Subgroups — C₆₁×#29861 #29533294

Order 15616.

712 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3080	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3038	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2923	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2063	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2051	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1998	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1849	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	—
11	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
13	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#490	—	2	2	no	—
15	C₆₁×#490	—	2	2	no	—
16	C₆₁×#479	—	2	2	no	—
17	C₆₁×#478	—	2	2	no	—
18	C₆₁×#460	—	2	2	no	—
19	C₆₁×#459	—	2	2	no	—
20	C₆₁×#433	—	2	2	no	—
21	C₆₁×#432	—	2	2	no	—
22	C₂×#108×C₆₁	C₂²	1	4	yes	—
23	C₆₁×#428	C₂²	1	2	yes	—
24	C₆₁×#424	C₂²	1	2	yes	—
25	C₆₁×#423	C₂²	1	1	yes	characteristic
26	C₆₁×#423	C₂²	1	1	yes	characteristic
27	C₆₁×#418	C₂²	1	2	yes	—
28	C₆₁×#417	C₂²	1	2	yes	—
29	C₂×#102×C₆₁	C₂²	1	4	yes	—
30	C₂×#100×C₆₁	C₂²	1	4	yes	—
31	C₆₁×#408	C₂²	1	2	yes	—
32	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
34	C₆₁×#343	C₂²	1	4	yes	—
35	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
36	C₂²×D₄×C₆₁	—	2	2	no	—
37	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
38	AGL(1,ℤ/8)×C₆₁	—	4	4	no	—
39	C₂×SD₁₆×C₆₁	—	2	2	no	—
40	C₂×D₈×C₆₁	—	2	2	no	—
41	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
42	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
43	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
45	#127×C₆₁	—	2	2	no	—
46	#127×C₆₁	—	2	1	no	—
47	#127×C₆₁	—	2	1	no	—
48	#125×C₆₁	—	2	2	no	—
49	#125×C₆₁	—	2	1	no	—
50	#125×C₆₁	—	2	1	no	—
51	C₄×Q₈×C₆₁	D₄	1	2	yes	—
52	C₄×D₄×C₆₁	—	2	2	no	—
53	C₄×D₄×C₆₁	D₄	1	2	yes	—
54	#117×C₆₁	—	2	1	no	—
55	#117×C₆₁	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₆₁	—	2	1	no	—
59	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
60	#108×C₆₁	—	4	4	no	—
61	#107×C₆₁	—	2	2	no	—
62	#106×C₆₁	—	2	2	no	—
63	C₄⋊C₈×C₆₁	—	2	4	no	—
64	C₄⋊C₈×C₆₁	—	2	2	no	—
65	C₄⋊C₈×C₆₁	—	2	2	no	—
66	#103×C₆₁	—	2	2	no	—
67	#102×C₆₁	—	2	2	no	—
68	#102×C₆₁	—	2	8	no	—
69	#102×C₆₁	—	2	4	no	—
70	#100×C₆₁	—	4	4	no	—
71	C₂²⋊C₈×C₆₁	—	2	4	no	—
72	#97×C₆₁	—	2	1	no	—
73	#97×C₆₁	D₄	1	2	yes	—
74	#97×C₆₁	D₄	1	2	yes	—
75	C₄×C₈×C₆₁	—	2	1	no	—
76	C₂⁴×C₆₁	—	2	2	no	—
77	#41×C₆₁	—	2	2	no	—
78	#41×C₆₁	—	2	2	no	—
79	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₂×D₄×C₆₁	—	2	1	no	—
81	C₂×D₄×C₆₁	—	2	2	no	—
82	C₂×D₄×C₆₁	—	2	2	no	—
83	C₂×D₄×C₆₁	—	2	4	no	—
84	C₂×D₄×C₆₁	—	2	4	no	—
85	C₂×D₄×C₆₁	—	4	2	no	—
86	C₂×D₄×C₆₁	—	4	2	no	—
87	C₂×D₄×C₆₁	—	8	4	no	—
88	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₁	—	2	1	no	—
90	C₂²×C₄×C₆₁	—	2	2	no	—
91	C₂²×C₄×C₆₁	—	2	2	no	—
92	C₂²×C₄×C₆₁	#41	1	2	yes	—
93	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
94	SD₁₆×C₆₁	—	4	4	no	—
95	D₈×C₆₁	—	4	4	no	—
96	M₁₆×C₆₁	—	2	2	no	—
97	M₁₆×C₆₁	—	2	4	no	—
98	M₁₆×C₆₁	—	2	4	no	—
99	M₁₆×C₆₁	—	2	2	no	—
100	C₂×C₈×C₆₁	—	2	2	no	—
101	C₂×C₈×C₆₁	—	2	2	no	—
102	C₂×C₈×C₆₁	—	2	2	no	—
103	C₂×C₈×C₆₁	—	2	2	no	—
104	C₂×C₈×C₆₁	—	2	2	no	—
105	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
106	C₄⋊C₄×C₆₁	—	2	2	no	—
107	C₄⋊C₄×C₆₁	—	2	2	no	—
108	C₄⋊C₄×C₆₁	—	2	2	no	—
109	C₄⋊C₄×C₆₁	—	4	2	no	—
110	C₄⋊C₄×C₆₁	—	4	2	no	—
111	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄×C₆₁	—	2	1	no	—
113	C₂²⋊C₄×C₆₁	—	2	2	no	—
114	C₄²×C₆₁	—	2	2	no	—
115	C₄²×C₆₁	—	2	1	no	—
116	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₆₁	C₂×D₄	1	2	yes	—
119	C₂³×C₆₁	—	2	1	no	—
120	C₂³×C₆₁	—	4	2	no	—
121	C₂³×C₆₁	—	4	4	no	—
122	C₂³×C₆₁	—	8	4	no	—
123	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
124	Q₈×C₆₁	—	2	1	no	—
125	Q₈×C₆₁	—	2	1	no	—
126	D₄×C₆₁	—	2	1	no	—
127	D₄×C₆₁	—	2	1	no	—
128	D₄×C₆₁	—	4	2	no	—
129	D₄×C₆₁	—	4	2	no	—
130	D₄×C₆₁	—	4	2	no	—
131	D₄×C₆₁	—	4	4	no	—
132	D₄×C₆₁	—	8	4	no	—
133	D₄×C₆₁	—	8	4	no	—
134	C₂×C₄×C₆₁	—	2	2	no	—
135	C₂×C₄×C₆₁	—	2	1	no	—
136	C₂×C₄×C₆₁	—	4	2	no	—
137	C₂×C₄×C₆₁	—	4	2	no	—
138	C₂×C₄×C₆₁	—	4	2	no	—
139	C₂×C₄×C₆₁	—	8	2	no	—
140	C₂×C₄×C₆₁	—	8	2	no	—
141	C₂×C₄×C₆₁	#121	1	2	yes	—
142	C₂×C₄×C₆₁	#121	1	2	yes	—
143	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
146	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
147	C₈×C₆₁	—	2	4	no	—
148	C₈×C₆₁	—	4	2	no	—
149	C₈×C₆₁	—	4	2	no	—
150	C₈×C₆₁	—	4	2	no	—
151	#29861	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₆₁	—	2	2	no	—
153	C₂²×C₆₁	—	4	1	no	—
154	C₂²×C₆₁	—	4	2	no	—
155	C₂²×C₆₁	—	8	2	no	—
156	C₂²×C₆₁	—	8	4	no	—
157	C₂²×C₆₁	—	16	4	no	—
158	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
159	C₂²×C₆₁	#542	1	2	yes	—
160	C₄×C₆₁	—	2	2	no	—
161	C₄×C₆₁	—	2	2	no	—
162	C₄×C₆₁	—	4	1	no	—
163	C₄×C₆₁	—	4	2	no	—
164	C₄×C₆₁	—	8	2	no	—
165	C₄×C₆₁	#535	1	1	yes	characteristic
166	C₄×C₆₁	#536	1	1	yes	characteristic
167	C₄×C₆₁	#537	1	2	yes	—
168	C₂×#490	C₂×C₆₁	1	1	yes	characteristic, complemented
169	#3080	C₂×C₆₁	1	1	yes	characteristic, complemented
170	#3038	C₂×C₆₁	1	1	yes	characteristic
171	#2923	C₂×C₆₁	1	1	yes	characteristic, complemented
172	#2063	C₂×C₆₁	1	4	yes	complemented
173	#2051	C₂×C₆₁	1	4	yes	complemented
174	#1998	C₂×C₆₁	1	1	yes	characteristic, complemented
175	#1965	C₂×C₆₁	1	1	yes	characteristic, complemented
176	#1849	C₂×C₆₁	1	1	yes	characteristic, complemented
177	C₂×C₆₁	—	2	2	no	—
178	C₂×C₆₁	—	8	1	no	—
179	C₂×C₆₁	—	16	2	no	—
180	C₂×C₆₁	#2620	1	1	yes	characteristic
181	C₂×C₆₁	#3068	1	2	yes	—
182	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	—
183	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
184	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
185	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
186	#490	—	2	2	no	—
187	#490	—	2	2	no	—
188	#479	—	2	2	no	—
189	#478	—	2	2	no	—
190	#460	—	2	2	no	—
191	#459	—	2	2	no	—
192	#433	—	2	2	no	—
193	#432	—	2	2	no	—
194	C₂×#108	C₂²×C₆₁	1	4	yes	—
195	#428	C₂²×C₆₁	1	2	yes	—
196	#424	C₂²×C₆₁	1	2	yes	—
197	#423	C₂²×C₆₁	1	1	yes	characteristic
198	#423	C₂²×C₆₁	1	1	yes	characteristic
199	#418	C₂²×C₆₁	1	2	yes	—
200	#417	C₂²×C₆₁	1	2	yes	—
201	C₂×#102	C₂²×C₆₁	1	4	yes	—
202	C₂×#100	C₂²×C₆₁	1	4	yes	—
203	#408	C₂²×C₆₁	1	2	yes	—
204	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
205	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
206	#343	C₂²×C₆₁	1	4	yes	—
207	C₆₁	#29861	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
209	C₂²×D₄	—	2	2	no	—
210	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
211	AGL(1,ℤ/8)	—	4	4	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
215	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
216	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
217	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
218	#127	—	2	2	no	—
219	#127	—	2	1	no	—
220	#127	—	2	1	no	—
221	#125	—	2	2	no	—
222	#125	—	2	1	no	—
223	#125	—	2	1	no	—
224	C₄×Q₈	D₄×C₆₁	1	2	yes	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	D₄×C₆₁	1	2	yes	—
227	#117	—	2	1	no	—
228	#117	C₂³×C₆₁	1	1	yes	characteristic
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
233	#108	—	4	4	no	—
234	#107	—	2	2	no	—
235	#106	—	2	2	no	—
236	C₄⋊C₈	—	2	4	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	2	no	—
239	#103	—	2	2	no	—
240	#102	—	2	2	no	—
241	#102	—	2	8	no	—
242	#102	—	2	4	no	—
243	#100	—	4	4	no	—
244	C₂²⋊C₈	—	2	4	no	—
245	#97	—	2	1	no	—
246	#97	D₄×C₆₁	1	2	yes	—
247	#97	D₄×C₆₁	1	2	yes	—
248	C₄×C₈	—	2	1	no	—
249	C₂⁴	—	2	2	no	—
250	#41	—	2	2	no	—
251	#41	—	2	2	no	—
252	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	8	4	no	—
261	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	#41×C₆₁	1	2	yes	—
266	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
267	SD₁₆	—	4	4	no	—
268	D₈	—	4	4	no	—
269	M₁₆	—	2	2	no	—
270	M₁₆	—	2	4	no	—
271	M₁₆	—	2	4	no	—
272	M₁₆	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	1	no	—
289	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₆₁	1	2	yes	—
292	C₂³	—	2	1	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	4	no	—
295	C₂³	—	8	4	no	—
296	C₂³	C₂×#41×C₆₁	1	1	yes	characteristic
297	Q₈	—	2	1	no	—
298	Q₈	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	4	no	—
305	D₄	—	8	4	no	—
306	D₄	—	8	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	#121×C₆₁	1	2	yes	—
315	C₂×C₄	#121×C₆₁	1	2	yes	—
316	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
318	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
319	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	2	no	—
323	C₈	—	4	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	8	4	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	16	4	no	—
330	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
331	C₂²	C₆₁×#542	1	2	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	8	2	no	—
337	C₄	C₆₁×#535	1	1	yes	characteristic
338	C₄	C₆₁×#536	1	1	yes	characteristic
339	C₄	C₆₁×#537	1	2	yes	—
340	C₂	—	2	2	no	—
341	C₂	—	8	1	no	—
342	C₂	—	16	2	no	—
343	C₂	C₆₁×#2620	1	1	yes	characteristic
344	C₂	C₆₁×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₆₁×#3038	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#490	C₂	semidirect
2. C₆₁×#3080	C₂	semidirect
4. C₆₁×#2923	C₂	semidirect
5. C₆₁×#2063	C₂	semidirect
6. C₆₁×#2051	C₂	semidirect
7. C₆₁×#1998	C₂	semidirect
8. C₆₁×#1965	C₂	semidirect
9. C₆₁×#1849	C₂	semidirect
151. #29861	C₆₁	direct (N × H)
168. C₂×#490	C₂×C₆₁	semidirect
169. #3080	C₂×C₆₁	semidirect
171. #2923	C₂×C₆₁	semidirect
172. #2063	C₂×C₆₁	semidirect
173. #2051	C₂×C₆₁	semidirect
174. #1998	C₂×C₆₁	semidirect
175. #1965	C₂×C₆₁	semidirect
176. #1849	C₂×C₆₁	semidirect
207. C₆₁	#29861	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×#490	2
C₆₁×#3080	2
C₆₁×#3038	2
C₆₁×#2923	2
C₆₁×#2063	2
C₆₁×#2051	2
C₆₁×#1998	2
C₆₁×#1965	2
C₆₁×#1849	2
#29861	61

Subgroups — C61×#29861 #29533294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29861 #29533294