Subgroups — C₂×C₆₁×#3231 #29564277

Order 15616.

1778 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#533	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#3231	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₆₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₆₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₆₁×#485	C₂	1	4	yes	maximal
9	C₂×C₆₁×#479	C₂	1	4	yes	maximal, complemented
10	C₂×C₆₁×#431	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#533	—	2	4	no	—
12	C₆₁×#533	—	2	4	no	—
13	C₆₁×#533	C₂²	1	8	yes	—
14	C₆₁×#533	C₂²	1	8	yes	—
15	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×#125×C₆₁	—	2	1	no	—
20	C₂×#125×C₆₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
22	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
23	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
24	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
26	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
27	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
28	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
29	C₂×#117×C₆₁	—	2	1	no	—
30	C₂×#117×C₆₁	C₂²	1	2	yes	—
31	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#500	C₂²	1	8	yes	complemented
33	C₆₁×#498	C₂²	1	8	yes	complemented
34	C₆₁×#487	C₂²	1	8	yes	complemented
35	C₆₁×#487	C₂²	1	8	yes	complemented
36	C₆₁×#485	C₂²	1	32	yes	—
37	C₆₁×#479	C₂²	1	32	yes	complemented
38	C₆₁×#431	C₂²	1	8	yes	—
39	C₂×#107×C₆₁	C₂²	1	4	yes	—
40	C₂×#106×C₆₁	C₂²	1	4	yes	—
41	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
42	C₂×#103×C₆₁	C₂²	1	4	yes	—
43	C₂×#103×C₆₁	C₂²	1	4	yes	—
44	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
47	C₂²×Q₈×C₆₁	—	2	1	no	—
48	C₂²×Q₈×C₆₁	—	2	1	no	—
49	C₂²×Q₈×C₆₁	—	2	1	no	—
50	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
51	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₆₁	—	2	2	no	—
57	C₄⋊Q₈×C₆₁	—	2	2	no	—
58	C₄⋊Q₈×C₆₁	—	2	2	no	—
59	C₄⋊Q₈×C₆₁	—	2	2	no	—
60	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
61	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
62	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
63	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
64	#125×C₆₁	—	2	4	no	—
65	#125×C₆₁	—	2	8	no	—
66	#125×C₆₁	C₂³	1	8	yes	—
67	C₂²⋊Q₈×C₆₁	—	2	4	no	—
68	C₂²⋊Q₈×C₆₁	—	2	8	no	—
69	C₂²⋊Q₈×C₆₁	—	2	8	no	—
70	C₂²⋊Q₈×C₆₁	—	2	8	no	—
71	C₂²⋊Q₈×C₆₁	C₂³	1	8	yes	—
72	C₄×Q₈×C₆₁	—	2	4	no	—
73	C₄×Q₈×C₆₁	—	2	8	no	—
74	C₄×Q₈×C₆₁	—	2	8	no	—
75	C₄×Q₈×C₆₁	—	2	8	no	—
76	C₄×Q₈×C₆₁	C₂³	1	8	yes	—
77	#117×C₆₁	—	2	4	no	—
78	#117×C₆₁	—	2	8	no	—
79	#117×C₆₁	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
86	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
88	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
89	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
94	C₂×C₄²×C₆₁	—	2	2	no	—
95	C₂×C₄²×C₆₁	—	2	2	no	—
96	C₂×C₄²×C₆₁	D₄	1	4	yes	—
97	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
101	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
102	#107×C₆₁	C₂³	1	16	yes	—
103	#106×C₆₁	C₂³	1	16	yes	—
104	C₄⋊C₈×C₆₁	C₂³	1	8	yes	—
105	#103×C₆₁	C₂³	1	16	yes	—
106	#103×C₆₁	C₂³	1	16	yes	—
107	C₂²⋊C₈×C₆₁	C₂³	1	8	yes	—
108	C₄×C₈×C₆₁	C₂³	1	4	yes	—
109	C₄×C₈×C₆₁	C₂³	1	4	yes	—
110	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
111	C₂×Q₈×C₆₁	—	2	2	no	—
112	C₂×Q₈×C₆₁	—	2	4	no	—
113	C₂×Q₈×C₆₁	—	2	4	no	—
114	C₂×Q₈×C₆₁	—	2	4	no	—
115	C₂×Q₈×C₆₁	—	2	8	no	—
116	C₂×Q₈×C₆₁	—	4	4	no	—
117	C₂×Q₈×C₆₁	—	4	4	no	—
118	C₂×Q₈×C₆₁	—	4	4	no	—
119	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₆₁	—	2	4	no	—
121	C₂²×C₄×C₆₁	—	2	4	no	—
122	C₂²×C₄×C₆₁	—	2	4	no	—
123	C₂²×C₄×C₆₁	—	2	2	no	—
124	C₂²×C₄×C₆₁	—	2	2	no	—
125	C₂²×C₄×C₆₁	—	2	2	no	—
126	C₂²×C₄×C₆₁	—	2	2	no	—
127	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
130	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
136	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
137	C₂×C₈×C₆₁	—	2	4	no	—
138	C₂×C₈×C₆₁	—	2	4	no	—
139	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
140	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
141	C₄⋊C₄×C₆₁	—	2	4	no	—
142	C₄⋊C₄×C₆₁	—	2	4	no	—
143	C₄⋊C₄×C₆₁	—	2	8	no	—
144	C₄⋊C₄×C₆₁	—	2	8	no	—
145	C₄⋊C₄×C₆₁	—	2	4	no	—
146	C₄⋊C₄×C₆₁	—	2	4	no	—
147	C₄⋊C₄×C₆₁	—	2	4	no	—
148	C₄⋊C₄×C₆₁	—	2	8	no	—
149	C₄⋊C₄×C₆₁	—	2	8	no	—
150	C₄⋊C₄×C₆₁	—	2	2	no	—
151	C₄⋊C₄×C₆₁	—	2	2	no	—
152	C₄⋊C₄×C₆₁	—	2	2	no	—
153	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
154	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
155	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
156	C₂²⋊C₄×C₆₁	—	2	8	no	—
157	C₂²⋊C₄×C₆₁	—	2	8	no	—
158	C₄²×C₆₁	—	2	8	no	—
159	C₄²×C₆₁	—	2	8	no	—
160	C₄²×C₆₁	C₂×D₄	1	4	yes	—
161	C₄²×C₆₁	C₂×D₄	1	4	yes	—
162	C₄²×C₆₁	C₂×D₄	1	2	yes	—
163	C₄²×C₆₁	C₂×D₄	1	2	yes	—
164	C₄²×C₆₁	C₂⁴	1	2	yes	—
165	C₄²×C₆₁	C₂⁴	1	2	yes	—
166	C₂³×C₆₁	—	2	2	no	—
167	C₂³×C₆₁	—	2	2	no	—
168	C₂³×C₆₁	—	2	2	no	—
169	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
170	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
171	Q₈×C₆₁	—	2	8	no	—
172	Q₈×C₆₁	—	4	4	no	—
173	Q₈×C₆₁	—	4	8	no	—
174	Q₈×C₆₁	—	4	8	no	—
175	Q₈×C₆₁	—	4	8	no	—
176	C₂×C₄×C₆₁	—	2	2	no	—
177	C₂×C₄×C₆₁	—	2	2	no	—
178	C₂×C₄×C₆₁	—	2	2	no	—
179	C₂×C₄×C₆₁	—	2	4	no	—
180	C₂×C₄×C₆₁	—	2	8	no	—
181	C₂×C₄×C₆₁	—	2	8	no	—
182	C₂×C₄×C₆₁	—	2	4	no	—
183	C₂×C₄×C₆₁	—	2	4	no	—
184	C₂×C₄×C₆₁	—	2	4	no	—
185	C₂×C₄×C₆₁	—	2	4	no	—
186	C₂×C₄×C₆₁	—	2	4	no	—
187	C₂×C₄×C₆₁	—	4	4	no	—
188	C₂×C₄×C₆₁	—	4	4	no	—
189	C₂×C₄×C₆₁	—	4	4	no	—
190	C₂×C₄×C₆₁	—	4	4	no	—
191	C₂×C₄×C₆₁	#135	1	4	yes	—
192	C₂×C₄×C₆₁	#137	1	4	yes	—
193	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
195	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
196	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
198	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
199	C₂×C₄×C₆₁	2₋¹⁺⁴	1	4	yes	—
200	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
201	C₈×C₆₁	—	2	4	no	—
202	C₈×C₆₁	—	2	4	no	—
203	C₂×#3231	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂²×C₆₁	—	2	2	no	—
205	C₂²×C₆₁	—	2	2	no	—
206	C₂²×C₆₁	—	2	2	no	—
207	C₂²×C₆₁	—	4	4	no	—
208	C₂²×C₆₁	#536	1	2	yes	—
209	C₂²×C₆₁	C₂×#135	1	2	yes	—
210	C₂²×C₆₁	C₂×#137	1	2	yes	—
211	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
212	C₄×C₆₁	—	2	2	no	—
213	C₄×C₆₁	—	2	2	no	—
214	C₄×C₆₁	—	2	2	no	—
215	C₄×C₆₁	—	4	4	no	—
216	C₄×C₆₁	—	4	4	no	—
217	C₄×C₆₁	—	4	4	no	—
218	C₄×C₆₁	—	4	4	no	—
219	C₄×C₆₁	#536	1	2	yes	—
220	C₄×C₆₁	C₂×#135	1	4	yes	—
221	C₄×C₆₁	C₂×#137	1	4	yes	—
222	C₄×C₆₁	C₂×2₋¹⁺⁴	1	2	yes	—
223	C₂×#533	C₂×C₆₁	1	1	yes	characteristic
224	C₂×#533	C₂×C₆₁	1	1	yes	characteristic
225	#3231	C₂×C₆₁	1	16	yes	direct product factor
226	C₂×#500	C₂×C₆₁	1	1	yes	characteristic, complemented
227	C₂×#498	C₂×C₆₁	1	1	yes	characteristic, complemented
228	C₂×#487	C₂×C₆₁	1	1	yes	characteristic, complemented
229	C₂×#487	C₂×C₆₁	1	1	yes	characteristic, complemented
230	C₂×#485	C₂×C₆₁	1	4	yes	—
231	C₂×#479	C₂×C₆₁	1	4	yes	complemented
232	C₂×#431	C₂×C₆₁	1	1	yes	characteristic
233	C₂×C₆₁	—	4	2	no	—
234	C₂×C₆₁	#3231	1	4	yes	direct product factor
235	C₂×C₆₁	C₂×#536	1	1	yes	characteristic
236	C₂×C₆₁	C₂²×#135	1	1	yes	characteristic
237	C₂×C₆₁	C₂²×#137	1	1	yes	characteristic
238	#533	—	2	4	no	—
239	#533	—	2	4	no	—
240	#533	C₂²×C₆₁	1	8	yes	—
241	#533	C₂²×C₆₁	1	8	yes	—
242	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
243	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
244	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
245	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
246	C₂×#125	—	2	1	no	—
247	C₂×#125	C₂²×C₆₁	1	2	yes	—
248	C₂×C₂²⋊Q₈	—	2	1	no	—
249	C₂×C₂²⋊Q₈	—	2	1	no	—
250	C₂×C₂²⋊Q₈	—	2	1	no	—
251	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
252	C₂×C₄×Q₈	—	2	1	no	—
253	C₂×C₄×Q₈	—	2	1	no	—
254	C₂×C₄×Q₈	—	2	1	no	—
255	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
256	C₂×#117	—	2	1	no	—
257	C₂×#117	C₂²×C₆₁	1	2	yes	—
258	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
259	#500	C₂²×C₆₁	1	8	yes	complemented
260	#498	C₂²×C₆₁	1	8	yes	complemented
261	#487	C₂²×C₆₁	1	8	yes	complemented
262	#487	C₂²×C₆₁	1	8	yes	complemented
263	#485	C₂²×C₆₁	1	32	yes	—
264	#479	C₂²×C₆₁	1	32	yes	complemented
265	#431	C₂²×C₆₁	1	8	yes	—
266	C₂×#107	C₂²×C₆₁	1	4	yes	—
267	C₂×#106	C₂²×C₆₁	1	4	yes	—
268	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
269	C₂×#103	C₂²×C₆₁	1	4	yes	—
270	C₂×#103	C₂²×C₆₁	1	4	yes	—
271	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
272	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
273	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
274	C₆₁	C₂×#3231	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²×Q₈	—	2	1	no	—
276	C₂²×Q₈	—	2	1	no	—
277	C₂²×Q₈	—	2	1	no	—
278	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
279	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
280	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
283	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
284	C₄⋊Q₈	—	2	2	no	—
285	C₄⋊Q₈	—	2	2	no	—
286	C₄⋊Q₈	—	2	2	no	—
287	C₄⋊Q₈	—	2	2	no	—
288	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
289	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
290	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
291	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
292	#125	—	2	8	no	—
293	#125	—	2	4	no	—
294	#125	C₂³×C₆₁	1	8	yes	—
295	C₂²⋊Q₈	—	2	8	no	—
296	C₂²⋊Q₈	—	2	8	no	—
297	C₂²⋊Q₈	—	2	8	no	—
298	C₂²⋊Q₈	—	2	4	no	—
299	C₂²⋊Q₈	C₂³×C₆₁	1	8	yes	—
300	C₄×Q₈	—	2	8	no	—
301	C₄×Q₈	—	2	8	no	—
302	C₄×Q₈	—	2	8	no	—
303	C₄×Q₈	—	2	4	no	—
304	C₄×Q₈	C₂³×C₆₁	1	8	yes	—
305	#117	—	2	8	no	—
306	#117	—	2	4	no	—
307	#117	C₂³×C₆₁	1	8	yes	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
318	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
319	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₄²	—	2	2	no	—
323	C₂×C₄²	—	2	2	no	—
324	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
325	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
326	C₂×C₄²	D₄×C₆₁	1	4	yes	—
327	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
328	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
329	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
330	#107	C₂³×C₆₁	1	16	yes	—
331	#106	C₂³×C₆₁	1	16	yes	—
332	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
333	#103	C₂³×C₆₁	1	16	yes	—
334	#103	C₂³×C₆₁	1	16	yes	—
335	C₂²⋊C₈	C₂³×C₆₁	1	8	yes	—
336	C₄×C₈	C₂³×C₆₁	1	4	yes	—
337	C₄×C₈	C₂³×C₆₁	1	4	yes	—
338	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	2	4	no	—
341	C₂×Q₈	—	2	4	no	—
342	C₂×Q₈	—	2	4	no	—
343	C₂×Q₈	—	2	8	no	—
344	C₂×Q₈	—	4	4	no	—
345	C₂×Q₈	—	4	4	no	—
346	C₂×Q₈	—	4	4	no	—
347	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
358	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
360	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
361	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
363	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
364	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
365	C₂×C₈	—	2	4	no	—
366	C₂×C₈	—	2	4	no	—
367	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
368	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
369	C₄⋊C₄	—	2	4	no	—
370	C₄⋊C₄	—	2	4	no	—
371	C₄⋊C₄	—	2	8	no	—
372	C₄⋊C₄	—	2	8	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
382	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
383	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
384	C₂²⋊C₄	—	2	8	no	—
385	C₂²⋊C₄	—	2	8	no	—
386	C₄²	—	2	8	no	—
387	C₄²	—	2	8	no	—
388	C₄²	C₂×D₄×C₆₁	1	4	yes	—
389	C₄²	C₂×D₄×C₆₁	1	4	yes	—
390	C₄²	C₂×D₄×C₆₁	1	2	yes	—
391	C₄²	C₂×D₄×C₆₁	1	2	yes	—
392	C₄²	C₂⁴×C₆₁	1	2	yes	—
393	C₄²	C₂⁴×C₆₁	1	2	yes	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
398	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
399	Q₈	—	2	8	no	—
400	Q₈	—	4	4	no	—
401	Q₈	—	4	8	no	—
402	Q₈	—	4	8	no	—
403	Q₈	—	4	8	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	8	no	—
409	C₂×C₄	—	2	8	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	4	4	no	—
416	C₂×C₄	—	4	4	no	—
417	C₂×C₄	—	4	4	no	—
418	C₂×C₄	—	4	4	no	—
419	C₂×C₄	#135×C₆₁	1	4	yes	—
420	C₂×C₄	#137×C₆₁	1	4	yes	—
421	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
422	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
423	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
424	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
425	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
426	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
427	C₂×C₄	2₋¹⁺⁴×C₆₁	1	4	yes	—
428	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
429	C₈	—	2	4	no	—
430	C₈	—	2	4	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	4	4	no	—
435	C₂²	C₆₁×#536	1	2	yes	—
436	C₂²	C₂×#135×C₆₁	1	2	yes	—
437	C₂²	C₂×#137×C₆₁	1	2	yes	—
438	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
439	C₄	—	2	2	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	2	no	—
442	C₄	—	4	4	no	—
443	C₄	—	4	4	no	—
444	C₄	—	4	4	no	—
445	C₄	—	4	4	no	—
446	C₄	C₆₁×#536	1	2	yes	—
447	C₄	C₂×#135×C₆₁	1	4	yes	—
448	C₄	C₂×#137×C₆₁	1	4	yes	—
449	C₄	C₂×2₋¹⁺⁴×C₆₁	1	2	yes	—
450	C₂	—	4	2	no	—
451	C₂	C₆₁×#3231	1	4	yes	direct product factor
452	C₂	C₂×C₆₁×#536	1	1	yes	characteristic
453	C₂	C₂²×#135×C₆₁	1	1	yes	characteristic
454	C₂	C₂²×#137×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₂⁴×C₆₁	C₂×D₄
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
3. C₆₁×#3231	C₂	direct (N × H)
4. C₂×C₆₁×#500	C₂	semidirect
5. C₂×C₆₁×#498	C₂	semidirect
6. C₂×C₆₁×#487	C₂	semidirect
7. C₂×C₆₁×#487	C₂	semidirect
9. C₂×C₆₁×#479	C₂	semidirect
32. C₆₁×#500	C₂²	semidirect
33. C₆₁×#498	C₂²	semidirect
34. C₆₁×#487	C₂²	semidirect
35. C₆₁×#487	C₂²	semidirect
37. C₆₁×#479	C₂²	semidirect
203. C₂×#3231	C₆₁	direct (N × H)
225. #3231	C₂×C₆₁	direct (N × H)
226. C₂×#500	C₂×C₆₁	semidirect
227. C₂×#498	C₂×C₆₁	semidirect
228. C₂×#487	C₂×C₆₁	semidirect
229. C₂×#487	C₂×C₆₁	semidirect
231. C₂×#479	C₂×C₆₁	semidirect
234. C₂×C₆₁	#3231	direct (N × H)
259. #500	C₂²×C₆₁	semidirect
260. #498	C₂²×C₆₁	semidirect
261. #487	C₂²×C₆₁	semidirect
262. #487	C₂²×C₆₁	semidirect
264. #479	C₂²×C₆₁	semidirect
274. C₆₁	C₂×#3231	direct (N × H)
451. C₂	C₆₁×#3231	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×#533	2
C₆₁×#3231	2
C₂×C₆₁×#500	2
C₂×C₆₁×#498	2
C₂×C₆₁×#487	2
C₂×C₆₁×#485	2
C₂×C₆₁×#479	2
C₂×C₆₁×#431	2
C₂×#3231	61

Subgroups — C2×C61×#3231 #29564277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3231 #29564277