Subgroups — C₂×C₆₁×#1807 #29524089

Order 15616.

1790 conjugacy classes of subgroups.

The subgroup lattice has 294 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×#106×C₆₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₆₁×#380	C₂	1	2	yes	maximal
4	C₆₁×#1807	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₆₁×#340	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₆₁	—	2	8	no	—
8	C₂³×C₈×C₆₁	C₄	1	8	yes	complemented
9	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₆₁	C₂²	1	4	yes	—
11	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
13	C₂²×C₂²⋊C₄×C₆₁	—	2	2	no	—
14	C₂×#106×C₆₁	—	2	8	no	—
15	C₂×#106×C₆₁	C₂²	1	8	yes	complemented
16	C₆₁×#380	—	2	4	no	—
17	C₆₁×#380	C₂²	1	8	yes	—
18	C₂×#95×C₆₁	—	2	2	no	—
19	C₆₁×#340	C₂²	1	8	yes	complemented
20	C₂⁵×C₆₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₁	—	2	4	no	—
22	C₂³×C₄×C₆₁	—	2	4	no	—
23	C₂³×C₄×C₆₁	—	2	4	no	—
24	C₂³×C₄×C₆₁	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₆₁	D₄	1	2	yes	—
26	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₁	Q₈	1	4	yes	—
28	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
29	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	—	2	4	no	—
31	C₂²×C₈×C₆₁	—	2	32	no	—
32	C₂²×C₈×C₆₁	—	2	16	no	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₆₁	C₂×C₄	1	16	yes	complemented
36	C₂²×C₈×C₆₁	D₄	1	8	yes	—
37	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
40	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
41	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
43	C₂×C₂²⋊C₄×C₆₁	—	2	8	no	—
44	C₂×C₂²⋊C₄×C₆₁	—	4	8	no	—
45	#106×C₆₁	—	2	16	no	—
46	#95×C₆₁	—	2	8	no	—
47	C₂⁴×C₆₁	—	2	4	no	—
48	C₂⁴×C₆₁	—	2	4	no	—
49	C₂⁴×C₆₁	C₄⋊C₄	1	4	yes	—
50	C₂⁴×C₆₁	SD₁₆	1	8	yes	—
51	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
52	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	—	2	4	no	—
54	C₂²×C₄×C₆₁	—	2	8	no	—
55	C₂²×C₄×C₆₁	—	2	8	no	—
56	C₂²×C₄×C₆₁	—	2	8	no	—
57	C₂²×C₄×C₆₁	—	2	16	no	—
58	C₂²×C₄×C₆₁	—	2	4	no	—
59	C₂²×C₄×C₆₁	—	2	4	no	—
60	C₂²×C₄×C₆₁	—	2	8	no	—
61	C₂²×C₄×C₆₁	—	4	8	no	—
62	C₂²×C₄×C₆₁	—	4	8	no	—
63	C₂²×C₄×C₆₁	—	4	8	no	—
64	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₆₁	C₄⋊C₄	1	8	yes	—
66	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₆₁	#41	1	4	yes	—
74	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₆₁	—	2	8	no	—
76	C₂×C₈×C₆₁	—	2	4	no	—
77	C₂×C₈×C₆₁	—	2	32	no	—
78	C₂×C₈×C₆₁	C₂²⋊C₄	1	16	yes	complemented
79	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₆₁	C₂×D₄	1	8	yes	—
82	C₄⋊C₄×C₆₁	—	2	8	no	—
83	C₄⋊C₄×C₆₁	—	2	8	no	—
84	C₄⋊C₄×C₆₁	—	4	8	no	—
85	C₂²⋊C₄×C₆₁	—	4	16	no	—
86	C₂³×C₆₁	—	2	4	no	—
87	C₂³×C₆₁	—	2	4	no	—
88	C₂³×C₆₁	—	2	8	no	—
89	C₂³×C₆₁	—	2	8	no	—
90	C₂³×C₆₁	—	2	32	no	—
91	C₂³×C₆₁	#106	1	16	yes	complemented
92	C₂³×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
93	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₆₁	#121	1	4	yes	—
95	C₂³×C₆₁	C₂×SD₁₆	1	8	yes	—
96	C₂³×C₆₁	C₂×SD₁₆	1	4	yes	—
97	C₂³×C₆₁	#135	1	4	yes	—
98	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	—	2	4	no	—
100	C₂×C₄×C₆₁	—	2	8	no	—
101	C₂×C₄×C₆₁	—	2	4	no	—
102	C₂×C₄×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	—	2	8	no	—
104	C₂×C₄×C₆₁	—	2	16	no	—
105	C₂×C₄×C₆₁	—	4	8	no	—
106	C₂×C₄×C₆₁	—	4	8	no	—
107	C₂×C₄×C₆₁	—	4	8	no	—
108	C₂×C₄×C₆₁	—	8	16	no	—
109	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₆₁	#117	1	4	yes	—
113	C₂×C₄×C₆₁	#121	1	4	yes	—
114	C₂×C₄×C₆₁	C₂²⋊Q₈	1	8	yes	—
115	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
117	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
119	C₈×C₆₁	—	2	4	no	—
120	C₈×C₆₁	C₂×C₂²⋊C₄	1	8	yes	complemented
121	C₂×#1807	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×C₆₁	—	2	32	no	—
123	C₂²×C₆₁	—	2	4	no	—
124	C₂²×C₆₁	—	2	4	no	—
125	C₂²×C₆₁	—	2	16	no	—
126	C₂²×C₆₁	#380	1	4	yes	—
127	C₂²×C₆₁	C₂×#106	1	4	yes	—
128	C₂²×C₆₁	C₂×#106	1	8	yes	complemented
129	C₂²×C₆₁	#427	1	4	yes	—
130	C₂²×C₆₁	#465	1	16	yes	—
131	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₆₁	C₂×#121	1	2	yes	—
133	C₂²×C₆₁	C₂²×SD₁₆	1	2	yes	—
134	C₂²×C₆₁	C₂×#135	1	2	yes	—
135	C₄×C₆₁	—	2	4	no	—
136	C₄×C₆₁	—	8	8	no	—
137	C₄×C₆₁	#380	1	4	yes	—
138	C₄×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
140	C₄×C₆₁	C₂×C₂²⋊Q₈	1	2	yes	—
141	C₂⁴×C₈	C₂×C₆₁	1	1	yes	characteristic
142	C₂²×#106	C₂×C₆₁	1	2	yes	complemented
143	C₂×#380	C₂×C₆₁	1	2	yes	—
144	#1807	C₂×C₆₁	1	8	yes	direct product factor
145	C₂×#340	C₂×C₆₁	1	2	yes	complemented
146	C₂×C₆₁	—	2	8	no	—
147	C₂×C₆₁	#1807	1	8	yes	direct product factor
148	C₂×C₆₁	C₂×#380	1	1	yes	characteristic
149	C₂×C₆₁	C₂²×#106	1	1	yes	characteristic
150	C₂×C₆₁	C₂×#427	1	1	yes	characteristic
151	C₂×C₆₁	C₂×#465	1	4	yes	—
152	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
153	C₂³×C₈	—	2	8	no	—
154	C₂³×C₈	C₄×C₆₁	1	8	yes	complemented
155	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
156	C₂³×C₈	C₂²×C₆₁	1	4	yes	—
157	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
159	C₂²×C₂²⋊C₄	—	2	2	no	—
160	C₂×#106	—	2	8	no	—
161	C₂×#106	C₂²×C₆₁	1	8	yes	complemented
162	#380	—	2	4	no	—
163	#380	C₂²×C₆₁	1	8	yes	—
164	C₂×#95	—	2	2	no	—
165	#340	C₂²×C₆₁	1	8	yes	complemented
166	C₆₁	C₂×#1807	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁵	D₄×C₆₁	1	1	yes	characteristic
168	C₂³×C₄	—	2	4	no	—
169	C₂³×C₄	—	2	4	no	—
170	C₂³×C₄	—	2	4	no	—
171	C₂³×C₄	C₂×C₄×C₆₁	1	4	yes	—
172	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
173	C₂³×C₄	D₄×C₆₁	1	2	yes	—
174	C₂³×C₄	Q₈×C₆₁	1	4	yes	—
175	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
176	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	2	32	no	—
179	C₂²×C₈	—	2	16	no	—
180	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
181	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
182	C₂²×C₈	C₂×C₄×C₆₁	1	16	yes	complemented
183	C₂²×C₈	D₄×C₆₁	1	8	yes	—
184	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
185	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
186	C₂×C₄⋊C₄	—	2	8	no	—
187	C₂×C₄⋊C₄	—	2	8	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
190	C₂×C₂²⋊C₄	—	2	8	no	—
191	C₂×C₂²⋊C₄	—	4	8	no	—
192	#106	—	2	16	no	—
193	#95	—	2	8	no	—
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	C₄⋊C₄×C₆₁	1	4	yes	—
197	C₂⁴	SD₁₆×C₆₁	1	8	yes	—
198	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
199	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	16	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	4	8	no	—
209	C₂²×C₄	—	4	8	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
212	C₂²×C₄	C₄⋊C₄×C₆₁	1	8	yes	—
213	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
217	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×Q₈×C₆₁	1	4	yes	—
219	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
220	C₂²×C₄	#41×C₆₁	1	4	yes	—
221	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	32	no	—
225	C₂×C₈	C₂²⋊C₄×C₆₁	1	16	yes	complemented
226	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
227	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
228	C₂×C₈	C₂×D₄×C₆₁	1	8	yes	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	4	8	no	—
232	C₂²⋊C₄	—	4	16	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	8	no	—
235	C₂³	—	2	8	no	—
236	C₂³	—	2	32	no	—
237	C₂³	—	2	4	no	—
238	C₂³	#106×C₆₁	1	16	yes	complemented
239	C₂³	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
240	C₂³	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
241	C₂³	#121×C₆₁	1	4	yes	—
242	C₂³	C₂×SD₁₆×C₆₁	1	8	yes	—
243	C₂³	C₂×SD₁₆×C₆₁	1	4	yes	—
244	C₂³	#135×C₆₁	1	4	yes	—
245	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	16	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	8	16	no	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	4	yes	—
257	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
259	C₂×C₄	#117×C₆₁	1	4	yes	—
260	C₂×C₄	#121×C₆₁	1	4	yes	—
261	C₂×C₄	C₂²⋊Q₈×C₆₁	1	8	yes	—
262	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
264	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
265	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	C₂×C₂²⋊C₄×C₆₁	1	8	yes	complemented
268	C₂²	—	2	4	no	—
269	C₂²	—	2	16	no	—
270	C₂²	—	2	32	no	—
271	C₂²	—	2	4	no	—
272	C₂²	C₆₁×#380	1	4	yes	—
273	C₂²	C₂×#106×C₆₁	1	4	yes	—
274	C₂²	C₂×#106×C₆₁	1	8	yes	complemented
275	C₂²	C₆₁×#427	1	4	yes	—
276	C₂²	C₆₁×#465	1	16	yes	—
277	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
278	C₂²	C₂×#121×C₆₁	1	2	yes	—
279	C₂²	C₂²×SD₁₆×C₆₁	1	2	yes	—
280	C₂²	C₂×#135×C₆₁	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	8	8	no	—
283	C₄	C₆₁×#380	1	4	yes	—
284	C₄	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
285	C₄	C₂×#117×C₆₁	1	1	yes	characteristic
286	C₄	C₂×C₂²⋊Q₈×C₆₁	1	2	yes	—
287	C₂	—	2	8	no	—
288	C₂	C₆₁×#1807	1	8	yes	direct product factor
289	C₂	C₂×C₆₁×#380	1	1	yes	characteristic
290	C₂	C₂²×#106×C₆₁	1	1	yes	characteristic
291	C₂	C₂×C₆₁×#427	1	1	yes	characteristic
292	C₂	C₂×C₆₁×#465	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂²×#106×C₆₁	C₂	semidirect
4. C₆₁×#1807	C₂	direct (N × H)
5. C₂×C₆₁×#340	C₂	semidirect
8. C₂³×C₈×C₆₁	C₄	semidirect
15. C₂×#106×C₆₁	C₂²	semidirect
19. C₆₁×#340	C₂²	semidirect
35. C₂²×C₈×C₆₁	C₂×C₄	semidirect
78. C₂×C₈×C₆₁	C₂²⋊C₄	semidirect
91. C₂³×C₆₁	#106	semidirect
120. C₈×C₆₁	C₂×C₂²⋊C₄	semidirect
121. C₂×#1807	C₆₁	direct (N × H)
128. C₂²×C₆₁	C₂×#106	semidirect
142. C₂²×#106	C₂×C₆₁	semidirect
144. #1807	C₂×C₆₁	direct (N × H)
145. C₂×#340	C₂×C₆₁	semidirect
147. C₂×C₆₁	#1807	direct (N × H)
154. C₂³×C₈	C₄×C₆₁	semidirect
161. C₂×#106	C₂²×C₆₁	semidirect
165. #340	C₂²×C₆₁	semidirect
166. C₆₁	C₂×#1807	direct (N × H)
182. C₂²×C₈	C₂×C₄×C₆₁	semidirect
225. C₂×C₈	C₂²⋊C₄×C₆₁	semidirect
238. C₂³	#106×C₆₁	semidirect
267. C₈	C₂×C₂²⋊C₄×C₆₁	semidirect
274. C₂²	C₂×#106×C₆₁	semidirect
288. C₂	C₆₁×#1807	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×#106×C₆₁	2
C₂×C₆₁×#380	2
C₆₁×#1807	2
C₂×C₆₁×#340	2
C₂×#1807	61

Subgroups — C2×C61×#1807 #29524089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1807 #29524089