆₁	1	1	yes	characteristic
464	C₂²⋊C₄	—	2	3	no	—
465	C₂²⋊C₄	—	2	1	no	—
466	C₂²⋊C₄	—	2	3	no	—
467	C₂²⋊C₄	—	2	6	no	—
468	C₂²⋊C₄	—	2	6	no	—
469	C₂²⋊C₄	C₂×D₄×C₆₁	1	6	yes	—
470	C₄²	—	2	3	no	—
471	C₄²	—	2	3	no	—
472	C₄²	—	2	6	no	—
473	C₄²	—	2	3	no	—
474	C₄²	C₂×D₄×C₆₁	1	3	yes	—
475	C₄²	C₂×D₄×C₆₁	1	3	yes	—
476	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
477	C₄²	C₂⁴×C₆₁	1	3	yes	—
478	C₂³	—	2	1	no	—
479	C₂³	—	2	2	no	—
480	C₂³	C₂²×D₄×C₆₁	1	3	yes	—
481	Q₈	—	2	3	no	—
482	Q₈	—	2	12	no	—
483	Q₈	—	2	6	no	—
484	Q₈	—	2	6	no	—
485	Q₈	—	4	12	no	—
486	Q₈	#121×C₆₁	1	4	yes	—
487	Q₈	C₂²×D₄×C₆₁	1	6	yes	—
488	D₄	—	2	12	no	—
489	D₄	—	2	2	no	—
490	D₄	—	2	1	no	—
491	D₄	—	2	2	no	—
492	D₄	—	4	2	no	—
493	D₄	—	4	12	no	—
494	D₄	—	4	2	no	—
495	D₄	#121×C₆₁	1	12	yes	—
496	D₄	C₂²×D₄×C₆₁	1	2	yes	—
497	C₂×C₄	—	2	3	no	—
498	C₂×C₄	—	2	12	no	—
499	C₂×C₄	—	2	6	no	—
500	C₂×C₄	—	2	12	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	3	no	—
503	C₂×C₄	—	2	6	no	—
504	C₂×C₄	—	4	3	no	—
505	C₂×C₄	—	4	12	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	12	no	—
508	C₂×C₄	—	8	3	no	—
509	C₂×C₄	—	8	3	no	—
510	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
511	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
512	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
513	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
514	C₂×C₄	C₂²×D₄×C₆₁	1	6	yes	—
515	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
516	C₂×C₄	C₂×#41×C₆₁	1	6	yes	—
517	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
518	C₈	—	2	2	no	—
519	C₈	—	2	6	no	—
520	C₂²	—	2	3	no	—
521	C₂²	—	2	3	no	—
522	C₂²	—	2	2	no	—
523	C₂²	—	4	1	no	—
524	C₂²	—	4	1	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	C₂×#121×C₆₁	1	6	yes	—
529	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
530	C₄	—	2	3	no	—
531	C₄	—	2	1	no	—
532	C₄	—	2	3	no	—
533	C₄	—	2	6	no	—
534	C₄	—	4	1	no	—
535	C₄	—	4	3	no	—
536	C₄	—	4	6	no	—
537	C₄	C₂×#121×C₆₁	1	6	yes	—
538	C₄	C₆₁×#578	1	2	yes	complemented
539	C₄	C₂³×D₄×C₆₁	1	1	yes	characteristic
540	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
541	C₂	—	2	6	no	—
542	C₂	—	4	2	no	—
543	C₂	—	8	2	no	—
544	C₂	C₂²×#121×C₆₁	1	1	yes	characteristic
545	C₂	C₂×C₆₁×#578	1	1	yes	characteristic
546	C₂	C₆₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×#41×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#578	C₂	semidirect
2. C₆₁×#3499	C₂	semidirect
3. C₄×2₋¹⁺⁴×C₆₁	C₂	semidirect
4. C₆₁×#3046	C₂	semidirect
5. C₆₁×#3044	C₂	semidirect
6. C₆₁×#3039	C₂	semidirect
7. C₆₁×#3037	C₂	semidirect
8. C₆₁×#3035	C₂	semidirect
9. C₂×C₆₁×#462	C₂	semidirect
10. C₆₁×#2906	C₂	semidirect
11. C₆₁×#2898	C₂	semidirect
58. 2₋¹⁺⁴×C₆₁	D₄	semidirect
246. #60432	C₆₁	direct (N × H)
265. C₄×C₆₁	#578	semidirect
268. C₂×#578	C₂×C₆₁	semidirect
269. #3499	C₂×C₆₁	semidirect
270. C₄×2₋¹⁺⁴	C₂×C₆₁	semidirect
271. #3046	C₂×C₆₁	semidirect
272. #3044	C₂×C₆₁	semidirect
273. #3039	C₂×C₆₁	semidirect
274. #3037	C₂×C₆₁	semidirect
275. #3035	C₂×C₆₁	semidirect
276. C₂×#462	C₂×C₆₁	semidirect
277. #2906	C₂×C₆₁	semidirect
278. #2898	C₂×C₆₁	semidirect
328. C₆₁	#60432	direct (N × H)
332. 2₋¹⁺⁴	D₄×C₆₁	semidirect
538. C₄	C₆₁×#578	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#578	2
C₆₁×#3499	2
C₄×2₋¹⁺⁴×C₆₁	2
C₆₁×#3046	2
C₆₁×#3044	2
C₆₁×#3039	2
C₆₁×#3037	2
C₆₁×#3035	2
C₂×C₆₁×#462	2
C₆₁×#2906	2
C₆₁×#2898	2
#60432	61

Subgroups — C61×#60432 #29563865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#60432 #29563865