Subgroups — C₆₁×#33757 #29537190

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3050	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2237	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2234	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2217	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2189	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2186	C₂	1	2	yes	maximal, complemented
9	C₂×C₆₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₆₁	C₂²	1	2	yes	—
11	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#525	—	2	2	no	—
13	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic, complemented
14	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	complemented
15	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#467	C₂²	1	2	yes	—
19	C₆₁×#467	C₂²	1	2	yes	—
20	C₆₁×#466	C₂²	1	2	yes	—
21	C₆₁×#466	C₂²	1	2	yes	—
22	C₆₁×#465	—	2	2	no	—
23	C₆₁×#465	—	2	2	no	—
24	C₆₁×#427	C₂²	1	2	yes	—
25	C₆₁×#416	—	2	2	no	—
26	C₆₁×#367	C₂²	1	2	yes	complemented
27	C₆₁×#366	C₂²	1	1	yes	characteristic, complemented
28	C₆₁×#366	C₂²	1	1	yes	characteristic, complemented
29	C₆₁×#358	C₂²	1	2	yes	complemented
30	C₆₁×#358	C₂²	1	2	yes	complemented
31	C₆₁×#357	C₂²	1	2	yes	complemented
32	C₆₁×#357	C₂²	1	2	yes	complemented
33	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₂×#41×C₆₁	—	2	2	no	—
36	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
37	#135×C₆₁	—	4	2	no	—
38	#135×C₆₁	—	4	2	no	—
39	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₆₁	—	2	2	no	—
41	C₂×D₈×C₆₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₁	—	4	1	no	—
43	C₂²×C₈×C₆₁	—	4	1	no	—
44	C₂²×C₈×C₆₁	D₄	1	2	yes	—
45	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	#123×C₆₁	—	4	2	no	—
49	C₂²⋊Q₈×C₆₁	—	2	2	no	—
50	C₂²⋊Q₈×C₆₁	—	2	2	no	—
51	#121×C₆₁	—	2	2	no	—
52	#121×C₆₁	—	2	2	no	—
53	#120×C₆₁	—	4	2	no	—
54	C₄×D₄×C₆₁	—	4	2	no	—
55	C₄×D₄×C₆₁	—	4	2	no	—
56	#117×C₆₁	—	2	2	no	—
57	Q₃₂×C₆₁	—	4	1	no	—
58	SD₃₂×C₆₁	—	4	2	no	—
59	D₁₆×C₆₁	—	4	1	no	—
60	C₂×C₁₆×C₆₁	—	4	1	no	—
61	C₂×C₁₆×C₆₁	—	4	1	no	—
62	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
64	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
65	#107×C₆₁	C₂³	1	2	yes	—
66	#107×C₆₁	C₂³	1	2	yes	—
67	#106×C₆₁	—	2	2	no	—
68	#103×C₆₁	—	2	2	no	—
69	#103×C₆₁	—	2	2	no	—
70	#102×C₆₁	—	2	2	no	—
71	#102×C₆₁	—	2	2	no	—
72	C₂⁴×C₆₁	D₈	1	1	yes	characteristic
73	#41×C₆₁	—	4	4	no	—
74	#41×C₆₁	—	4	4	no	—
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×D₄×C₆₁	—	2	2	no	—
77	C₂×D₄×C₆₁	—	4	2	no	—
78	C₂×D₄×C₆₁	—	4	2	no	—
79	C₂×D₄×C₆₁	—	4	2	no	—
80	C₂²×C₄×C₆₁	—	4	1	no	—
81	C₂²×C₄×C₆₁	—	4	1	no	—
82	C₂²×C₄×C₆₁	—	4	2	no	—
83	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₁	D₈	1	2	yes	—
85	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
87	Q₁₆×C₆₁	—	2	2	no	—
88	Q₁₆×C₆₁	—	2	2	no	—
89	SD₁₆×C₆₁	—	4	4	no	—
90	D₈×C₆₁	—	2	2	no	—
91	D₈×C₆₁	—	2	2	no	—
92	C₂×C₈×C₆₁	—	2	1	no	—
93	C₂×C₈×C₆₁	—	2	1	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	2	2	no	—
96	C₂×C₈×C₆₁	—	4	2	no	—
97	C₂×C₈×C₆₁	—	4	1	no	—
98	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₆₁	—	2	2	no	—
102	C₄⋊C₄×C₆₁	—	2	2	no	—
103	C₄⋊C₄×C₆₁	—	4	2	no	—
104	C₄⋊C₄×C₆₁	—	4	2	no	—
105	C₂²⋊C₄×C₆₁	—	4	2	no	—
106	C₂²⋊C₄×C₆₁	—	4	2	no	—
107	C₂²⋊C₄×C₆₁	—	4	2	no	—
108	C₂²⋊C₄×C₆₁	—	4	2	no	—
109	C₂²⋊C₄×C₆₁	—	4	2	no	—
110	C₄²×C₆₁	—	4	2	no	—
111	C₁₆×C₆₁	—	2	1	no	—
112	C₁₆×C₆₁	—	2	2	no	—
113	C₁₆×C₆₁	—	2	1	no	—
114	C₂³×C₆₁	—	2	2	no	—
115	C₂³×C₆₁	—	4	1	no	—
116	C₂³×C₆₁	—	4	1	no	—
117	C₂³×C₆₁	—	4	2	no	—
118	C₂³×C₆₁	C₂×D₈	1	2	yes	—
119	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₆₁	—	4	2	no	—
121	Q₈×C₆₁	—	4	2	no	—
122	D₄×C₆₁	—	4	2	no	—
123	D₄×C₆₁	—	4	2	no	—
124	D₄×C₆₁	—	8	4	no	—
125	D₄×C₆₁	—	8	4	no	—
126	D₄×C₆₁	—	8	4	no	—
127	C₂×C₄×C₆₁	—	2	1	no	—
128	C₂×C₄×C₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	2	1	no	—
130	C₂×C₄×C₆₁	—	2	2	no	—
131	C₂×C₄×C₆₁	—	4	1	no	—
132	C₂×C₄×C₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	—	4	2	no	—
134	C₂×C₄×C₆₁	—	4	2	no	—
135	C₂×C₄×C₆₁	—	4	2	no	—
136	C₂×C₄×C₆₁	—	8	4	no	—
137	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₆₁	—	2	1	no	—
141	C₈×C₆₁	—	2	2	no	—
142	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
143	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
144	#33757	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₆₁	—	2	1	no	—
146	C₂²×C₆₁	—	2	1	no	—
147	C₂²×C₆₁	—	2	2	no	—
148	C₂²×C₆₁	—	2	2	no	—
149	C₂²×C₆₁	—	2	2	no	—
150	C₂²×C₆₁	—	4	4	no	—
151	C₂²×C₆₁	—	8	2	no	—
152	C₂²×C₆₁	—	8	2	no	—
153	C₂²×C₆₁	—	8	2	no	—
154	C₂²×C₆₁	#508	1	2	yes	—
155	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
156	C₄×C₆₁	—	2	2	no	—
157	C₄×C₆₁	—	2	1	no	—
158	C₄×C₆₁	—	8	2	no	—
159	C₄×C₆₁	—	8	2	no	—
160	C₄×C₆₁	—	8	2	no	—
161	C₄×C₆₁	#534	1	1	yes	characteristic
162	C₄×C₆₁	#575	1	1	yes	characteristic
163	#3050	C₂×C₆₁	1	2	yes	complemented
164	#2237	C₂×C₆₁	1	2	yes	complemented
165	#2234	C₂×C₆₁	1	2	yes	complemented
166	#2217	C₂×C₆₁	1	2	yes	complemented
167	#2216	C₂×C₆₁	1	1	yes	characteristic, complemented
168	#2215	C₂×C₆₁	1	1	yes	characteristic, complemented
169	#2189	C₂×C₆₁	1	2	yes	complemented
170	#2186	C₂×C₆₁	1	2	yes	complemented
171	C₂×C₂²⋊C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
172	C₂×C₆₁	—	2	2	no	—
173	C₂×C₆₁	—	4	2	no	—
174	C₂×C₆₁	—	16	2	no	—
175	C₂×C₆₁	#3186	1	1	yes	characteristic
176	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
177	C₂×C₆₁	#3414	1	1	yes	characteristic
178	C₂×#135	C₂²×C₆₁	1	2	yes	—
179	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
180	#525	—	2	2	no	—
181	C₂×Q₃₂	C₂²×C₆₁	1	1	yes	characteristic, complemented
182	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	complemented
183	C₂×D₁₆	C₂²×C₆₁	1	1	yes	characteristic, complemented
184	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
185	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
186	#467	C₂²×C₆₁	1	2	yes	—
187	#467	C₂²×C₆₁	1	2	yes	—
188	#466	C₂²×C₆₁	1	2	yes	—
189	#466	C₂²×C₆₁	1	2	yes	—
190	#465	—	2	2	no	—
191	#465	—	2	2	no	—
192	#427	C₂²×C₆₁	1	2	yes	—
193	#416	—	2	2	no	—
194	#367	C₂²×C₆₁	1	2	yes	complemented
195	#366	C₂²×C₆₁	1	1	yes	characteristic, complemented
196	#366	C₂²×C₆₁	1	1	yes	characteristic, complemented
197	#358	C₂²×C₆₁	1	2	yes	complemented
198	#358	C₂²×C₆₁	1	2	yes	complemented
199	#357	C₂²×C₆₁	1	2	yes	complemented
200	#357	C₂²×C₆₁	1	2	yes	complemented
201	C₂²⋊C₁₆	C₂²×C₆₁	1	2	yes	—
202	C₂²⋊C₁₆	C₂²×C₆₁	1	2	yes	—
203	C₆₁	#33757	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×#41	—	2	2	no	—
205	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
206	#135	—	4	2	no	—
207	#135	—	4	2	no	—
208	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×D₈	C₂³×C₆₁	1	2	yes	—
211	C₂²×C₈	—	4	1	no	—
212	C₂²×C₈	—	4	1	no	—
213	C₂²×C₈	D₄×C₆₁	1	2	yes	—
214	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
216	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
217	#123	—	4	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	#121	—	2	2	no	—
221	#121	—	2	2	no	—
222	#120	—	4	2	no	—
223	C₄×D₄	—	4	2	no	—
224	C₄×D₄	—	4	2	no	—
225	#117	—	2	2	no	—
226	Q₃₂	—	4	1	no	—
227	SD₃₂	—	4	2	no	—
228	D₁₆	—	4	1	no	—
229	C₂×C₁₆	—	4	1	no	—
230	C₂×C₁₆	—	4	1	no	—
231	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
232	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
233	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
234	#107	C₂³×C₆₁	1	2	yes	—
235	#107	C₂³×C₆₁	1	2	yes	—
236	#106	—	2	2	no	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	C₂⁴	D₈×C₆₁	1	1	yes	characteristic
242	#41	—	4	4	no	—
243	#41	—	4	4	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
253	C₂²×C₄	D₈×C₆₁	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	SD₁₆	—	4	4	no	—
259	D₈	—	2	2	no	—
260	D₈	—	2	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
269	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₁₆	—	2	1	no	—
281	C₁₆	—	2	2	no	—
282	C₁₆	—	2	1	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	2	no	—
287	C₂³	C₂×D₈×C₆₁	1	2	yes	—
288	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
307	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	2	no	—
311	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
312	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	C₆₁×#508	1	2	yes	—
323	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	C₆₁×#534	1	1	yes	characteristic
330	C₄	C₆₁×#575	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	4	2	no	—
333	C₂	—	16	2	no	—
334	C₂	C₆₁×#3186	1	1	yes	characteristic
335	C₂	C₂×C₆₁×#508	1	1	yes	characteristic
336	C₂	C₆₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3050	C₂	semidirect
2. C₆₁×#2237	C₂	semidirect
3. C₆₁×#2234	C₂	semidirect
4. C₆₁×#2217	C₂	semidirect
5. C₆₁×#2216	C₂	semidirect
6. C₆₁×#2215	C₂	semidirect
7. C₆₁×#2189	C₂	semidirect
8. C₆₁×#2186	C₂	semidirect
9. C₂×C₆₁×C₂²⋊C₁₆	C₂	semidirect
13. C₂×Q₃₂×C₆₁	C₂²	semidirect
14. C₂×SD₃₂×C₆₁	C₂²	semidirect
15. C₂×D₁₆×C₆₁	C₂²	semidirect
26. C₆₁×#367	C₂²	semidirect
27. C₆₁×#366	C₂²	semidirect
28. C₆₁×#366	C₂²	semidirect
29. C₆₁×#358	C₂²	semidirect
30. C₆₁×#358	C₂²	semidirect
31. C₆₁×#357	C₂²	semidirect
32. C₆₁×#357	C₂²	semidirect
144. #33757	C₆₁	direct (N × H)
163. #3050	C₂×C₆₁	semidirect
164. #2237	C₂×C₆₁	semidirect
165. #2234	C₂×C₆₁	semidirect
166. #2217	C₂×C₆₁	semidirect
167. #2216	C₂×C₆₁	semidirect
168. #2215	C₂×C₆₁	semidirect
169. #2189	C₂×C₆₁	semidirect
170. #2186	C₂×C₆₁	semidirect
171. C₂×C₂²⋊C₁₆	C₂×C₆₁	semidirect
181. C₂×Q₃₂	C₂²×C₆₁	semidirect
182. C₂×SD₃₂	C₂²×C₆₁	semidirect
183. C₂×D₁₆	C₂²×C₆₁	semidirect
194. #367	C₂²×C₆₁	semidirect
195. #366	C₂²×C₆₁	semidirect
196. #366	C₂²×C₆₁	semidirect
197. #358	C₂²×C₆₁	semidirect
198. #358	C₂²×C₆₁	semidirect
199. #357	C₂²×C₆₁	semidirect
200. #357	C₂²×C₆₁	semidirect
203. C₆₁	#33757	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×#3050	2
C₆₁×#2237	2
C₆₁×#2234	2
C₆₁×#2217	2
C₆₁×#2216	2
C₆₁×#2215	2
C₆₁×#2189	2
C₆₁×#2186	2
C₂×C₆₁×C₂²⋊C₁₆	2
#33757	61

Subgroups — C61×#33757 #29537190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33757 #29537190