Subgroups — C₆₁×#33350 #29536783

Order 15616.

1240 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3485	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2883	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2123	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2121	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2120	C₂	1	2	yes	maximal, complemented
7	C₂×C₆₁×#352	C₂	1	4	yes	maximal, complemented
8	C₂×H(5,𝔽₂)×C₆₁	C₂²	1	1	yes	characteristic
9	C₂²×#41×C₆₁	C₄	1	1	yes	characteristic, complemented
10	C₆₁×#568	—	2	1	no	—
11	C₆₁×#547	—	2	1	no	—
12	C₆₁×#547	—	2	2	no	—
13	C₆₁×#547	—	2	1	no	—
14	C₆₁×#546	—	2	1	no	—
15	C₆₁×#546	—	2	2	no	—
16	C₆₁×#546	—	2	1	no	—
17	C₆₁×#536	C₄	1	1	yes	characteristic, complemented
18	C₆₁×#536	C₂²	1	1	yes	characteristic
19	C₆₁×#534	C₄	1	2	yes	complemented
20	C₆₁×#534	C₂²	1	2	yes	—
21	C₆₁×#525	C₄	1	1	yes	characteristic, complemented
22	C₆₁×#525	C₂²	1	1	yes	characteristic
23	C₂×#123×C₆₁	C₄	1	2	yes	complemented
24	C₂×#123×C₆₁	C₂²	1	2	yes	—
25	C₆₁×#518	—	2	1	no	—
26	C₆₁×#518	C₄	1	1	yes	characteristic, complemented
27	C₆₁×#413	—	2	2	no	—
28	C₆₁×#413	C₂²	1	2	yes	—
29	C₂×#100×C₆₁	—	2	1	no	—
30	C₂×#100×C₆₁	C₂²	1	4	yes	—
31	C₆₁×#410	—	2	1	no	—
32	C₆₁×#410	C₂²	1	2	yes	—
33	C₂×#99×C₆₁	—	2	2	no	—
34	C₂×#99×C₆₁	C₂²	1	4	yes	—
35	C₆₁×#352	C₂²	1	4	yes	complemented
36	C₆₁×#352	C₂²	1	8	yes	complemented
37	C₆₁×AΣL(1,𝔽₁₆)	C₂²	1	4	yes	complemented
38	H(5,𝔽₂)×C₆₁	—	4	4	no	—
39	C₂×#41×C₆₁	—	2	1	no	—
40	C₂×#41×C₆₁	—	2	2	no	—
41	C₂×#41×C₆₁	—	2	1	no	—
42	C₂×#41×C₆₁	—	4	4	no	—
43	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×#41×C₆₁	D₄	1	2	yes	—
45	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
46	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
47	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
49	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
50	C₂²×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	complemented
52	C₂²×D₄×C₆₁	D₄	1	2	yes	—
53	C₂²×D₄×C₆₁	D₄	1	1	yes	characteristic
54	C₂²×D₄×C₆₁	D₄	1	1	yes	characteristic
55	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	complemented
57	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
58	#131×C₆₁	—	4	4	no	—
59	C₂×M₁₆×C₆₁	—	2	2	no	—
60	C₂×M₁₆×C₆₁	—	2	4	no	—
61	C₄⋊Q₈×C₆₁	—	2	1	no	—
62	C₄⋊Q₈×C₆₁	—	2	1	no	—
63	#124×C₆₁	—	2	1	no	—
64	#124×C₆₁	—	2	1	no	—
65	#123×C₆₁	—	2	2	no	—
66	#123×C₆₁	—	2	2	no	—
67	#123×C₆₁	—	2	2	no	—
68	#123×C₆₁	—	2	2	no	—
69	#123×C₆₁	—	2	2	no	—
70	#123×C₆₁	—	2	2	no	—
71	#123×C₆₁	C₂×C₄	1	2	yes	complemented
72	#123×C₆₁	C₂×C₄	1	2	yes	complemented
73	#123×C₆₁	C₂×C₄	1	2	yes	complemented
74	#123×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
75	#123×C₆₁	C₂×C₄	1	2	yes	complemented
76	#123×C₆₁	C₂³	1	2	yes	—
77	#123×C₆₁	C₂³	1	1	yes	characteristic
78	C₂²⋊Q₈×C₆₁	—	2	1	no	—
79	C₂²⋊Q₈×C₆₁	—	2	1	no	—
80	C₂²⋊Q₈×C₆₁	—	2	2	no	—
81	C₂²⋊Q₈×C₆₁	—	2	1	no	—
82	C₂²⋊Q₈×C₆₁	—	2	1	no	—
83	#121×C₆₁	—	2	2	no	—
84	#121×C₆₁	—	2	2	no	—
85	#121×C₆₁	—	2	1	no	—
86	#121×C₆₁	—	2	2	no	—
87	#121×C₆₁	—	2	2	no	—
88	#121×C₆₁	—	2	1	no	—
89	#120×C₆₁	—	2	2	no	—
90	#120×C₆₁	—	2	2	no	—
91	#120×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
92	#120×C₆₁	C₂×C₄	1	2	yes	complemented
93	#120×C₆₁	C₂³	1	1	yes	characteristic
94	C₄×D₄×C₆₁	—	2	2	no	—
95	C₄×D₄×C₆₁	—	2	1	no	—
96	C₄×D₄×C₆₁	—	2	1	no	—
97	C₄×D₄×C₆₁	—	2	2	no	—
98	C₄×D₄×C₆₁	—	2	1	no	—
99	C₄×D₄×C₆₁	—	2	1	no	—
100	C₄×D₄×C₆₁	—	4	4	no	—
101	#117×C₆₁	—	2	2	no	—
102	#117×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
103	#117×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
104	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
108	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
109	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	complemented
110	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	complemented
111	#101×C₆₁	—	2	2	no	—
112	#100×C₆₁	—	2	4	no	—
113	#100×C₆₁	C₂³	1	4	yes	—
114	#99×C₆₁	—	2	2	no	—
115	#99×C₆₁	—	2	4	no	—
116	#99×C₆₁	C₂³	1	4	yes	—
117	C₂⁴×C₆₁	C₂²⋊C₄	1	2	yes	—
118	C₂⁴×C₆₁	C₂²×C₄	1	2	yes	—
119	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
120	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
121	#41×C₆₁	—	2	4	no	—
122	#41×C₆₁	—	4	2	no	—
123	#41×C₆₁	—	4	4	no	—
124	#41×C₆₁	—	4	2	no	—
125	#41×C₆₁	—	4	2	no	—
126	#41×C₆₁	—	4	8	no	—
127	#41×C₆₁	—	4	4	no	—
128	#41×C₆₁	—	4	4	no	—
129	C₂×Q₈×C₆₁	—	2	1	no	—
130	C₂×Q₈×C₆₁	—	2	1	no	—
131	C₂×Q₈×C₆₁	—	4	2	no	—
132	C₂×Q₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
133	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
134	C₂×D₄×C₆₁	—	2	4	no	—
135	C₂×D₄×C₆₁	—	2	4	no	—
136	C₂×D₄×C₆₁	—	2	2	no	—
137	C₂×D₄×C₆₁	—	2	2	no	—
138	C₂×D₄×C₆₁	—	2	2	no	—
139	C₂×D₄×C₆₁	—	2	1	no	—
140	C₂×D₄×C₆₁	—	2	1	no	—
141	C₂×D₄×C₆₁	—	4	2	no	—
142	C₂×D₄×C₆₁	—	4	4	no	—
143	C₂×D₄×C₆₁	—	4	4	no	—
144	C₂×D₄×C₆₁	—	4	4	no	—
145	C₂×D₄×C₆₁	—	4	4	no	—
146	C₂×D₄×C₆₁	C₂²⋊C₄	1	2	yes	—
147	C₂×D₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
148	C₂×D₄×C₆₁	C₂²⋊C₄	1	2	yes	—
149	C₂×D₄×C₆₁	C₂²⋊C₄	1	2	yes	—
150	C₂×D₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
151	C₂×D₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
152	C₂×D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
153	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
154	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
155	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
156	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
157	C₂×D₄×C₆₁	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₆₁	—	2	2	no	—
159	C₂²×C₄×C₆₁	—	2	2	no	—
160	C₂²×C₄×C₆₁	—	2	2	no	—
161	C₂²×C₄×C₆₁	—	2	2	no	—
162	C₂²×C₄×C₆₁	—	2	1	no	—
163	C₂²×C₄×C₆₁	—	2	1	no	—
164	C₂²×C₄×C₆₁	—	4	2	no	—
165	C₂²×C₄×C₆₁	—	4	4	no	—
166	C₂²×C₄×C₆₁	—	4	4	no	—
167	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
168	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
169	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
172	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
175	M₁₆×C₆₁	—	2	4	no	—
176	M₁₆×C₆₁	—	4	4	no	—
177	M₁₆×C₆₁	—	4	2	no	—
178	C₂×C₈×C₆₁	—	4	4	no	—
179	C₂×C₈×C₆₁	—	4	2	no	—
180	C₄⋊C₄×C₆₁	—	2	1	no	—
181	C₄⋊C₄×C₆₁	—	2	2	no	—
182	C₄⋊C₄×C₆₁	—	2	2	no	—
183	C₄⋊C₄×C₆₁	—	2	2	no	—
184	C₄⋊C₄×C₆₁	—	2	2	no	—
185	C₄⋊C₄×C₆₁	—	2	1	no	—
186	C₄⋊C₄×C₆₁	—	2	1	no	—
187	C₄⋊C₄×C₆₁	—	2	1	no	—
188	C₄⋊C₄×C₆₁	—	2	2	no	—
189	C₄⋊C₄×C₆₁	—	4	2	no	—
190	C₂²⋊C₄×C₆₁	—	2	2	no	—
191	C₂²⋊C₄×C₆₁	—	2	1	no	—
192	C₂²⋊C₄×C₆₁	—	2	2	no	—
193	C₂²⋊C₄×C₆₁	—	2	2	no	—
194	C₂²⋊C₄×C₆₁	—	2	2	no	—
195	C₂²⋊C₄×C₆₁	—	2	2	no	—
196	C₂²⋊C₄×C₆₁	—	2	2	no	—
197	C₂²⋊C₄×C₆₁	—	2	1	no	—
198	C₂²⋊C₄×C₆₁	—	2	2	no	—
199	C₂²⋊C₄×C₆₁	—	2	1	no	—
200	C₂²⋊C₄×C₆₁	—	2	2	no	—
201	C₂²⋊C₄×C₆₁	—	2	1	no	—
202	C₂²⋊C₄×C₆₁	—	2	4	no	—
203	C₂²⋊C₄×C₆₁	—	4	4	no	—
204	C₂²⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
205	C₂²⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
206	C₂²⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
207	C₄²×C₆₁	—	2	1	no	—
208	C₄²×C₆₁	—	2	1	no	—
209	C₄²×C₆₁	—	4	2	no	—
210	C₂³×C₆₁	—	2	4	no	—
211	C₂³×C₆₁	—	2	4	no	—
212	C₂³×C₆₁	—	2	1	no	—
213	C₂³×C₆₁	—	4	2	no	—
214	C₂³×C₆₁	—	4	1	no	—
215	C₂³×C₆₁	—	8	2	no	—
216	C₂³×C₆₁	—	8	1	no	—
217	C₂³×C₆₁	—	8	2	no	—
218	C₂³×C₆₁	—	8	2	no	—
219	C₂³×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
220	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
221	C₂³×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
222	C₂³×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
223	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
224	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₆₁	—	2	2	no	—
226	Q₈×C₆₁	—	4	1	no	—
227	Q₈×C₆₁	—	4	2	no	—
228	Q₈×C₆₁	—	4	2	no	—
229	D₄×C₆₁	—	2	4	no	—
230	D₄×C₆₁	—	2	4	no	—
231	D₄×C₆₁	—	2	4	no	—
232	D₄×C₆₁	—	2	2	no	—
233	D₄×C₆₁	—	2	2	no	—
234	D₄×C₆₁	—	4	1	no	—
235	D₄×C₆₁	—	4	2	no	—
236	D₄×C₆₁	—	4	2	no	—
237	D₄×C₆₁	—	4	1	no	—
238	D₄×C₆₁	—	4	2	no	—
239	D₄×C₆₁	—	4	1	no	—
240	D₄×C₆₁	—	4	4	no	—
241	D₄×C₆₁	—	4	4	no	—
242	D₄×C₆₁	—	8	2	no	—
243	D₄×C₆₁	—	8	1	no	—
244	D₄×C₆₁	—	8	1	no	—
245	D₄×C₆₁	#99	1	4	yes	—
246	C₂×C₄×C₆₁	—	2	4	no	—
247	C₂×C₄×C₆₁	—	2	2	no	—
248	C₂×C₄×C₆₁	—	2	2	no	—
249	C₂×C₄×C₆₁	—	2	2	no	—
250	C₂×C₄×C₆₁	—	2	1	no	—
251	C₂×C₄×C₆₁	—	2	2	no	—
252	C₂×C₄×C₆₁	—	2	1	no	—
253	C₂×C₄×C₆₁	—	2	1	no	—
254	C₂×C₄×C₆₁	—	2	2	no	—
255	C₂×C₄×C₆₁	—	4	2	no	—
256	C₂×C₄×C₆₁	—	4	2	no	—
257	C₂×C₄×C₆₁	—	4	2	no	—
258	C₂×C₄×C₆₁	—	4	4	no	—
259	C₂×C₄×C₆₁	—	4	4	no	—
260	C₂×C₄×C₆₁	—	4	2	no	—
261	C₂×C₄×C₆₁	—	4	1	no	—
262	C₂×C₄×C₆₁	—	4	2	no	—
263	C₂×C₄×C₆₁	—	4	4	no	—
264	C₂×C₄×C₆₁	—	8	2	no	—
265	C₂×C₄×C₆₁	—	8	2	no	—
266	C₂×C₄×C₆₁	—	16	4	no	—
267	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
270	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
272	C₈×C₆₁	—	4	4	no	—
273	#33350	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂²×C₆₁	—	2	1	no	—
275	C₂²×C₆₁	—	2	1	no	—
276	C₂²×C₆₁	—	2	1	no	—
277	C₂²×C₆₁	—	2	2	no	—
278	C₂²×C₆₁	—	2	2	no	—
279	C₂²×C₆₁	—	4	2	no	—
280	C₂²×C₆₁	—	4	2	no	—
281	C₂²×C₆₁	—	4	2	no	—
282	C₂²×C₆₁	—	4	1	no	—
283	C₂²×C₆₁	—	4	1	no	—
284	C₂²×C₆₁	—	4	1	no	—
285	C₂²×C₆₁	—	4	1	no	—
286	C₂²×C₆₁	—	4	1	no	—
287	C₂²×C₆₁	—	4	1	no	—
288	C₂²×C₆₁	—	8	4	no	—
289	C₂²×C₆₁	—	8	4	no	—
290	C₂²×C₆₁	—	8	2	no	—
291	C₂²×C₆₁	—	16	2	no	—
292	C₂²×C₆₁	C₂×#99	1	4	yes	—
293	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
294	C₄×C₆₁	—	2	1	no	—
295	C₄×C₆₁	—	2	1	no	—
296	C₄×C₆₁	—	2	1	no	—
297	C₄×C₆₁	—	2	2	no	—
298	C₄×C₆₁	—	4	1	no	—
299	C₄×C₆₁	—	4	2	no	—
300	C₄×C₆₁	—	4	1	no	—
301	C₄×C₆₁	—	4	1	no	—
302	C₄×C₆₁	—	4	2	no	—
303	C₄×C₆₁	—	8	4	no	—
304	C₄×C₆₁	C₂×#99	1	2	yes	—
305	#3485	C₂×C₆₁	1	1	yes	characteristic
306	#2888	C₂×C₆₁	1	1	yes	characteristic, complemented
307	#2883	C₂×C₆₁	1	1	yes	characteristic, complemented
308	#2123	C₂×C₆₁	1	2	yes	complemented
309	#2121	C₂×C₆₁	1	4	yes	complemented
310	#2120	C₂×C₆₁	1	2	yes	complemented
311	C₂×#352	C₂×C₆₁	1	4	yes	complemented
312	C₂×C₆₁	—	2	1	no	—
313	C₂×C₆₁	—	2	4	no	—
314	C₂×C₆₁	—	4	1	no	—
315	C₂×C₆₁	—	4	1	no	—
316	C₂×C₆₁	—	4	2	no	—
317	C₂×C₆₁	—	4	2	no	—
318	C₂×C₆₁	—	4	1	no	—
319	C₂×C₆₁	—	8	1	no	—
320	C₂×C₆₁	C₂²×#99	1	1	yes	characteristic
321	C₂×H(5,𝔽₂)	C₂²×C₆₁	1	1	yes	characteristic
322	C₂²×#41	C₄×C₆₁	1	1	yes	characteristic, complemented
323	#568	—	2	1	no	—
324	#547	—	2	1	no	—
325	#547	—	2	2	no	—
326	#547	—	2	1	no	—
327	#546	—	2	1	no	—
328	#546	—	2	2	no	—
329	#546	—	2	1	no	—
330	#536	C₄×C₆₁	1	1	yes	characteristic, complemented
331	#536	C₂²×C₆₁	1	1	yes	characteristic
332	#534	C₄×C₆₁	1	2	yes	complemented
333	#534	C₂²×C₆₁	1	2	yes	—
334	#525	C₄×C₆₁	1	1	yes	characteristic, complemented
335	#525	C₂²×C₆₁	1	1	yes	characteristic
336	C₂×#123	C₄×C₆₁	1	2	yes	complemented
337	C₂×#123	C₂²×C₆₁	1	2	yes	—
338	#518	—	2	1	no	—
339	#518	C₄×C₆₁	1	1	yes	characteristic, complemented
340	#413	—	2	2	no	—
341	#413	C₂²×C₆₁	1	2	yes	—
342	C₂×#100	—	2	1	no	—
343	C₂×#100	C₂²×C₆₁	1	4	yes	—
344	#410	—	2	1	no	—
345	#410	C₂²×C₆₁	1	2	yes	—
346	C₂×#99	—	2	2	no	—
347	C₂×#99	C₂²×C₆₁	1	4	yes	—
348	#352	C₂²×C₆₁	1	4	yes	complemented
349	#352	C₂²×C₆₁	1	8	yes	complemented
350	AΣL(1,𝔽₁₆)	C₂²×C₆₁	1	4	yes	complemented
351	C₆₁	#33350	1	1	yes	characteristic, direct product factor, Sylow
352	H(5,𝔽₂)	—	4	4	no	—
353	C₂×#41	—	2	1	no	—
354	C₂×#41	—	2	2	no	—
355	C₂×#41	—	2	1	no	—
356	C₂×#41	—	4	4	no	—
357	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
358	C₂×#41	D₄×C₆₁	1	2	yes	—
359	C₂×#41	D₄×C₆₁	1	1	yes	characteristic
360	C₂×#41	D₄×C₆₁	1	1	yes	characteristic
361	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
362	C₂²×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
363	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	—
364	C₂²×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
365	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	complemented
366	C₂²×D₄	D₄×C₆₁	1	2	yes	—
367	C₂²×D₄	D₄×C₆₁	1	1	yes	characteristic
368	C₂²×D₄	D₄×C₆₁	1	1	yes	characteristic
369	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
370	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	complemented
371	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
372	#131	—	4	4	no	—
373	C₂×M₁₆	—	2	2	no	—
374	C₂×M₁₆	—	2	4	no	—
375	C₄⋊Q₈	—	2	1	no	—
376	C₄⋊Q₈	—	2	1	no	—
377	#124	—	2	1	no	—
378	#124	—	2	1	no	—
379	#123	—	2	2	no	—
380	#123	—	2	2	no	—
381	#123	—	2	2	no	—
382	#123	—	2	2	no	—
383	#123	—	2	2	no	—
384	#123	—	2	2	no	—
385	#123	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
386	#123	C₂×C₄×C₆₁	1	2	yes	complemented
387	#123	C₂×C₄×C₆₁	1	2	yes	complemented
388	#123	C₂×C₄×C₆₁	1	2	yes	complemented
389	#123	C₂×C₄×C₆₁	1	2	yes	complemented
390	#123	C₂³×C₆₁	1	1	yes	characteristic
391	#123	C₂³×C₆₁	1	2	yes	—
392	C₂²⋊Q₈	—	2	1	no	—
393	C₂²⋊Q₈	—	2	1	no	—
394	C₂²⋊Q₈	—	2	2	no	—
395	C₂²⋊Q₈	—	2	1	no	—
396	C₂²⋊Q₈	—	2	1	no	—
397	#121	—	2	1	no	—
398	#121	—	2	2	no	—
399	#121	—	2	2	no	—
400	#121	—	2	1	no	—
401	#121	—	2	2	no	—
402	#121	—	2	2	no	—
403	#120	—	2	2	no	—
404	#120	—	2	2	no	—
405	#120	C₂×C₄×C₆₁	1	2	yes	complemented
406	#120	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
407	#120	C₂³×C₆₁	1	1	yes	characteristic
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	1	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	4	4	no	—
415	#117	—	2	2	no	—
416	#117	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
417	#117	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
418	C₂×C₄⋊C₄	—	2	2	no	—
419	C₂×C₄⋊C₄	—	2	2	no	—
420	C₂×C₂²⋊C₄	—	2	2	no	—
421	C₂×C₂²⋊C₄	—	2	2	no	—
422	C₂×C₂²⋊C₄	—	2	4	no	—
423	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	complemented
424	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	complemented
425	#101	—	2	2	no	—
426	#100	—	2	4	no	—
427	#100	C₂³×C₆₁	1	4	yes	—
428	#99	—	2	2	no	—
429	#99	—	2	4	no	—
430	#99	C₂³×C₆₁	1	4	yes	—
431	C₂⁴	C₂²⋊C₄×C₆₁	1	2	yes	—
432	C₂⁴	C₂²×C₄×C₆₁	1	2	yes	—
433	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
434	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
435	#41	—	2	4	no	—
436	#41	—	4	2	no	—
437	#41	—	4	4	no	—
438	#41	—	4	2	no	—
439	#41	—	4	2	no	—
440	#41	—	4	8	no	—
441	#41	—	4	4	no	—
442	#41	—	4	4	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	4	2	no	—
446	C₂×Q₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
447	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	4	2	no	—
456	C₂×D₄	—	4	4	no	—
457	C₂×D₄	—	4	4	no	—
458	C₂×D₄	—	4	4	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	C₂²⋊C₄×C₆₁	1	2	yes	—
461	C₂×D₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
462	C₂×D₄	C₂²⋊C₄×C₆₁	1	2	yes	—
463	C₂×D₄	C₂²⋊C₄×C₆₁	1	2	yes	—
464	C₂×D₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
465	C₂×D₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
466	C₂×D₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
467	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
468	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
469	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
470	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
471	C₂×D₄	C₂⁴×C₆₁	1	1	yes	characteristic
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	4	2	no	—
479	C₂²×C₄	—	4	4	no	—
480	C₂²×C₄	—	4	4	no	—
481	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
482	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
483	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
484	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
485	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
486	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
488	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
489	M₁₆	—	2	4	no	—
490	M₁₆	—	4	4	no	—
491	M₁₆	—	4	2	no	—
492	C₂×C₈	—	4	4	no	—
493	C₂×C₈	—	4	2	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	4	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	1	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	1	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	1	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	4	4	no	—
518	C₂²⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
519	C₂²⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
520	C₂²⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
521	C₄²	—	2	1	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	4	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	2	1	no	—
527	C₂³	—	4	2	no	—
528	C₂³	—	4	1	no	—
529	C₂³	—	8	2	no	—
530	C₂³	—	8	1	no	—
531	C₂³	—	8	2	no	—
532	C₂³	—	8	2	no	—
533	C₂³	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
534	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
535	C₂³	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
536	C₂³	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
537	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
538	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
539	Q₈	—	2	2	no	—
540	Q₈	—	4	1	no	—
541	Q₈	—	4	2	no	—
542	Q₈	—	4	2	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	4	no	—
545	D₄	—	2	4	no	—
546	D₄	—	2	2	no	—
547	D₄	—	2	2	no	—
548	D₄	—	4	1	no	—
549	D₄	—	4	2	no	—
550	D₄	—	4	2	no	—
551	D₄	—	4	1	no	—
552	D₄	—	4	2	no	—
553	D₄	—	4	1	no	—
554	D₄	—	4	4	no	—
555	D₄	—	4	4	no	—
556	D₄	—	8	2	no	—
557	D₄	—	8	1	no	—
558	D₄	—	8	1	no	—
559	D₄	#99×C₆₁	1	4	yes	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	—	4	2	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	2	no	—
577	C₂×C₄	—	4	4	no	—
578	C₂×C₄	—	8	2	no	—
579	C₂×C₄	—	8	2	no	—
580	C₂×C₄	—	16	4	no	—
581	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
582	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
583	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
584	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
586	C₈	—	4	4	no	—
587	C₂²	—	2	1	no	—
588	C₂²	—	2	1	no	—
589	C₂²	—	2	1	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	1	no	—
596	C₂²	—	4	1	no	—
597	C₂²	—	4	1	no	—
598	C₂²	—	4	1	no	—
599	C₂²	—	4	1	no	—
600	C₂²	—	4	1	no	—
601	C₂²	—	8	4	no	—
602	C₂²	—	8	4	no	—
603	C₂²	—	8	2	no	—
604	C₂²	—	16	2	no	—
605	C₂²	C₂×#99×C₆₁	1	4	yes	—
606	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	2	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	2	no	—
613	C₄	—	4	1	no	—
614	C₄	—	4	1	no	—
615	C₄	—	4	2	no	—
616	C₄	—	8	4	no	—
617	C₄	C₂×#99×C₆₁	1	2	yes	—
618	C₂	—	2	1	no	—
619	C₂	—	2	4	no	—
620	C₂	—	4	1	no	—
621	C₂	—	4	1	no	—
622	C₂	—	4	2	no	—
623	C₂	—	4	2	no	—
624	C₂	—	4	1	no	—
625	C₂	—	8	1	no	—
626	C₂	C₂²×#99×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₁	C₂²×#99
commutator	C₂³	C₂³×C₄×C₆₁
Frattini, cosocle	C₂×D₄	C₂⁴×C₆₁
prime radical	C₆₁×#3485	C₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×D₄×C₆₁ ◁ C₂×H(5,𝔽₂)×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2888	C₂	semidirect
3. C₆₁×#2883	C₂	semidirect
4. C₆₁×#2123	C₂	semidirect
5. C₆₁×#2121	C₂	semidirect
6. C₆₁×#2120	C₂	semidirect
7. C₂×C₆₁×#352	C₂	semidirect
9. C₂²×#41×C₆₁	C₄	semidirect
17. C₆₁×#536	C₄	semidirect
19. C₆₁×#534	C₄	semidirect
21. C₆₁×#525	C₄	semidirect
23. C₂×#123×C₆₁	C₄	semidirect
26. C₆₁×#518	C₄	semidirect
35. C₆₁×#352	C₂²	semidirect
36. C₆₁×#352	C₂²	semidirect
37. C₆₁×AΣL(1,𝔽₁₆)	C₂²	semidirect
48. C₂²×Q₈×C₆₁	C₂×C₄	semidirect
51. C₂²×D₄×C₆₁	C₂×C₄	semidirect
56. C₂³×C₄×C₆₁	C₂×C₄	semidirect
57. C₂³×C₄×C₆₁	C₂×C₄	semidirect
71. #123×C₆₁	C₂×C₄	semidirect
72. #123×C₆₁	C₂×C₄	semidirect
73. #123×C₆₁	C₂×C₄	semidirect
74. #123×C₆₁	C₂×C₄	semidirect
75. #123×C₆₁	C₂×C₄	semidirect
91. #120×C₆₁	C₂×C₄	semidirect
92. #120×C₆₁	C₂×C₄	semidirect
102. #117×C₆₁	C₂×C₄	semidirect
103. #117×C₆₁	C₂×C₄	semidirect
109. C₂×C₂²⋊C₄×C₆₁	C₂×C₄	semidirect
110. C₂×C₂²⋊C₄×C₆₁	C₂×C₄	semidirect
273. #33350	C₆₁	direct (N × H)
306. #2888	C₂×C₆₁	semidirect
307. #2883	C₂×C₆₁	semidirect
308. #2123	C₂×C₆₁	semidirect
309. #2121	C₂×C₆₁	semidirect
310. #2120	C₂×C₆₁	semidirect
311. C₂×#352	C₂×C₆₁	semidirect
322. C₂²×#41	C₄×C₆₁	semidirect
330. #536	C₄×C₆₁	semidirect
332. #534	C₄×C₆₁	semidirect
334. #525	C₄×C₆₁	semidirect
336. C₂×#123	C₄×C₆₁	semidirect
339. #518	C₄×C₆₁	semidirect
348. #352	C₂²×C₆₁	semidirect
349. #352	C₂²×C₆₁	semidirect
350. AΣL(1,𝔽₁₆)	C₂²×C₆₁	semidirect
351. C₆₁	#33350	direct (N × H)
362. C₂²×Q₈	C₂×C₄×C₆₁	semidirect
365. C₂²×D₄	C₂×C₄×C₆₁	semidirect
370. C₂³×C₄	C₂×C₄×C₆₁	semidirect
371. C₂³×C₄	C₂×C₄×C₆₁	semidirect
385. #123	C₂×C₄×C₆₁	semidirect
386. #123	C₂×C₄×C₆₁	semidirect
387. #123	C₂×C₄×C₆₁	semidirect
388. #123	C₂×C₄×C₆₁	semidirect
389. #123	C₂×C₄×C₆₁	semidirect
405. #120	C₂×C₄×C₆₁	semidirect
406. #120	C₂×C₄×C₆₁	semidirect
416. #117	C₂×C₄×C₆₁	semidirect
417. #117	C₂×C₄×C₆₁	semidirect
423. C₂×C₂²⋊C₄	C₂×C₄×C₆₁	semidirect
424. C₂×C₂²⋊C₄	C₂×C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3485	2
C₆₁×#2888	2
C₆₁×#2883	2
C₆₁×#2123	2
C₆₁×#2121	2
C₆₁×#2120	2
C₂×C₆₁×#352	2
#33350	61

Subgroups — C61×#33350 #29536783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33350 #29536783