Subgroups — C₆₁×#19052 #29522485

Order 15616.

1078 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3549	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3214	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3150	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1659	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1655	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1536	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1530	C₂	1	2	yes	maximal, complemented
8	C₂×#135×C₆₁	—	2	2	no	—
9	C₆₁×H(3,𝔽₄)	C₂²	1	4	yes	—
10	C₆₁×#555	—	2	2	no	—
11	C₆₁×#555	—	2	1	no	—
12	D₄²×C₆₁	—	2	2	no	—
13	D₄²×C₆₁	—	2	1	no	—
14	C₆₁×#535	—	2	2	no	—
15	C₆₁×#535	—	2	4	no	—
16	C₆₁×#532	C₂²	1	2	yes	—
17	C₆₁×#532	C₂²	1	1	yes	characteristic
18	C₆₁×#532	C₂²	1	1	yes	characteristic, complemented
19	C₆₁×#497	—	2	1	no	—
20	C₆₁×#497	C₂²	1	2	yes	complemented
21	C₆₁×#496	—	2	1	no	—
22	C₆₁×#496	C₂²	1	2	yes	complemented
23	C₆₁×#496	C₂²	1	4	yes	complemented
24	C₆₁×#489	C₂²	1	4	yes	complemented
25	C₆₁×#461	—	2	2	no	—
26	C₆₁×#459	—	2	2	no	—
27	C₆₁×#449	—	2	4	no	—
28	C₆₁×#432	C₂²	1	2	yes	complemented
29	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic, complemented
30	C₆₁×#329	C₂²	1	8	yes	complemented
31	C₆₁×#329	C₂²	1	4	yes	complemented
32	C₂×#41×C₆₁	—	2	1	no	—
33	C₂×#41×C₆₁	D₄	1	2	yes	—
34	C₂×#41×C₆₁	D₄	1	4	yes	—
35	C₂²×D₄×C₆₁	—	2	4	no	—
36	C₂²×D₄×C₆₁	—	2	2	no	—
37	#135×C₆₁	—	4	8	no	—
38	C₂×Q₁₆×C₆₁	—	2	2	no	—
39	C₂×SD₁₆×C₆₁	—	2	2	no	—
40	C₂×SD₁₆×C₆₁	—	2	2	no	—
41	C₂×SD₁₆×C₆₁	—	2	4	no	—
42	C₂×D₈×C₆₁	—	2	2	no	—
43	C₂×D₈×C₆₁	—	2	4	no	—
44	C₂²×C₈×C₆₁	—	2	1	no	—
45	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
46	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
48	#127×C₆₁	—	2	4	no	—
49	#127×C₆₁	D₄	1	2	yes	—
50	#127×C₆₁	D₄	1	1	yes	characteristic
51	#127×C₆₁	C₂³	1	1	yes	characteristic
52	#124×C₆₁	—	2	8	no	—
53	#124×C₆₁	—	2	4	no	—
54	#124×C₆₁	—	2	1	no	—
55	#124×C₆₁	C₂³	1	4	yes	—
56	#124×C₆₁	C₂³	1	2	yes	—
57	#121×C₆₁	—	2	4	no	—
58	#121×C₆₁	—	2	4	no	—
59	#121×C₆₁	—	2	4	no	—
60	#121×C₆₁	—	2	2	no	—
61	#121×C₆₁	—	2	2	no	—
62	#121×C₆₁	—	2	2	no	—
63	#121×C₆₁	—	2	2	no	—
64	#120×C₆₁	—	2	4	no	—
65	#120×C₆₁	—	2	8	no	—
66	#120×C₆₁	—	2	4	no	—
67	C₄×D₄×C₆₁	—	2	2	no	—
68	C₄×D₄×C₆₁	—	2	2	no	—
69	C₄×D₄×C₆₁	—	2	2	no	—
70	C₄×D₄×C₆₁	—	2	2	no	—
71	#117×C₆₁	—	2	4	no	—
72	#117×C₆₁	—	2	2	no	—
73	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₆₁	—	2	2	no	—
75	#103×C₆₁	—	2	4	no	—
76	#102×C₆₁	—	2	4	no	—
77	C₂²⋊C₈×C₆₁	—	2	2	no	—
78	#97×C₆₁	—	2	1	no	—
79	#97×C₆₁	C₂³	1	2	yes	—
80	#97×C₆₁	C₂³	1	4	yes	—
81	C₂⁴×C₆₁	—	2	4	no	—
82	#41×C₆₁	—	4	4	no	—
83	#41×C₆₁	—	4	4	no	—
84	#41×C₆₁	—	8	4	no	—
85	C₂×Q₈×C₆₁	—	2	1	no	—
86	C₂×Q₈×C₆₁	C₂×D₄	1	4	yes	—
87	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
88	C₂×D₄×C₆₁	—	2	4	no	—
89	C₂×D₄×C₆₁	—	2	8	no	—
90	C₂×D₄×C₆₁	—	2	2	no	—
91	C₂×D₄×C₆₁	—	2	4	no	—
92	C₂×D₄×C₆₁	—	2	2	no	—
93	C₂×D₄×C₆₁	—	2	2	no	—
94	C₂×D₄×C₆₁	—	2	2	no	—
95	C₂×D₄×C₆₁	—	2	2	no	—
96	C₂×D₄×C₆₁	—	2	1	no	—
97	C₂×D₄×C₆₁	—	4	1	no	—
98	C₂×D₄×C₆₁	—	8	8	no	—
99	C₂×D₄×C₆₁	—	8	4	no	—
100	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
101	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₁	—	2	4	no	—
103	C₂²×C₄×C₆₁	—	2	2	no	—
104	C₂²×C₄×C₆₁	—	4	1	no	—
105	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
107	Q₁₆×C₆₁	—	4	4	no	—
108	SD₁₆×C₆₁	—	4	4	no	—
109	SD₁₆×C₆₁	—	4	4	no	—
110	SD₁₆×C₆₁	—	8	4	no	—
111	D₈×C₆₁	—	4	4	no	—
112	D₈×C₆₁	—	8	4	no	—
113	C₂×C₈×C₆₁	—	2	4	no	—
114	C₂×C₈×C₆₁	—	2	2	no	—
115	C₂×C₈×C₆₁	—	2	4	no	—
116	C₄⋊C₄×C₆₁	—	2	2	no	—
117	C₄⋊C₄×C₆₁	—	2	2	no	—
118	C₄⋊C₄×C₆₁	—	2	2	no	—
119	C₄⋊C₄×C₆₁	—	2	2	no	—
120	C₂²⋊C₄×C₆₁	—	2	8	no	—
121	C₂²⋊C₄×C₆₁	—	2	4	no	—
122	C₂²⋊C₄×C₆₁	—	2	4	no	—
123	C₂²⋊C₄×C₆₁	—	2	4	no	—
124	C₂²⋊C₄×C₆₁	—	2	4	no	—
125	C₂²⋊C₄×C₆₁	—	4	2	no	—
126	C₄²×C₆₁	—	2	4	no	—
127	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₁	C₂×D₄	1	2	yes	—
129	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
130	C₂³×C₆₁	—	2	2	no	—
131	C₂³×C₆₁	—	2	4	no	—
132	C₂³×C₆₁	—	4	1	no	—
133	C₂³×C₆₁	—	8	4	no	—
134	C₂³×C₆₁	—	8	8	no	—
135	C₂³×C₆₁	#120	1	1	yes	characteristic
136	Q₈×C₆₁	—	4	2	no	—
137	Q₈×C₆₁	—	4	4	no	—
138	Q₈×C₆₁	—	4	2	no	—
139	D₄×C₆₁	—	4	2	no	—
140	D₄×C₆₁	—	4	4	no	—
141	D₄×C₆₁	—	4	4	no	—
142	D₄×C₆₁	—	4	4	no	—
143	D₄×C₆₁	—	4	2	no	—
144	D₄×C₆₁	—	4	2	no	—
145	D₄×C₆₁	—	4	2	no	—
146	D₄×C₆₁	—	4	4	no	—
147	D₄×C₆₁	—	8	2	no	—
148	D₄×C₆₁	—	8	4	no	—
149	D₄×C₆₁	—	8	4	no	—
150	D₄×C₆₁	—	8	8	no	—
151	D₄×C₆₁	—	8	2	no	—
152	C₂×C₄×C₆₁	—	2	2	no	—
153	C₂×C₄×C₆₁	—	2	2	no	—
154	C₂×C₄×C₆₁	—	2	4	no	—
155	C₂×C₄×C₆₁	—	4	2	no	—
156	C₂×C₄×C₆₁	—	4	8	no	—
157	C₂×C₄×C₆₁	—	4	4	no	—
158	C₂×C₄×C₆₁	—	4	1	no	—
159	C₂×C₄×C₆₁	—	8	2	no	—
160	C₂×C₄×C₆₁	#120	1	2	yes	—
161	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
162	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
163	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
164	C₈×C₆₁	—	2	4	no	—
165	C₈×C₆₁	—	4	4	no	—
166	#19052	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂²×C₆₁	—	2	1	no	—
168	C₂²×C₆₁	—	2	2	no	—
169	C₂²×C₆₁	—	4	2	no	—
170	C₂²×C₆₁	—	4	4	no	—
171	C₂²×C₆₁	—	4	4	no	—
172	C₂²×C₆₁	—	4	4	no	—
173	C₂²×C₆₁	—	4	4	no	—
174	C₂²×C₆₁	—	8	1	no	—
175	C₂²×C₆₁	—	8	2	no	—
176	C₂²×C₆₁	—	16	8	no	—
177	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
178	C₄×C₆₁	—	2	1	no	—
179	C₄×C₆₁	—	2	2	no	—
180	C₄×C₆₁	—	2	2	no	—
181	C₄×C₆₁	—	4	2	no	—
182	C₄×C₆₁	—	4	4	no	—
183	C₄×C₆₁	—	8	1	no	—
184	C₄×C₆₁	#453	1	2	yes	—
185	#3549	C₂×C₆₁	1	1	yes	characteristic, complemented
186	#3214	C₂×C₆₁	1	2	yes	complemented
187	#3150	C₂×C₆₁	1	1	yes	characteristic, complemented
188	#1659	C₂×C₆₁	1	4	yes	complemented
189	#1655	C₂×C₆₁	1	4	yes	complemented
190	#1536	C₂×C₆₁	1	1	yes	characteristic, complemented
191	#1530	C₂×C₆₁	1	2	yes	complemented
192	C₂×C₆₁	—	4	1	no	—
193	C₂×C₆₁	—	8	2	no	—
194	C₂×C₆₁	—	8	4	no	—
195	C₂×C₆₁	—	16	1	no	—
196	C₂×C₆₁	C₂×#453	1	1	yes	characteristic
197	C₂×C₆₁	#3019	1	2	yes	—
198	C₂×#135	—	2	2	no	—
199	H(3,𝔽₄)	C₂²×C₆₁	1	4	yes	—
200	#555	—	2	2	no	—
201	#555	—	2	1	no	—
202	D₄²	—	2	2	no	—
203	D₄²	—	2	1	no	—
204	#535	—	2	2	no	—
205	#535	—	2	4	no	—
206	#532	C₂²×C₆₁	1	1	yes	characteristic
207	#532	C₂²×C₆₁	1	2	yes	—
208	#532	C₂²×C₆₁	1	1	yes	characteristic, complemented
209	#497	—	2	1	no	—
210	#497	C₂²×C₆₁	1	2	yes	complemented
211	#496	—	2	1	no	—
212	#496	C₂²×C₆₁	1	2	yes	complemented
213	#496	C₂²×C₆₁	1	4	yes	complemented
214	#489	C₂²×C₆₁	1	4	yes	complemented
215	#461	—	2	2	no	—
216	#459	—	2	2	no	—
217	#449	—	2	4	no	—
218	#432	C₂²×C₆₁	1	2	yes	complemented
219	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic, complemented
220	#329	C₂²×C₆₁	1	4	yes	complemented
221	#329	C₂²×C₆₁	1	8	yes	complemented
222	C₆₁	#19052	1	1	yes	characteristic, direct product factor, Sylow
223	C₂×#41	—	2	1	no	—
224	C₂×#41	D₄×C₆₁	1	2	yes	—
225	C₂×#41	D₄×C₆₁	1	4	yes	—
226	C₂²×D₄	—	2	4	no	—
227	C₂²×D₄	—	2	2	no	—
228	#135	—	4	8	no	—
229	C₂×Q₁₆	—	2	2	no	—
230	C₂×SD₁₆	—	2	2	no	—
231	C₂×SD₁₆	—	2	2	no	—
232	C₂×SD₁₆	—	2	4	no	—
233	C₂×D₈	—	2	2	no	—
234	C₂×D₈	—	2	4	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
237	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
238	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
239	#127	—	2	4	no	—
240	#127	D₄×C₆₁	1	2	yes	—
241	#127	D₄×C₆₁	1	1	yes	characteristic
242	#127	C₂³×C₆₁	1	1	yes	characteristic
243	#124	—	2	8	no	—
244	#124	—	2	4	no	—
245	#124	—	2	1	no	—
246	#124	C₂³×C₆₁	1	4	yes	—
247	#124	C₂³×C₆₁	1	2	yes	—
248	#121	—	2	4	no	—
249	#121	—	2	4	no	—
250	#121	—	2	4	no	—
251	#121	—	2	2	no	—
252	#121	—	2	2	no	—
253	#121	—	2	2	no	—
254	#121	—	2	2	no	—
255	#120	—	2	4	no	—
256	#120	—	2	8	no	—
257	#120	—	2	4	no	—
258	C₄×D₄	—	2	2	no	—
259	C₄×D₄	—	2	2	no	—
260	C₄×D₄	—	2	2	no	—
261	C₄×D₄	—	2	2	no	—
262	#117	—	2	4	no	—
263	#117	—	2	2	no	—
264	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
265	C₄⋊C₈	—	2	2	no	—
266	#103	—	2	4	no	—
267	#102	—	2	4	no	—
268	C₂²⋊C₈	—	2	2	no	—
269	#97	—	2	1	no	—
270	#97	C₂³×C₆₁	1	2	yes	—
271	#97	C₂³×C₆₁	1	4	yes	—
272	C₂⁴	—	2	4	no	—
273	#41	—	4	4	no	—
274	#41	—	4	4	no	—
275	#41	—	8	4	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	C₂×D₄×C₆₁	1	4	yes	—
278	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
279	C₂×D₄	—	2	4	no	—
280	C₂×D₄	—	2	8	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	4	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	8	8	no	—
290	C₂×D₄	—	8	4	no	—
291	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
292	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
297	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
298	Q₁₆	—	4	4	no	—
299	SD₁₆	—	4	4	no	—
300	SD₁₆	—	4	4	no	—
301	SD₁₆	—	8	4	no	—
302	D₈	—	4	4	no	—
303	D₈	—	8	4	no	—
304	C₂×C₈	—	2	4	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	4	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₂²⋊C₄	—	2	8	no	—
312	C₂²⋊C₄	—	2	4	no	—
313	C₂²⋊C₄	—	2	4	no	—
314	C₂²⋊C₄	—	2	4	no	—
315	C₂²⋊C₄	—	2	4	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₄²	—	2	4	no	—
318	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
319	C₄²	C₂×D₄×C₆₁	1	2	yes	—
320	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	—	2	4	no	—
323	C₂³	—	4	1	no	—
324	C₂³	—	8	4	no	—
325	C₂³	—	8	8	no	—
326	C₂³	#120×C₆₁	1	1	yes	characteristic
327	Q₈	—	4	2	no	—
328	Q₈	—	4	4	no	—
329	Q₈	—	4	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	4	no	—
332	D₄	—	4	4	no	—
333	D₄	—	4	4	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	4	no	—
338	D₄	—	8	2	no	—
339	D₄	—	8	4	no	—
340	D₄	—	8	4	no	—
341	D₄	—	8	8	no	—
342	D₄	—	8	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	8	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	#120×C₆₁	1	2	yes	—
352	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
353	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
354	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
355	C₈	—	2	4	no	—
356	C₈	—	4	4	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	4	no	—
361	C₂²	—	4	4	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	4	4	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	16	8	no	—
367	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	2	no	—
370	C₄	—	2	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	4	4	no	—
373	C₄	—	8	1	no	—
374	C₄	C₆₁×#453	1	2	yes	—
375	C₂	—	4	1	no	—
376	C₂	—	8	2	no	—
377	C₂	—	8	4	no	—
378	C₂	—	16	1	no	—
379	C₂	C₂×C₆₁×#453	1	1	yes	characteristic
380	C₂	C₆₁×#3019	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
LCS	G ▷ C₄² ▷ C₂² ▷ 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. C₆₁×#3549	C₂	semidirect
2. C₆₁×#3214	C₂	semidirect
3. C₆₁×#3150	C₂	semidirect
4. C₆₁×#1659	C₂	semidirect
5. C₆₁×#1655	C₂	semidirect
6. C₆₁×#1536	C₂	semidirect
7. C₆₁×#1530	C₂	semidirect
18. C₆₁×#532	C₂²	semidirect
20. C₆₁×#497	C₂²	semidirect
22. C₆₁×#496	C₂²	semidirect
23. C₆₁×#496	C₂²	semidirect
24. C₆₁×#489	C₂²	semidirect
28. C₆₁×#432	C₂²	semidirect
29. C₂×#97×C₆₁	C₂²	semidirect
30. C₆₁×#329	C₂²	semidirect
31. C₆₁×#329	C₂²	semidirect
166. #19052	C₆₁	direct (N × H)
185. #3549	C₂×C₆₁	semidirect
186. #3214	C₂×C₆₁	semidirect
187. #3150	C₂×C₆₁	semidirect
188. #1659	C₂×C₆₁	semidirect
189. #1655	C₂×C₆₁	semidirect
190. #1536	C₂×C₆₁	semidirect
191. #1530	C₂×C₆₁	semidirect
208. #532	C₂²×C₆₁	semidirect
210. #497	C₂²×C₆₁	semidirect
212. #496	C₂²×C₆₁	semidirect
213. #496	C₂²×C₆₁	semidirect
214. #489	C₂²×C₆₁	semidirect
218. #432	C₂²×C₆₁	semidirect
219. C₂×#97	C₂²×C₆₁	semidirect
220. #329	C₂²×C₆₁	semidirect
221. #329	C₂²×C₆₁	semidirect
222. C₆₁	#19052	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×#3549	2
C₆₁×#3214	2
C₆₁×#3150	2
C₆₁×#1659	2
C₆₁×#1655	2
C₆₁×#1536	2
C₆₁×#1530	2
#19052	61

Subgroups — C61×#19052 #29522485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19052 #29522485