Subgroups — C₆₁×#25805 #29529238

Order 15616.

1550 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#576	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₆₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₁×#453	C₂	1	2	yes	maximal, complemented
6	C₆₁×#3008	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2018	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2002	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1974	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#576	—	2	4	no	—
14	C₆₁×#576	—	2	4	no	—
15	C₂×AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
16	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	—
17	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	—
18	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	complemented
19	C₂×#135×C₆₁	—	2	1	no	—
20	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×D₈×C₆₁	—	2	1	no	—
22	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
24	C₂²×D₈×C₆₁	C₂²	1	2	yes	complemented
25	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#535	C₂²	1	2	yes	complemented
27	C₂×#127×C₆₁	C₂²	1	2	yes	complemented
28	C₆₁×#496	—	2	2	no	—
29	C₆₁×#495	—	2	1	no	—
30	C₆₁×#493	—	2	4	no	—
31	C₆₁×#493	—	2	1	no	—
32	C₆₁×#472	—	2	4	no	—
33	C₆₁×#453	—	2	8	no	—
34	C₆₁×#449	—	2	4	no	—
35	C₆₁×#447	—	2	4	no	—
36	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic, complemented
37	C₂×#104×C₆₁	C₂²	1	1	yes	characteristic
38	C₂×#104×C₆₁	C₂²	1	2	yes	complemented
39	C₆₁×#418	C₂²	1	2	yes	complemented
40	C₆₁×#416	C₂²	1	1	yes	characteristic
41	C₂×#100×C₆₁	C₂²	1	2	yes	complemented
42	C₂×#100×C₆₁	C₂²	1	2	yes	complemented
43	C₆₁×#408	C₂²	1	2	yes	complemented
44	C₆₁×#405	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
46	C₆₁×#341	C₂²	1	1	yes	characteristic, complemented
47	C₆₁×#339	C₂²	1	1	yes	characteristic, complemented
48	H(5,𝔽₂)×C₆₁	—	2	4	no	—
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	C₂×#41×C₆₁	—	2	2	no	—
53	C₂×#41×C₆₁	—	2	2	no	—
54	C₂×#41×C₆₁	C₂³	1	2	yes	—
55	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₆₁	—	2	2	no	—
57	C₂²×D₄×C₆₁	—	2	2	no	—
58	C₂²×D₄×C₆₁	—	4	2	no	—
59	C₂²×D₄×C₆₁	—	4	2	no	—
60	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
61	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
62	AGL(1,ℤ/8)×C₆₁	—	2	16	no	—
63	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
64	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
65	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
66	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
67	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
68	AGL(1,ℤ/8)×C₆₁	—	4	2	no	—
69	#135×C₆₁	—	2	8	no	—
70	#135×C₆₁	—	2	2	no	—
71	#135×C₆₁	—	2	2	no	—
72	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic, complemented
73	C₂×SD₁₆×C₆₁	—	2	2	no	—
74	C₂×SD₁₆×C₆₁	—	2	2	no	—
75	C₂×SD₁₆×C₆₁	—	2	2	no	—
76	C₂×SD₁₆×C₆₁	D₄	1	2	yes	complemented
77	C₂×D₈×C₆₁	—	2	8	no	—
78	C₂×D₈×C₆₁	—	2	2	no	—
79	C₂×D₈×C₆₁	—	2	4	no	—
80	C₂×D₈×C₆₁	—	2	2	no	—
81	C₂×D₈×C₆₁	—	2	2	no	—
82	C₂×D₈×C₆₁	—	2	1	no	—
83	C₂×D₈×C₆₁	—	2	1	no	—
84	C₂×D₈×C₆₁	—	2	2	no	—
85	C₂×D₈×C₆₁	—	2	4	no	—
86	C₂×D₈×C₆₁	—	2	2	no	—
87	C₂×D₈×C₆₁	—	2	2	no	—
88	C₂×D₈×C₆₁	—	2	1	no	—
89	C₂×D₈×C₆₁	—	2	1	no	—
90	C₂×D₈×C₆₁	—	4	2	no	—
91	C₂×D₈×C₆₁	D₄	1	1	yes	characteristic, complemented
92	C₂×D₈×C₆₁	D₄	1	4	yes	complemented
93	#131×C₆₁	—	2	2	no	—
94	#131×C₆₁	D₄	1	4	yes	complemented
95	C₂×M₁₆×C₆₁	—	2	1	no	—
96	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
97	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
98	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₆₁	—	2	1	no	—
100	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
101	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
102	#127×C₆₁	—	2	4	no	—
103	#127×C₆₁	—	2	2	no	—
104	#127×C₆₁	—	2	4	no	—
105	#124×C₆₁	—	2	2	no	—
106	#121×C₆₁	—	2	4	no	—
107	#120×C₆₁	—	4	2	no	—
108	#117×C₆₁	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
110	#108×C₆₁	—	2	2	no	—
111	#104×C₆₁	—	2	4	no	—
112	#104×C₆₁	D₄	1	4	yes	complemented
113	#103×C₆₁	D₄	1	2	yes	complemented
114	#102×C₆₁	—	2	4	no	—
115	#102×C₆₁	D₄	1	2	yes	complemented
116	#100×C₆₁	—	2	4	no	—
117	#100×C₆₁	—	2	4	no	—
118	C₂²⋊C₈×C₆₁	—	2	4	no	—
119	#97×C₆₁	—	2	1	no	—
120	C₄×C₈×C₆₁	—	2	2	no	—
121	C₄×C₈×C₆₁	—	2	1	no	—
122	C₂⁴×C₆₁	—	2	2	no	—
123	C₂⁴×C₆₁	—	2	2	no	—
124	C₂⁴×C₆₁	—	4	2	no	—
125	#41×C₆₁	—	2	1	no	—
126	#41×C₆₁	—	2	1	no	—
127	#41×C₆₁	—	2	8	no	—
128	#41×C₆₁	—	2	2	no	—
129	#41×C₆₁	—	2	2	no	—
130	#41×C₆₁	—	2	2	no	—
131	#41×C₆₁	—	2	2	no	—
132	#41×C₆₁	—	4	4	no	—
133	#41×C₆₁	—	4	4	no	—
134	#41×C₆₁	—	4	4	no	—
135	#41×C₆₁	C₂×D₄	1	2	yes	—
136	#41×C₆₁	C₂×D₄	1	2	yes	—
137	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
138	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
139	C₂×D₄×C₆₁	—	2	1	no	—
140	C₂×D₄×C₆₁	—	2	8	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	2	no	—
145	C₂×D₄×C₆₁	—	2	2	no	—
146	C₂×D₄×C₆₁	—	2	2	no	—
147	C₂×D₄×C₆₁	—	2	2	no	—
148	C₂×D₄×C₆₁	—	2	2	no	—
149	C₂×D₄×C₆₁	—	2	2	no	—
150	C₂×D₄×C₆₁	—	2	2	no	—
151	C₂×D₄×C₆₁	—	2	2	no	—
152	C₂×D₄×C₆₁	—	2	2	no	—
153	C₂×D₄×C₆₁	—	4	4	no	—
154	C₂×D₄×C₆₁	—	4	2	no	—
155	C₂×D₄×C₆₁	—	4	2	no	—
156	C₂×D₄×C₆₁	—	4	8	no	—
157	C₂×D₄×C₆₁	—	4	4	no	—
158	C₂×D₄×C₆₁	—	4	4	no	—
159	C₂×D₄×C₆₁	—	4	8	no	—
160	C₂×D₄×C₆₁	—	4	8	no	—
161	C₂×D₄×C₆₁	—	4	4	no	—
162	C₂×D₄×C₆₁	—	8	8	no	—
163	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
164	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
165	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₁	—	2	1	no	—
167	C₂²×C₄×C₆₁	—	2	1	no	—
168	C₂²×C₄×C₆₁	—	2	2	no	—
169	C₂²×C₄×C₆₁	—	4	2	no	—
170	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
171	Q₁₆×C₆₁	—	2	2	no	—
172	SD₁₆×C₆₁	—	2	8	no	—
173	SD₁₆×C₆₁	—	2	4	no	—
174	SD₁₆×C₆₁	—	4	4	no	—
175	SD₁₆×C₆₁	—	4	4	no	—
176	D₈×C₆₁	—	2	8	no	—
177	D₈×C₆₁	—	2	2	no	—
178	D₈×C₆₁	—	2	4	no	—
179	D₈×C₆₁	—	2	4	no	—
180	D₈×C₆₁	—	4	8	no	—
181	D₈×C₆₁	—	4	4	no	—
182	D₈×C₆₁	—	4	4	no	—
183	D₈×C₆₁	—	4	2	no	—
184	D₈×C₆₁	—	4	2	no	—
185	M₁₆×C₆₁	—	2	1	no	—
186	M₁₆×C₆₁	—	2	1	no	—
187	M₁₆×C₆₁	—	2	4	no	—
188	M₁₆×C₆₁	—	2	2	no	—
189	M₁₆×C₆₁	—	2	2	no	—
190	M₁₆×C₆₁	C₂×D₄	1	2	yes	—
191	M₁₆×C₆₁	C₂×D₄	1	2	yes	—
192	C₂×C₈×C₆₁	—	2	1	no	—
193	C₂×C₈×C₆₁	—	2	1	no	—
194	C₂×C₈×C₆₁	—	2	1	no	—
195	C₂×C₈×C₆₁	—	2	1	no	—
196	C₂×C₈×C₆₁	—	2	4	no	—
197	C₂×C₈×C₆₁	—	2	2	no	—
198	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
199	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
200	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
201	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
202	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
203	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
204	C₂²⋊C₄×C₆₁	—	2	1	no	—
205	C₂²⋊C₄×C₆₁	—	4	4	no	—
206	C₄²×C₆₁	—	2	1	no	—
207	C₄²×C₆₁	—	2	1	no	—
208	C₄²×C₆₁	C₂×D₄	1	2	yes	—
209	C₂³×C₆₁	—	2	1	no	—
210	C₂³×C₆₁	—	2	2	no	—
211	C₂³×C₆₁	—	2	4	no	—
212	C₂³×C₆₁	—	4	8	no	—
213	C₂³×C₆₁	—	4	2	no	—
214	C₂³×C₆₁	—	4	2	no	—
215	C₂³×C₆₁	—	4	2	no	—
216	C₂³×C₆₁	—	4	2	no	—
217	C₂³×C₆₁	—	4	2	no	—
218	C₂³×C₆₁	—	8	4	no	—
219	C₂³×C₆₁	—	8	4	no	—
220	C₂³×C₆₁	—	16	4	no	—
221	C₂³×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
222	Q₈×C₆₁	—	2	1	no	—
223	Q₈×C₆₁	—	2	2	no	—
224	Q₈×C₆₁	—	2	2	no	—
225	Q₈×C₆₁	#127	1	2	yes	—
226	D₄×C₆₁	—	2	2	no	—
227	D₄×C₆₁	—	2	2	no	—
228	D₄×C₆₁	—	2	2	no	—
229	D₄×C₆₁	—	2	4	no	—
230	D₄×C₆₁	—	2	1	no	—
231	D₄×C₆₁	—	2	4	no	—
232	D₄×C₆₁	—	2	2	no	—
233	D₄×C₆₁	—	4	8	no	—
234	D₄×C₆₁	—	4	8	no	—
235	D₄×C₆₁	—	4	2	no	—
236	D₄×C₆₁	—	4	2	no	—
237	D₄×C₆₁	—	4	2	no	—
238	D₄×C₆₁	—	4	2	no	—
239	D₄×C₆₁	—	4	2	no	—
240	D₄×C₆₁	—	4	2	no	—
241	D₄×C₆₁	—	4	2	no	—
242	D₄×C₆₁	—	4	2	no	—
243	D₄×C₆₁	—	8	16	no	—
244	D₄×C₆₁	—	16	4	no	—
245	D₄×C₆₁	#127	1	2	yes	—
246	C₂×C₄×C₆₁	—	2	4	no	—
247	C₂×C₄×C₆₁	—	2	2	no	—
248	C₂×C₄×C₆₁	—	2	1	no	—
249	C₂×C₄×C₆₁	—	2	2	no	—
250	C₂×C₄×C₆₁	—	2	2	no	—
251	C₂×C₄×C₆₁	—	2	2	no	—
252	C₂×C₄×C₆₁	—	2	2	no	—
253	C₂×C₄×C₆₁	—	4	8	no	—
254	C₂×C₄×C₆₁	—	4	2	no	—
255	C₂×C₄×C₆₁	—	4	2	no	—
256	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
262	C₈×C₆₁	—	2	2	no	—
263	C₈×C₆₁	—	2	2	no	—
264	C₈×C₆₁	—	2	2	no	—
265	C₈×C₆₁	—	4	2	no	—
266	C₈×C₆₁	#120	1	2	yes	—
267	C₈×C₆₁	#120	1	2	yes	—
268	#25805	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
269	C₂²×C₆₁	—	2	2	no	—
270	C₂²×C₆₁	—	2	2	no	—
271	C₂²×C₆₁	—	4	2	no	—
272	C₂²×C₆₁	—	4	2	no	—
273	C₂²×C₆₁	—	4	4	no	—
274	C₂²×C₆₁	—	4	4	no	—
275	C₂²×C₆₁	—	8	4	no	—
276	C₂²×C₆₁	—	8	4	no	—
277	C₂²×C₆₁	—	8	4	no	—
278	C₂²×C₆₁	—	8	2	no	—
279	C₂²×C₆₁	—	8	4	no	—
280	C₂²×C₆₁	—	16	2	no	—
281	C₂²×C₆₁	—	16	2	no	—
282	C₂²×C₆₁	—	16	8	no	—
283	C₂²×C₆₁	#535	1	1	yes	characteristic
284	C₂²×C₆₁	D₄²	1	1	yes	characteristic
285	C₂²×C₆₁	D₄²	1	1	yes	characteristic
286	C₄×C₆₁	—	2	4	no	—
287	C₄×C₆₁	—	2	2	no	—
288	C₄×C₆₁	—	4	2	no	—
289	C₄×C₆₁	—	4	2	no	—
290	C₄×C₆₁	C₂×#120	1	1	yes	characteristic
291	C₄×C₆₁	C₂×#127	1	1	yes	characteristic
292	C₄×C₆₁	D₄²	1	1	yes	characteristic
293	C₄×C₆₁	D₄²	1	1	yes	characteristic
294	C₂×#576	C₂×C₆₁	1	1	yes	characteristic, complemented
295	#3155	C₂×C₆₁	1	1	yes	characteristic, complemented
296	C₂×#493	C₂×C₆₁	1	1	yes	characteristic, complemented
297	C₂×#472	C₂×C₆₁	1	1	yes	characteristic, complemented
298	C₂×#453	C₂×C₆₁	1	2	yes	complemented
299	#3008	C₂×C₆₁	1	2	yes	complemented
300	#2018	C₂×C₆₁	1	2	yes	complemented
301	#2002	C₂×C₆₁	1	2	yes	complemented
302	#1978	C₂×C₆₁	1	1	yes	characteristic, complemented
303	#1974	C₂×C₆₁	1	1	yes	characteristic, complemented
304	#1764	C₂×C₆₁	1	1	yes	characteristic, complemented
305	C₂×C₆₁	—	2	1	no	—
306	C₂×C₆₁	—	2	1	no	—
307	C₂×C₆₁	—	4	2	no	—
308	C₂×C₆₁	—	8	2	no	—
309	C₂×C₆₁	—	8	4	no	—
310	C₂×C₆₁	—	16	2	no	—
311	C₂×C₆₁	#2433	1	1	yes	characteristic
312	C₂×C₆₁	#3292	1	2	yes	—
313	C₂×H(5,𝔽₂)	C₂²×C₆₁	1	2	yes	—
314	#576	—	2	4	no	—
315	#576	—	2	4	no	—
316	C₂×AGL(1,ℤ/8)	—	2	2	no	—
317	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	—
318	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	—
319	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	complemented
320	C₂×#135	—	2	1	no	—
321	C₂×#135	C₂²×C₆₁	1	1	yes	characteristic
322	C₂²×D₈	—	2	1	no	—
323	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
324	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
325	C₂²×D₈	C₂²×C₆₁	1	2	yes	complemented
326	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
327	#535	C₂²×C₆₁	1	2	yes	complemented
328	C₂×#127	C₂²×C₆₁	1	2	yes	complemented
329	#496	—	2	2	no	—
330	#495	—	2	1	no	—
331	#493	—	2	4	no	—
332	#493	—	2	1	no	—
333	#472	—	2	4	no	—
334	#453	—	2	8	no	—
335	#449	—	2	4	no	—
336	#447	—	2	4	no	—
337	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic, complemented
338	C₂×#104	C₂²×C₆₁	1	1	yes	characteristic
339	C₂×#104	C₂²×C₆₁	1	2	yes	complemented
340	#418	C₂²×C₆₁	1	2	yes	complemented
341	#416	C₂²×C₆₁	1	1	yes	characteristic
342	C₂×#100	C₂²×C₆₁	1	2	yes	complemented
343	C₂×#100	C₂²×C₆₁	1	2	yes	complemented
344	#408	C₂²×C₆₁	1	2	yes	complemented
345	#405	C₂²×C₆₁	1	1	yes	characteristic, complemented
346	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
347	#341	C₂²×C₆₁	1	1	yes	characteristic, complemented
348	#339	C₂²×C₆₁	1	1	yes	characteristic, complemented
349	C₆₁	#25805	1	1	yes	characteristic, direct product factor, Sylow
350	H(5,𝔽₂)	—	2	4	no	—
351	H(5,𝔽₂)	—	2	4	no	—
352	H(5,𝔽₂)	—	2	4	no	—
353	H(5,𝔽₂)	—	4	2	no	—
354	C₂×#41	—	2	2	no	—
355	C₂×#41	—	2	2	no	—
356	C₂×#41	C₂³×C₆₁	1	2	yes	—
357	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
358	C₂²×D₄	—	2	2	no	—
359	C₂²×D₄	—	2	2	no	—
360	C₂²×D₄	—	4	2	no	—
361	C₂²×D₄	—	4	2	no	—
362	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
363	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
364	AGL(1,ℤ/8)	—	2	16	no	—
365	AGL(1,ℤ/8)	—	2	4	no	—
366	AGL(1,ℤ/8)	—	2	4	no	—
367	AGL(1,ℤ/8)	—	2	4	no	—
368	AGL(1,ℤ/8)	—	2	4	no	—
369	AGL(1,ℤ/8)	—	2	4	no	—
370	AGL(1,ℤ/8)	—	4	2	no	—
371	#135	—	2	8	no	—
372	#135	—	2	2	no	—
373	#135	—	2	2	no	—
374	C₂×Q₁₆	D₄×C₆₁	1	1	yes	characteristic, complemented
375	C₂×SD₁₆	—	2	2	no	—
376	C₂×SD₁₆	—	2	2	no	—
377	C₂×SD₁₆	—	2	2	no	—
378	C₂×SD₁₆	D₄×C₆₁	1	2	yes	complemented
379	C₂×D₈	—	2	8	no	—
380	C₂×D₈	—	2	2	no	—
381	C₂×D₈	—	2	4	no	—
382	C₂×D₈	—	2	2	no	—
383	C₂×D₈	—	2	2	no	—
384	C₂×D₈	—	2	1	no	—
385	C₂×D₈	—	2	1	no	—
386	C₂×D₈	—	2	2	no	—
387	C₂×D₈	—	2	4	no	—
388	C₂×D₈	—	2	2	no	—
389	C₂×D₈	—	2	2	no	—
390	C₂×D₈	—	2	1	no	—
391	C₂×D₈	—	2	1	no	—
392	C₂×D₈	—	4	2	no	—
393	C₂×D₈	D₄×C₆₁	1	1	yes	characteristic, complemented
394	C₂×D₈	D₄×C₆₁	1	4	yes	complemented
395	#131	—	2	2	no	—
396	#131	D₄×C₆₁	1	4	yes	complemented
397	C₂×M₁₆	—	2	1	no	—
398	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
399	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
400	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
401	C₂²×C₈	—	2	1	no	—
402	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
403	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
404	#127	—	2	4	no	—
405	#127	—	2	2	no	—
406	#127	—	2	4	no	—
407	#124	—	2	2	no	—
408	#121	—	2	4	no	—
409	#120	—	4	2	no	—
410	#117	C₂³×C₆₁	1	1	yes	characteristic
411	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
412	#108	—	2	2	no	—
413	#104	—	2	4	no	—
414	#104	D₄×C₆₁	1	4	yes	complemented
415	#103	D₄×C₆₁	1	2	yes	complemented
416	#102	—	2	4	no	—
417	#102	D₄×C₆₁	1	2	yes	complemented
418	#100	—	2	4	no	—
419	#100	—	2	4	no	—
420	C₂²⋊C₈	—	2	4	no	—
421	#97	—	2	1	no	—
422	C₄×C₈	—	2	2	no	—
423	C₄×C₈	—	2	1	no	—
424	C₂⁴	—	2	2	no	—
425	C₂⁴	—	2	2	no	—
426	C₂⁴	—	4	2	no	—
427	#41	—	2	1	no	—
428	#41	—	2	1	no	—
429	#41	—	2	8	no	—
430	#41	—	2	2	no	—
431	#41	—	2	2	no	—
432	#41	—	2	2	no	—
433	#41	—	2	2	no	—
434	#41	—	4	4	no	—
435	#41	—	4	4	no	—
436	#41	—	4	4	no	—
437	#41	C₂×D₄×C₆₁	1	2	yes	—
438	#41	C₂×D₄×C₆₁	1	2	yes	—
439	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
440	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	8	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	2	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	2	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	4	4	no	—
456	C₂×D₄	—	4	2	no	—
457	C₂×D₄	—	4	2	no	—
458	C₂×D₄	—	4	8	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	—	4	4	no	—
461	C₂×D₄	—	4	8	no	—
462	C₂×D₄	—	4	8	no	—
463	C₂×D₄	—	4	4	no	—
464	C₂×D₄	—	8	8	no	—
465	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
466	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
467	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	4	2	no	—
472	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
473	Q₁₆	—	2	2	no	—
474	SD₁₆	—	2	8	no	—
475	SD₁₆	—	2	4	no	—
476	SD₁₆	—	4	4	no	—
477	SD₁₆	—	4	4	no	—
478	D₈	—	2	8	no	—
479	D₈	—	2	2	no	—
480	D₈	—	2	4	no	—
481	D₈	—	2	4	no	—
482	D₈	—	4	8	no	—
483	D₈	—	4	4	no	—
484	D₈	—	4	4	no	—
485	D₈	—	4	2	no	—
486	D₈	—	4	2	no	—
487	M₁₆	—	2	1	no	—
488	M₁₆	—	2	1	no	—
489	M₁₆	—	2	4	no	—
490	M₁₆	—	2	2	no	—
491	M₁₆	—	2	2	no	—
492	M₁₆	C₂×D₄×C₆₁	1	2	yes	—
493	M₁₆	C₂×D₄×C₆₁	1	2	yes	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	4	no	—
499	C₂×C₈	—	2	2	no	—
500	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
501	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
502	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
503	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
504	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
505	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	4	4	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	C₂×D₄×C₆₁	1	2	yes	—
511	C₂³	—	2	1	no	—
512	C₂³	—	2	2	no	—
513	C₂³	—	2	4	no	—
514	C₂³	—	4	8	no	—
515	C₂³	—	4	2	no	—
516	C₂³	—	4	2	no	—
517	C₂³	—	4	2	no	—
518	C₂³	—	4	2	no	—
519	C₂³	—	4	2	no	—
520	C₂³	—	8	4	no	—
521	C₂³	—	8	4	no	—
522	C₂³	—	16	4	no	—
523	C₂³	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
524	Q₈	—	2	1	no	—
525	Q₈	—	2	2	no	—
526	Q₈	—	2	2	no	—
527	Q₈	#127×C₆₁	1	2	yes	—
528	D₄	—	2	2	no	—
529	D₄	—	2	2	no	—
530	D₄	—	2	1	no	—
531	D₄	—	2	4	no	—
532	D₄	—	2	2	no	—
533	D₄	—	2	2	no	—
534	D₄	—	2	4	no	—
535	D₄	—	4	8	no	—
536	D₄	—	4	8	no	—
537	D₄	—	4	2	no	—
538	D₄	—	4	2	no	—
539	D₄	—	4	2	no	—
540	D₄	—	4	2	no	—
541	D₄	—	4	2	no	—
542	D₄	—	4	2	no	—
543	D₄	—	4	2	no	—
544	D₄	—	4	2	no	—
545	D₄	—	8	16	no	—
546	D₄	—	16	4	no	—
547	D₄	#127×C₆₁	1	2	yes	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	4	8	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	4	2	no	—
558	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
560	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
561	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
562	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
563	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
564	C₈	—	2	2	no	—
565	C₈	—	2	2	no	—
566	C₈	—	2	2	no	—
567	C₈	—	4	2	no	—
568	C₈	#120×C₆₁	1	2	yes	—
569	C₈	#120×C₆₁	1	2	yes	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	4	2	no	—
575	C₂²	—	4	4	no	—
576	C₂²	—	8	4	no	—
577	C₂²	—	8	4	no	—
578	C₂²	—	8	4	no	—
579	C₂²	—	8	4	no	—
580	C₂²	—	8	2	no	—
581	C₂²	—	16	2	no	—
582	C₂²	—	16	2	no	—
583	C₂²	—	16	8	no	—
584	C₂²	C₆₁×#535	1	1	yes	characteristic
585	C₂²	D₄²×C₆₁	1	1	yes	characteristic
586	C₂²	D₄²×C₆₁	1	1	yes	characteristic
587	C₄	—	2	4	no	—
588	C₄	—	2	2	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	2	no	—
591	C₄	C₂×#120×C₆₁	1	1	yes	characteristic
592	C₄	C₂×#127×C₆₁	1	1	yes	characteristic
593	C₄	D₄²×C₆₁	1	1	yes	characteristic
594	C₄	D₄²×C₆₁	1	1	yes	characteristic
595	C₂	—	2	1	no	—
596	C₂	—	2	1	no	—
597	C₂	—	4	2	no	—
598	C₂	—	8	2	no	—
599	C₂	—	8	4	no	—
600	C₂	—	16	2	no	—
601	C₂	C₆₁×#2433	1	1	yes	characteristic
602	C₂	C₆₁×#3292	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#576	C₂	semidirect
2. C₆₁×#3155	C₂	semidirect
3. C₂×C₆₁×#493	C₂	semidirect
4. C₂×C₆₁×#472	C₂	semidirect
5. C₂×C₆₁×#453	C₂	semidirect
6. C₆₁×#3008	C₂	semidirect
7. C₆₁×#2018	C₂	semidirect
8. C₆₁×#2002	C₂	semidirect
9. C₆₁×#1978	C₂	semidirect
10. C₆₁×#1974	C₂	semidirect
11. C₆₁×#1764	C₂	semidirect
18. C₂×AGL(1,ℤ/8)×C₆₁	C₂²	semidirect
23. C₂²×D₈×C₆₁	C₂²	semidirect
24. C₂²×D₈×C₆₁	C₂²	semidirect
26. C₆₁×#535	C₂²	semidirect
27. C₂×#127×C₆₁	C₂²	semidirect
36. C₂×#108×C₆₁	C₂²	semidirect
38. C₂×#104×C₆₁	C₂²	semidirect
39. C₆₁×#418	C₂²	semidirect
41. C₂×#100×C₆₁	C₂²	semidirect
42. C₂×#100×C₆₁	C₂²	semidirect
43. C₆₁×#408	C₂²	semidirect
44. C₆₁×#405	C₂²	semidirect
45. C₂×C₄×C₈×C₆₁	C₂²	semidirect
46. C₆₁×#341	C₂²	semidirect
47. C₆₁×#339	C₂²	semidirect
72. C₂×Q₁₆×C₆₁	D₄	semidirect
76. C₂×SD₁₆×C₆₁	D₄	semidirect
91. C₂×D₈×C₆₁	D₄	semidirect
92. C₂×D₈×C₆₁	D₄	semidirect
94. #131×C₆₁	D₄	semidirect
112. #104×C₆₁	D₄	semidirect
113. #103×C₆₁	D₄	semidirect
115. #102×C₆₁	D₄	semidirect
268. #25805	C₆₁	direct (N × H)
294. C₂×#576	C₂×C₆₁	semidirect
295. #3155	C₂×C₆₁	semidirect
296. C₂×#493	C₂×C₆₁	semidirect
297. C₂×#472	C₂×C₆₁	semidirect
298. C₂×#453	C₂×C₆₁	semidirect
299. #3008	C₂×C₆₁	semidirect
300. #2018	C₂×C₆₁	semidirect
301. #2002	C₂×C₆₁	semidirect
302. #1978	C₂×C₆₁	semidirect
303. #1974	C₂×C₆₁	semidirect
304. #1764	C₂×C₆₁	semidirect
319. C₂×AGL(1,ℤ/8)	C₂²×C₆₁	semidirect
324. C₂²×D₈	C₂²×C₆₁	semidirect
325. C₂²×D₈	C₂²×C₆₁	semidirect
327. #535	C₂²×C₆₁	semidirect
328. C₂×#127	C₂²×C₆₁	semidirect
337. C₂×#108	C₂²×C₆₁	semidirect
339. C₂×#104	C₂²×C₆₁	semidirect
340. #418	C₂²×C₆₁	semidirect
342. C₂×#100	C₂²×C₆₁	semidirect
343. C₂×#100	C₂²×C₆₁	semidirect
344. #408	C₂²×C₆₁	semidirect
345. #405	C₂²×C₆₁	semidirect
346. C₂×C₄×C₈	C₂²×C₆₁	semidirect
347. #341	C₂²×C₆₁	semidirect
348. #339	C₂²×C₆₁	semidirect
349. C₆₁	#25805	direct (N × H)
374. C₂×Q₁₆	D₄×C₆₁	semidirect
378. C₂×SD₁₆	D₄×C₆₁	semidirect
393. C₂×D₈	D₄×C₆₁	semidirect
394. C₂×D₈	D₄×C₆₁	semidirect
396. #131	D₄×C₆₁	semidirect
414. #104	D₄×C₆₁	semidirect
415. #103	D₄×C₆₁	semidirect
417. #102	D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#576	2
C₆₁×#3155	2
C₂×C₆₁×#493	2
C₂×C₆₁×#472	2
C₂×C₆₁×#453	2
C₆₁×#3008	2
C₆₁×#2018	2
C₆₁×#2002	2
C₆₁×#1978	2
C₆₁×#1974	2
C₆₁×#1764	2
#25805	61

Subgroups — C61×#25805 #29529238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25805 #29529238