Subgroups — C₂×C₆₁×#3251 #29564334

Order 15616.

2178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3251	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₆₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₁×#478	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₆₁×#469	C₂	1	2	yes	maximal, complemented
8	C₂×C₆₁×#468	C₂	1	2	yes	maximal, complemented
9	C₂×C₆₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₆₁×#463	C₂	1	2	yes	maximal, complemented
11	C₂×C₆₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₆₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₁	—	2	1	no	—
14	C₂²×SD₁₆×C₆₁	C₂²	1	2	yes	—
15	C₂²×D₈×C₆₁	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
17	C₆₁×#532	—	2	4	no	—
18	C₆₁×#532	C₂²	1	8	yes	complemented
19	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
21	C₆₁×#529	—	2	4	no	—
22	C₆₁×#529	C₂²	1	8	yes	complemented
23	C₂×#126×C₆₁	—	2	1	no	—
24	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#124×C₆₁	—	2	1	no	—
26	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#123×C₆₁	—	2	1	no	—
28	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×#121×C₆₁	—	2	1	no	—
30	C₂×#121×C₆₁	C₂²	1	2	yes	—
31	C₂×#121×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₆₁	—	2	1	no	—
34	C₂×C₄×D₄×C₆₁	—	2	1	no	—
35	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×#117×C₆₁	—	2	1	no	—
38	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
39	C₆₁×#478	C₂²	1	16	yes	complemented
40	C₆₁×#475	C₂²	1	8	yes	complemented
41	C₆₁×#474	C₂²	1	8	yes	complemented
42	C₆₁×#469	C₂²	1	16	yes	complemented
43	C₆₁×#468	C₂²	1	16	yes	complemented
44	C₆₁×#463	C₂²	1	16	yes	complemented
45	C₆₁×#460	C₂²	1	8	yes	complemented
46	C₆₁×#459	C₂²	1	8	yes	complemented
47	C₂×#107×C₆₁	C₂²	1	2	yes	—
48	C₂×#106×C₆₁	C₂²	1	2	yes	—
49	C₆₁×#423	C₂²	1	8	yes	complemented
50	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
51	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
52	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
53	C₂×#103×C₆₁	C₂²	1	2	yes	—
54	C₂×#102×C₆₁	C₂²	1	2	yes	—
55	C₂×#102×C₆₁	C₂²	1	2	yes	—
56	C₂×#102×C₆₁	C₂²	1	2	yes	—
57	C₂×#41×C₆₁	—	2	8	no	—
58	C₂×#41×C₆₁	—	4	8	no	—
59	C₂²×Q₈×C₆₁	—	2	1	no	—
60	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₆₁	—	2	2	no	—
62	C₂²×D₄×C₆₁	—	2	1	no	—
63	C₂²×D₄×C₆₁	—	2	1	no	—
64	C₂²×D₄×C₆₁	—	2	1	no	—
65	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₁	—	2	2	no	—
68	C₂³×C₄×C₆₁	—	2	1	no	—
69	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
70	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
71	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
72	C₂×SD₁₆×C₆₁	—	4	4	no	—
73	C₂×SD₁₆×C₆₁	C₂³	1	8	yes	—
74	C₂×D₈×C₆₁	—	4	4	no	—
75	C₂×D₈×C₆₁	C₂³	1	8	yes	—
76	C₂×M₁₆×C₆₁	—	4	4	no	—
77	C₂×M₁₆×C₆₁	C₂³	1	8	yes	—
78	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
79	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₆₁	—	2	2	no	—
81	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
82	#127×C₆₁	—	2	2	no	—
83	#127×C₆₁	C₂³	1	4	yes	—
84	#126×C₆₁	—	2	8	no	—
85	#125×C₆₁	—	2	2	no	—
86	#125×C₆₁	C₂³	1	4	yes	—
87	#124×C₆₁	—	2	8	no	—
88	#124×C₆₁	—	2	2	no	—
89	#124×C₆₁	C₂³	1	4	yes	—
90	#123×C₆₁	—	2	8	no	—
91	C₂²⋊Q₈×C₆₁	—	2	2	no	—
92	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
93	#121×C₆₁	—	2	4	no	—
94	#121×C₆₁	—	2	8	no	—
95	#121×C₆₁	—	2	2	no	—
96	#121×C₆₁	C₂³	1	8	yes	—
97	#121×C₆₁	C₂³	1	4	yes	—
98	C₄×Q₈×C₆₁	—	2	2	no	—
99	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
100	C₄×D₄×C₆₁	—	2	8	no	—
101	C₄×D₄×C₆₁	—	2	4	no	—
102	C₄×D₄×C₆₁	—	2	8	no	—
103	C₄×D₄×C₆₁	—	2	2	no	—
104	C₄×D₄×C₆₁	C₂³	1	8	yes	—
105	C₄×D₄×C₆₁	C₂³	1	4	yes	—
106	#117×C₆₁	—	2	8	no	—
107	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
108	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
109	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
110	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
111	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
112	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
122	C₂×C₄²×C₆₁	—	2	1	no	—
123	C₂×C₄²×C₆₁	D₄	1	4	yes	—
124	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
128	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
129	#107×C₆₁	C₂³	1	8	yes	—
130	#106×C₆₁	C₂³	1	8	yes	—
131	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
132	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
133	C₄⋊C₈×C₆₁	C₂³	1	8	yes	—
134	#103×C₆₁	C₂³	1	8	yes	—
135	#102×C₆₁	C₂³	1	8	yes	—
136	#102×C₆₁	C₂³	1	8	yes	—
137	#102×C₆₁	C₂³	1	8	yes	—
138	C₂⁴×C₆₁	—	2	1	no	—
139	C₂⁴×C₆₁	—	2	2	no	—
140	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
141	#41×C₆₁	—	4	32	no	—
142	C₂×Q₈×C₆₁	—	2	1	no	—
143	C₂×Q₈×C₆₁	—	2	4	no	—
144	C₂×Q₈×C₆₁	—	2	2	no	—
145	C₂×Q₈×C₆₁	—	2	2	no	—
146	C₂×Q₈×C₆₁	—	4	4	no	—
147	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
148	C₂×D₄×C₆₁	—	2	4	no	—
149	C₂×D₄×C₆₁	—	2	8	no	—
150	C₂×D₄×C₆₁	—	2	1	no	—
151	C₂×D₄×C₆₁	—	2	2	no	—
152	C₂×D₄×C₆₁	—	2	4	no	—
153	C₂×D₄×C₆₁	—	2	4	no	—
154	C₂×D₄×C₆₁	—	2	2	no	—
155	C₂×D₄×C₆₁	—	2	4	no	—
156	C₂×D₄×C₆₁	—	2	4	no	—
157	C₂×D₄×C₆₁	—	2	2	no	—
158	C₂×D₄×C₆₁	—	4	4	no	—
159	C₂×D₄×C₆₁	—	4	4	no	—
160	C₂×D₄×C₆₁	—	4	8	no	—
161	C₂×D₄×C₆₁	—	4	4	no	—
162	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
163	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₆₁	—	2	4	no	—
165	C₂²×C₄×C₆₁	—	2	2	no	—
166	C₂²×C₄×C₆₁	—	2	2	no	—
167	C₂²×C₄×C₆₁	—	2	1	no	—
168	C₂²×C₄×C₆₁	—	2	1	no	—
169	C₂²×C₄×C₆₁	—	2	1	no	—
170	C₂²×C₄×C₆₁	—	2	2	no	—
171	C₂²×C₄×C₆₁	—	2	4	no	—
172	C₂²×C₄×C₆₁	—	2	8	no	—
173	C₂²×C₄×C₆₁	—	4	2	no	—
174	C₂²×C₄×C₆₁	—	4	8	no	—
175	C₂²×C₄×C₆₁	—	4	4	no	—
176	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
178	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
186	SD₁₆×C₆₁	—	4	8	no	—
187	D₈×C₆₁	—	4	8	no	—
188	M₁₆×C₆₁	—	4	8	no	—
189	C₂×C₈×C₆₁	—	2	4	no	—
190	C₂×C₈×C₆₁	—	2	4	no	—
191	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
192	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
193	C₄⋊C₄×C₆₁	—	2	4	no	—
194	C₄⋊C₄×C₆₁	—	2	4	no	—
195	C₄⋊C₄×C₆₁	—	2	4	no	—
196	C₄⋊C₄×C₆₁	—	2	4	no	—
197	C₄⋊C₄×C₆₁	—	2	4	no	—
198	C₄⋊C₄×C₆₁	—	2	2	no	—
199	C₄⋊C₄×C₆₁	—	2	1	no	—
200	C₄⋊C₄×C₆₁	—	2	1	no	—
201	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
202	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
203	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
204	C₂²⋊C₄×C₆₁	—	2	4	no	—
205	C₂²⋊C₄×C₆₁	—	2	4	no	—
206	C₂²⋊C₄×C₆₁	—	2	4	no	—
207	C₂²⋊C₄×C₆₁	—	2	8	no	—
208	C₂²⋊C₄×C₆₁	—	2	4	no	—
209	C₂²⋊C₄×C₆₁	—	2	8	no	—
210	C₂²⋊C₄×C₆₁	—	2	4	no	—
211	C₄²×C₆₁	—	2	4	no	—
212	C₄²×C₆₁	C₂×D₄	1	4	yes	—
213	C₄²×C₆₁	C₂×D₄	1	2	yes	—
214	C₄²×C₆₁	C₂×D₄	1	4	yes	—
215	C₄²×C₆₁	C₂×D₄	1	2	yes	—
216	C₄²×C₆₁	C₂⁴	1	2	yes	—
217	C₄²×C₆₁	C₂⁴	1	2	yes	—
218	C₂³×C₆₁	—	2	2	no	—
219	C₂³×C₆₁	—	2	2	no	—
220	C₂³×C₆₁	—	2	2	no	—
221	C₂³×C₆₁	—	2	2	no	—
222	C₂³×C₆₁	—	2	4	no	—
223	C₂³×C₆₁	—	4	2	no	—
224	C₂³×C₆₁	—	4	4	no	—
225	C₂³×C₆₁	—	4	4	no	—
226	C₂³×C₆₁	—	4	2	no	—
227	C₂³×C₆₁	—	4	4	no	—
228	C₂³×C₆₁	—	4	2	no	—
229	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
230	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
231	Q₈×C₆₁	—	2	2	no	—
232	Q₈×C₆₁	—	2	2	no	—
233	Q₈×C₆₁	—	4	2	no	—
234	Q₈×C₆₁	—	4	8	no	—
235	D₄×C₆₁	—	2	2	no	—
236	D₄×C₆₁	—	2	4	no	—
237	D₄×C₆₁	—	2	2	no	—
238	D₄×C₆₁	—	2	4	no	—
239	D₄×C₆₁	—	4	8	no	—
240	D₄×C₆₁	—	4	8	no	—
241	D₄×C₆₁	—	4	16	no	—
242	D₄×C₆₁	—	4	2	no	—
243	D₄×C₆₁	—	4	4	no	—
244	D₄×C₆₁	—	4	8	no	—
245	C₂×C₄×C₆₁	—	2	2	no	—
246	C₂×C₄×C₆₁	—	2	2	no	—
247	C₂×C₄×C₆₁	—	2	2	no	—
248	C₂×C₄×C₆₁	—	2	2	no	—
249	C₂×C₄×C₆₁	—	2	4	no	—
250	C₂×C₄×C₆₁	—	2	8	no	—
251	C₂×C₄×C₆₁	—	2	2	no	—
252	C₂×C₄×C₆₁	—	2	2	no	—
253	C₂×C₄×C₆₁	—	2	2	no	—
254	C₂×C₄×C₆₁	—	2	2	no	—
255	C₂×C₄×C₆₁	—	2	2	no	—
256	C₂×C₄×C₆₁	—	2	4	no	—
257	C₂×C₄×C₆₁	—	4	4	no	—
258	C₂×C₄×C₆₁	—	4	16	no	—
259	C₂×C₄×C₆₁	—	4	2	no	—
260	C₂×C₄×C₆₁	—	4	2	no	—
261	C₂×C₄×C₆₁	—	4	4	no	—
262	C₂×C₄×C₆₁	—	4	2	no	—
263	C₂×C₄×C₆₁	—	4	8	no	—
264	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
265	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
269	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
272	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
273	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
274	C₈×C₆₁	—	2	4	no	—
275	C₈×C₆₁	—	2	4	no	—
276	C₂×#3251	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
277	C₂²×C₆₁	—	2	2	no	—
278	C₂²×C₆₁	—	2	2	no	—
279	C₂²×C₆₁	—	2	2	no	—
280	C₂²×C₆₁	—	4	4	no	—
281	C₂²×C₆₁	—	4	2	no	—
282	C₂²×C₆₁	—	4	4	no	—
283	C₂²×C₆₁	—	4	4	no	—
284	C₂²×C₆₁	—	4	2	no	—
285	C₂²×C₆₁	—	4	4	no	—
286	C₂²×C₆₁	—	4	2	no	—
287	C₂²×C₆₁	—	8	8	no	—
288	C₂²×C₆₁	—	8	4	no	—
289	C₂²×C₆₁	#537	1	2	yes	—
290	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
291	C₂²×C₆₁	#575	1	2	yes	—
292	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
293	C₄×C₆₁	—	2	2	no	—
294	C₄×C₆₁	—	2	2	no	—
295	C₄×C₆₁	—	2	2	no	—
296	C₄×C₆₁	—	2	2	no	—
297	C₄×C₆₁	—	4	2	no	—
298	C₄×C₆₁	—	4	2	no	—
299	C₄×C₆₁	—	4	4	no	—
300	C₄×C₆₁	—	4	2	no	—
301	C₄×C₆₁	#537	1	2	yes	—
302	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
303	C₄×C₆₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
304	C₄×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
305	C₂×#532	C₂×C₆₁	1	1	yes	characteristic, complemented
306	C₂×#529	C₂×C₆₁	1	1	yes	characteristic, complemented
307	#3251	C₂×C₆₁	1	16	yes	direct product factor
308	C₂×#475	C₂×C₆₁	1	1	yes	characteristic, complemented
309	C₂×#478	C₂×C₆₁	1	2	yes	complemented
310	C₂×#474	C₂×C₆₁	1	1	yes	characteristic, complemented
311	C₂×#469	C₂×C₆₁	1	2	yes	complemented
312	C₂×#468	C₂×C₆₁	1	2	yes	complemented
313	C₂×#460	C₂×C₆₁	1	1	yes	characteristic, complemented
314	C₂×#463	C₂×C₆₁	1	2	yes	complemented
315	C₂×#459	C₂×C₆₁	1	1	yes	characteristic, complemented
316	C₂×#423	C₂×C₆₁	1	1	yes	characteristic, complemented
317	C₂×C₆₁	—	4	2	no	—
318	C₂×C₆₁	—	8	4	no	—
319	C₂×C₆₁	—	8	2	no	—
320	C₂×C₆₁	#3251	1	4	yes	direct product factor
321	C₂×C₆₁	C₂×#537	1	1	yes	characteristic
322	C₂×C₆₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
323	C₂×C₆₁	C₂×#575	1	1	yes	characteristic
324	C₂²×#41	—	2	1	no	—
325	C₂²×SD₁₆	C₂²×C₆₁	1	2	yes	—
326	C₂²×D₈	C₂²×C₆₁	1	2	yes	—
327	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
328	#532	—	2	4	no	—
329	#532	C₂²×C₆₁	1	8	yes	complemented
330	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
331	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
332	#529	—	2	4	no	—
333	#529	C₂²×C₆₁	1	8	yes	complemented
334	C₂×#126	—	2	1	no	—
335	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
336	C₂×#124	—	2	1	no	—
337	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
338	C₂×#123	—	2	1	no	—
339	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
340	C₂×#121	—	2	1	no	—
341	C₂×#121	C₂²×C₆₁	1	1	yes	characteristic
342	C₂×#121	C₂²×C₆₁	1	2	yes	—
343	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
344	C₂×C₄×D₄	—	2	1	no	—
345	C₂×C₄×D₄	—	2	1	no	—
346	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
347	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
348	C₂×#117	—	2	1	no	—
349	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
350	#478	C₂²×C₆₁	1	16	yes	complemented
351	#475	C₂²×C₆₁	1	8	yes	complemented
352	#474	C₂²×C₆₁	1	8	yes	complemented
353	#469	C₂²×C₆₁	1	16	yes	complemented
354	#468	C₂²×C₆₁	1	16	yes	complemented
355	#463	C₂²×C₆₁	1	16	yes	complemented
356	#460	C₂²×C₆₁	1	8	yes	complemented
357	#459	C₂²×C₆₁	1	8	yes	complemented
358	C₂×#107	C₂²×C₆₁	1	2	yes	—
359	C₂×#106	C₂²×C₆₁	1	2	yes	—
360	#423	C₂²×C₆₁	1	8	yes	complemented
361	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
362	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
363	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
364	C₂×#103	C₂²×C₆₁	1	2	yes	—
365	C₂×#102	C₂²×C₆₁	1	2	yes	—
366	C₂×#102	C₂²×C₆₁	1	2	yes	—
367	C₂×#102	C₂²×C₆₁	1	2	yes	—
368	C₆₁	C₂×#3251	1	1	yes	characteristic, direct product factor, Sylow
369	C₂×#41	—	2	8	no	—
370	C₂×#41	—	4	8	no	—
371	C₂²×Q₈	—	2	1	no	—
372	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
373	C₂²×D₄	—	2	1	no	—
374	C₂²×D₄	—	2	2	no	—
375	C₂²×D₄	—	2	1	no	—
376	C₂²×D₄	—	2	1	no	—
377	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
378	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
379	C₂³×C₄	—	2	1	no	—
380	C₂³×C₄	—	2	2	no	—
381	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
382	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
383	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
384	C₂×SD₁₆	—	4	4	no	—
385	C₂×SD₁₆	C₂³×C₆₁	1	8	yes	—
386	C₂×D₈	—	4	4	no	—
387	C₂×D₈	C₂³×C₆₁	1	8	yes	—
388	C₂×M₁₆	—	4	4	no	—
389	C₂×M₁₆	C₂³×C₆₁	1	8	yes	—
390	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
391	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
392	C₄⋊Q₈	—	2	2	no	—
393	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
394	#127	—	2	2	no	—
395	#127	C₂³×C₆₁	1	4	yes	—
396	#126	—	2	8	no	—
397	#125	—	2	2	no	—
398	#125	C₂³×C₆₁	1	4	yes	—
399	#124	—	2	8	no	—
400	#124	—	2	2	no	—
401	#124	C₂³×C₆₁	1	4	yes	—
402	#123	—	2	8	no	—
403	C₂²⋊Q₈	—	2	2	no	—
404	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
405	#121	—	2	8	no	—
406	#121	—	2	2	no	—
407	#121	—	2	4	no	—
408	#121	C₂³×C₆₁	1	4	yes	—
409	#121	C₂³×C₆₁	1	8	yes	—
410	C₄×Q₈	—	2	2	no	—
411	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
412	C₄×D₄	—	2	8	no	—
413	C₄×D₄	—	2	8	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	4	no	—
416	C₄×D₄	C₂³×C₆₁	1	4	yes	—
417	C₄×D₄	C₂³×C₆₁	1	8	yes	—
418	#117	—	2	8	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	1	no	—
430	C₂×C₂²⋊C₄	—	2	2	no	—
431	C₂×C₂²⋊C₄	—	2	1	no	—
432	C₂×C₂²⋊C₄	—	2	2	no	—
433	C₂×C₂²⋊C₄	—	2	1	no	—
434	C₂×C₄²	—	2	1	no	—
435	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
436	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
437	C₂×C₄²	D₄×C₆₁	1	4	yes	—
438	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
441	#107	C₂³×C₆₁	1	8	yes	—
442	#106	C₂³×C₆₁	1	8	yes	—
443	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
444	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
445	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
446	#103	C₂³×C₆₁	1	8	yes	—
447	#102	C₂³×C₆₁	1	8	yes	—
448	#102	C₂³×C₆₁	1	8	yes	—
449	#102	C₂³×C₆₁	1	8	yes	—
450	C₂⁴	—	2	1	no	—
451	C₂⁴	—	2	2	no	—
452	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
453	#41	—	4	32	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	4	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	—	4	4	no	—
459	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	8	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	4	4	no	—
471	C₂×D₄	—	4	8	no	—
472	C₂×D₄	—	4	4	no	—
473	C₂×D₄	—	4	4	no	—
474	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
475	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	8	no	—
485	C₂²×C₄	—	4	2	no	—
486	C₂²×C₄	—	4	8	no	—
487	C₂²×C₄	—	4	4	no	—
488	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
490	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
491	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
494	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
495	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
498	SD₁₆	—	4	8	no	—
499	D₈	—	4	8	no	—
500	M₁₆	—	4	8	no	—
501	C₂×C₈	—	2	4	no	—
502	C₂×C₈	—	2	4	no	—
503	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
504	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
514	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
515	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	8	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	—	2	8	no	—
522	C₂²⋊C₄	—	2	4	no	—
523	C₄²	—	2	4	no	—
524	C₄²	C₂×D₄×C₆₁	1	4	yes	—
525	C₄²	C₂×D₄×C₆₁	1	2	yes	—
526	C₄²	C₂×D₄×C₆₁	1	4	yes	—
527	C₄²	C₂×D₄×C₆₁	1	2	yes	—
528	C₄²	C₂⁴×C₆₁	1	2	yes	—
529	C₄²	C₂⁴×C₆₁	1	2	yes	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	4	2	no	—
536	C₂³	—	4	4	no	—
537	C₂³	—	4	4	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	4	4	no	—
540	C₂³	—	4	2	no	—
541	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
542	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
543	Q₈	—	2	2	no	—
544	Q₈	—	2	2	no	—
545	Q₈	—	4	2	no	—
546	Q₈	—	4	8	no	—
547	D₄	—	2	2	no	—
548	D₄	—	2	4	no	—
549	D₄	—	2	2	no	—
550	D₄	—	2	4	no	—
551	D₄	—	4	8	no	—
552	D₄	—	4	8	no	—
553	D₄	—	4	16	no	—
554	D₄	—	4	2	no	—
555	D₄	—	4	4	no	—
556	D₄	—	4	8	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	8	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	4	4	no	—
570	C₂×C₄	—	4	16	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	2	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	8	no	—
576	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	4	yes	—
577	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
578	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
580	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
582	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
583	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
584	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
585	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
586	C₈	—	2	4	no	—
587	C₈	—	2	4	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	4	4	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	4	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	2	no	—
597	C₂²	—	4	4	no	—
598	C₂²	—	8	8	no	—
599	C₂²	—	8	4	no	—
600	C₂²	C₆₁×#537	1	2	yes	—
601	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	2	yes	—
602	C₂²	C₆₁×#575	1	2	yes	—
603	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	2	no	—
612	C₄	C₆₁×#537	1	2	yes	—
613	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	4	yes	—
614	C₄	C₂×H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
615	C₄	C₂×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
616	C₂	—	4	2	no	—
617	C₂	—	8	4	no	—
618	C₂	—	8	2	no	—
619	C₂	C₆₁×#3251	1	4	yes	direct product factor
620	C₂	C₂×C₆₁×#537	1	1	yes	characteristic
621	C₂	C₂²×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
622	C₂	C₂×C₆₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#532	C₂	semidirect
2. C₂×C₆₁×#529	C₂	semidirect
3. C₆₁×#3251	C₂	direct (N × H)
4. C₂×C₆₁×#475	C₂	semidirect
5. C₂×C₆₁×#478	C₂	semidirect
6. C₂×C₆₁×#474	C₂	semidirect
7. C₂×C₆₁×#469	C₂	semidirect
8. C₂×C₆₁×#468	C₂	semidirect
9. C₂×C₆₁×#460	C₂	semidirect
10. C₂×C₆₁×#463	C₂	semidirect
11. C₂×C₆₁×#459	C₂	semidirect
12. C₂×C₆₁×#423	C₂	semidirect
18. C₆₁×#532	C₂²	semidirect
22. C₆₁×#529	C₂²	semidirect
39. C₆₁×#478	C₂²	semidirect
40. C₆₁×#475	C₂²	semidirect
41. C₆₁×#474	C₂²	semidirect
42. C₆₁×#469	C₂²	semidirect
43. C₆₁×#468	C₂²	semidirect
44. C₆₁×#463	C₂²	semidirect
45. C₆₁×#460	C₂²	semidirect
46. C₆₁×#459	C₂²	semidirect
49. C₆₁×#423	C₂²	semidirect
276. C₂×#3251	C₆₁	direct (N × H)
305. C₂×#532	C₂×C₆₁	semidirect
306. C₂×#529	C₂×C₆₁	semidirect
307. #3251	C₂×C₆₁	direct (N × H)
308. C₂×#475	C₂×C₆₁	semidirect
309. C₂×#478	C₂×C₆₁	semidirect
310. C₂×#474	C₂×C₆₁	semidirect
311. C₂×#469	C₂×C₆₁	semidirect
312. C₂×#468	C₂×C₆₁	semidirect
313. C₂×#460	C₂×C₆₁	semidirect
314. C₂×#463	C₂×C₆₁	semidirect
315. C₂×#459	C₂×C₆₁	semidirect
316. C₂×#423	C₂×C₆₁	semidirect
320. C₂×C₆₁	#3251	direct (N × H)
329. #532	C₂²×C₆₁	semidirect
333. #529	C₂²×C₆₁	semidirect
350. #478	C₂²×C₆₁	semidirect
351. #475	C₂²×C₆₁	semidirect
352. #474	C₂²×C₆₁	semidirect
353. #469	C₂²×C₆₁	semidirect
354. #468	C₂²×C₆₁	semidirect
355. #463	C₂²×C₆₁	semidirect
356. #460	C₂²×C₆₁	semidirect
357. #459	C₂²×C₆₁	semidirect
360. #423	C₂²×C₆₁	semidirect
368. C₆₁	C₂×#3251	direct (N × H)
619. C₂	C₆₁×#3251	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#532	2
C₂×C₆₁×#529	2
C₆₁×#3251	2
C₂×C₆₁×#475	2
C₂×C₆₁×#478	2
C₂×C₆₁×#474	2
C₂×C₆₁×#469	2
C₂×C₆₁×#468	2
C₂×C₆₁×#460	2
C₂×C₆₁×#463	2
C₂×C₆₁×#459	2
C₂×C₆₁×#423	2
C₂×#3251	61

Subgroups — C2×C61×#3251 #29564334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3251 #29564334