Subgroups — C₂×C₆₁×#3210 #29564213

Order 15616.

2186 conjugacy classes of subgroups.

The subgroup lattice has 502 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, complemented
2	C₂×C₆₁×#532	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#3210	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₆₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₆₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₆₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₆₁×#464	C₂	1	4	yes	maximal, complemented
9	C₂×C₆₁×#452	C₂	1	4	yes	maximal, complemented
10	C₂×C₆₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₆₁	C₂²	1	2	yes	—
12	C₂²×Q₁₆×C₆₁	C₂²	1	4	yes	—
13	C₂²×SD₁₆×C₆₁	C₂²	1	4	yes	—
14	C₆₁×#533	—	2	4	no	—
15	C₆₁×#533	C₂²	1	8	yes	complemented
16	C₆₁×#532	—	2	4	no	—
17	C₆₁×#532	C₂²	1	8	yes	—
18	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×#125×C₆₁	—	2	1	no	—
23	C₂×#124×C₆₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
25	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
26	C₂×#121×C₆₁	—	2	1	no	—
27	C₂×#121×C₆₁	—	2	1	no	—
28	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
29	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₆₁	—	2	1	no	—
31	C₂×C₄×D₄×C₆₁	—	2	1	no	—
32	C₂×#117×C₆₁	—	2	1	no	—
33	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
34	C₆₁×#494	C₂²	1	8	yes	complemented
35	C₆₁×#492	C₂²	1	8	yes	complemented
36	C₆₁×#487	C₂²	1	8	yes	complemented
37	C₆₁×#486	C₂²	1	8	yes	complemented
38	C₆₁×#464	C₂²	1	32	yes	complemented
39	C₆₁×#452	C₂²	1	32	yes	complemented
40	C₆₁×#431	C₂²	1	8	yes	complemented
41	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
42	C₂×#103×C₆₁	C₂²	1	4	yes	—
43	C₂×#102×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	C₂×#41×C₆₁	—	2	4	no	—
48	C₂×#41×C₆₁	—	4	8	no	—
49	C₂×#41×C₆₁	C₂³	1	8	yes	—
50	C₂²×Q₈×C₆₁	—	2	1	no	—
51	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
52	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
53	C₂²×D₄×C₆₁	—	2	2	no	—
54	C₂²×D₄×C₆₁	—	2	1	no	—
55	C₂²×D₄×C₆₁	—	2	1	no	—
56	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₁	—	2	2	no	—
58	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×Q₁₆×C₆₁	—	4	8	no	—
62	C₂×Q₁₆×C₆₁	C₂³	1	16	yes	—
63	C₂×SD₁₆×C₆₁	—	4	8	no	—
64	C₂×SD₁₆×C₆₁	C₂³	1	16	yes	—
65	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
66	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
67	C₄⋊Q₈×C₆₁	—	2	2	no	—
68	C₄⋊Q₈×C₆₁	—	2	2	no	—
69	C₄⋊Q₈×C₆₁	—	2	2	no	—
70	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
71	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
72	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
73	#127×C₆₁	—	2	2	no	—
74	#127×C₆₁	C₂³	1	4	yes	—
75	#125×C₆₁	—	2	8	no	—
76	#124×C₆₁	—	2	4	no	—
77	#124×C₆₁	C₂³	1	8	yes	—
78	C₂²⋊Q₈×C₆₁	—	2	4	no	—
79	C₂²⋊Q₈×C₆₁	—	2	8	no	—
80	C₂²⋊Q₈×C₆₁	C₂³	1	8	yes	—
81	#121×C₆₁	—	2	8	no	—
82	#121×C₆₁	—	2	8	no	—
83	C₄×Q₈×C₆₁	—	2	4	no	—
84	C₄×Q₈×C₆₁	—	2	8	no	—
85	C₄×Q₈×C₆₁	C₂³	1	8	yes	—
86	C₄×D₄×C₆₁	—	2	8	no	—
87	C₄×D₄×C₆₁	—	2	8	no	—
88	#117×C₆₁	—	2	8	no	—
89	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
90	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
100	C₂×C₄²×C₆₁	—	2	2	no	—
101	C₂×C₄²×C₆₁	D₄	1	4	yes	—
102	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
104	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
106	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
107	C₄⋊C₈×C₆₁	C₂³	1	8	yes	—
108	#103×C₆₁	C₂³	1	16	yes	—
109	#102×C₆₁	C₂³	1	16	yes	—
110	C₂²⋊C₈×C₆₁	C₂³	1	8	yes	—
111	C₄×C₈×C₆₁	C₂³	1	4	yes	—
112	C₄×C₈×C₆₁	C₂³	1	4	yes	—
113	C₂⁴×C₆₁	—	2	2	no	—
114	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
115	#41×C₆₁	—	4	16	no	—
116	#41×C₆₁	—	4	16	no	—
117	C₂×Q₈×C₆₁	—	2	2	no	—
118	C₂×Q₈×C₆₁	—	2	4	no	—
119	C₂×Q₈×C₆₁	—	2	2	no	—
120	C₂×Q₈×C₆₁	—	2	8	no	—
121	C₂×Q₈×C₆₁	—	2	8	no	—
122	C₂×Q₈×C₆₁	—	4	4	no	—
123	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
124	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
125	C₂×D₄×C₆₁	—	2	8	no	—
126	C₂×D₄×C₆₁	—	2	4	no	—
127	C₂×D₄×C₆₁	—	2	4	no	—
128	C₂×D₄×C₆₁	—	2	2	no	—
129	C₂×D₄×C₆₁	—	2	8	no	—
130	C₂×D₄×C₆₁	—	4	8	no	—
131	C₂×D₄×C₆₁	—	4	4	no	—
132	C₂×D₄×C₆₁	—	4	4	no	—
133	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₆₁	—	2	4	no	—
135	C₂²×C₄×C₆₁	—	2	4	no	—
136	C₂²×C₄×C₆₁	—	2	4	no	—
137	C₂²×C₄×C₆₁	—	2	2	no	—
138	C₂²×C₄×C₆₁	—	2	2	no	—
139	C₂²×C₄×C₆₁	—	2	2	no	—
140	C₂²×C₄×C₆₁	—	2	8	no	—
141	C₂²×C₄×C₆₁	—	4	8	no	—
142	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
144	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
152	Q₁₆×C₆₁	—	4	16	no	—
153	SD₁₆×C₆₁	—	4	16	no	—
154	C₂×C₈×C₆₁	—	2	4	no	—
155	C₂×C₈×C₆₁	—	2	4	no	—
156	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
157	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
158	C₄⋊C₄×C₆₁	—	2	4	no	—
159	C₄⋊C₄×C₆₁	—	2	4	no	—
160	C₄⋊C₄×C₆₁	—	2	4	no	—
161	C₄⋊C₄×C₆₁	—	2	4	no	—
162	C₄⋊C₄×C₆₁	—	2	4	no	—
163	C₄⋊C₄×C₆₁	—	2	8	no	—
164	C₄⋊C₄×C₆₁	—	2	8	no	—
165	C₄⋊C₄×C₆₁	—	2	2	no	—
166	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
167	C₂²⋊C₄×C₆₁	—	2	8	no	—
168	C₂²⋊C₄×C₆₁	—	2	8	no	—
169	C₂²⋊C₄×C₆₁	—	2	8	no	—
170	C₄²×C₆₁	—	2	8	no	—
171	C₄²×C₆₁	C₂×D₄	1	4	yes	—
172	C₄²×C₆₁	C₂×D₄	1	4	yes	—
173	C₄²×C₆₁	C₂×D₄	1	2	yes	—
174	C₄²×C₆₁	C₂×D₄	1	2	yes	—
175	C₄²×C₆₁	C₂⁴	1	2	yes	—
176	C₄²×C₆₁	C₂⁴	1	2	yes	—
177	C₂³×C₆₁	—	2	2	no	—
178	C₂³×C₆₁	—	2	2	no	—
179	C₂³×C₆₁	—	2	2	no	—
180	C₂³×C₆₁	—	2	4	no	—
181	C₂³×C₆₁	—	4	4	no	—
182	C₂³×C₆₁	—	4	4	no	—
183	C₂³×C₆₁	—	4	4	no	—
184	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
185	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
186	Q₈×C₆₁	—	2	8	no	—
187	Q₈×C₆₁	—	2	8	no	—
188	Q₈×C₆₁	—	4	4	no	—
189	Q₈×C₆₁	—	4	4	no	—
190	Q₈×C₆₁	—	4	8	no	—
191	D₄×C₆₁	—	2	8	no	—
192	D₄×C₆₁	—	4	16	no	—
193	D₄×C₆₁	—	4	4	no	—
194	D₄×C₆₁	—	4	8	no	—
195	D₄×C₆₁	—	4	8	no	—
196	C₂×C₄×C₆₁	—	2	2	no	—
197	C₂×C₄×C₆₁	—	2	2	no	—
198	C₂×C₄×C₆₁	—	2	2	no	—
199	C₂×C₄×C₆₁	—	2	4	no	—
200	C₂×C₄×C₆₁	—	2	8	no	—
201	C₂×C₄×C₆₁	—	2	8	no	—
202	C₂×C₄×C₆₁	—	2	4	no	—
203	C₂×C₄×C₆₁	—	2	4	no	—
204	C₂×C₄×C₆₁	—	2	4	no	—
205	C₂×C₄×C₆₁	—	2	4	no	—
206	C₂×C₄×C₆₁	—	4	16	no	—
207	C₂×C₄×C₆₁	—	4	4	no	—
208	C₂×C₄×C₆₁	—	4	4	no	—
209	C₂×C₄×C₆₁	—	4	4	no	—
210	C₂×C₄×C₆₁	#135	1	4	yes	—
211	C₂×C₄×C₆₁	#137	1	4	yes	—
212	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₆₁	H(5,𝔽₂)	1	4	yes	—
219	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
220	C₈×C₆₁	—	2	4	no	—
221	C₈×C₆₁	—	2	4	no	—
222	C₂×#3210	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂²×C₆₁	—	2	2	no	—
224	C₂²×C₆₁	—	2	2	no	—
225	C₂²×C₆₁	—	2	2	no	—
226	C₂²×C₆₁	—	4	4	no	—
227	C₂²×C₆₁	—	4	4	no	—
228	C₂²×C₆₁	—	4	4	no	—
229	C₂²×C₆₁	—	4	4	no	—
230	C₂²×C₆₁	—	8	8	no	—
231	C₂²×C₆₁	#535	1	2	yes	—
232	C₂²×C₆₁	C₂×#135	1	2	yes	—
233	C₂²×C₆₁	C₂×#137	1	2	yes	—
234	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
235	C₄×C₆₁	—	2	2	no	—
236	C₄×C₆₁	—	2	2	no	—
237	C₄×C₆₁	—	2	2	no	—
238	C₄×C₆₁	—	4	4	no	—
239	C₄×C₆₁	—	4	4	no	—
240	C₄×C₆₁	—	4	4	no	—
241	C₄×C₆₁	#535	1	2	yes	—
242	C₄×C₆₁	C₂×#135	1	4	yes	—
243	C₄×C₆₁	C₂×#137	1	4	yes	—
244	C₄×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
245	C₂×#533	C₂×C₆₁	1	1	yes	characteristic, complemented
246	C₂×#532	C₂×C₆₁	1	1	yes	characteristic
247	#3210	C₂×C₆₁	1	16	yes	direct product factor
248	C₂×#494	C₂×C₆₁	1	1	yes	characteristic, complemented
249	C₂×#492	C₂×C₆₁	1	1	yes	characteristic, complemented
250	C₂×#487	C₂×C₆₁	1	1	yes	characteristic, complemented
251	C₂×#486	C₂×C₆₁	1	1	yes	characteristic, complemented
252	C₂×#464	C₂×C₆₁	1	4	yes	complemented
253	C₂×#452	C₂×C₆₁	1	4	yes	complemented
254	C₂×#431	C₂×C₆₁	1	1	yes	characteristic, complemented
255	C₂×C₆₁	—	4	2	no	—
256	C₂×C₆₁	—	8	4	no	—
257	C₂×C₆₁	#3210	1	4	yes	direct product factor
258	C₂×C₆₁	C₂×#535	1	1	yes	characteristic
259	C₂×C₆₁	C₂²×#135	1	1	yes	characteristic
260	C₂×C₆₁	C₂²×#137	1	1	yes	characteristic
261	C₂²×#41	C₂²×C₆₁	1	2	yes	—
262	C₂²×Q₁₆	C₂²×C₆₁	1	4	yes	—
263	C₂²×SD₁₆	C₂²×C₆₁	1	4	yes	—
264	#533	—	2	4	no	—
265	#533	C₂²×C₆₁	1	8	yes	complemented
266	#532	—	2	4	no	—
267	#532	C₂²×C₆₁	1	8	yes	—
268	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
269	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
270	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
271	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
272	C₂×#125	—	2	1	no	—
273	C₂×#124	C₂²×C₆₁	1	2	yes	—
274	C₂×C₂²⋊Q₈	—	2	1	no	—
275	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
276	C₂×#121	—	2	1	no	—
277	C₂×#121	—	2	1	no	—
278	C₂×C₄×Q₈	—	2	1	no	—
279	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
280	C₂×C₄×D₄	—	2	1	no	—
281	C₂×C₄×D₄	—	2	1	no	—
282	C₂×#117	—	2	1	no	—
283	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
284	#494	C₂²×C₆₁	1	8	yes	complemented
285	#492	C₂²×C₆₁	1	8	yes	complemented
286	#487	C₂²×C₆₁	1	8	yes	complemented
287	#486	C₂²×C₆₁	1	8	yes	complemented
288	#464	C₂²×C₆₁	1	32	yes	complemented
289	#452	C₂²×C₆₁	1	32	yes	complemented
290	#431	C₂²×C₆₁	1	8	yes	complemented
291	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
292	C₂×#103	C₂²×C₆₁	1	4	yes	—
293	C₂×#102	C₂²×C₆₁	1	4	yes	—
294	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
295	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
296	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
297	C₆₁	C₂×#3210	1	1	yes	characteristic, direct product factor, Sylow
298	C₂×#41	—	2	4	no	—
299	C₂×#41	—	4	8	no	—
300	C₂×#41	C₂³×C₆₁	1	8	yes	—
301	C₂²×Q₈	—	2	1	no	—
302	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
303	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
304	C₂²×D₄	—	2	2	no	—
305	C₂²×D₄	—	2	1	no	—
306	C₂²×D₄	—	2	1	no	—
307	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
308	C₂³×C₄	—	2	2	no	—
309	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
310	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
311	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
312	C₂×Q₁₆	—	4	8	no	—
313	C₂×Q₁₆	C₂³×C₆₁	1	16	yes	—
314	C₂×SD₁₆	—	4	8	no	—
315	C₂×SD₁₆	C₂³×C₆₁	1	16	yes	—
316	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
317	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
318	C₄⋊Q₈	—	2	2	no	—
319	C₄⋊Q₈	—	2	2	no	—
320	C₄⋊Q₈	—	2	2	no	—
321	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
322	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
323	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
324	#127	—	2	2	no	—
325	#127	C₂³×C₆₁	1	4	yes	—
326	#125	—	2	8	no	—
327	#124	—	2	4	no	—
328	#124	C₂³×C₆₁	1	8	yes	—
329	C₂²⋊Q₈	—	2	8	no	—
330	C₂²⋊Q₈	—	2	4	no	—
331	C₂²⋊Q₈	C₂³×C₆₁	1	8	yes	—
332	#121	—	2	8	no	—
333	#121	—	2	8	no	—
334	C₄×Q₈	—	2	8	no	—
335	C₄×Q₈	—	2	4	no	—
336	C₄×Q₈	C₂³×C₆₁	1	8	yes	—
337	C₄×D₄	—	2	8	no	—
338	C₄×D₄	—	2	8	no	—
339	#117	—	2	8	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	2	no	—
347	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₂²⋊C₄	—	2	2	no	—
351	C₂×C₄²	—	2	2	no	—
352	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
353	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
354	C₂×C₄²	D₄×C₆₁	1	4	yes	—
355	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
357	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
358	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
359	#103	C₂³×C₆₁	1	16	yes	—
360	#102	C₂³×C₆₁	1	16	yes	—
361	C₂²⋊C₈	C₂³×C₆₁	1	8	yes	—
362	C₄×C₈	C₂³×C₆₁	1	4	yes	—
363	C₄×C₈	C₂³×C₆₁	1	4	yes	—
364	C₂⁴	—	2	2	no	—
365	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
366	#41	—	4	16	no	—
367	#41	—	4	16	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	8	no	—
372	C₂×Q₈	—	2	8	no	—
373	C₂×Q₈	—	4	4	no	—
374	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
375	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
376	C₂×D₄	—	2	8	no	—
377	C₂×D₄	—	2	4	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	8	no	—
381	C₂×D₄	—	4	8	no	—
382	C₂×D₄	—	4	4	no	—
383	C₂×D₄	—	4	4	no	—
384	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	8	no	—
392	C₂²×C₄	—	4	8	no	—
393	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
396	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
398	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
399	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
402	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
403	Q₁₆	—	4	16	no	—
404	SD₁₆	—	4	16	no	—
405	C₂×C₈	—	2	4	no	—
406	C₂×C₈	—	2	4	no	—
407	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
408	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
418	C₂²⋊C₄	—	2	8	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₄²	—	2	8	no	—
422	C₄²	C₂×D₄×C₆₁	1	4	yes	—
423	C₄²	C₂×D₄×C₆₁	1	4	yes	—
424	C₄²	C₂×D₄×C₆₁	1	2	yes	—
425	C₄²	C₂×D₄×C₆₁	1	2	yes	—
426	C₄²	C₂⁴×C₆₁	1	2	yes	—
427	C₄²	C₂⁴×C₆₁	1	2	yes	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	4	4	no	—
433	C₂³	—	4	4	no	—
434	C₂³	—	4	4	no	—
435	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
436	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
437	Q₈	—	2	8	no	—
438	Q₈	—	2	8	no	—
439	Q₈	—	4	4	no	—
440	Q₈	—	4	4	no	—
441	Q₈	—	4	8	no	—
442	D₄	—	2	8	no	—
443	D₄	—	4	16	no	—
444	D₄	—	4	4	no	—
445	D₄	—	4	8	no	—
446	D₄	—	4	8	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	8	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	4	16	no	—
458	C₂×C₄	—	4	4	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	#135×C₆₁	1	4	yes	—
462	C₂×C₄	#137×C₆₁	1	4	yes	—
463	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
464	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
465	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
466	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
467	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
468	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
469	C₂×C₄	H(5,𝔽₂)×C₆₁	1	4	yes	—
470	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
471	C₈	—	2	4	no	—
472	C₈	—	2	4	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	8	8	no	—
481	C₂²	C₆₁×#535	1	2	yes	—
482	C₂²	C₂×#135×C₆₁	1	2	yes	—
483	C₂²	C₂×#137×C₆₁	1	2	yes	—
484	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	2	2	no	—
488	C₄	—	4	4	no	—
489	C₄	—	4	4	no	—
490	C₄	—	4	4	no	—
491	C₄	C₆₁×#535	1	2	yes	—
492	C₄	C₂×#135×C₆₁	1	4	yes	—
493	C₄	C₂×#137×C₆₁	1	4	yes	—
494	C₄	C₂×H(5,𝔽₂)×C₆₁	1	2	yes	—
495	C₂	—	4	2	no	—
496	C₂	—	8	4	no	—
497	C₂	C₆₁×#3210	1	4	yes	direct product factor
498	C₂	C₂×C₆₁×#535	1	1	yes	characteristic
499	C₂	C₂²×#135×C₆₁	1	1	yes	characteristic
500	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	C₂×C₆₁×#532	C₂
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×#533	C₂	semidirect
3. C₆₁×#3210	C₂	direct (N × H)
4. C₂×C₆₁×#494	C₂	semidirect
5. C₂×C₆₁×#492	C₂	semidirect
6. C₂×C₆₁×#487	C₂	semidirect
7. C₂×C₆₁×#486	C₂	semidirect
8. C₂×C₆₁×#464	C₂	semidirect
9. C₂×C₆₁×#452	C₂	semidirect
10. C₂×C₆₁×#431	C₂	semidirect
15. C₆₁×#533	C₂²	semidirect
34. C₆₁×#494	C₂²	semidirect
35. C₆₁×#492	C₂²	semidirect
36. C₆₁×#487	C₂²	semidirect
37. C₆₁×#486	C₂²	semidirect
38. C₆₁×#464	C₂²	semidirect
39. C₆₁×#452	C₂²	semidirect
40. C₆₁×#431	C₂²	semidirect
222. C₂×#3210	C₆₁	direct (N × H)
245. C₂×#533	C₂×C₆₁	semidirect
247. #3210	C₂×C₆₁	direct (N × H)
248. C₂×#494	C₂×C₆₁	semidirect
249. C₂×#492	C₂×C₆₁	semidirect
250. C₂×#487	C₂×C₆₁	semidirect
251. C₂×#486	C₂×C₆₁	semidirect
252. C₂×#464	C₂×C₆₁	semidirect
253. C₂×#452	C₂×C₆₁	semidirect
254. C₂×#431	C₂×C₆₁	semidirect
257. C₂×C₆₁	#3210	direct (N × H)
265. #533	C₂²×C₆₁	semidirect
284. #494	C₂²×C₆₁	semidirect
285. #492	C₂²×C₆₁	semidirect
286. #487	C₂²×C₆₁	semidirect
287. #486	C₂²×C₆₁	semidirect
288. #464	C₂²×C₆₁	semidirect
289. #452	C₂²×C₆₁	semidirect
290. #431	C₂²×C₆₁	semidirect
297. C₆₁	C₂×#3210	direct (N × H)
497. C₂	C₆₁×#3210	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
C₂×C₆₁×#532	2
C₆₁×#3210	2
C₂×C₆₁×#494	2
C₂×C₆₁×#492	2
C₂×C₆₁×#487	2
C₂×C₆₁×#486	2
C₂×C₆₁×#464	2
C₂×C₆₁×#452	2
C₂×C₆₁×#431	2
C₂×#3210	61

Subgroups — C2×C61×#3210 #29564213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3210 #29564213