Subgroups — C₂×C₆₁×#1306 #29511544

Order 15616.

942 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#407	C₂	1	2	yes	maximal
2	C₂×C₆₁×#381	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1306	C₂	1	4	yes	maximal, direct product factor
4	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₆₁	—	2	2	no	—
6	C₂²×C₂²⋊C₄×C₆₁	C₄	1	2	yes	—
7	C₆₁×#407	—	2	2	no	—
8	C₆₁×#407	—	2	2	no	—
9	C₆₁×#407	C₄	1	4	yes	complemented
10	C₆₁×#407	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₈×C₆₁	—	2	2	no	—
12	C₂×C₂²⋊C₈×C₆₁	C₄	1	4	yes	complemented
13	C₆₁×#381	—	2	2	no	—
14	C₆₁×#381	C₄	1	2	yes	—
15	C₆₁×#381	C₂²	1	2	yes	—
16	C₂×#95×C₆₁	—	2	2	no	—
17	C₂⁵×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₆₁	—	2	2	no	—
19	C₂³×C₄×C₆₁	—	4	4	no	—
20	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₆₁	D₄	1	2	yes	—
24	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₁	D₄	1	2	yes	—
27	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₆₁	—	2	4	no	—
31	C₂×M₁₆×C₆₁	—	2	4	no	—
32	C₂×M₁₆×C₆₁	—	4	8	no	—
33	C₂²×C₈×C₆₁	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₆₁	—	4	4	no	—
36	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
37	C₂²⋊C₈×C₆₁	—	2	4	no	—
38	C₂²⋊C₈×C₆₁	—	2	4	no	—
39	C₂²⋊C₈×C₆₁	—	2	4	no	—
40	C₂²⋊C₈×C₆₁	C₂×C₄	1	8	yes	complemented
41	#95×C₆₁	—	2	4	no	—
42	#95×C₆₁	—	2	4	no	—
43	C₂⁴×C₆₁	—	2	4	no	—
44	C₂⁴×C₆₁	—	4	4	no	—
45	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
46	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
47	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	—	2	8	no	—
53	C₂²×C₄×C₆₁	—	2	4	no	—
54	C₂²×C₄×C₆₁	—	4	4	no	—
55	C₂²×C₄×C₆₁	—	4	4	no	—
56	C₂²×C₄×C₆₁	—	4	2	no	—
57	C₂²×C₄×C₆₁	—	4	2	no	—
58	C₂²×C₄×C₆₁	—	4	2	no	—
59	C₂²×C₄×C₆₁	—	4	4	no	—
60	C₂²×C₄×C₆₁	—	4	2	no	—
61	C₂²×C₄×C₆₁	—	4	2	no	—
62	C₂²×C₄×C₆₁	—	4	2	no	—
63	C₂²×C₄×C₆₁	—	4	8	no	—
64	C₂²×C₄×C₆₁	—	4	4	no	—
65	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
66	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
81	M₁₆×C₆₁	—	4	8	no	—
82	M₁₆×C₆₁	—	4	8	no	—
83	C₂×C₈×C₆₁	—	2	8	no	—
84	C₂×C₈×C₆₁	—	4	8	no	—
85	C₂²⋊C₄×C₆₁	—	4	4	no	—
86	C₂²⋊C₄×C₆₁	—	4	4	no	—
87	C₂³×C₆₁	—	2	4	no	—
88	C₂³×C₆₁	—	2	2	no	—
89	C₂³×C₆₁	—	2	2	no	—
90	C₂³×C₆₁	—	2	2	no	—
91	C₂³×C₆₁	—	2	2	no	—
92	C₂³×C₆₁	—	2	4	no	—
93	C₂³×C₆₁	—	2	4	no	—
94	C₂³×C₆₁	—	2	2	no	—
95	C₂³×C₆₁	—	2	2	no	—
96	C₂³×C₆₁	—	4	2	no	—
97	C₂³×C₆₁	—	4	2	no	—
98	C₂³×C₆₁	—	4	2	no	—
99	C₂³×C₆₁	—	4	2	no	—
100	C₂³×C₆₁	—	4	16	no	—
101	C₂³×C₆₁	#95	1	1	yes	characteristic
102	C₂³×C₆₁	#95	1	1	yes	characteristic
103	C₂³×C₆₁	#95	1	1	yes	characteristic
104	C₂³×C₆₁	#95	1	1	yes	characteristic
105	C₂³×C₆₁	#99	1	4	yes	—
106	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
107	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂³×C₆₁	C₂×C₄⋊C₄	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	4	no	—
115	C₂×C₄×C₆₁	—	4	2	no	—
116	C₂×C₄×C₆₁	—	4	2	no	—
117	C₂×C₄×C₆₁	—	4	2	no	—
118	C₂×C₄×C₆₁	—	4	1	no	—
119	C₂×C₄×C₆₁	—	4	1	no	—
120	C₂×C₄×C₆₁	—	4	2	no	—
121	C₂×C₄×C₆₁	—	4	2	no	—
122	C₂×C₄×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	—	4	1	no	—
124	C₂×C₄×C₆₁	—	4	1	no	—
125	C₂×C₄×C₆₁	—	4	8	no	—
126	C₂×C₄×C₆₁	—	4	4	no	—
127	C₂×C₄×C₆₁	—	8	8	no	—
128	C₂×C₄×C₆₁	#95	1	2	yes	—
129	C₂×C₄×C₆₁	#95	1	2	yes	—
130	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
132	C₈×C₆₁	—	4	8	no	—
133	C₂×#1306	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₆₁	—	2	8	no	—
135	C₂²×C₆₁	—	2	2	no	—
136	C₂²×C₆₁	—	2	2	no	—
137	C₂²×C₆₁	—	2	1	no	—
138	C₂²×C₆₁	—	2	1	no	—
139	C₂²×C₆₁	—	2	1	no	—
140	C₂²×C₆₁	—	2	1	no	—
141	C₂²×C₆₁	—	2	2	no	—
142	C₂²×C₆₁	—	2	2	no	—
143	C₂²×C₆₁	—	2	2	no	—
144	C₂²×C₆₁	—	2	2	no	—
145	C₂²×C₆₁	—	4	4	no	—
146	C₂²×C₆₁	—	4	4	no	—
147	C₂²×C₆₁	—	4	4	no	—
148	C₂²×C₆₁	—	4	4	no	—
149	C₂²×C₆₁	—	4	4	no	—
150	C₂²×C₆₁	—	4	4	no	—
151	C₂²×C₆₁	#342	1	2	yes	—
152	C₂²×C₆₁	#342	1	2	yes	—
153	C₂²×C₆₁	#344	1	4	yes	—
154	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
155	C₂²×C₆₁	C₂×#99	1	2	yes	—
156	C₄×C₆₁	—	2	2	no	—
157	C₄×C₆₁	—	2	2	no	—
158	C₄×C₆₁	—	2	2	no	—
159	C₄×C₆₁	—	2	2	no	—
160	C₄×C₆₁	—	8	4	no	—
161	C₂×#407	C₂×C₆₁	1	2	yes	—
162	C₂×#381	C₂×C₆₁	1	1	yes	characteristic
163	#1306	C₂×C₆₁	1	4	yes	direct product factor
164	C₂×C₆₁	—	2	2	no	—
165	C₂×C₆₁	—	2	2	no	—
166	C₂×C₆₁	—	4	1	no	—
167	C₂×C₆₁	—	4	1	no	—
168	C₂×C₆₁	—	4	1	no	—
169	C₂×C₆₁	—	4	1	no	—
170	C₂×C₆₁	#1306	1	4	yes	direct product factor
171	C₂×C₆₁	C₂×#342	1	1	yes	characteristic
172	C₂×C₆₁	C₂×#344	1	2	yes	—
173	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
174	C₂²×M₁₆	—	2	2	no	—
175	C₂²×C₂²⋊C₄	C₄×C₆₁	1	2	yes	—
176	#407	—	2	2	no	—
177	#407	—	2	2	no	—
178	#407	C₄×C₆₁	1	4	yes	complemented
179	#407	C₂²×C₆₁	1	4	yes	—
180	C₂×C₂²⋊C₈	—	2	2	no	—
181	C₂×C₂²⋊C₈	C₄×C₆₁	1	4	yes	complemented
182	#381	—	2	2	no	—
183	#381	C₄×C₆₁	1	2	yes	—
184	#381	C₂²×C₆₁	1	2	yes	—
185	C₂×#95	—	2	2	no	—
186	C₆₁	C₂×#1306	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁵	C₂×C₄×C₆₁	1	1	yes	characteristic
188	C₂³×C₄	—	2	2	no	—
189	C₂³×C₄	—	4	4	no	—
190	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
191	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
192	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
193	C₂³×C₄	D₄×C₆₁	1	2	yes	—
194	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
195	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
196	C₂³×C₄	D₄×C₆₁	1	2	yes	—
197	C₂³×C₄	Q₈×C₆₁	1	1	yes	characteristic
198	C₂³×C₄	Q₈×C₆₁	1	1	yes	characteristic
199	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
200	C₂×M₁₆	—	2	4	no	—
201	C₂×M₁₆	—	2	4	no	—
202	C₂×M₁₆	—	4	8	no	—
203	C₂²×C₈	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	4	4	no	—
206	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
207	C₂²⋊C₈	—	2	4	no	—
208	C₂²⋊C₈	—	2	4	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	C₂×C₄×C₆₁	1	8	yes	complemented
211	#95	—	2	4	no	—
212	#95	—	2	4	no	—
213	C₂⁴	—	2	4	no	—
214	C₂⁴	—	4	4	no	—
215	C₂⁴	C₄²×C₆₁	1	1	yes	characteristic
216	C₂⁴	C₄²×C₆₁	1	1	yes	characteristic
217	C₂⁴	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
218	C₂⁴	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
219	C₂⁴	C₄⋊C₄×C₆₁	1	1	yes	characteristic
220	C₂⁴	C₄⋊C₄×C₆₁	1	1	yes	characteristic
221	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	8	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	C₄²×C₆₁	1	2	yes	—
236	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
237	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
238	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
239	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
240	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
241	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
242	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
243	C₂²×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
244	C₂²×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
245	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
246	C₂²×C₄	C₄⋊C₄×C₆₁	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
251	M₁₆	—	4	8	no	—
252	M₁₆	—	4	8	no	—
253	C₂×C₈	—	2	8	no	—
254	C₂×C₈	—	4	8	no	—
255	C₂²⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	4	4	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	16	no	—
271	C₂³	#95×C₆₁	1	1	yes	characteristic
272	C₂³	#95×C₆₁	1	1	yes	characteristic
273	C₂³	#95×C₆₁	1	1	yes	characteristic
274	C₂³	#95×C₆₁	1	1	yes	characteristic
275	C₂³	#99×C₆₁	1	4	yes	—
276	C₂³	C₂×C₄²×C₆₁	1	1	yes	characteristic
277	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
278	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	8	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	8	no	—
298	C₂×C₄	#95×C₆₁	1	2	yes	—
299	C₂×C₄	#95×C₆₁	1	2	yes	—
300	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
301	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
302	C₈	—	4	8	no	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	4	no	—
320	C₂²	C₆₁×#342	1	2	yes	—
321	C₂²	C₆₁×#342	1	2	yes	—
322	C₂²	C₆₁×#344	1	4	yes	—
323	C₂²	C₂×#95×C₆₁	1	1	yes	characteristic
324	C₂²	C₂×#99×C₆₁	1	2	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	8	4	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	2	no	—
332	C₂	—	4	1	no	—
333	C₂	—	4	1	no	—
334	C₂	—	4	1	no	—
335	C₂	—	4	1	no	—
336	C₂	C₆₁×#1306	1	4	yes	direct product factor
337	C₂	C₂×C₆₁×#342	1	1	yes	characteristic
338	C₂	C₂×C₆₁×#344	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	#95
commutator	C₂³	C₂×C₄²×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₂⁵×C₆₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂⁴×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1306	C₂	direct (N × H)
9. C₆₁×#407	C₄	semidirect
12. C₂×C₂²⋊C₈×C₆₁	C₄	semidirect
40. C₂²⋊C₈×C₆₁	C₂×C₄	semidirect
133. C₂×#1306	C₆₁	direct (N × H)
163. #1306	C₂×C₆₁	direct (N × H)
170. C₂×C₆₁	#1306	direct (N × H)
178. #407	C₄×C₆₁	semidirect
181. C₂×C₂²⋊C₈	C₄×C₆₁	semidirect
186. C₆₁	C₂×#1306	direct (N × H)
210. C₂²⋊C₈	C₂×C₄×C₆₁	semidirect
336. C₂	C₆₁×#1306	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×#407	2
C₂×C₆₁×#381	2
C₆₁×#1306	2
C₂×#1306	61

Subgroups — C2×C61×#1306 #29511544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1306 #29511544