Subgroups — C₂×C₆₁×#1321 #29514870

Order 15616.

518 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#408	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#349	C₂	1	2	yes	maximal, complemented
3	C₆₁×#1321	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#41×C₆₁	C₄	1	1	yes	characteristic
5	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₆₁	—	2	2	no	—
8	C₆₁×#408	—	2	2	no	—
9	C₆₁×#408	C₄	1	2	yes	—
10	C₆₁×#408	C₂²	1	2	yes	—
11	C₂×C₂²⋊C₈×C₆₁	—	2	2	no	—
12	C₆₁×#349	—	2	2	no	—
13	C₆₁×#349	C₂²	1	4	yes	complemented
14	C₂×#41×C₆₁	—	2	1	no	—
15	C₂×#41×C₆₁	—	2	1	no	—
16	C₂×#41×C₆₁	—	4	1	no	—
17	C₂×#41×C₆₁	—	4	1	no	—
18	C₂×#41×C₆₁	—	4	1	no	—
19	C₂×#41×C₆₁	—	4	1	no	—
20	C₂×#41×C₆₁	C₈	1	2	yes	—
21	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×Q₈×C₆₁	C₈	1	1	yes	characteristic
23	C₂²×D₄×C₆₁	—	2	1	no	—
24	C₂²×D₄×C₆₁	C₈	1	1	yes	characteristic
25	C₂³×C₄×C₆₁	—	2	1	no	—
26	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₆₁	—	4	1	no	—
28	C₂×M₁₆×C₆₁	—	4	1	no	—
29	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
31	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	—	2	1	no	—
33	C₂²×C₈×C₆₁	—	2	1	no	—
34	C₂²×C₈×C₆₁	—	4	1	no	—
35	C₂²×C₈×C₆₁	—	4	1	no	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₆₁	D₄	1	2	yes	—
38	M₃₂×C₆₁	—	2	4	no	—
39	M₃₂×C₆₁	—	4	2	no	—
40	C₂×C₁₆×C₆₁	—	4	2	no	—
41	C₂²⋊C₈×C₆₁	—	2	4	no	—
42	C₂²⋊C₈×C₆₁	—	2	4	no	—
43	C₂⁴×C₆₁	—	2	1	no	—
44	C₂⁴×C₆₁	M₁₆	1	1	yes	characteristic
45	#41×C₆₁	—	4	2	no	—
46	#41×C₆₁	—	4	2	no	—
47	#41×C₆₁	—	4	2	no	—
48	#41×C₆₁	—	4	2	no	—
49	#41×C₆₁	—	4	2	no	—
50	#41×C₆₁	—	4	2	no	—
51	#41×C₆₁	—	4	2	no	—
52	#41×C₆₁	—	4	2	no	—
53	C₂×Q₈×C₆₁	—	2	1	no	—
54	C₂×Q₈×C₆₁	—	4	1	no	—
55	C₂×Q₈×C₆₁	—	4	1	no	—
56	C₂×Q₈×C₆₁	C₂×C₈	1	2	yes	—
57	C₂×D₄×C₆₁	—	2	2	no	—
58	C₂×D₄×C₆₁	—	2	2	no	—
59	C₂×D₄×C₆₁	—	2	1	no	—
60	C₂×D₄×C₆₁	—	4	1	no	—
61	C₂×D₄×C₆₁	—	4	1	no	—
62	C₂×D₄×C₆₁	—	4	2	no	—
63	C₂×D₄×C₆₁	—	4	2	no	—
64	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₆₁	—	2	1	no	—
66	C₂²×C₄×C₆₁	—	2	1	no	—
67	C₂²×C₄×C₆₁	—	2	1	no	—
68	C₂²×C₄×C₆₁	—	2	1	no	—
69	C₂²×C₄×C₆₁	—	2	1	no	—
70	C₂²×C₄×C₆₁	—	2	2	no	—
71	C₂²×C₄×C₆₁	—	2	2	no	—
72	C₂²×C₄×C₆₁	—	4	1	no	—
73	C₂²×C₄×C₆₁	—	4	1	no	—
74	C₂²×C₄×C₆₁	—	4	1	no	—
75	C₂²×C₄×C₆₁	—	4	1	no	—
76	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	M₁₆×C₆₁	—	4	1	no	—
83	M₁₆×C₆₁	—	4	2	no	—
84	M₁₆×C₆₁	—	4	1	no	—
85	C₂×C₈×C₆₁	—	2	2	no	—
86	C₂×C₈×C₆₁	—	2	2	no	—
87	C₂×C₈×C₆₁	—	2	2	no	—
88	C₂×C₈×C₆₁	—	4	2	no	—
89	C₂×C₈×C₆₁	—	4	1	no	—
90	C₂×C₈×C₆₁	—	4	1	no	—
91	C₂×C₈×C₆₁	—	4	2	no	—
92	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
93	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
94	C₁₆×C₆₁	—	4	4	no	—
95	C₂³×C₆₁	—	2	1	no	—
96	C₂³×C₆₁	—	2	1	no	—
97	C₂³×C₆₁	—	2	2	no	—
98	C₂³×C₆₁	—	4	2	no	—
99	C₂³×C₆₁	—	4	4	no	—
100	C₂³×C₆₁	—	4	1	no	—
101	C₂³×C₆₁	—	4	1	no	—
102	C₂³×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
103	C₂³×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
104	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
105	Q₈×C₆₁	—	4	2	no	—
106	Q₈×C₆₁	—	4	2	no	—
107	D₄×C₆₁	—	4	2	no	—
108	D₄×C₆₁	—	4	2	no	—
109	D₄×C₆₁	—	4	4	no	—
110	D₄×C₆₁	—	4	4	no	—
111	C₂×C₄×C₆₁	—	2	1	no	—
112	C₂×C₄×C₆₁	—	2	1	no	—
113	C₂×C₄×C₆₁	—	2	1	no	—
114	C₂×C₄×C₆₁	—	2	1	no	—
115	C₂×C₄×C₆₁	—	2	1	no	—
116	C₂×C₄×C₆₁	—	2	1	no	—
117	C₂×C₄×C₆₁	—	2	1	no	—
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	—	2	1	no	—
120	C₂×C₄×C₆₁	—	2	1	no	—
121	C₂×C₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	—	4	1	no	—
123	C₂×C₄×C₆₁	—	4	1	no	—
124	C₂×C₄×C₆₁	—	4	2	no	—
125	C₂×C₄×C₆₁	—	4	2	no	—
126	C₂×C₄×C₆₁	—	4	2	no	—
127	C₂×C₄×C₆₁	—	4	2	no	—
128	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
131	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
132	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
133	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
136	C₈×C₆₁	—	2	2	no	—
137	C₈×C₆₁	—	2	2	no	—
138	C₈×C₆₁	—	4	2	no	—
139	C₂×#1321	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₆₁	—	2	2	no	—
141	C₂²×C₆₁	—	2	1	no	—
142	C₂²×C₆₁	—	2	1	no	—
143	C₂²×C₆₁	—	4	1	no	—
144	C₂²×C₆₁	—	4	1	no	—
145	C₂²×C₆₁	—	4	2	no	—
146	C₂²×C₆₁	—	4	4	no	—
147	C₂²×C₆₁	—	4	2	no	—
148	C₂²×C₆₁	—	4	2	no	—
149	C₂²×C₆₁	—	8	2	no	—
150	C₂²×C₆₁	—	8	2	no	—
151	C₂²×C₆₁	#323	1	1	yes	characteristic
152	C₂²×C₆₁	#323	1	1	yes	characteristic
153	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₄×C₆₁	—	2	1	no	—
155	C₄×C₆₁	—	2	1	no	—
156	C₄×C₆₁	—	4	2	no	—
157	C₄×C₆₁	#323	1	1	yes	characteristic
158	C₄×C₆₁	#323	1	1	yes	characteristic
159	C₄×C₆₁	C₂×#99	1	1	yes	characteristic
160	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
161	C₂×#408	C₂×C₆₁	1	1	yes	characteristic
162	C₂×#349	C₂×C₆₁	1	2	yes	complemented
163	#1321	C₂×C₆₁	1	4	yes	direct product factor
164	C₂×C₆₁	—	2	1	no	—
165	C₂×C₆₁	—	2	1	no	—
166	C₂×C₆₁	—	4	1	no	—
167	C₂×C₆₁	—	4	1	no	—
168	C₂×C₆₁	—	8	2	no	—
169	C₂×C₆₁	#1321	1	2	yes	direct product factor
170	C₂×C₆₁	C₂×#323	1	1	yes	characteristic
171	C₂²×#41	C₄×C₆₁	1	1	yes	characteristic
172	C₂²×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
173	C₂³×C₈	C₄×C₆₁	1	1	yes	characteristic
174	C₂×M₃₂	—	2	2	no	—
175	#408	—	2	2	no	—
176	#408	C₄×C₆₁	1	2	yes	—
177	#408	C₂²×C₆₁	1	2	yes	—
178	C₂×C₂²⋊C₈	—	2	2	no	—
179	#349	—	2	2	no	—
180	#349	C₂²×C₆₁	1	4	yes	complemented
181	C₆₁	C₂×#1321	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×#41	—	2	1	no	—
183	C₂×#41	—	2	1	no	—
184	C₂×#41	—	4	1	no	—
185	C₂×#41	—	4	1	no	—
186	C₂×#41	—	4	1	no	—
187	C₂×#41	—	4	1	no	—
188	C₂×#41	C₈×C₆₁	1	2	yes	—
189	C₂×#41	C₂×C₄×C₆₁	1	2	yes	—
190	C₂²×Q₈	C₈×C₆₁	1	1	yes	characteristic
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	C₈×C₆₁	1	1	yes	characteristic
193	C₂³×C₄	—	2	1	no	—
194	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
195	C₂×M₁₆	—	4	1	no	—
196	C₂×M₁₆	—	4	1	no	—
197	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
198	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
199	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	4	1	no	—
203	C₂²×C₈	—	4	1	no	—
204	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
205	C₂²×C₈	D₄×C₆₁	1	2	yes	—
206	M₃₂	—	2	4	no	—
207	M₃₂	—	4	2	no	—
208	C₂×C₁₆	—	4	2	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₂⁴	—	2	1	no	—
212	C₂⁴	M₁₆×C₆₁	1	1	yes	characteristic
213	#41	—	4	2	no	—
214	#41	—	4	2	no	—
215	#41	—	4	2	no	—
216	#41	—	4	2	no	—
217	#41	—	4	2	no	—
218	#41	—	4	2	no	—
219	#41	—	4	2	no	—
220	#41	—	4	2	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×Q₈	—	4	1	no	—
224	C₂×Q₈	C₂×C₈×C₆₁	1	2	yes	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	C₂×C₈×C₆₁	1	2	yes	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
248	C₂²×C₄	M₁₆×C₆₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	2	no	—
252	M₁₆	—	4	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
261	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
262	C₁₆	—	4	4	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	4	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
271	C₂³	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
272	C₂³	C₂×M₁₆×C₆₁	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
297	C₂×C₄	C₂²⋊C₈×C₆₁	1	1	yes	characteristic
298	C₂×C₄	#99×C₆₁	1	1	yes	characteristic
299	C₂×C₄	#99×C₆₁	1	1	yes	characteristic
300	C₂×C₄	#100×C₆₁	1	1	yes	characteristic
301	C₂×C₄	#100×C₆₁	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₆₁×#323	1	1	yes	characteristic
319	C₂²	C₆₁×#323	1	1	yes	characteristic
320	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	2	no	—
324	C₄	C₆₁×#323	1	1	yes	characteristic
325	C₄	C₆₁×#323	1	1	yes	characteristic
326	C₄	C₂×#99×C₆₁	1	1	yes	characteristic
327	C₄	C₂×#100×C₆₁	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	4	1	no	—
331	C₂	—	4	1	no	—
332	C₂	—	8	2	no	—
333	C₂	C₆₁×#1321	1	2	yes	direct product factor
334	C₂	C₂×C₆₁×#323	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#99
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂×M₁₆	C₂³×C₆₁
socle	C₂²×C₆₁	#323
prime radical	C₂²×#41×C₆₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×M₁₆×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₁×#349	C₂	semidirect
3. C₆₁×#1321	C₂	direct (N × H)
13. C₆₁×#349	C₂²	semidirect
139. C₂×#1321	C₆₁	direct (N × H)
162. C₂×#349	C₂×C₆₁	semidirect
163. #1321	C₂×C₆₁	direct (N × H)
169. C₂×C₆₁	#1321	direct (N × H)
180. #349	C₂²×C₆₁	semidirect
181. C₆₁	C₂×#1321	direct (N × H)
333. C₂	C₆₁×#1321	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×#408	2
C₂×C₆₁×#349	2
C₆₁×#1321	2
C₂×#1321	61

Subgroups — C2×C61×#1321 #29514870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1321 #29514870