Subgroups — C₂×C₆₁×#3356 #29564682

Order 15616.

1866 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#554	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#3356	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₆₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₁×#485	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#483	C₂	1	2	yes	maximal, complemented
7	C₂×C₆₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₆₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₆₁×#470	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×#436	C₂	1	1	yes	maximal, characteristic
12	C₂×C₆₁×#428	C₂	1	2	yes	maximal, complemented
13	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#556	—	2	4	no	—
16	C₆₁×#556	C₂²	1	8	yes	—
17	C₆₁×#554	—	2	4	no	—
18	C₆₁×#554	C₂²	1	8	yes	—
19	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#126×C₆₁	—	2	2	no	—
22	C₂×#125×C₆₁	—	2	1	no	—
23	C₂×#124×C₆₁	—	2	1	no	—
24	C₂×C₂²⋊Q₈×C₆₁	—	2	2	no	—
25	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
28	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₁	—	2	1	no	—
31	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×#117×C₆₁	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
34	C₆₁×#498	C₂²	1	8	yes	complemented
35	C₆₁×#485	C₂²	1	16	yes	complemented
36	C₆₁×#483	C₂²	1	16	yes	complemented
37	C₆₁×#477	C₂²	1	8	yes	complemented
38	C₆₁×#475	C₂²	1	8	yes	complemented
39	C₆₁×#470	C₂²	1	16	yes	complemented
40	C₄×Q₁₆×C₆₁	C₂²	1	8	yes	complemented
41	C₆₁×#436	C₂²	1	8	yes	—
42	C₆₁×#428	C₂²	1	16	yes	complemented
43	C₂×#107×C₆₁	C₂²	1	2	yes	—
44	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
45	C₂×#106×C₆₁	C₂²	1	2	yes	—
46	C₂×#106×C₆₁	C₂²	1	2	yes	—
47	C₂×#106×C₆₁	C₂²	1	2	yes	—
48	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
49	C₂×#103×C₆₁	C₂²	1	2	yes	—
50	C₂×#103×C₆₁	C₂²	1	2	yes	—
51	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
52	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
53	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₆₁	—	2	1	no	—
56	C₂²×Q₈×C₆₁	—	2	1	no	—
57	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
58	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₆₁	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₆₁	—	2	2	no	—
61	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
62	C₂×Q₁₆×C₆₁	—	2	4	no	—
63	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
64	C₂×M₁₆×C₆₁	—	2	8	no	—
65	C₂×M₁₆×C₆₁	C₂³	1	8	yes	—
66	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
67	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₆₁	—	2	2	no	—
70	C₄⋊Q₈×C₆₁	—	2	2	no	—
71	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
72	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
73	#126×C₆₁	—	2	16	no	—
74	#125×C₆₁	—	2	8	no	—
75	#124×C₆₁	—	2	8	no	—
76	C₂²⋊Q₈×C₆₁	—	2	4	no	—
77	C₂²⋊Q₈×C₆₁	—	2	4	no	—
78	C₂²⋊Q₈×C₆₁	—	2	16	no	—
79	C₂²⋊Q₈×C₆₁	C₂³	1	8	yes	—
80	C₂²⋊Q₈×C₆₁	C₂³	1	8	yes	—
81	C₄×Q₈×C₆₁	—	2	8	no	—
82	C₄×Q₈×C₆₁	—	2	2	no	—
83	C₄×Q₈×C₆₁	—	2	2	no	—
84	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
85	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
86	C₄×D₄×C₆₁	—	2	8	no	—
87	C₄×D₄×C₆₁	D₄	1	4	yes	—
88	C₄×D₄×C₆₁	C₂³	1	4	yes	—
89	#117×C₆₁	—	2	4	no	—
90	#117×C₆₁	C₂³	1	8	yes	—
91	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
94	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
97	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
103	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
105	C₂×C₂²⋊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	D₄	1	2	yes	—
109	C₂×C₄²×C₆₁	—	2	2	no	—
110	C₂×C₄²×C₆₁	—	2	1	no	—
111	C₂×C₄²×C₆₁	—	2	1	no	—
112	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
113	#107×C₆₁	C₂³	1	4	yes	—
114	#107×C₆₁	C₂³	1	8	yes	—
115	#106×C₆₁	C₂³	1	8	yes	—
116	#106×C₆₁	C₂³	1	8	yes	—
117	#106×C₆₁	C₂³	1	8	yes	—
118	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
119	#103×C₆₁	C₂³	1	4	yes	—
120	#103×C₆₁	C₂³	1	4	yes	—
121	#103×C₆₁	C₂³	1	8	yes	—
122	#103×C₆₁	C₂³	1	8	yes	—
123	C₂²⋊C₈×C₆₁	C₂³	1	8	yes	—
124	C₄×C₈×C₆₁	C₂³	1	4	yes	—
125	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
126	C₂×Q₈×C₆₁	—	2	1	no	—
127	C₂×Q₈×C₆₁	—	2	1	no	—
128	C₂×Q₈×C₆₁	—	2	4	no	—
129	C₂×Q₈×C₆₁	—	2	4	no	—
130	C₂×Q₈×C₆₁	—	2	2	no	—
131	C₂×Q₈×C₆₁	—	2	2	no	—
132	C₂×Q₈×C₆₁	—	2	2	no	—
133	C₂×Q₈×C₆₁	—	2	2	no	—
134	C₂×Q₈×C₆₁	—	4	4	no	—
135	C₂×Q₈×C₆₁	—	8	2	no	—
136	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
137	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
138	C₂×D₄×C₆₁	—	2	4	no	—
139	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₆₁	—	2	4	no	—
141	C₂²×C₄×C₆₁	—	2	4	no	—
142	C₂²×C₄×C₆₁	—	2	1	no	—
143	C₂²×C₄×C₆₁	—	2	1	no	—
144	C₂²×C₄×C₆₁	—	2	1	no	—
145	C₂²×C₄×C₆₁	—	2	2	no	—
146	C₂²×C₄×C₆₁	—	2	1	no	—
147	C₂²×C₄×C₆₁	—	2	2	no	—
148	C₂²×C₄×C₆₁	—	2	8	no	—
149	C₂²×C₄×C₆₁	—	8	4	no	—
150	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
151	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
156	Q₁₆×C₆₁	—	2	8	no	—
157	M₁₆×C₆₁	—	2	16	no	—
158	C₂×C₈×C₆₁	—	2	4	no	—
159	C₂×C₈×C₆₁	—	2	2	no	—
160	C₂×C₈×C₆₁	#41	1	4	yes	—
161	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
162	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
163	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
164	C₄⋊C₄×C₆₁	—	2	1	no	—
165	C₄⋊C₄×C₆₁	—	2	8	no	—
166	C₄⋊C₄×C₆₁	—	2	2	no	—
167	C₄⋊C₄×C₆₁	—	2	4	no	—
168	C₄⋊C₄×C₆₁	—	2	8	no	—
169	C₄⋊C₄×C₆₁	—	2	4	no	—
170	C₄⋊C₄×C₆₁	—	2	4	no	—
171	C₄⋊C₄×C₆₁	—	2	8	no	—
172	C₄⋊C₄×C₆₁	—	2	4	no	—
173	C₄⋊C₄×C₆₁	—	2	1	no	—
174	C₄⋊C₄×C₆₁	—	2	2	no	—
175	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
176	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
177	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
178	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
179	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
180	C₂²⋊C₄×C₆₁	—	2	8	no	—
181	C₂²⋊C₄×C₆₁	—	2	8	no	—
182	C₂²⋊C₄×C₆₁	—	2	8	no	—
183	C₂²⋊C₄×C₆₁	C₂×D₄	1	8	yes	—
184	C₄²×C₆₁	—	2	8	no	—
185	C₄²×C₆₁	—	2	4	no	—
186	C₄²×C₆₁	—	2	4	no	—
187	C₄²×C₆₁	C₂×D₄	1	2	yes	—
188	C₄²×C₆₁	C₂⁴	1	2	yes	—
189	C₂³×C₆₁	—	2	4	no	—
190	C₂³×C₆₁	—	2	4	no	—
191	C₂³×C₆₁	—	2	4	no	—
192	C₂³×C₆₁	C₂²×D₄	1	4	yes	—
193	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₆₁	—	2	2	no	—
195	Q₈×C₆₁	—	2	2	no	—
196	Q₈×C₆₁	—	2	2	no	—
197	Q₈×C₆₁	—	2	2	no	—
198	Q₈×C₆₁	—	4	2	no	—
199	Q₈×C₆₁	—	4	2	no	—
200	Q₈×C₆₁	—	4	8	no	—
201	Q₈×C₆₁	—	8	4	no	—
202	D₄×C₆₁	—	2	8	no	—
203	C₂×C₄×C₆₁	—	2	4	no	—
204	C₂×C₄×C₆₁	—	2	2	no	—
205	C₂×C₄×C₆₁	—	2	8	no	—
206	C₂×C₄×C₆₁	—	2	4	no	—
207	C₂×C₄×C₆₁	—	2	4	no	—
208	C₂×C₄×C₆₁	—	2	2	no	—
209	C₂×C₄×C₆₁	—	2	2	no	—
210	C₂×C₄×C₆₁	—	2	2	no	—
211	C₂×C₄×C₆₁	—	2	4	no	—
212	C₂×C₄×C₆₁	—	2	2	no	—
213	C₂×C₄×C₆₁	—	2	4	no	—
214	C₂×C₄×C₆₁	—	4	2	no	—
215	C₂×C₄×C₆₁	—	4	2	no	—
216	C₂×C₄×C₆₁	—	4	4	no	—
217	C₂×C₄×C₆₁	—	4	2	no	—
218	C₂×C₄×C₆₁	—	4	2	no	—
219	C₂×C₄×C₆₁	—	4	4	no	—
220	C₂×C₄×C₆₁	—	8	8	no	—
221	C₂×C₄×C₆₁	#137	1	4	yes	—
222	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
227	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
228	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
229	C₈×C₆₁	—	2	4	no	—
230	C₈×C₆₁	—	2	2	no	—
231	C₈×C₆₁	C₂×#41	1	4	yes	—
232	C₂×#3356	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂²×C₆₁	—	2	4	no	—
234	C₂²×C₆₁	—	2	4	no	—
235	C₂²×C₆₁	—	2	4	no	—
236	C₂²×C₆₁	—	4	8	no	—
237	C₂²×C₆₁	#547	1	2	yes	—
238	C₂²×C₆₁	C₂×#137	1	2	yes	—
239	C₂²×C₆₁	#577	1	2	yes	—
240	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
241	C₄×C₆₁	—	2	4	no	—
242	C₄×C₆₁	—	2	2	no	—
243	C₄×C₆₁	—	4	2	no	—
244	C₄×C₆₁	—	4	2	no	—
245	C₄×C₆₁	—	4	4	no	—
246	C₄×C₆₁	—	4	2	no	—
247	C₄×C₆₁	—	4	2	no	—
248	C₄×C₆₁	—	4	4	no	—
249	C₄×C₆₁	#547	1	2	yes	—
250	C₄×C₆₁	C₂×#137	1	4	yes	—
251	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
252	C₄×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
253	C₂×#556	C₂×C₆₁	1	1	yes	characteristic
254	C₂×#554	C₂×C₆₁	1	1	yes	characteristic
255	#3356	C₂×C₆₁	1	16	yes	direct product factor
256	C₂×#498	C₂×C₆₁	1	1	yes	characteristic, complemented
257	C₂×#485	C₂×C₆₁	1	2	yes	complemented
258	C₂×#483	C₂×C₆₁	1	2	yes	complemented
259	C₂×#477	C₂×C₆₁	1	1	yes	characteristic, complemented
260	C₂×#475	C₂×C₆₁	1	1	yes	characteristic, complemented
261	C₂×#470	C₂×C₆₁	1	2	yes	complemented
262	C₂×C₄×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
263	C₂×#436	C₂×C₆₁	1	1	yes	characteristic
264	C₂×#428	C₂×C₆₁	1	2	yes	complemented
265	C₂×C₆₁	—	4	4	no	—
266	C₂×C₆₁	#3356	1	4	yes	direct product factor
267	C₂×C₆₁	C₂×#547	1	1	yes	characteristic
268	C₂×C₆₁	C₂²×#137	1	1	yes	characteristic
269	C₂×C₆₁	C₂×#577	1	1	yes	characteristic
270	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
271	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
272	#556	—	2	4	no	—
273	#556	C₂²×C₆₁	1	8	yes	—
274	#554	—	2	4	no	—
275	#554	C₂²×C₆₁	1	8	yes	—
276	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
277	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
278	C₂×#126	—	2	2	no	—
279	C₂×#125	—	2	1	no	—
280	C₂×#124	—	2	1	no	—
281	C₂×C₂²⋊Q₈	—	2	2	no	—
282	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
283	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
284	C₂×C₄×Q₈	—	2	1	no	—
285	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
286	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
287	C₂×C₄×D₄	—	2	1	no	—
288	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
289	C₂×#117	C₂²×C₆₁	1	2	yes	—
290	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
291	#498	C₂²×C₆₁	1	8	yes	complemented
292	#485	C₂²×C₆₁	1	16	yes	complemented
293	#483	C₂²×C₆₁	1	16	yes	complemented
294	#477	C₂²×C₆₁	1	8	yes	complemented
295	#475	C₂²×C₆₁	1	8	yes	complemented
296	#470	C₂²×C₆₁	1	16	yes	complemented
297	C₄×Q₁₆	C₂²×C₆₁	1	8	yes	complemented
298	#436	C₂²×C₆₁	1	8	yes	—
299	#428	C₂²×C₆₁	1	16	yes	complemented
300	C₂×#107	C₂²×C₆₁	1	2	yes	—
301	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
302	C₂×#106	C₂²×C₆₁	1	2	yes	—
303	C₂×#106	C₂²×C₆₁	1	2	yes	—
304	C₂×#106	C₂²×C₆₁	1	2	yes	—
305	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
306	C₂×#103	C₂²×C₆₁	1	2	yes	—
307	C₂×#103	C₂²×C₆₁	1	2	yes	—
308	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
309	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
310	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
311	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
312	C₆₁	C₂×#3356	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²×Q₈	—	2	1	no	—
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
316	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
317	C₂²×D₄	D₄×C₆₁	1	1	yes	characteristic
318	C₂³×C₄	—	2	2	no	—
319	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
320	C₂×Q₁₆	—	2	4	no	—
321	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
322	C₂×M₁₆	—	2	8	no	—
323	C₂×M₁₆	C₂³×C₆₁	1	8	yes	—
324	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
325	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
327	C₄⋊Q₈	—	2	2	no	—
328	C₄⋊Q₈	—	2	2	no	—
329	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
330	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
331	#126	—	2	16	no	—
332	#125	—	2	8	no	—
333	#124	—	2	8	no	—
334	C₂²⋊Q₈	—	2	4	no	—
335	C₂²⋊Q₈	—	2	4	no	—
336	C₂²⋊Q₈	—	2	16	no	—
337	C₂²⋊Q₈	C₂³×C₆₁	1	8	yes	—
338	C₂²⋊Q₈	C₂³×C₆₁	1	8	yes	—
339	C₄×Q₈	—	2	8	no	—
340	C₄×Q₈	—	2	2	no	—
341	C₄×Q₈	—	2	2	no	—
342	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
343	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
344	C₄×D₄	—	2	8	no	—
345	C₄×D₄	D₄×C₆₁	1	4	yes	—
346	C₄×D₄	C₂³×C₆₁	1	4	yes	—
347	#117	—	2	4	no	—
348	#117	C₂³×C₆₁	1	8	yes	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	4	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	2	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	—	2	2	no	—
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
359	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
360	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
361	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
362	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
363	C₂×C₂²⋊C₄	—	2	2	no	—
364	C₂×C₂²⋊C₄	—	2	2	no	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
367	C₂×C₄²	—	2	2	no	—
368	C₂×C₄²	—	2	1	no	—
369	C₂×C₄²	—	2	1	no	—
370	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
371	#107	C₂³×C₆₁	1	4	yes	—
372	#107	C₂³×C₆₁	1	8	yes	—
373	#106	C₂³×C₆₁	1	8	yes	—
374	#106	C₂³×C₆₁	1	8	yes	—
375	#106	C₂³×C₆₁	1	8	yes	—
376	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
377	#103	C₂³×C₆₁	1	4	yes	—
378	#103	C₂³×C₆₁	1	4	yes	—
379	#103	C₂³×C₆₁	1	8	yes	—
380	#103	C₂³×C₆₁	1	8	yes	—
381	C₂²⋊C₈	C₂³×C₆₁	1	8	yes	—
382	C₄×C₈	C₂³×C₆₁	1	4	yes	—
383	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
384	C₂×Q₈	—	2	1	no	—
385	C₂×Q₈	—	2	1	no	—
386	C₂×Q₈	—	2	4	no	—
387	C₂×Q₈	—	2	4	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	2	no	—
392	C₂×Q₈	—	4	4	no	—
393	C₂×Q₈	—	8	2	no	—
394	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
395	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	—	8	4	no	—
408	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
409	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
410	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
411	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
413	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
414	Q₁₆	—	2	8	no	—
415	M₁₆	—	2	16	no	—
416	C₂×C₈	—	2	4	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	#41×C₆₁	1	4	yes	—
419	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
420	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
421	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	8	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	8	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	8	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
434	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
435	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
436	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
437	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
438	C₂²⋊C₄	—	2	8	no	—
439	C₂²⋊C₄	—	2	8	no	—
440	C₂²⋊C₄	—	2	8	no	—
441	C₂²⋊C₄	C₂×D₄×C₆₁	1	8	yes	—
442	C₄²	—	2	8	no	—
443	C₄²	—	2	4	no	—
444	C₄²	—	2	4	no	—
445	C₄²	C₂×D₄×C₆₁	1	2	yes	—
446	C₄²	C₂⁴×C₆₁	1	2	yes	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	4	no	—
450	C₂³	C₂²×D₄×C₆₁	1	4	yes	—
451	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
452	Q₈	—	2	2	no	—
453	Q₈	—	2	2	no	—
454	Q₈	—	2	2	no	—
455	Q₈	—	2	2	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	4	2	no	—
458	Q₈	—	4	8	no	—
459	Q₈	—	8	4	no	—
460	D₄	—	2	8	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	8	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	8	8	no	—
479	C₂×C₄	#137×C₆₁	1	4	yes	—
480	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
481	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
484	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
485	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
486	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
487	C₈	—	2	4	no	—
488	C₈	—	2	2	no	—
489	C₈	C₂×#41×C₆₁	1	4	yes	—
490	C₂²	—	2	4	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	4	8	no	—
494	C₂²	C₆₁×#547	1	2	yes	—
495	C₂²	C₂×#137×C₆₁	1	2	yes	—
496	C₂²	C₆₁×#577	1	2	yes	—
497	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	2	4	no	—
500	C₄	—	4	4	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	4	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	C₆₁×#547	1	2	yes	—
507	C₄	C₂×#137×C₆₁	1	4	yes	—
508	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
509	C₄	C₂×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
510	C₂	—	4	4	no	—
511	C₂	C₆₁×#3356	1	4	yes	direct product factor
512	C₂	C₂×C₆₁×#547	1	1	yes	characteristic
513	C₂	C₂²×#137×C₆₁	1	1	yes	characteristic
514	C₂	C₂×C₆₁×#577	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₆₁×#3356	C₂	direct (N × H)
4. C₂×C₆₁×#498	C₂	semidirect
5. C₂×C₆₁×#485	C₂	semidirect
6. C₂×C₆₁×#483	C₂	semidirect
7. C₂×C₆₁×#477	C₂	semidirect
8. C₂×C₆₁×#475	C₂	semidirect
9. C₂×C₆₁×#470	C₂	semidirect
10. C₂×C₄×Q₁₆×C₆₁	C₂	semidirect
12. C₂×C₆₁×#428	C₂	semidirect
34. C₆₁×#498	C₂²	semidirect
35. C₆₁×#485	C₂²	semidirect
36. C₆₁×#483	C₂²	semidirect
37. C₆₁×#477	C₂²	semidirect
38. C₆₁×#475	C₂²	semidirect
39. C₆₁×#470	C₂²	semidirect
40. C₄×Q₁₆×C₆₁	C₂²	semidirect
42. C₆₁×#428	C₂²	semidirect
232. C₂×#3356	C₆₁	direct (N × H)
255. #3356	C₂×C₆₁	direct (N × H)
256. C₂×#498	C₂×C₆₁	semidirect
257. C₂×#485	C₂×C₆₁	semidirect
258. C₂×#483	C₂×C₆₁	semidirect
259. C₂×#477	C₂×C₆₁	semidirect
260. C₂×#475	C₂×C₆₁	semidirect
261. C₂×#470	C₂×C₆₁	semidirect
262. C₂×C₄×Q₁₆	C₂×C₆₁	semidirect
264. C₂×#428	C₂×C₆₁	semidirect
266. C₂×C₆₁	#3356	direct (N × H)
291. #498	C₂²×C₆₁	semidirect
292. #485	C₂²×C₆₁	semidirect
293. #483	C₂²×C₆₁	semidirect
294. #477	C₂²×C₆₁	semidirect
295. #475	C₂²×C₆₁	semidirect
296. #470	C₂²×C₆₁	semidirect
297. C₄×Q₁₆	C₂²×C₆₁	semidirect
299. #428	C₂²×C₆₁	semidirect
312. C₆₁	C₂×#3356	direct (N × H)
511. C₂	C₆₁×#3356	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×#556	2
C₂×C₆₁×#554	2
C₆₁×#3356	2
C₂×C₆₁×#498	2
C₂×C₆₁×#485	2
C₂×C₆₁×#483	2
C₂×C₆₁×#477	2
C₂×C₆₁×#475	2
C₂×C₆₁×#470	2
C₂×C₄×Q₁₆×C₆₁	2
C₂×C₆₁×#436	2
C₂×C₆₁×#428	2
C₂×#3356	61

Subgroups — C2×C61×#3356 #29564682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3356 #29564682