Subgroups — C₆₁×#18714 #29522147

Order 15616.

1132 conjugacy classes of subgroups.

The subgroup lattice has 360 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×#3205	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1624	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1619	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1543	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1484	C₂	1	2	yes	maximal, complemented
8	C₂×AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
9	C₆₁×H(3,𝔽₄)	—	2	2	no	—
10	C₆₁×#555	—	2	2	no	—
11	C₆₁×#555	C₂²	1	2	yes	complemented
12	D₄²×C₆₁	—	2	2	no	—
13	D₄²×C₆₁	C₂²	1	2	yes	complemented
14	C₆₁×#535	—	2	2	no	—
15	C₆₁×#535	—	2	4	no	—
16	C₆₁×#532	C₂²	1	1	yes	characteristic
17	C₆₁×#532	C₂²	1	2	yes	—
18	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic, complemented
19	C₆₁×#496	—	2	2	no	—
20	C₆₁×#493	—	2	2	no	—
21	C₆₁×#493	C₂²	1	2	yes	—
22	C₆₁×#492	C₂²	1	2	yes	complemented
23	C₆₁×#486	—	2	2	no	—
24	C₆₁×#459	C₂²	1	8	yes	complemented
25	C₆₁×#447	—	2	4	no	—
26	C₆₁×#431	C₂²	1	2	yes	complemented
27	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
28	C₆₁×#333	C₂²	1	2	yes	complemented
29	C₆₁×#331	C₂²	1	2	yes	complemented
30	C₆₁×D₄⋊C₈	C₂²	1	8	yes	complemented
31	C₂×#41×C₆₁	—	2	2	no	—
32	C₂×#41×C₆₁	D₄	1	2	yes	—
33	C₂²×D₄×C₆₁	—	2	2	no	—
34	C₂²×D₄×C₆₁	—	2	4	no	—
35	C₂²×D₄×C₆₁	—	2	1	no	—
36	C₂²×D₄×C₆₁	—	2	2	no	—
37	AGL(1,ℤ/8)×C₆₁	—	4	8	no	—
38	C₂×SD₁₆×C₆₁	—	2	4	no	—
39	C₂×D₈×C₆₁	—	2	4	no	—
40	C₂×D₈×C₆₁	—	2	8	no	—
41	C₂×M₁₆×C₆₁	—	2	1	no	—
42	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
43	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
44	#127×C₆₁	—	2	4	no	—
45	#127×C₆₁	—	2	2	no	—
46	#127×C₆₁	—	2	1	no	—
47	#127×C₆₁	D₄	1	2	yes	—
48	#127×C₆₁	C₂³	1	2	yes	—
49	#127×C₆₁	C₂³	1	1	yes	characteristic
50	#124×C₆₁	—	2	8	no	—
51	#124×C₆₁	—	2	4	no	—
52	#124×C₆₁	—	2	2	no	—
53	#124×C₆₁	D₄	1	2	yes	—
54	#121×C₆₁	—	2	4	no	—
55	#121×C₆₁	—	2	4	no	—
56	#121×C₆₁	—	2	2	no	—
57	#121×C₆₁	—	2	2	no	—
58	#121×C₆₁	—	2	4	no	—
59	#121×C₆₁	D₄	1	4	yes	—
60	#120×C₆₁	—	2	4	no	—
61	#120×C₆₁	—	2	8	no	—
62	#120×C₆₁	—	2	4	no	—
63	C₄×D₄×C₆₁	—	2	2	no	—
64	C₄×D₄×C₆₁	—	2	2	no	—
65	C₄×D₄×C₆₁	C₂³	1	4	yes	—
66	#117×C₆₁	—	2	2	no	—
67	#117×C₆₁	—	2	4	no	—
68	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
70	#102×C₆₁	—	2	8	no	—
71	C₂²⋊C₈×C₆₁	—	2	2	no	—
72	#97×C₆₁	—	2	1	no	—
73	C₄×C₈×C₆₁	—	2	2	no	—
74	C₄×C₈×C₆₁	C₂³	1	2	yes	—
75	C₂⁴×C₆₁	—	2	4	no	—
76	C₂⁴×C₆₁	—	4	1	no	—
77	#41×C₆₁	—	4	4	no	—
78	#41×C₆₁	—	8	4	no	—
79	C₂×Q₈×C₆₁	—	2	2	no	—
80	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
81	C₂×D₄×C₆₁	—	2	4	no	—
82	C₂×D₄×C₆₁	—	2	8	no	—
83	C₂×D₄×C₆₁	—	2	2	no	—
84	C₂×D₄×C₆₁	—	2	4	no	—
85	C₂×D₄×C₆₁	—	2	2	no	—
86	C₂×D₄×C₆₁	—	2	2	no	—
87	C₂×D₄×C₆₁	—	2	2	no	—
88	C₂×D₄×C₆₁	—	2	4	no	—
89	C₂×D₄×C₆₁	—	2	2	no	—
90	C₂×D₄×C₆₁	—	4	4	no	—
91	C₂×D₄×C₆₁	—	4	2	no	—
92	C₂×D₄×C₆₁	—	4	1	no	—
93	C₂×D₄×C₆₁	—	8	4	no	—
94	C₂×D₄×C₆₁	—	8	8	no	—
95	C₂×D₄×C₆₁	—	8	4	no	—
96	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
97	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₆₁	—	2	2	no	—
99	C₂²×C₄×C₆₁	—	2	4	no	—
100	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
102	SD₁₆×C₆₁	—	4	8	no	—
103	D₈×C₆₁	—	4	8	no	—
104	D₈×C₆₁	—	8	8	no	—
105	M₁₆×C₆₁	—	2	4	no	—
106	C₂×C₈×C₆₁	—	2	2	no	—
107	C₂×C₈×C₆₁	—	2	4	no	—
108	C₄⋊C₄×C₆₁	—	2	2	no	—
109	C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
111	C₂²⋊C₄×C₆₁	—	2	8	no	—
112	C₂²⋊C₄×C₆₁	—	2	4	no	—
113	C₂²⋊C₄×C₆₁	—	2	4	no	—
114	C₂²⋊C₄×C₆₁	—	2	4	no	—
115	C₂²⋊C₄×C₆₁	—	2	4	no	—
116	C₄²×C₆₁	—	2	4	no	—
117	C₄²×C₆₁	C₂×D₄	1	2	yes	—
118	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
120	C₂³×C₆₁	—	2	4	no	—
121	C₂³×C₆₁	—	2	2	no	—
122	C₂³×C₆₁	—	4	2	no	—
123	C₂³×C₆₁	—	8	2	no	—
124	C₂³×C₆₁	—	8	4	no	—
125	C₂³×C₆₁	—	8	8	no	—
126	C₂³×C₆₁	—	8	4	no	—
127	C₂³×C₆₁	#120	1	1	yes	characteristic
128	Q₈×C₆₁	—	4	2	no	—
129	Q₈×C₆₁	—	8	2	no	—
130	D₄×C₆₁	—	2	8	no	—
131	D₄×C₆₁	—	4	2	no	—
132	D₄×C₆₁	—	4	4	no	—
133	D₄×C₆₁	—	4	4	no	—
134	D₄×C₆₁	—	4	8	no	—
135	D₄×C₆₁	—	4	4	no	—
136	D₄×C₆₁	—	8	2	no	—
137	D₄×C₆₁	—	8	4	no	—
138	D₄×C₆₁	—	8	2	no	—
139	D₄×C₆₁	—	8	4	no	—
140	D₄×C₆₁	—	8	8	no	—
141	D₄×C₆₁	—	8	4	no	—
142	D₄×C₆₁	—	8	2	no	—
143	C₂×C₄×C₆₁	—	2	4	no	—
144	C₂×C₄×C₆₁	—	2	2	no	—
145	C₂×C₄×C₆₁	—	2	2	no	—
146	C₂×C₄×C₆₁	—	2	4	no	—
147	C₂×C₄×C₆₁	—	4	8	no	—
148	C₂×C₄×C₆₁	—	4	4	no	—
149	C₂×C₄×C₆₁	#120	1	2	yes	—
150	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
151	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
153	C₈×C₆₁	—	2	4	no	—
154	C₈×C₆₁	—	4	4	no	—
155	#18714	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂²×C₆₁	—	2	2	no	—
157	C₂²×C₆₁	—	2	1	no	—
158	C₂²×C₆₁	—	4	4	no	—
159	C₂²×C₆₁	—	4	4	no	—
160	C₂²×C₆₁	—	4	4	no	—
161	C₂²×C₆₁	—	4	2	no	—
162	C₂²×C₆₁	—	4	4	no	—
163	C₂²×C₆₁	—	8	4	no	—
164	C₂²×C₆₁	—	8	2	no	—
165	C₂²×C₆₁	—	16	4	no	—
166	C₂²×C₆₁	—	16	8	no	—
167	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
168	C₄×C₆₁	—	2	2	no	—
169	C₄×C₆₁	—	2	1	no	—
170	C₄×C₆₁	—	4	4	no	—
171	C₄×C₆₁	—	4	2	no	—
172	C₄×C₆₁	#453	1	2	yes	—
173	C₄×C₆₁	#576	1	4	yes	—
174	#3549	C₂×C₆₁	1	1	yes	characteristic, complemented
175	#3205	C₂×C₆₁	1	2	yes	complemented
176	#3151	C₂×C₆₁	1	1	yes	characteristic, complemented
177	#1624	C₂×C₆₁	1	4	yes	complemented
178	#1619	C₂×C₆₁	1	4	yes	complemented
179	#1543	C₂×C₆₁	1	1	yes	characteristic, complemented
180	#1484	C₂×C₆₁	1	2	yes	complemented
181	C₂×C₆₁	—	4	1	no	—
182	C₂×C₆₁	—	8	4	no	—
183	C₂×C₆₁	—	8	2	no	—
184	C₂×C₆₁	—	16	2	no	—
185	C₂×C₆₁	#3008	1	2	yes	—
186	C₂×C₆₁	C₂×#453	1	1	yes	characteristic
187	C₂×AGL(1,ℤ/8)	—	2	2	no	—
188	H(3,𝔽₄)	—	2	2	no	—
189	#555	—	2	2	no	—
190	#555	C₂²×C₆₁	1	2	yes	complemented
191	D₄²	—	2	2	no	—
192	D₄²	C₂²×C₆₁	1	2	yes	complemented
193	#535	—	2	2	no	—
194	#535	—	2	4	no	—
195	#532	C₂²×C₆₁	1	1	yes	characteristic
196	#532	C₂²×C₆₁	1	2	yes	—
197	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic, complemented
198	#496	—	2	2	no	—
199	#493	—	2	2	no	—
200	#493	C₂²×C₆₁	1	2	yes	—
201	#492	C₂²×C₆₁	1	2	yes	complemented
202	#486	—	2	2	no	—
203	#459	C₂²×C₆₁	1	8	yes	complemented
204	#447	—	2	4	no	—
205	#431	C₂²×C₆₁	1	2	yes	complemented
206	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic, complemented
207	#333	C₂²×C₆₁	1	2	yes	complemented
208	#331	C₂²×C₆₁	1	2	yes	complemented
209	D₄⋊C₈	C₂²×C₆₁	1	8	yes	complemented
210	C₆₁	#18714	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×#41	—	2	2	no	—
212	C₂×#41	D₄×C₆₁	1	2	yes	—
213	C₂²×D₄	—	2	2	no	—
214	C₂²×D₄	—	2	4	no	—
215	C₂²×D₄	—	2	1	no	—
216	C₂²×D₄	—	2	2	no	—
217	AGL(1,ℤ/8)	—	4	8	no	—
218	C₂×SD₁₆	—	2	4	no	—
219	C₂×D₈	—	2	4	no	—
220	C₂×D₈	—	2	8	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
223	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
224	#127	—	2	4	no	—
225	#127	—	2	2	no	—
226	#127	—	2	1	no	—
227	#127	D₄×C₆₁	1	2	yes	—
228	#127	C₂³×C₆₁	1	2	yes	—
229	#127	C₂³×C₆₁	1	1	yes	characteristic
230	#124	—	2	8	no	—
231	#124	—	2	4	no	—
232	#124	—	2	2	no	—
233	#124	D₄×C₆₁	1	2	yes	—
234	#121	—	2	4	no	—
235	#121	—	2	4	no	—
236	#121	—	2	2	no	—
237	#121	—	2	2	no	—
238	#121	—	2	4	no	—
239	#121	D₄×C₆₁	1	4	yes	—
240	#120	—	2	4	no	—
241	#120	—	2	8	no	—
242	#120	—	2	4	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	—	2	2	no	—
245	C₄×D₄	C₂³×C₆₁	1	4	yes	—
246	#117	—	2	2	no	—
247	#117	—	2	4	no	—
248	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
249	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
250	#102	—	2	8	no	—
251	C₂²⋊C₈	—	2	2	no	—
252	#97	—	2	1	no	—
253	C₄×C₈	—	2	2	no	—
254	C₄×C₈	C₂³×C₆₁	1	2	yes	—
255	C₂⁴	—	2	4	no	—
256	C₂⁴	—	4	1	no	—
257	#41	—	4	4	no	—
258	#41	—	8	4	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	2	8	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	4	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	4	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	4	4	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	8	4	no	—
274	C₂×D₄	—	8	8	no	—
275	C₂×D₄	—	8	4	no	—
276	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
277	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
281	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
282	SD₁₆	—	4	8	no	—
283	D₈	—	4	8	no	—
284	D₈	—	8	8	no	—
285	M₁₆	—	2	4	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	4	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
291	C₂²⋊C₄	—	2	8	no	—
292	C₂²⋊C₄	—	2	4	no	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₂²⋊C₄	—	2	4	no	—
295	C₂²⋊C₄	—	2	4	no	—
296	C₄²	—	2	4	no	—
297	C₄²	C₂×D₄×C₆₁	1	2	yes	—
298	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
299	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
300	C₂³	—	2	4	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	8	2	no	—
304	C₂³	—	8	4	no	—
305	C₂³	—	8	8	no	—
306	C₂³	—	8	4	no	—
307	C₂³	#120×C₆₁	1	1	yes	characteristic
308	Q₈	—	4	2	no	—
309	Q₈	—	8	2	no	—
310	D₄	—	2	8	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	8	no	—
315	D₄	—	4	4	no	—
316	D₄	—	8	2	no	—
317	D₄	—	8	4	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	4	no	—
320	D₄	—	8	8	no	—
321	D₄	—	8	4	no	—
322	D₄	—	8	2	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	4	8	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	#120×C₆₁	1	2	yes	—
330	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
333	C₈	—	2	4	no	—
334	C₈	—	4	4	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	16	4	no	—
345	C₂²	—	16	8	no	—
346	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	2	1	no	—
349	C₄	—	4	4	no	—
350	C₄	—	4	2	no	—
351	C₄	C₆₁×#453	1	2	yes	—
352	C₄	C₆₁×#576	1	4	yes	—
353	C₂	—	4	1	no	—
354	C₂	—	8	4	no	—
355	C₂	—	8	2	no	—
356	C₂	—	16	2	no	—
357	C₂	C₆₁×#3008	1	2	yes	—
358	C₂	C₂×C₆₁×#453	1	1	yes	characteristic

	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×#3205	C₂	semidirect
3. C₆₁×#3151	C₂	semidirect
4. C₆₁×#1624	C₂	semidirect
5. C₆₁×#1619	C₂	semidirect
6. C₆₁×#1543	C₂	semidirect
7. C₆₁×#1484	C₂	semidirect
11. C₆₁×#555	C₂²	semidirect
13. D₄²×C₆₁	C₂²	semidirect
18. C₂×#127×C₆₁	C₂²	semidirect
22. C₆₁×#492	C₂²	semidirect
24. C₆₁×#459	C₂²	semidirect
26. C₆₁×#431	C₂²	semidirect
27. C₄×M₁₆×C₆₁	C₂²	semidirect
28. C₆₁×#333	C₂²	semidirect
29. C₆₁×#331	C₂²	semidirect
30. C₆₁×D₄⋊C₈	C₂²	semidirect
155. #18714	C₆₁	direct (N × H)
174. #3549	C₂×C₆₁	semidirect
175. #3205	C₂×C₆₁	semidirect
176. #3151	C₂×C₆₁	semidirect
177. #1624	C₂×C₆₁	semidirect
178. #1619	C₂×C₆₁	semidirect
179. #1543	C₂×C₆₁	semidirect
180. #1484	C₂×C₆₁	semidirect
190. #555	C₂²×C₆₁	semidirect
192. D₄²	C₂²×C₆₁	semidirect
197. C₂×#127	C₂²×C₆₁	semidirect
201. #492	C₂²×C₆₁	semidirect
203. #459	C₂²×C₆₁	semidirect
205. #431	C₂²×C₆₁	semidirect
206. C₄×M₁₆	C₂²×C₆₁	semidirect
207. #333	C₂²×C₆₁	semidirect
208. #331	C₂²×C₆₁	semidirect
209. D₄⋊C₈	C₂²×C₆₁	semidirect
210. C₆₁	#18714	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×#3205	2
C₆₁×#3151	2
C₆₁×#1624	2
C₆₁×#1619	2
C₆₁×#1543	2
C₆₁×#1484	2
#18714	61

Subgroups — C61×#18714 #29522147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18714 #29522147