Subgroups — C₆₁×#33961 #29537394

Order 15616.

520 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3396	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3393	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2249	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2211	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2210	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2206	C₂	1	1	yes	maximal, characteristic
9	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×SD₃₂×C₆₁	C₂	1	2	yes	maximal, complemented
12	C₆₁×#559	—	2	1	no	—
13	C₆₁×#556	—	2	1	no	—
14	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	—
15	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
16	C₂×SD₃₂×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
18	C₂×D₁₆×C₆₁	C₂²	1	2	yes	—
19	C₆₁×#495	C₂²	1	2	yes	—
20	C₆₁×#495	C₂²	1	2	yes	—
21	C₆₁×#494	C₂²	1	1	yes	characteristic
22	C₆₁×#493	C₂²	1	1	yes	characteristic
23	C₆₁×#464	—	2	2	no	—
24	C₆₁×#463	—	2	2	no	—
25	C₆₁×#461	—	2	1	no	—
26	C₆₁×#460	—	2	1	no	—
27	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
29	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
30	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₆₁	C₂²	1	2	yes	—
32	C₆₁×#367	C₂²	1	1	yes	characteristic
33	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₆₁×#358	C₂²	1	2	yes	—
36	C₆₁×#358	C₂²	1	1	yes	characteristic
37	C₆₁×#357	C₂²	1	2	yes	—
38	C₆₁×#357	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₆₁	—	2	1	no	—
42	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
43	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×SD₁₆×C₆₁	—	2	2	no	—
45	C₂×SD₁₆×C₆₁	—	2	2	no	—
46	C₂×D₈×C₆₁	—	2	1	no	—
47	C₂×D₈×C₆₁	C₂³	1	2	yes	—
48	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₁	—	2	1	no	—
50	#127×C₆₁	—	2	1	no	—
51	#126×C₆₁	—	4	2	no	—
52	#124×C₆₁	—	2	2	no	—
53	#124×C₆₁	—	2	2	no	—
54	#124×C₆₁	—	4	1	no	—
55	#123×C₆₁	—	4	2	no	—
56	C₂²⋊Q₈×C₆₁	—	4	1	no	—
57	C₄×Q₈×C₆₁	—	2	2	no	—
58	C₄×Q₈×C₆₁	—	2	1	no	—
59	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
60	C₄×D₄×C₆₁	—	2	1	no	—
61	C₄×D₄×C₆₁	—	2	2	no	—
62	C₄×D₄×C₆₁	—	4	1	no	—
63	#117×C₆₁	—	4	1	no	—
64	Q₃₂×C₆₁	—	4	2	no	—
65	SD₃₂×C₆₁	—	2	4	no	—
66	SD₃₂×C₆₁	—	2	2	no	—
67	SD₃₂×C₆₁	—	4	2	no	—
68	D₁₆×C₆₁	—	4	2	no	—
69	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
70	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
72	#107×C₆₁	C₂³	1	1	yes	characteristic
73	#107×C₆₁	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
76	#103×C₆₁	—	2	2	no	—
77	#103×C₆₁	—	2	1	no	—
78	#102×C₆₁	—	2	2	no	—
79	#102×C₆₁	—	2	1	no	—
80	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₆₁	C₂³	1	2	yes	—
82	C₂×Q₈×C₆₁	—	2	1	no	—
83	C₂×Q₈×C₆₁	—	2	2	no	—
84	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
85	C₂×D₄×C₆₁	—	2	1	no	—
86	C₂×D₄×C₆₁	—	2	2	no	—
87	C₂×D₄×C₆₁	—	4	1	no	—
88	C₂×D₄×C₆₁	—	4	1	no	—
89	C₂²×C₄×C₆₁	—	4	1	no	—
90	C₂²×C₄×C₆₁	—	4	2	no	—
91	Q₁₆×C₆₁	—	2	2	no	—
92	Q₁₆×C₆₁	—	2	1	no	—
93	Q₁₆×C₆₁	—	2	2	no	—
94	Q₁₆×C₆₁	—	2	1	no	—
95	Q₁₆×C₆₁	—	8	1	no	—
96	SD₁₆×C₆₁	—	8	2	no	—
97	SD₁₆×C₆₁	—	8	2	no	—
98	D₈×C₆₁	—	2	2	no	—
99	D₈×C₆₁	—	2	1	no	—
100	D₈×C₆₁	—	2	1	no	—
101	D₈×C₆₁	—	2	2	no	—
102	D₈×C₆₁	—	8	1	no	—
103	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₆₁	—	2	1	no	—
107	C₄⋊C₄×C₆₁	—	2	1	no	—
108	C₄⋊C₄×C₆₁	—	4	1	no	—
109	C₄⋊C₄×C₆₁	—	4	2	no	—
110	C₄⋊C₄×C₆₁	—	4	1	no	—
111	C₄⋊C₄×C₆₁	—	4	1	no	—
112	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
113	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
114	C₂²⋊C₄×C₆₁	—	4	2	no	—
115	C₂²⋊C₄×C₆₁	—	4	2	no	—
116	C₂²⋊C₄×C₆₁	—	4	1	no	—
117	C₂²⋊C₄×C₆₁	—	4	2	no	—
118	C₂²⋊C₄×C₆₁	—	4	2	no	—
119	C₂²⋊C₄×C₆₁	—	4	2	no	—
120	C₄²×C₆₁	—	4	2	no	—
121	C₄²×C₆₁	—	4	1	no	—
122	C₄²×C₆₁	C₂×D₄	1	2	yes	—
123	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₆₁	—	2	2	no	—
125	C₁₆×C₆₁	—	2	1	no	—
126	C₁₆×C₆₁	#41	1	2	yes	—
127	C₂³×C₆₁	—	4	1	no	—
128	C₂³×C₆₁	—	4	2	no	—
129	Q₈×C₆₁	—	2	2	no	—
130	Q₈×C₆₁	—	4	2	no	—
131	Q₈×C₆₁	—	4	1	no	—
132	Q₈×C₆₁	—	4	1	no	—
133	Q₈×C₆₁	—	4	2	no	—
134	D₄×C₆₁	—	4	1	no	—
135	D₄×C₆₁	—	4	2	no	—
136	D₄×C₆₁	—	4	2	no	—
137	D₄×C₆₁	—	4	1	no	—
138	D₄×C₆₁	—	8	2	no	—
139	D₄×C₆₁	—	16	1	no	—
140	C₂×C₄×C₆₁	—	4	1	no	—
141	C₂×C₄×C₆₁	—	4	1	no	—
142	C₂×C₄×C₆₁	—	4	1	no	—
143	C₂×C₄×C₆₁	—	4	2	no	—
144	C₂×C₄×C₆₁	—	8	2	no	—
145	C₂×C₄×C₆₁	—	16	2	no	—
146	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₆₁	—	2	1	no	—
152	C₈×C₆₁	—	2	2	no	—
153	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
154	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
155	#33961	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂²×C₆₁	—	8	2	no	—
157	C₂²×C₆₁	—	8	1	no	—
158	C₂²×C₆₁	—	8	1	no	—
159	C₂²×C₆₁	—	8	2	no	—
160	C₂²×C₆₁	—	8	2	no	—
161	C₂²×C₆₁	—	8	1	no	—
162	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₆₁	—	2	2	no	—
164	C₄×C₆₁	—	2	2	no	—
165	C₄×C₆₁	—	2	1	no	—
166	C₄×C₆₁	—	8	1	no	—
167	C₄×C₆₁	—	8	2	no	—
168	C₄×C₆₁	—	8	1	no	—
169	C₄×C₆₁	—	8	1	no	—
170	C₄×C₆₁	#508	1	2	yes	—
171	C₄×C₆₁	#550	1	1	yes	characteristic
172	C₄×C₆₁	#576	1	1	yes	characteristic
173	#3396	C₂×C₆₁	1	1	yes	characteristic, complemented
174	#3393	C₂×C₆₁	1	1	yes	characteristic, complemented
175	#2249	C₂×C₆₁	1	2	yes	complemented
176	#2248	C₂×C₆₁	1	1	yes	characteristic, complemented
177	#2211	C₂×C₆₁	1	2	yes	complemented
178	#2210	C₂×C₆₁	1	2	yes	complemented
179	#2209	C₂×C₆₁	1	1	yes	characteristic, complemented
180	#2206	C₂×C₆₁	1	1	yes	characteristic
181	Q₈×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
182	C₄×SD₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
183	C₄×SD₃₂	C₂×C₆₁	1	2	yes	complemented
184	C₂×C₆₁	—	16	1	no	—
185	C₂×C₆₁	—	16	2	no	—
186	C₂×C₆₁	#3391	1	1	yes	characteristic
187	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
188	C₂×C₆₁	#3416	1	1	yes	characteristic
189	#559	—	2	1	no	—
190	#556	—	2	1	no	—
191	C₂×Q₃₂	C₂²×C₆₁	1	2	yes	—
192	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
193	C₂×SD₃₂	C₂²×C₆₁	1	1	yes	characteristic
194	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
195	C₂×D₁₆	C₂²×C₆₁	1	2	yes	—
196	#495	C₂²×C₆₁	1	2	yes	—
197	#495	C₂²×C₆₁	1	2	yes	—
198	#494	C₂²×C₆₁	1	1	yes	characteristic
199	#493	C₂²×C₆₁	1	1	yes	characteristic
200	#464	—	2	2	no	—
201	#463	—	2	2	no	—
202	#461	—	2	1	no	—
203	#460	—	2	1	no	—
204	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
205	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
206	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
207	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
208	C₄×D₈	C₂²×C₆₁	1	2	yes	—
209	#367	C₂²×C₆₁	1	1	yes	characteristic
210	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
211	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
212	#358	C₂²×C₆₁	1	2	yes	—
213	#358	C₂²×C₆₁	1	1	yes	characteristic
214	#357	C₂²×C₆₁	1	2	yes	—
215	#357	C₂²×C₆₁	1	1	yes	characteristic
216	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
217	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
218	C₆₁	#33961	1	1	yes	characteristic, direct product factor, Sylow
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
221	C₂×Q₁₆	C₂³×C₆₁	1	1	yes	characteristic
222	C₂×SD₁₆	—	2	2	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	C₂³×C₆₁	1	2	yes	—
226	C₂×D₈	C₂³×C₆₁	1	1	yes	characteristic
227	C₄⋊Q₈	—	2	1	no	—
228	#127	—	2	1	no	—
229	#126	—	4	2	no	—
230	#124	—	2	2	no	—
231	#124	—	2	2	no	—
232	#124	—	4	1	no	—
233	#123	—	4	2	no	—
234	C₂²⋊Q₈	—	4	1	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	2	2	no	—
240	C₄×D₄	—	4	1	no	—
241	#117	—	4	1	no	—
242	Q₃₂	—	4	2	no	—
243	SD₃₂	—	2	4	no	—
244	SD₃₂	—	2	2	no	—
245	SD₃₂	—	4	2	no	—
246	D₁₆	—	4	2	no	—
247	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
248	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
249	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
250	#107	C₂³×C₆₁	1	1	yes	characteristic
251	#107	C₂³×C₆₁	1	1	yes	characteristic
252	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
253	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
254	#103	—	2	2	no	—
255	#103	—	2	1	no	—
256	#102	—	2	2	no	—
257	#102	—	2	1	no	—
258	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
259	C₄×C₈	C₂³×C₆₁	1	2	yes	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	2	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	2	2	no	—
272	Q₁₆	—	2	1	no	—
273	Q₁₆	—	8	1	no	—
274	SD₁₆	—	8	2	no	—
275	SD₁₆	—	8	2	no	—
276	D₈	—	2	2	no	—
277	D₈	—	2	1	no	—
278	D₈	—	2	1	no	—
279	D₈	—	2	2	no	—
280	D₈	—	8	1	no	—
281	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
283	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
291	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₄²	—	4	2	no	—
299	C₄²	—	4	1	no	—
300	C₄²	C₂×D₄×C₆₁	1	2	yes	—
301	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
302	C₁₆	—	2	2	no	—
303	C₁₆	—	2	1	no	—
304	C₁₆	#41×C₆₁	1	2	yes	—
305	C₂³	—	4	1	no	—
306	C₂³	—	4	2	no	—
307	Q₈	—	2	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	2	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	8	2	no	—
317	D₄	—	16	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	16	2	no	—
324	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
325	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
326	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
327	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	2	2	no	—
331	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
332	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
333	C₂²	—	8	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	C₆₁×#508	1	2	yes	—
348	C₄	C₆₁×#550	1	1	yes	characteristic
349	C₄	C₆₁×#576	1	1	yes	characteristic
350	C₂	—	16	1	no	—
351	C₂	—	16	2	no	—
352	C₂	C₆₁×#3391	1	1	yes	characteristic
353	C₂	C₂×C₆₁×#508	1	1	yes	characteristic
354	C₂	C₆₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₆₁×#2206	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×Q₈×C₆₁ ◁ C₈×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3396	C₂	semidirect
2. C₆₁×#3393	C₂	semidirect
3. C₆₁×#2249	C₂	semidirect
4. C₆₁×#2248	C₂	semidirect
5. C₆₁×#2211	C₂	semidirect
6. C₆₁×#2210	C₂	semidirect
7. C₆₁×#2209	C₂	semidirect
9. Q₈×C₁₆×C₆₁	C₂	semidirect
10. C₄×SD₃₂×C₆₁	C₂	semidirect
11. C₄×SD₃₂×C₆₁	C₂	semidirect
155. #33961	C₆₁	direct (N × H)
173. #3396	C₂×C₆₁	semidirect
174. #3393	C₂×C₆₁	semidirect
175. #2249	C₂×C₆₁	semidirect
176. #2248	C₂×C₆₁	semidirect
177. #2211	C₂×C₆₁	semidirect
178. #2210	C₂×C₆₁	semidirect
179. #2209	C₂×C₆₁	semidirect
181. Q₈×C₁₆	C₂×C₆₁	semidirect
182. C₄×SD₃₂	C₂×C₆₁	semidirect
183. C₄×SD₃₂	C₂×C₆₁	semidirect
218. C₆₁	#33961	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×#3396	2
C₆₁×#3393	2
C₆₁×#2249	2
C₆₁×#2248	2
C₆₁×#2211	2
C₆₁×#2210	2
C₆₁×#2209	2
C₆₁×#2206	2
Q₈×C₁₆×C₆₁	2
C₄×SD₃₂×C₆₁	2
#33961	61

Subgroups — C61×#33961 #29537394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33961 #29537394