Subgroups — C₆₁×#7679 #29511112

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2858	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1270	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
6	C₂×#127×C₆₁	C₄	1	1	yes	characteristic
7	C₂×#124×C₆₁	—	2	1	no	—
8	C₂×#124×C₆₁	—	2	1	no	—
9	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
10	C₂×#97×C₆₁	—	2	1	no	—
11	C₂×C₄×C₈×C₆₁	—	2	1	no	—
12	C₆₁×#390	—	2	1	no	—
13	C₆₁×#390	—	2	1	no	—
14	C₆₁×#390	C₄	1	2	yes	—
15	C₆₁×#390	C₂²	1	2	yes	—
16	C₄³×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#336	—	2	1	no	—
18	C₆₁×#336	C₂²	1	2	yes	—
19	C₆₁×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₆₁	—	2	1	no	—
21	C₂²×D₄×C₆₁	—	2	1	no	—
22	C₂²×D₄×C₆₁	—	2	1	no	—
23	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₁	—	2	2	no	—
25	C₂²×C₈×C₆₁	—	2	2	no	—
26	C₄⋊Q₈×C₆₁	—	4	2	no	—
27	#127×C₆₁	—	4	2	no	—
28	#124×C₆₁	—	4	4	no	—
29	#124×C₆₁	—	4	2	no	—
30	#124×C₆₁	—	4	2	no	—
31	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
34	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₁	—	2	4	no	—
45	#97×C₆₁	—	4	2	no	—
46	C₄×C₈×C₆₁	—	2	4	no	—
47	C₂⁴×C₆₁	—	2	2	no	—
48	C₂×Q₈×C₆₁	—	4	2	no	—
49	C₂×Q₈×C₆₁	—	4	2	no	—
50	C₂×Q₈×C₆₁	—	4	2	no	—
51	C₂×D₄×C₆₁	—	2	4	no	—
52	C₂×D₄×C₆₁	—	4	2	no	—
53	C₂×D₄×C₆₁	—	4	2	no	—
54	C₂×D₄×C₆₁	—	4	2	no	—
55	C₂×D₄×C₆₁	—	4	2	no	—
56	C₂×D₄×C₆₁	—	4	2	no	—
57	C₂×D₄×C₆₁	—	4	2	no	—
58	C₂×D₄×C₆₁	—	4	2	no	—
59	C₂×D₄×C₆₁	—	4	2	no	—
60	C₂²×C₄×C₆₁	—	2	2	no	—
61	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₁	—	2	4	no	—
69	C₂×C₈×C₆₁	—	2	4	no	—
70	C₂×C₈×C₆₁	—	4	2	no	—
71	C₂×C₈×C₆₁	—	4	2	no	—
72	C₂×C₈×C₆₁	—	4	2	no	—
73	C₂×C₈×C₆₁	—	4	2	no	—
74	C₄⋊C₄×C₆₁	—	4	2	no	—
75	C₄⋊C₄×C₆₁	—	4	2	no	—
76	C₂²⋊C₄×C₆₁	—	4	4	no	—
77	C₂²⋊C₄×C₆₁	—	4	4	no	—
78	C₂²⋊C₄×C₆₁	—	4	2	no	—
79	C₂²⋊C₄×C₆₁	—	4	2	no	—
80	C₂²⋊C₄×C₆₁	—	4	4	no	—
81	C₄²×C₆₁	—	2	2	no	—
82	C₄²×C₆₁	—	2	2	no	—
83	C₄²×C₆₁	—	2	2	no	—
84	C₄²×C₆₁	—	2	2	no	—
85	C₄²×C₆₁	—	2	2	no	—
86	C₄²×C₆₁	—	4	1	no	—
87	C₄²×C₆₁	D₈	1	2	yes	—
88	C₄²×C₆₁	SD₁₆	1	2	yes	—
89	C₂³×C₆₁	—	4	2	no	—
90	C₂³×C₆₁	—	4	2	no	—
91	C₂³×C₆₁	—	4	2	no	—
92	C₂³×C₆₁	—	4	2	no	—
93	C₂³×C₆₁	—	4	2	no	—
94	C₂³×C₆₁	—	4	2	no	—
95	C₂³×C₆₁	—	4	2	no	—
96	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	Q₈×C₆₁	—	8	4	no	—
98	D₄×C₆₁	—	8	4	no	—
99	D₄×C₆₁	—	8	4	no	—
100	D₄×C₆₁	—	8	4	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	—	2	1	no	—
105	C₂×C₄×C₆₁	—	2	1	no	—
106	C₂×C₄×C₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	1	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	—	4	2	no	—
115	C₂×C₄×C₆₁	—	4	2	no	—
116	C₂×C₄×C₆₁	—	4	2	no	—
117	C₂×C₄×C₆₁	#100	1	2	yes	—
118	C₂×C₄×C₆₁	#102	1	2	yes	—
119	C₂×C₄×C₆₁	#102	1	2	yes	—
120	C₂×C₄×C₆₁	#104	1	2	yes	—
121	C₂×C₄×C₆₁	#104	1	2	yes	—
122	C₂×C₄×C₆₁	#131	1	2	yes	—
123	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
126	C₈×C₆₁	—	4	4	no	—
127	C₈×C₆₁	—	4	4	no	—
128	#7679	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₆₁	—	8	2	no	—
130	C₂²×C₆₁	—	8	2	no	—
131	C₂²×C₆₁	—	8	2	no	—
132	C₂²×C₆₁	—	8	2	no	—
133	C₂²×C₆₁	—	8	2	no	—
134	C₂²×C₆₁	—	8	2	no	—
135	C₂²×C₆₁	—	8	2	no	—
136	C₂²×C₆₁	#408	1	1	yes	characteristic
137	C₂²×C₆₁	#410	1	1	yes	characteristic
138	C₂²×C₆₁	C₂×#100	1	1	yes	characteristic
139	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
140	C₂²×C₆₁	#418	1	1	yes	characteristic
141	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
142	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
143	C₄×C₆₁	—	2	2	no	—
144	C₄×C₆₁	—	2	2	no	—
145	C₄×C₆₁	—	2	2	no	—
146	C₄×C₆₁	—	2	1	no	—
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	2	2	no	—
149	C₄×C₆₁	—	4	1	no	—
150	C₄×C₆₁	—	8	2	no	—
151	C₄×C₆₁	#329	1	4	yes	—
152	#2858	C₂×C₆₁	1	1	yes	characteristic
153	#1767	C₂×C₆₁	1	1	yes	characteristic, complemented
154	#1744	C₂×C₆₁	1	1	yes	characteristic, complemented
155	#1270	C₂×C₆₁	1	4	yes	complemented
156	C₂×C₆₁	—	16	2	no	—
157	C₂×C₆₁	#1469	1	1	yes	characteristic
158	C₂×C₆₁	#1484	1	1	yes	characteristic
159	C₂×C₆₁	#1492	1	1	yes	characteristic
160	C₂×C₆₁	#1502	1	1	yes	characteristic
161	C₂×C₆₁	#1505	1	1	yes	characteristic
162	C₂×C₆₁	C₂×#329	1	1	yes	characteristic
163	C₂×C₆₁	#1545	1	1	yes	characteristic
164	C₂×C₄⋊Q₈	C₄×C₆₁	1	1	yes	characteristic
165	C₂×#127	C₄×C₆₁	1	1	yes	characteristic
166	C₂×#124	—	2	1	no	—
167	C₂×#124	—	2	1	no	—
168	C₂×C₄⋊C₈	—	2	1	no	—
169	C₂×#97	—	2	1	no	—
170	C₂×C₄×C₈	—	2	1	no	—
171	#390	—	2	1	no	—
172	#390	—	2	1	no	—
173	#390	C₄×C₆₁	1	2	yes	—
174	#390	C₂²×C₆₁	1	2	yes	—
175	C₄³	C₂²×C₆₁	1	1	yes	characteristic
176	#336	—	2	1	no	—
177	#336	C₂²×C₆₁	1	2	yes	—
178	#336	C₂²×C₆₁	1	2	yes	—
179	C₆₁	#7679	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	—	2	1	no	—
183	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₄⋊Q₈	—	4	2	no	—
187	#127	—	4	2	no	—
188	#124	—	4	4	no	—
189	#124	—	4	2	no	—
190	#124	—	4	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
198	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
202	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
204	C₄⋊C₈	—	2	4	no	—
205	#97	—	4	2	no	—
206	C₄×C₈	—	2	4	no	—
207	C₂⁴	—	2	2	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	D₈×C₆₁	1	2	yes	—
248	C₄²	SD₁₆×C₆₁	1	2	yes	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
257	Q₈	—	8	4	no	—
258	D₄	—	8	4	no	—
259	D₄	—	8	4	no	—
260	D₄	—	8	4	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	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	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	#100×C₆₁	1	2	yes	—
278	C₂×C₄	#102×C₆₁	1	2	yes	—
279	C₂×C₄	#102×C₆₁	1	2	yes	—
280	C₂×C₄	#104×C₆₁	1	2	yes	—
281	C₂×C₄	#104×C₆₁	1	2	yes	—
282	C₂×C₄	#131×C₆₁	1	2	yes	—
283	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
285	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
286	C₈	—	4	4	no	—
287	C₈	—	4	4	no	—
288	C₂²	—	8	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	C₆₁×#408	1	1	yes	characteristic
296	C₂²	C₆₁×#410	1	1	yes	characteristic
297	C₂²	C₂×#100×C₆₁	1	1	yes	characteristic
298	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
299	C₂²	C₆₁×#418	1	1	yes	characteristic
300	C₂²	C₂×#104×C₆₁	1	1	yes	characteristic
301	C₂²	C₂×#104×C₆₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	8	2	no	—
310	C₄	C₆₁×#329	1	4	yes	—
311	C₂	—	16	2	no	—
312	C₂	C₆₁×#1469	1	1	yes	characteristic
313	C₂	C₆₁×#1484	1	1	yes	characteristic
314	C₂	C₆₁×#1492	1	1	yes	characteristic
315	C₂	C₆₁×#1502	1	1	yes	characteristic
316	C₂	C₆₁×#1505	1	1	yes	characteristic
317	C₂	C₂×C₆₁×#329	1	1	yes	characteristic
318	C₂	C₆₁×#1545	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂×#127×C₆₁	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#1767	C₂	semidirect
3. C₆₁×#1744	C₂	semidirect
4. C₆₁×#1270	C₂	semidirect
128. #7679	C₆₁	direct (N × H)
153. #1767	C₂×C₆₁	semidirect
154. #1744	C₂×C₆₁	semidirect
155. #1270	C₂×C₆₁	semidirect
179. C₆₁	#7679	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#2858	2
C₆₁×#1767	2
C₆₁×#1744	2
C₆₁×#1270	2
#7679	61

Subgroups — C61×#7679 #29511112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7679 #29511112