Subgroups — C₆₁×#8671 #29512104

Order 15616.

408 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	C₈×#41×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₄×#108×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1489	C₂	1	2	yes	maximal
5	C₆₁×#1299	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
8	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
9	C₈×D₄×C₆₁	—	2	1	no	—
10	C₈×D₄×C₆₁	—	2	1	no	—
11	C₈×D₄×C₆₁	C₄	1	1	yes	characteristic
12	C₈×D₄×C₆₁	C₄	1	1	yes	characteristic
13	C₆₁×#431	—	2	1	no	—
14	C₆₁×#431	C₂²	1	1	yes	characteristic
15	C₂×#108×C₆₁	—	2	1	no	—
16	C₆₁×#423	C₂²	1	2	yes	—
17	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₁	—	2	1	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#341	—	2	1	no	—
21	C₆₁×#335	—	2	1	no	—
22	C₆₁×#335	—	2	1	no	—
23	C₆₁×Q₈⋊C₈	—	2	2	no	—
24	C₆₁×D₄⋊C₈	—	2	2	no	—
25	C₂×#41×C₆₁	Q₈	1	1	yes	characteristic
26	C₂×M₁₆×C₆₁	—	2	2	no	—
27	C₂×M₁₆×C₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	—	2	1	no	—
29	C₂²×C₈×C₆₁	—	2	1	no	—
30	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
34	C₄×D₄×C₆₁	—	2	1	no	—
35	C₄×D₄×C₆₁	—	2	1	no	—
36	C₄×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
38	#117×C₆₁	—	2	1	no	—
39	#117×C₆₁	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	—	2	1	no	—
41	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	#108×C₆₁	—	2	4	no	—
43	C₄⋊C₈×C₆₁	—	2	1	no	—
44	C₄⋊C₈×C₆₁	—	2	1	no	—
45	C₄⋊C₈×C₆₁	—	2	2	no	—
46	C₄⋊C₈×C₆₁	—	2	2	no	—
47	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂²⋊C₈×C₆₁	—	2	1	no	—
50	C₂²⋊C₈×C₆₁	—	2	1	no	—
51	C₂²⋊C₈×C₆₁	—	2	1	no	—
52	#97×C₆₁	—	2	2	no	—
53	C₄×C₈×C₆₁	—	2	1	no	—
54	C₄×C₈×C₆₁	—	2	1	no	—
55	C₄×C₈×C₆₁	—	2	1	no	—
56	C₄×C₈×C₆₁	—	2	2	no	—
57	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
61	#41×C₆₁	—	2	2	no	—
62	#41×C₆₁	—	2	2	no	—
63	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
64	C₂×D₄×C₆₁	—	2	1	no	—
65	C₂×D₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
72	M₁₆×C₆₁	—	2	4	no	—
73	M₁₆×C₆₁	—	4	2	no	—
74	M₁₆×C₆₁	—	8	2	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	1	no	—
77	C₂×C₈×C₆₁	—	2	1	no	—
78	C₂×C₈×C₆₁	—	4	1	no	—
79	C₂×C₈×C₆₁	—	4	2	no	—
80	C₂×C₈×C₆₁	—	4	2	no	—
81	C₂×C₈×C₆₁	—	4	2	no	—
82	C₂×C₈×C₆₁	—	4	2	no	—
83	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₆₁	—	2	1	no	—
88	C₄⋊C₄×C₆₁	—	2	1	no	—
89	C₄⋊C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
91	C₂²⋊C₄×C₆₁	—	2	1	no	—
92	C₂²⋊C₄×C₆₁	—	2	1	no	—
93	C₂²⋊C₄×C₆₁	—	2	1	no	—
94	C₄²×C₆₁	—	2	1	no	—
95	C₄²×C₆₁	—	2	1	no	—
96	C₄²×C₆₁	—	2	1	no	—
97	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₁	#41	1	1	yes	characteristic
101	C₂³×C₆₁	—	2	1	no	—
102	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
103	Q₈×C₆₁	—	2	1	no	—
104	Q₈×C₆₁	#108	1	2	yes	complemented
105	D₄×C₆₁	—	2	4	no	—
106	D₄×C₆₁	—	2	1	no	—
107	D₄×C₆₁	#108	1	2	yes	complemented
108	C₂×C₄×C₆₁	—	2	4	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	2	1	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	—	4	1	no	—
119	C₂×C₄×C₆₁	—	4	1	no	—
120	C₂×C₄×C₆₁	—	4	1	no	—
121	C₂×C₄×C₆₁	—	4	1	no	—
122	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
125	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
126	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
128	C₈×C₆₁	—	2	1	no	—
129	C₈×C₆₁	—	2	1	no	—
130	C₈×C₆₁	—	2	1	no	—
131	C₈×C₆₁	—	4	4	no	—
132	C₈×C₆₁	—	4	1	no	—
133	C₈×C₆₁	—	4	1	no	—
134	C₈×C₆₁	—	8	2	no	—
135	C₈×C₆₁	#104	1	2	yes	—
136	#8671	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₆₁	—	2	1	no	—
138	C₂²×C₆₁	—	2	1	no	—
139	C₂²×C₆₁	—	2	2	no	—
140	C₂²×C₆₁	—	2	1	no	—
141	C₂²×C₆₁	—	4	1	no	—
142	C₂²×C₆₁	—	4	1	no	—
143	C₂²×C₆₁	#380	1	1	yes	characteristic
144	C₄×C₆₁	—	2	1	no	—
145	C₄×C₆₁	—	2	1	no	—
146	C₄×C₆₁	—	2	2	no	—
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	4	1	no	—
149	C₄×C₆₁	—	4	1	no	—
150	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
151	C₄×C₆₁	#421	1	1	yes	characteristic
152	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
153	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
154	C₄×C₆₁	#472	1	1	yes	characteristic
155	C₄×C₆₁	#473	1	1	yes	characteristic
156	C₈×#41	C₂×C₆₁	1	1	yes	characteristic
157	C₄×#108	C₂×C₆₁	1	1	yes	characteristic, complemented
158	#1567	C₂×C₆₁	1	1	yes	characteristic, complemented
159	#1489	C₂×C₆₁	1	2	yes	—
160	#1299	C₂×C₆₁	1	2	yes	complemented
161	C₂×C₆₁	—	4	1	no	—
162	C₂×C₆₁	—	4	2	no	—
163	C₂×C₆₁	#1806	1	1	yes	characteristic
164	C₂×C₆₁	#1810	1	1	yes	characteristic
165	C₂×C₆₁	#1816	1	1	yes	characteristic
166	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
167	C₈×Q₈	C₄×C₆₁	1	1	yes	characteristic
168	C₈×Q₈	C₄×C₆₁	1	1	yes	characteristic
169	C₈×D₄	—	2	1	no	—
170	C₈×D₄	—	2	1	no	—
171	C₈×D₄	C₄×C₆₁	1	1	yes	characteristic
172	C₈×D₄	C₄×C₆₁	1	1	yes	characteristic
173	#431	—	2	1	no	—
174	#431	C₂²×C₆₁	1	1	yes	characteristic
175	C₂×#108	—	2	1	no	—
176	#423	C₂²×C₆₁	1	2	yes	—
177	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
178	C₂×C₄×C₈	—	2	1	no	—
179	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
180	#341	—	2	1	no	—
181	#335	—	2	1	no	—
182	#335	—	2	1	no	—
183	Q₈⋊C₈	—	2	2	no	—
184	D₄⋊C₈	—	2	2	no	—
185	C₆₁	#8671	1	1	yes	characteristic, direct product factor, Sylow
186	C₂×#41	Q₈×C₆₁	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
192	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
193	C₄×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
194	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
198	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
199	#117	—	2	1	no	—
200	#117	Q₈×C₆₁	1	1	yes	characteristic
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
203	#108	—	2	4	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
209	C₄⋊C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	2	1	no	—
212	C₂²⋊C₈	—	2	1	no	—
213	#97	—	2	2	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	2	2	no	—
218	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
219	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
220	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
221	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
222	#41	—	2	2	no	—
223	#41	—	2	2	no	—
224	C₂×Q₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
232	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	8	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
245	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
246	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
247	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
251	C₄⋊C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
261	C₄²	#41×C₆₁	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	#108×C₆₁	1	2	yes	complemented
266	D₄	—	2	4	no	—
267	D₄	—	2	1	no	—
268	D₄	#108×C₆₁	1	2	yes	complemented
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
285	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
286	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
287	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
288	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	4	4	no	—
293	C₈	—	4	1	no	—
294	C₈	—	4	1	no	—
295	C₈	—	8	2	no	—
296	C₈	#104×C₆₁	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	C₆₁×#380	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	C₂×#104×C₆₁	1	1	yes	characteristic
311	C₄	C₆₁×#421	1	1	yes	characteristic
312	C₄	C₂×#108×C₆₁	1	1	yes	characteristic
313	C₄	C₂×#108×C₆₁	1	1	yes	characteristic
314	C₄	C₆₁×#472	1	1	yes	characteristic
315	C₄	C₆₁×#473	1	1	yes	characteristic
316	C₂	—	4	1	no	—
317	C₂	—	4	2	no	—
318	C₂	C₆₁×#1806	1	1	yes	characteristic
319	C₂	C₆₁×#1810	1	1	yes	characteristic
320	C₂	C₆₁×#1816	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#121
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#380
prime radical	C₂×#41×C₆₁	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#108×C₆₁	C₂	semidirect
3. C₆₁×#1567	C₂	semidirect
5. C₆₁×#1299	C₂	semidirect
104. Q₈×C₆₁	#108	semidirect
107. D₄×C₆₁	#108	semidirect
136. #8671	C₆₁	direct (N × H)
157. C₄×#108	C₂×C₆₁	semidirect
158. #1567	C₂×C₆₁	semidirect
160. #1299	C₂×C₆₁	semidirect
185. C₆₁	#8671	direct (N × H)
265. Q₈	#108×C₆₁	semidirect
268. D₄	#108×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₆₁	2
C₄×#108×C₆₁	2
C₆₁×#1567	2
C₆₁×#1489	2
C₆₁×#1299	2
#8671	61

Subgroups — C61×#8671 #29512104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8671 #29512104