Subgroups — C₆₁×#21473 #29524906

Order 15616.

1486 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×H(5,𝔽₂)×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#453	C₂	1	1	yes	maximal, characteristic
3	C₄×AGL(1,ℤ/8)×C₆₁	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2328	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1887	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1883	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1860	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1856	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1792	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1759	C₂	1	2	yes	maximal, complemented
11	C₄×#100×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	—
14	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#520	—	2	2	no	—
16	C₆₁×#520	C₂²	1	2	yes	—
17	C₆₁×#518	—	2	2	no	—
18	C₆₁×#518	—	2	2	no	—
19	C₆₁×#518	C₂²	1	1	yes	characteristic
20	C₄×#41×C₆₁	—	2	2	no	—
21	C₄×#41×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₆₁	—	2	2	no	—
23	C₂×C₄×D₄×C₆₁	—	2	2	no	—
24	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic, complemented
26	C₆₁×#453	C₄	1	8	yes	complemented
27	C₆₁×#442	—	2	2	no	—
28	C₆₁×#440	—	2	2	no	—
29	C₄×SD₁₆×C₆₁	—	2	2	no	—
30	C₄×D₈×C₆₁	—	2	2	no	—
31	C₆₁×#428	C₂²	1	2	yes	complemented
32	C₂×#104×C₆₁	C₂²	1	2	yes	—
33	C₂×#104×C₆₁	C₂²	1	2	yes	—
34	C₆₁×#418	C₂²	1	2	yes	complemented
35	C₆₁×#417	C₂²	1	2	yes	—
36	C₂×#100×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×#100×C₆₁	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₆₁	C₂²	1	2	yes	complemented
39	C₆₁×#390	C₂²	1	1	yes	characteristic
40	C₆₁×#390	C₂²	1	1	yes	characteristic, complemented
41	C₆₁×#386	—	2	2	no	—
42	C₆₁×#386	—	2	2	no	—
43	C₆₁×#382	C₂²	1	1	yes	characteristic, complemented
44	C₆₁×#376	C₂²	1	1	yes	characteristic, complemented
45	C₆₁×#343	C₂²	1	2	yes	complemented
46	C₆₁×#339	C₂²	1	2	yes	complemented
47	C₆₁×#339	C₂²	1	2	yes	complemented
48	H(5,𝔽₂)×C₆₁	—	2	4	no	—
49	H(5,𝔽₂)×C₆₁	—	2	2	no	—
50	H(5,𝔽₂)×C₆₁	C₂×C₄	1	4	yes	complemented
51	C₂×#41×C₆₁	—	2	2	no	—
52	C₂×#41×C₆₁	C₂³	1	2	yes	—
53	C₂²×D₄×C₆₁	—	2	2	no	—
54	C₂²×D₄×C₆₁	—	2	2	no	—
55	C₂²×D₄×C₆₁	—	2	2	no	—
56	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₁	—	2	2	no	—
59	C₂³×C₄×C₆₁	—	2	1	no	—
60	C₂³×C₄×C₆₁	D₄	1	2	yes	—
61	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
62	AGL(1,ℤ/8)×C₆₁	C₂×C₄	1	8	yes	complemented
63	C₂×SD₁₆×C₆₁	—	2	2	no	—
64	C₂×D₈×C₆₁	—	2	2	no	—
65	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
66	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
67	#127×C₆₁	—	2	2	no	—
68	#127×C₆₁	C₂×C₄	1	4	yes	complemented
69	C₄×Q₈×C₆₁	—	2	2	no	—
70	C₄×Q₈×C₆₁	D₄	1	2	yes	complemented
71	C₄×Q₈×C₆₁	D₄	1	2	yes	complemented
72	C₄×D₄×C₆₁	—	2	4	no	—
73	C₄×D₄×C₆₁	—	2	4	no	—
74	C₄×D₄×C₆₁	—	2	4	no	—
75	C₄×D₄×C₆₁	—	2	4	no	—
76	C₄×D₄×C₆₁	—	2	2	no	—
77	C₄×D₄×C₆₁	—	2	2	no	—
78	C₄×D₄×C₆₁	—	2	4	no	—
79	C₄×D₄×C₆₁	—	2	2	no	—
80	C₄×D₄×C₆₁	—	2	2	no	—
81	C₄×D₄×C₆₁	—	2	2	no	—
82	C₄×D₄×C₆₁	—	2	2	no	—
83	C₄×D₄×C₆₁	—	2	2	no	—
84	C₄×D₄×C₆₁	—	2	2	no	—
85	C₄×D₄×C₆₁	—	2	2	no	—
86	C₄×D₄×C₆₁	D₄	1	2	yes	complemented
87	C₄×D₄×C₆₁	D₄	1	2	yes	complemented
88	#117×C₆₁	—	2	4	no	—
89	#117×C₆₁	—	2	2	no	—
90	#117×C₆₁	—	2	2	no	—
91	#117×C₆₁	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
93	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
98	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
101	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
102	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
103	C₂×C₄²×C₆₁	—	2	2	no	—
104	C₂×C₄²×C₆₁	—	2	2	no	—
105	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
108	#107×C₆₁	—	2	2	no	—
109	#106×C₆₁	—	2	2	no	—
110	#104×C₆₁	—	2	4	no	—
111	#104×C₆₁	C₂×C₄	1	8	yes	complemented
112	#103×C₆₁	—	2	2	no	—
113	#102×C₆₁	—	2	2	no	—
114	#102×C₆₁	—	2	2	no	—
115	#102×C₆₁	—	2	2	no	—
116	#100×C₆₁	—	2	2	no	—
117	#100×C₆₁	C₂×C₄	1	4	yes	complemented
118	#97×C₆₁	—	2	2	no	—
119	C₄×C₈×C₆₁	—	2	2	no	—
120	#95×C₆₁	—	2	1	no	—
121	#95×C₆₁	—	2	1	no	—
122	#95×C₆₁	—	2	2	no	—
123	C₂⁴×C₆₁	—	2	2	no	—
124	C₂⁴×C₆₁	—	2	1	no	—
125	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
126	#41×C₆₁	—	2	8	no	—
127	#41×C₆₁	—	2	8	no	—
128	#41×C₆₁	—	2	2	no	—
129	#41×C₆₁	—	2	2	no	—
130	#41×C₆₁	C₂²×C₄	1	4	yes	—
131	#41×C₆₁	#41	1	4	yes	—
132	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
133	C₂×D₄×C₆₁	—	2	4	no	—
134	C₂×D₄×C₆₁	—	2	8	no	—
135	C₂×D₄×C₆₁	—	2	8	no	—
136	C₂×D₄×C₆₁	—	2	8	no	—
137	C₂×D₄×C₆₁	—	2	2	no	—
138	C₂×D₄×C₆₁	—	2	8	no	—
139	C₂×D₄×C₆₁	—	2	8	no	—
140	C₂×D₄×C₆₁	—	2	2	no	—
141	C₂×D₄×C₆₁	—	2	2	no	—
142	C₂×D₄×C₆₁	—	2	2	no	—
143	C₂×D₄×C₆₁	—	2	2	no	—
144	C₂×D₄×C₆₁	—	2	1	no	—
145	C₂×D₄×C₆₁	—	8	2	no	—
146	C₂×D₄×C₆₁	—	8	2	no	—
147	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
148	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	complemented
149	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₆₁	#41	1	2	yes	—
151	C₂²×C₄×C₆₁	—	2	2	no	—
152	C₂²×C₄×C₆₁	—	2	2	no	—
153	C₂²×C₄×C₆₁	—	2	1	no	—
154	C₂²×C₄×C₆₁	—	2	2	no	—
155	C₂²×C₄×C₆₁	—	2	4	no	—
156	C₂²×C₄×C₆₁	—	2	2	no	—
157	C₂²×C₄×C₆₁	—	2	2	no	—
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	1	no	—
162	C₂²×C₄×C₆₁	—	2	1	no	—
163	C₂²×C₄×C₆₁	—	4	2	no	—
164	C₂²×C₄×C₆₁	—	8	2	no	—
165	C₂²×C₄×C₆₁	—	8	4	no	—
166	C₂²×C₄×C₆₁	—	8	2	no	—
167	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
171	SD₁₆×C₆₁	—	2	8	no	—
172	D₈×C₆₁	—	2	8	no	—
173	M₁₆×C₆₁	—	2	2	no	—
174	M₁₆×C₆₁	—	2	2	no	—
175	M₁₆×C₆₁	—	2	2	no	—
176	M₁₆×C₆₁	C₂²×C₄	1	4	yes	complemented
177	C₂×C₈×C₆₁	—	2	2	no	—
178	C₂×C₈×C₆₁	—	2	2	no	—
179	C₄⋊C₄×C₆₁	—	2	4	no	—
180	C₄⋊C₄×C₆₁	—	2	2	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	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
186	C₂²⋊C₄×C₆₁	—	2	4	no	—
187	C₂²⋊C₄×C₆₁	—	2	4	no	—
188	C₂²⋊C₄×C₆₁	—	2	2	no	—
189	C₂²⋊C₄×C₆₁	—	2	2	no	—
190	C₂²⋊C₄×C₆₁	—	2	2	no	—
191	C₂²⋊C₄×C₆₁	—	2	4	no	—
192	C₂²⋊C₄×C₆₁	—	4	1	no	—
193	C₂²⋊C₄×C₆₁	—	4	1	no	—
194	C₂²⋊C₄×C₆₁	—	8	2	no	—
195	C₂²⋊C₄×C₆₁	—	8	2	no	—
196	C₄²×C₆₁	—	2	4	no	—
197	C₄²×C₆₁	—	2	2	no	—
198	C₄²×C₆₁	—	2	2	no	—
199	C₄²×C₆₁	—	2	2	no	—
200	C₄²×C₆₁	—	2	1	no	—
201	C₄²×C₆₁	—	2	2	no	—
202	C₄²×C₆₁	C₂²×C₄	1	2	yes	complemented
203	C₄²×C₆₁	C₂×D₄	1	2	yes	—
204	C₄²×C₆₁	C₂×D₄	1	2	yes	—
205	C₂³×C₆₁	—	2	8	no	—
206	C₂³×C₆₁	—	2	2	no	—
207	C₂³×C₆₁	—	2	2	no	—
208	C₂³×C₆₁	—	2	2	no	—
209	C₂³×C₆₁	—	2	2	no	—
210	C₂³×C₆₁	—	4	2	no	—
211	C₂³×C₆₁	—	4	2	no	—
212	C₂³×C₆₁	—	4	2	no	—
213	C₂³×C₆₁	—	4	2	no	—
214	C₂³×C₆₁	—	4	1	no	—
215	C₂³×C₆₁	—	4	1	no	—
216	C₂³×C₆₁	—	8	2	no	—
217	C₂³×C₆₁	—	8	2	no	—
218	C₂³×C₆₁	C₄×D₄	1	4	yes	—
219	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
220	Q₈×C₆₁	—	2	2	no	—
221	Q₈×C₆₁	C₄×D₄	1	4	yes	—
222	D₄×C₆₁	—	2	16	no	—
223	D₄×C₆₁	—	2	4	no	—
224	D₄×C₆₁	—	2	4	no	—
225	D₄×C₆₁	—	2	4	no	—
226	D₄×C₆₁	—	2	4	no	—
227	D₄×C₆₁	—	2	2	no	—
228	D₄×C₆₁	—	2	4	no	—
229	D₄×C₆₁	—	4	2	no	—
230	D₄×C₆₁	—	8	8	no	—
231	D₄×C₆₁	C₄×D₄	1	4	yes	—
232	C₂×C₄×C₆₁	—	2	8	no	—
233	C₂×C₄×C₆₁	—	2	2	no	—
234	C₂×C₄×C₆₁	—	2	4	no	—
235	C₂×C₄×C₆₁	—	2	4	no	—
236	C₂×C₄×C₆₁	—	2	2	no	—
237	C₂×C₄×C₆₁	—	2	2	no	—
238	C₂×C₄×C₆₁	—	2	2	no	—
239	C₂×C₄×C₆₁	—	2	2	no	—
240	C₂×C₄×C₆₁	—	4	1	no	—
241	C₂×C₄×C₆₁	—	4	1	no	—
242	C₂×C₄×C₆₁	—	4	4	no	—
243	C₂×C₄×C₆₁	—	4	4	no	—
244	C₂×C₄×C₆₁	—	4	1	no	—
245	C₂×C₄×C₆₁	—	4	1	no	—
246	C₂×C₄×C₆₁	—	4	2	no	—
247	C₂×C₄×C₆₁	—	4	1	no	—
248	C₂×C₄×C₆₁	—	4	1	no	—
249	C₂×C₄×C₆₁	—	4	2	no	—
250	C₂×C₄×C₆₁	—	4	1	no	—
251	C₂×C₄×C₆₁	—	4	1	no	—
252	C₂×C₄×C₆₁	—	8	2	no	—
253	C₂×C₄×C₆₁	—	8	2	no	—
254	C₂×C₄×C₆₁	—	8	4	no	—
255	C₂×C₄×C₆₁	—	8	2	no	—
256	C₂×C₄×C₆₁	—	8	4	no	—
257	C₂×C₄×C₆₁	—	8	2	no	—
258	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
259	C₂×C₄×C₆₁	#120	1	2	yes	—
260	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
261	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
262	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
264	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
265	C₈×C₆₁	—	2	4	no	—
266	C₈×C₆₁	—	4	2	no	—
267	#21473	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂²×C₆₁	—	2	1	no	—
269	C₂²×C₆₁	—	2	1	no	—
270	C₂²×C₆₁	—	2	4	no	—
271	C₂²×C₆₁	—	2	4	no	—
272	C₂²×C₆₁	—	4	4	no	—
273	C₂²×C₆₁	—	4	4	no	—
274	C₂²×C₆₁	—	4	2	no	—
275	C₂²×C₆₁	—	4	2	no	—
276	C₂²×C₆₁	—	4	2	no	—
277	C₂²×C₆₁	—	4	2	no	—
278	C₂²×C₆₁	—	4	2	no	—
279	C₂²×C₆₁	—	8	2	no	—
280	C₂²×C₆₁	—	8	2	no	—
281	C₂²×C₆₁	—	8	8	no	—
282	C₂²×C₆₁	—	8	1	no	—
283	C₂²×C₆₁	—	8	1	no	—
284	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
285	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
286	C₂²×C₆₁	#525	1	1	yes	characteristic
287	C₄×C₆₁	—	2	1	no	—
288	C₄×C₆₁	—	2	1	no	—
289	C₄×C₆₁	—	2	2	no	—
290	C₄×C₆₁	—	2	4	no	—
291	C₄×C₆₁	—	4	2	no	—
292	C₄×C₆₁	—	4	2	no	—
293	C₄×C₆₁	—	4	2	no	—
294	C₄×C₆₁	—	4	2	no	—
295	C₄×C₆₁	—	8	1	no	—
296	C₄×C₆₁	—	8	1	no	—
297	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
298	C₄×C₆₁	#525	1	2	yes	—
299	C₄×H(5,𝔽₂)	C₂×C₆₁	1	1	yes	characteristic, complemented
300	C₂×#453	C₂×C₆₁	1	1	yes	characteristic
301	C₄×AGL(1,ℤ/8)	C₂×C₆₁	1	2	yes	complemented
302	#2328	C₂×C₆₁	1	1	yes	characteristic, complemented
303	#1887	C₂×C₆₁	1	1	yes	characteristic, complemented
304	#1883	C₂×C₆₁	1	2	yes	complemented
305	#1860	C₂×C₆₁	1	2	yes	complemented
306	#1856	C₂×C₆₁	1	1	yes	characteristic, complemented
307	#1792	C₂×C₆₁	1	1	yes	characteristic, complemented
308	#1759	C₂×C₆₁	1	2	yes	complemented
309	C₄×#100	C₂×C₆₁	1	1	yes	characteristic, complemented
310	C₂×C₆₁	—	2	1	no	—
311	C₂×C₆₁	—	2	1	no	—
312	C₂×C₆₁	—	4	4	no	—
313	C₂×C₆₁	—	4	4	no	—
314	C₂×C₆₁	—	8	2	no	—
315	C₂×C₆₁	C₄×#120	1	1	yes	characteristic
316	C₂×C₆₁	#3019	1	1	yes	characteristic
317	C₂×C₆₁	#3021	1	1	yes	characteristic
318	C₂×H(5,𝔽₂)	C₂²×C₆₁	1	1	yes	characteristic
319	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	—
320	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
321	#520	—	2	2	no	—
322	#520	C₂²×C₆₁	1	2	yes	—
323	#518	—	2	2	no	—
324	#518	—	2	2	no	—
325	#518	C₂²×C₆₁	1	1	yes	characteristic
326	C₄×#41	—	2	2	no	—
327	C₄×#41	C₂²×C₆₁	1	2	yes	—
328	C₂×C₄×D₄	—	2	2	no	—
329	C₂×C₄×D₄	—	2	2	no	—
330	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
331	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic, complemented
332	#453	C₄×C₆₁	1	8	yes	complemented
333	#442	—	2	2	no	—
334	#440	—	2	2	no	—
335	C₄×SD₁₆	—	2	2	no	—
336	C₄×D₈	—	2	2	no	—
337	#428	C₂²×C₆₁	1	2	yes	complemented
338	C₂×#104	C₂²×C₆₁	1	2	yes	—
339	C₂×#104	C₂²×C₆₁	1	2	yes	—
340	#418	C₂²×C₆₁	1	2	yes	complemented
341	#417	C₂²×C₆₁	1	2	yes	—
342	C₂×#100	C₂²×C₆₁	1	1	yes	characteristic
343	C₂×#100	C₂²×C₆₁	1	1	yes	characteristic
344	C₄×M₁₆	C₂²×C₆₁	1	2	yes	complemented
345	#390	C₂²×C₆₁	1	1	yes	characteristic
346	#390	C₂²×C₆₁	1	1	yes	characteristic, complemented
347	#386	—	2	2	no	—
348	#386	—	2	2	no	—
349	#382	C₂²×C₆₁	1	1	yes	characteristic, complemented
350	#376	C₂²×C₆₁	1	1	yes	characteristic, complemented
351	#343	C₂²×C₆₁	1	2	yes	complemented
352	#339	C₂²×C₆₁	1	2	yes	complemented
353	#339	C₂²×C₆₁	1	2	yes	complemented
354	C₆₁	#21473	1	1	yes	characteristic, direct product factor, Sylow
355	H(5,𝔽₂)	—	2	4	no	—
356	H(5,𝔽₂)	—	2	2	no	—
357	H(5,𝔽₂)	C₂×C₄×C₆₁	1	4	yes	complemented
358	C₂×#41	—	2	2	no	—
359	C₂×#41	C₂³×C₆₁	1	2	yes	—
360	C₂²×D₄	—	2	2	no	—
361	C₂²×D₄	—	2	2	no	—
362	C₂²×D₄	—	2	2	no	—
363	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
364	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
365	C₂³×C₄	—	2	2	no	—
366	C₂³×C₄	—	2	1	no	—
367	C₂³×C₄	D₄×C₆₁	1	2	yes	—
368	AGL(1,ℤ/8)	—	2	4	no	—
369	AGL(1,ℤ/8)	C₂×C₄×C₆₁	1	8	yes	complemented
370	C₂×SD₁₆	—	2	2	no	—
371	C₂×D₈	—	2	2	no	—
372	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
373	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
374	#127	—	2	2	no	—
375	#127	C₂×C₄×C₆₁	1	4	yes	complemented
376	C₄×Q₈	—	2	2	no	—
377	C₄×Q₈	D₄×C₆₁	1	2	yes	complemented
378	C₄×Q₈	D₄×C₆₁	1	2	yes	complemented
379	C₄×D₄	—	2	4	no	—
380	C₄×D₄	—	2	4	no	—
381	C₄×D₄	—	2	4	no	—
382	C₄×D₄	—	2	4	no	—
383	C₄×D₄	—	2	2	no	—
384	C₄×D₄	—	2	2	no	—
385	C₄×D₄	—	2	4	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	D₄×C₆₁	1	2	yes	complemented
394	C₄×D₄	D₄×C₆₁	1	2	yes	complemented
395	#117	—	2	4	no	—
396	#117	—	2	2	no	—
397	#117	—	2	2	no	—
398	#117	C₂³×C₆₁	1	2	yes	—
399	C₂×C₄⋊C₄	—	2	2	no	—
400	C₂×C₄⋊C₄	—	2	2	no	—
401	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	—	2	2	no	—
404	C₂×C₂²⋊C₄	—	2	4	no	—
405	C₂×C₂²⋊C₄	—	2	2	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
410	C₂×C₄²	—	2	2	no	—
411	C₂×C₄²	—	2	2	no	—
412	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
413	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
415	#107	—	2	2	no	—
416	#106	—	2	2	no	—
417	#104	—	2	4	no	—
418	#104	C₂×C₄×C₆₁	1	8	yes	complemented
419	#103	—	2	2	no	—
420	#102	—	2	2	no	—
421	#102	—	2	2	no	—
422	#102	—	2	2	no	—
423	#100	—	2	2	no	—
424	#100	C₂×C₄×C₆₁	1	4	yes	complemented
425	#97	—	2	2	no	—
426	C₄×C₈	—	2	2	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	#95	—	2	2	no	—
430	C₂⁴	—	2	2	no	—
431	C₂⁴	—	2	1	no	—
432	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
433	#41	—	2	8	no	—
434	#41	—	2	8	no	—
435	#41	—	2	2	no	—
436	#41	—	2	2	no	—
437	#41	C₂²×C₄×C₆₁	1	4	yes	—
438	#41	#41×C₆₁	1	4	yes	—
439	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	8	no	—
442	C₂×D₄	—	2	8	no	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	8	no	—
446	C₂×D₄	—	2	8	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	—	8	2	no	—
453	C₂×D₄	—	8	2	no	—
454	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
455	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	complemented
456	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
457	C₂×D₄	#41×C₆₁	1	2	yes	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	—	8	2	no	—
472	C₂²×C₄	—	8	4	no	—
473	C₂²×C₄	—	8	2	no	—
474	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
475	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
478	SD₁₆	—	2	8	no	—
479	D₈	—	2	8	no	—
480	M₁₆	—	2	2	no	—
481	M₁₆	—	2	2	no	—
482	M₁₆	—	2	2	no	—
483	M₁₆	C₂²×C₄×C₆₁	1	4	yes	complemented
484	C₂×C₈	—	2	2	no	—
485	C₂×C₈	—	2	2	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
492	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	4	1	no	—
500	C₂²⋊C₄	—	4	1	no	—
501	C₂²⋊C₄	—	8	2	no	—
502	C₂²⋊C₄	—	8	2	no	—
503	C₄²	—	2	4	no	—
504	C₄²	—	2	2	no	—
505	C₄²	—	2	2	no	—
506	C₄²	—	2	2	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	2	no	—
509	C₄²	C₂²×C₄×C₆₁	1	2	yes	complemented
510	C₄²	C₂×D₄×C₆₁	1	2	yes	—
511	C₄²	C₂×D₄×C₆₁	1	2	yes	—
512	C₂³	—	2	8	no	—
513	C₂³	—	2	2	no	—
514	C₂³	—	2	2	no	—
515	C₂³	—	2	2	no	—
516	C₂³	—	2	2	no	—
517	C₂³	—	4	2	no	—
518	C₂³	—	4	2	no	—
519	C₂³	—	4	2	no	—
520	C₂³	—	4	2	no	—
521	C₂³	—	4	1	no	—
522	C₂³	—	4	1	no	—
523	C₂³	—	8	2	no	—
524	C₂³	—	8	2	no	—
525	C₂³	C₄×D₄×C₆₁	1	4	yes	—
526	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
527	Q₈	—	2	2	no	—
528	Q₈	C₄×D₄×C₆₁	1	4	yes	—
529	D₄	—	2	16	no	—
530	D₄	—	2	4	no	—
531	D₄	—	2	4	no	—
532	D₄	—	2	4	no	—
533	D₄	—	2	4	no	—
534	D₄	—	2	2	no	—
535	D₄	—	2	4	no	—
536	D₄	—	4	2	no	—
537	D₄	—	8	8	no	—
538	D₄	C₄×D₄×C₆₁	1	4	yes	—
539	C₂×C₄	—	2	8	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	4	no	—
550	C₂×C₄	—	4	4	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	8	2	no	—
560	C₂×C₄	—	8	2	no	—
561	C₂×C₄	—	8	4	no	—
562	C₂×C₄	—	8	2	no	—
563	C₂×C₄	—	8	4	no	—
564	C₂×C₄	—	8	2	no	—
565	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
566	C₂×C₄	#120×C₆₁	1	2	yes	—
567	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
568	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
569	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
570	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
571	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
572	C₈	—	2	4	no	—
573	C₈	—	4	2	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	2	no	—
584	C₂²	—	4	2	no	—
585	C₂²	—	8	2	no	—
586	C₂²	—	8	2	no	—
587	C₂²	—	8	8	no	—
588	C₂²	—	8	1	no	—
589	C₂²	—	8	1	no	—
590	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
591	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
592	C₂²	C₆₁×#525	1	1	yes	characteristic
593	C₄	—	2	4	no	—
594	C₄	—	2	1	no	—
595	C₄	—	2	1	no	—
596	C₄	—	2	2	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	2	no	—
600	C₄	—	4	2	no	—
601	C₄	—	8	1	no	—
602	C₄	—	8	1	no	—
603	C₄	C₂×C₄×D₄×C₆₁	1	2	yes	—
604	C₄	C₆₁×#525	1	2	yes	—
605	C₂	—	2	1	no	—
606	C₂	—	2	1	no	—
607	C₂	—	4	4	no	—
608	C₂	—	4	4	no	—
609	C₂	—	8	2	no	—
610	C₂	C₄×#120×C₆₁	1	1	yes	characteristic
611	C₂	C₆₁×#3019	1	1	yes	characteristic
612	C₂	C₆₁×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×C₆₁×#453	C₂
LCS	G ▷ C₂×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. C₄×H(5,𝔽₂)×C₆₁	C₂	semidirect
3. C₄×AGL(1,ℤ/8)×C₆₁	C₂	semidirect
4. C₆₁×#2328	C₂	semidirect
5. C₆₁×#1887	C₂	semidirect
6. C₆₁×#1883	C₂	semidirect
7. C₆₁×#1860	C₂	semidirect
8. C₆₁×#1856	C₂	semidirect
9. C₆₁×#1792	C₂	semidirect
10. C₆₁×#1759	C₂	semidirect
11. C₄×#100×C₆₁	C₂	semidirect
25. C₂×C₄×D₄×C₆₁	C₂²	semidirect
26. C₆₁×#453	C₄	semidirect
31. C₆₁×#428	C₂²	semidirect
34. C₆₁×#418	C₂²	semidirect
38. C₄×M₁₆×C₆₁	C₂²	semidirect
40. C₆₁×#390	C₂²	semidirect
43. C₆₁×#382	C₂²	semidirect
44. C₆₁×#376	C₂²	semidirect
45. C₆₁×#343	C₂²	semidirect
46. C₆₁×#339	C₂²	semidirect
47. C₆₁×#339	C₂²	semidirect
50. H(5,𝔽₂)×C₆₁	C₂×C₄	semidirect
62. AGL(1,ℤ/8)×C₆₁	C₂×C₄	semidirect
68. #127×C₆₁	C₂×C₄	semidirect
70. C₄×Q₈×C₆₁	D₄	semidirect
71. C₄×Q₈×C₆₁	D₄	semidirect
86. C₄×D₄×C₆₁	D₄	semidirect
87. C₄×D₄×C₆₁	D₄	semidirect
111. #104×C₆₁	C₂×C₄	semidirect
117. #100×C₆₁	C₂×C₄	semidirect
148. C₂×D₄×C₆₁	C₂²×C₄	semidirect
176. M₁₆×C₆₁	C₂²×C₄	semidirect
202. C₄²×C₆₁	C₂²×C₄	semidirect
267. #21473	C₆₁	direct (N × H)
299. C₄×H(5,𝔽₂)	C₂×C₆₁	semidirect
301. C₄×AGL(1,ℤ/8)	C₂×C₆₁	semidirect
302. #2328	C₂×C₆₁	semidirect
303. #1887	C₂×C₆₁	semidirect
304. #1883	C₂×C₆₁	semidirect
305. #1860	C₂×C₆₁	semidirect
306. #1856	C₂×C₆₁	semidirect
307. #1792	C₂×C₆₁	semidirect
308. #1759	C₂×C₆₁	semidirect
309. C₄×#100	C₂×C₆₁	semidirect
331. C₂×C₄×D₄	C₂²×C₆₁	semidirect
332. #453	C₄×C₆₁	semidirect
337. #428	C₂²×C₆₁	semidirect
340. #418	C₂²×C₆₁	semidirect
344. C₄×M₁₆	C₂²×C₆₁	semidirect
346. #390	C₂²×C₆₁	semidirect
349. #382	C₂²×C₆₁	semidirect
350. #376	C₂²×C₆₁	semidirect
351. #343	C₂²×C₆₁	semidirect
352. #339	C₂²×C₆₁	semidirect
353. #339	C₂²×C₆₁	semidirect
354. C₆₁	#21473	direct (N × H)
357. H(5,𝔽₂)	C₂×C₄×C₆₁	semidirect
369. AGL(1,ℤ/8)	C₂×C₄×C₆₁	semidirect
375. #127	C₂×C₄×C₆₁	semidirect
377. C₄×Q₈	D₄×C₆₁	semidirect
378. C₄×Q₈	D₄×C₆₁	semidirect
393. C₄×D₄	D₄×C₆₁	semidirect
394. C₄×D₄	D₄×C₆₁	semidirect
418. #104	C₂×C₄×C₆₁	semidirect
424. #100	C₂×C₄×C₆₁	semidirect
455. C₂×D₄	C₂²×C₄×C₆₁	semidirect
483. M₁₆	C₂²×C₄×C₆₁	semidirect
509. C₄²	C₂²×C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×H(5,𝔽₂)×C₆₁	2
C₂×C₆₁×#453	2
C₄×AGL(1,ℤ/8)×C₆₁	2
C₆₁×#2328	2
C₆₁×#1887	2
C₆₁×#1883	2
C₆₁×#1860	2
C₆₁×#1856	2
C₆₁×#1792	2
C₆₁×#1759	2
C₄×#100×C₆₁	2
#21473	61

Subgroups — C61×#21473 #29524906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#21473 #29524906