Subgroups — C₂×C₆₁×#1810 #29524098

Order 15616.

1566 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₆₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₆₁×#407	C₂	1	2	yes	maximal
4	C₆₁×#1810	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₆₁×#341	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₁	C₂²	1	4	yes	—
8	C₂³×C₈×C₆₁	—	2	8	no	—
9	C₂³×C₈×C₆₁	C₄	1	4	yes	—
10	C₂³×C₈×C₆₁	C₄	1	4	yes	—
11	C₂³×C₈×C₆₁	C₂²	1	4	yes	—
12	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₆₁	—	2	8	no	—
15	C₂×#108×C₆₁	C₂²	1	8	yes	complemented
16	C₆₁×#407	—	2	4	no	—
17	C₆₁×#407	C₂²	1	8	yes	—
18	C₂×C₂²⋊C₈×C₆₁	—	2	4	no	—
19	C₆₁×#341	C₂²	1	8	yes	complemented
20	C₂⁵×C₆₁	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₆₁	—	2	4	no	—
22	C₂³×C₄×C₆₁	—	2	4	no	—
23	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₆₁	D₄	1	2	yes	—
26	C₂³×C₄×C₆₁	D₄	1	2	yes	—
27	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₆₁	Q₈	1	2	yes	—
29	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
30	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₆₁	—	2	4	no	—
32	C₂×M₁₆×C₆₁	—	2	8	no	—
33	C₂×M₁₆×C₆₁	—	2	8	no	—
34	C₂×M₁₆×C₆₁	C₂³	1	8	yes	—
35	C₂²×C₈×C₆₁	—	2	4	no	—
36	C₂²×C₈×C₆₁	—	2	16	no	—
37	C₂²×C₈×C₆₁	—	2	16	no	—
38	C₂²×C₈×C₆₁	—	2	16	no	—
39	C₂²×C₈×C₆₁	—	2	4	no	—
40	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₆₁	C₂×C₄	1	16	yes	—
43	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₆₁	D₄	1	8	yes	—
46	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
48	#108×C₆₁	—	2	16	no	—
49	C₂²⋊C₈×C₆₁	—	2	16	no	—
50	C₂⁴×C₆₁	—	2	8	no	—
51	C₂⁴×C₆₁	—	2	4	no	—
52	C₂⁴×C₆₁	C₄⋊C₄	1	2	yes	—
53	C₂⁴×C₆₁	C₄⋊C₄	1	2	yes	—
54	C₂⁴×C₆₁	C₂×Q₈	1	2	yes	—
55	C₂⁴×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	—	2	4	no	—
57	C₂²×C₄×C₆₁	—	2	4	no	—
58	C₂²×C₄×C₆₁	—	2	4	no	—
59	C₂²×C₄×C₆₁	—	2	4	no	—
60	C₂²×C₄×C₆₁	—	2	4	no	—
61	C₂²×C₄×C₆₁	—	2	8	no	—
62	C₂²×C₄×C₆₁	—	2	8	no	—
63	C₂²×C₄×C₆₁	—	2	8	no	—
64	C₂²×C₄×C₆₁	—	2	4	no	—
65	C₂²×C₄×C₆₁	—	2	4	no	—
66	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₁	#41	1	4	yes	—
80	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₆₁	—	2	8	no	—
82	M₁₆×C₆₁	—	2	8	no	—
83	M₁₆×C₆₁	—	4	8	no	—
84	C₂×C₈×C₆₁	—	2	4	no	—
85	C₂×C₈×C₆₁	—	2	4	no	—
86	C₂×C₈×C₆₁	—	2	4	no	—
87	C₂×C₈×C₆₁	—	2	32	no	—
88	C₂×C₈×C₆₁	—	2	8	no	—
89	C₂×C₈×C₆₁	—	4	8	no	—
90	C₂×C₈×C₆₁	C₂²⋊C₄	1	8	yes	—
91	C₂×C₈×C₆₁	C₂²⋊C₄	1	8	yes	—
92	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₆₁	C₂×D₄	1	8	yes	—
95	C₂³×C₆₁	—	2	32	no	—
96	C₂³×C₆₁	—	2	4	no	—
97	C₂³×C₆₁	—	2	4	no	—
98	C₂³×C₆₁	—	2	4	no	—
99	C₂³×C₆₁	—	2	4	no	—
100	C₂³×C₆₁	—	2	4	no	—
101	C₂³×C₆₁	—	2	8	no	—
102	C₂³×C₆₁	—	2	8	no	—
103	C₂³×C₆₁	#108	1	8	yes	complemented
104	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
106	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₆₁	C₂²⋊Q₈	1	4	yes	—
108	C₂³×C₆₁	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	—	2	4	no	—
110	C₂×C₄×C₆₁	—	2	4	no	—
111	C₂×C₄×C₆₁	—	2	4	no	—
112	C₂×C₄×C₆₁	—	2	4	no	—
113	C₂×C₄×C₆₁	—	2	4	no	—
114	C₂×C₄×C₆₁	—	2	8	no	—
115	C₂×C₄×C₆₁	—	2	8	no	—
116	C₂×C₄×C₆₁	—	2	8	no	—
117	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	#117	1	2	yes	—
123	C₂×C₄×C₆₁	#117	1	2	yes	—
124	C₂×C₄×C₆₁	#121	1	4	yes	—
125	C₂×C₄×C₆₁	#121	1	4	yes	—
126	C₂×C₄×C₆₁	C₂²⋊Q₈	1	4	yes	—
127	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
131	C₈×C₆₁	—	2	4	no	—
132	C₈×C₆₁	—	4	8	no	—
133	C₈×C₆₁	C₂×C₂²⋊C₄	1	8	yes	—
134	C₂×#1810	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₆₁	—	2	16	no	—
136	C₂²×C₆₁	—	2	16	no	—
137	C₂²×C₆₁	—	2	16	no	—
138	C₂²×C₆₁	—	2	8	no	—
139	C₂²×C₆₁	—	2	2	no	—
140	C₂²×C₆₁	—	2	4	no	—
141	C₂²×C₆₁	—	2	2	no	—
142	C₂²×C₆₁	—	2	4	no	—
143	C₂²×C₆₁	#380	1	2	yes	—
144	C₂²×C₆₁	#380	1	2	yes	—
145	C₂²×C₆₁	C₂×#108	1	4	yes	—
146	C₂²×C₆₁	C₂×#108	1	8	yes	complemented
147	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	2	yes	—
149	C₄×C₆₁	—	2	4	no	—
150	C₄×C₆₁	#380	1	2	yes	—
151	C₄×C₆₁	#380	1	2	yes	—
152	C₄×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
154	C₄×C₆₁	C₂×#121	1	2	yes	—
155	C₂⁴×C₈	C₂×C₆₁	1	1	yes	characteristic
156	C₂²×#108	C₂×C₆₁	1	2	yes	complemented
157	C₂×#407	C₂×C₆₁	1	2	yes	—
158	#1810	C₂×C₆₁	1	8	yes	direct product factor
159	C₂×#341	C₂×C₆₁	1	2	yes	complemented
160	C₂×C₆₁	—	2	2	no	—
161	C₂×C₆₁	—	2	2	no	—
162	C₂×C₆₁	—	2	8	no	—
163	C₂×C₆₁	#1810	1	4	yes	direct product factor
164	C₂×C₆₁	C₂×#380	1	1	yes	characteristic
165	C₂×C₆₁	C₂²×#108	1	2	yes	—
166	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
167	C₂²×M₁₆	C₂²×C₆₁	1	4	yes	—
168	C₂³×C₈	—	2	8	no	—
169	C₂³×C₈	C₄×C₆₁	1	4	yes	—
170	C₂³×C₈	C₄×C₆₁	1	4	yes	—
171	C₂³×C₈	C₂²×C₆₁	1	4	yes	—
172	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
173	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
174	C₂×#108	—	2	8	no	—
175	C₂×#108	C₂²×C₆₁	1	8	yes	complemented
176	#407	—	2	4	no	—
177	#407	C₂²×C₆₁	1	8	yes	—
178	C₂×C₂²⋊C₈	—	2	4	no	—
179	#341	C₂²×C₆₁	1	8	yes	complemented
180	C₆₁	C₂×#1810	1	1	yes	characteristic, direct product factor, Sylow
181	C₂⁵	Q₈×C₆₁	1	1	yes	characteristic
182	C₂³×C₄	—	2	4	no	—
183	C₂³×C₄	—	2	4	no	—
184	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
185	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
186	C₂³×C₄	D₄×C₆₁	1	2	yes	—
187	C₂³×C₄	D₄×C₆₁	1	2	yes	—
188	C₂³×C₄	Q₈×C₆₁	1	1	yes	characteristic
189	C₂³×C₄	Q₈×C₆₁	1	2	yes	—
190	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
191	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	2	8	no	—
194	C₂×M₁₆	—	2	8	no	—
195	C₂×M₁₆	C₂³×C₆₁	1	8	yes	—
196	C₂²×C₈	—	2	4	no	—
197	C₂²×C₈	—	2	16	no	—
198	C₂²×C₈	—	2	16	no	—
199	C₂²×C₈	—	2	16	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
202	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
203	C₂²×C₈	C₂×C₄×C₆₁	1	16	yes	—
204	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
205	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
206	C₂²×C₈	D₄×C₆₁	1	8	yes	—
207	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
208	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
209	#108	—	2	16	no	—
210	C₂²⋊C₈	—	2	16	no	—
211	C₂⁴	—	2	8	no	—
212	C₂⁴	—	2	4	no	—
213	C₂⁴	C₄⋊C₄×C₆₁	1	2	yes	—
214	C₂⁴	C₄⋊C₄×C₆₁	1	2	yes	—
215	C₂⁴	C₂×Q₈×C₆₁	1	2	yes	—
216	C₂⁴	C₂×Q₈×C₆₁	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
228	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
229	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
230	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
235	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
236	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
239	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
240	C₂²×C₄	#41×C₆₁	1	4	yes	—
241	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
242	M₁₆	—	2	8	no	—
243	M₁₆	—	2	8	no	—
244	M₁₆	—	4	8	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	32	no	—
249	C₂×C₈	—	2	8	no	—
250	C₂×C₈	—	4	8	no	—
251	C₂×C₈	C₂²⋊C₄×C₆₁	1	8	yes	—
252	C₂×C₈	C₂²⋊C₄×C₆₁	1	8	yes	—
253	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
254	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
255	C₂×C₈	C₂×D₄×C₆₁	1	8	yes	—
256	C₂³	—	2	32	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	2	8	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	2	4	no	—
264	C₂³	#108×C₆₁	1	8	yes	complemented
265	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
266	C₂³	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
267	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
268	C₂³	C₂²⋊Q₈×C₆₁	1	4	yes	—
269	C₂³	C₂²×Q₈×C₆₁	1	1	yes	characteristic
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
279	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
281	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
283	C₂×C₄	#117×C₆₁	1	2	yes	—
284	C₂×C₄	#117×C₆₁	1	2	yes	—
285	C₂×C₄	#121×C₆₁	1	4	yes	—
286	C₂×C₄	#121×C₆₁	1	4	yes	—
287	C₂×C₄	C₂²⋊Q₈×C₆₁	1	4	yes	—
288	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
289	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
290	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
291	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
292	C₈	—	2	4	no	—
293	C₈	—	4	8	no	—
294	C₈	C₂×C₂²⋊C₄×C₆₁	1	8	yes	—
295	C₂²	—	2	16	no	—
296	C₂²	—	2	16	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	16	no	—
299	C₂²	—	2	8	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	C₆₁×#380	1	2	yes	—
304	C₂²	C₆₁×#380	1	2	yes	—
305	C₂²	C₂×#108×C₆₁	1	4	yes	—
306	C₂²	C₂×#108×C₆₁	1	8	yes	complemented
307	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
308	C₂²	C₂×C₂²⋊Q₈×C₆₁	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	C₆₁×#380	1	2	yes	—
311	C₄	C₆₁×#380	1	2	yes	—
312	C₄	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
313	C₄	C₂×#117×C₆₁	1	1	yes	characteristic
314	C₄	C₂×#121×C₆₁	1	2	yes	—
315	C₂	—	2	2	no	—
316	C₂	—	2	8	no	—
317	C₂	—	2	2	no	—
318	C₂	C₆₁×#1810	1	4	yes	direct product factor
319	C₂	C₂×C₆₁×#380	1	1	yes	characteristic
320	C₂	C₂²×#108×C₆₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁵×C₆₁	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₂⁴×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#108×C₆₁	C₂	semidirect
4. C₆₁×#1810	C₂	direct (N × H)
5. C₂×C₆₁×#341	C₂	semidirect
15. C₂×#108×C₆₁	C₂²	semidirect
19. C₆₁×#341	C₂²	semidirect
103. C₂³×C₆₁	#108	semidirect
134. C₂×#1810	C₆₁	direct (N × H)
146. C₂²×C₆₁	C₂×#108	semidirect
156. C₂²×#108	C₂×C₆₁	semidirect
158. #1810	C₂×C₆₁	direct (N × H)
159. C₂×#341	C₂×C₆₁	semidirect
163. C₂×C₆₁	#1810	direct (N × H)
175. C₂×#108	C₂²×C₆₁	semidirect
179. #341	C₂²×C₆₁	semidirect
180. C₆₁	C₂×#1810	direct (N × H)
264. C₂³	#108×C₆₁	semidirect
306. C₂²	C₂×#108×C₆₁	semidirect
318. C₂	C₆₁×#1810	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₆₁	2
C₂²×#108×C₆₁	2
C₂×C₆₁×#407	2
C₆₁×#1810	2
C₂×C₆₁×#341	2
C₂×#1810	61

Subgroups — C2×C61×#1810 #29524098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1810 #29524098