Subgroups — C₆₁×#19272 #29522705

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3079	C₂	1	2	yes	maximal
3	C₈×#41×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1676	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1675	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1548	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
9	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#500	—	2	1	no	—
11	C₆₁×#500	—	2	1	no	—
12	C₆₁×#500	C₂²	1	4	yes	—
13	C₆₁×#477	C₂²	1	4	yes	—
14	C₆₁×#476	C₂²	1	4	yes	—
15	C₆₁×#475	—	2	2	no	—
16	C₆₁×#474	—	2	2	no	—
17	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
18	C₈×D₄×C₆₁	—	2	2	no	—
19	C₈×D₄×C₆₁	C₂²	1	2	yes	—
20	C₆₁×#431	—	2	1	no	—
21	C₆₁×#431	C₂²	1	1	yes	characteristic
22	C₆₁×#428	—	2	2	no	—
23	C₂×#107×C₆₁	—	2	2	no	—
24	C₆₁×#423	C₂²	1	2	yes	—
25	C₆₁×#417	—	2	2	no	—
26	C₂×C₄×C₈×C₆₁	—	2	1	no	—
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#335	C₂²	1	4	yes	—
29	C₆₁×#333	C₂²	1	8	yes	—
30	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₆₁	—	2	2	no	—
32	C₂×M₁₆×C₆₁	—	2	2	no	—
33	C₂²×C₈×C₆₁	—	2	2	no	—
34	C₂²×C₈×C₆₁	D₄	1	2	yes	—
35	C₄⋊Q₈×C₆₁	—	2	2	no	—
36	C₄⋊Q₈×C₆₁	—	2	2	no	—
37	C₄⋊Q₈×C₆₁	—	2	2	no	—
38	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
39	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
41	C₄×D₄×C₆₁	—	2	1	no	—
42	C₄×D₄×C₆₁	—	2	1	no	—
43	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
44	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
45	#117×C₆₁	—	2	1	no	—
46	#117×C₆₁	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
49	C₂×C₄²×C₆₁	—	2	1	no	—
50	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
51	#107×C₆₁	—	2	4	no	—
52	#107×C₆₁	—	2	4	no	—
53	#107×C₆₁	—	2	4	no	—
54	#106×C₆₁	—	2	4	no	—
55	C₄⋊C₈×C₆₁	—	2	2	no	—
56	C₄⋊C₈×C₆₁	—	2	2	no	—
57	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
58	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
59	#103×C₆₁	—	2	4	no	—
60	#102×C₆₁	—	2	4	no	—
61	C₂²⋊C₈×C₆₁	—	2	1	no	—
62	C₂²⋊C₈×C₆₁	—	2	2	no	—
63	C₄×C₈×C₆₁	—	2	2	no	—
64	C₄×C₈×C₆₁	—	2	1	no	—
65	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₆₁	C₂³	1	2	yes	—
68	#41×C₆₁	—	2	2	no	—
69	#41×C₆₁	Q₁₆	1	4	yes	—
70	C₂×Q₈×C₆₁	—	2	4	no	—
71	C₂×Q₈×C₆₁	—	4	2	no	—
72	C₂×Q₈×C₆₁	—	8	4	no	—
73	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄×C₆₁	—	2	1	no	—
75	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₁	—	2	1	no	—
77	C₂²×C₄×C₆₁	—	2	1	no	—
78	C₂²×C₄×C₆₁	—	2	1	no	—
79	C₂²×C₄×C₆₁	—	4	2	no	—
80	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
83	M₁₆×C₆₁	—	4	4	no	—
84	C₂×C₈×C₆₁	—	2	4	no	—
85	C₂×C₈×C₆₁	—	2	2	no	—
86	C₂×C₈×C₆₁	—	2	4	no	—
87	C₂×C₈×C₆₁	—	4	4	no	—
88	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
90	C₄⋊C₄×C₆₁	—	2	1	no	—
91	C₄⋊C₄×C₆₁	—	2	1	no	—
92	C₄⋊C₄×C₆₁	—	2	4	no	—
93	C₄⋊C₄×C₆₁	—	2	4	no	—
94	C₄⋊C₄×C₆₁	—	4	2	no	—
95	C₄⋊C₄×C₆₁	—	4	2	no	—
96	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂²⋊C₄×C₆₁	—	2	1	no	—
99	C₂²⋊C₄×C₆₁	—	2	1	no	—
100	C₂²⋊C₄×C₆₁	—	2	1	no	—
101	C₄²×C₆₁	—	2	1	no	—
102	C₄²×C₆₁	—	2	1	no	—
103	C₄²×C₆₁	—	2	1	no	—
104	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₆₁	#41	1	1	yes	characteristic
107	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
108	C₂³×C₆₁	—	2	1	no	—
109	C₂³×C₆₁	#121	1	1	yes	characteristic
110	Q₈×C₆₁	—	2	1	no	—
111	Q₈×C₆₁	—	8	4	no	—
112	Q₈×C₆₁	—	8	4	no	—
113	Q₈×C₆₁	C₂×Q₁₆	1	2	yes	—
114	D₄×C₆₁	—	2	4	no	—
115	D₄×C₆₁	—	2	1	no	—
116	D₄×C₆₁	C₂×Q₁₆	1	2	yes	—
117	C₂×C₄×C₆₁	—	2	1	no	—
118	C₂×C₄×C₆₁	—	2	2	no	—
119	C₂×C₄×C₆₁	—	2	4	no	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	—	2	1	no	—
122	C₂×C₄×C₆₁	—	2	1	no	—
123	C₂×C₄×C₆₁	—	4	2	no	—
124	C₂×C₄×C₆₁	—	4	2	no	—
125	C₂×C₄×C₆₁	—	4	4	no	—
126	C₂×C₄×C₆₁	—	8	4	no	—
127	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
128	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
129	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
131	C₂×C₄×C₆₁	#135	1	2	yes	—
132	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
135	C₈×C₆₁	—	2	2	no	—
136	C₈×C₆₁	—	4	4	no	—
137	C₈×C₆₁	—	4	2	no	—
138	C₈×C₆₁	#137	1	4	yes	—
139	#19272	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₆₁	—	2	1	no	—
141	C₂²×C₆₁	—	2	1	no	—
142	C₂²×C₆₁	—	2	2	no	—
143	C₂²×C₆₁	—	4	1	no	—
144	C₂²×C₆₁	—	4	1	no	—
145	C₂²×C₆₁	#467	1	2	yes	—
146	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	2	2	no	—
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	—	4	1	no	—
151	C₄×C₆₁	—	4	1	no	—
152	C₄×C₆₁	—	8	4	no	—
153	C₄×C₆₁	#467	1	2	yes	—
154	C₄×C₆₁	#472	1	2	yes	—
155	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
156	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
157	C₄×C₆₁	C₂×#137	1	2	yes	—
158	C₂×#500	C₂×C₆₁	1	1	yes	characteristic, complemented
159	#3079	C₂×C₆₁	1	2	yes	—
160	C₈×#41	C₂×C₆₁	1	1	yes	characteristic
161	#1676	C₂×C₆₁	1	4	yes	complemented
162	#1675	C₂×C₆₁	1	4	yes	complemented
163	#1567	C₂×C₆₁	1	1	yes	characteristic, complemented
164	#1548	C₂×C₆₁	1	2	yes	complemented
165	C₂×C₆₁	—	2	2	no	—
166	C₂×C₆₁	—	4	2	no	—
167	C₂×C₆₁	#3039	1	1	yes	characteristic
168	C₂×C₆₁	C₂×#467	1	1	yes	characteristic
169	C₂×C₆₁	C₂×#472	1	1	yes	characteristic
170	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
171	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
172	#500	—	2	1	no	—
173	#500	—	2	1	no	—
174	#500	C₂²×C₆₁	1	4	yes	—
175	#477	C₂²×C₆₁	1	4	yes	—
176	#476	C₂²×C₆₁	1	4	yes	—
177	#475	—	2	2	no	—
178	#474	—	2	2	no	—
179	C₈×Q₈	C₂²×C₆₁	1	2	yes	—
180	C₈×D₄	—	2	2	no	—
181	C₈×D₄	C₂²×C₆₁	1	2	yes	—
182	#431	—	2	1	no	—
183	#431	C₂²×C₆₁	1	1	yes	characteristic
184	#428	—	2	2	no	—
185	C₂×#107	—	2	2	no	—
186	#423	C₂²×C₆₁	1	2	yes	—
187	#417	—	2	2	no	—
188	C₂×C₄×C₈	—	2	1	no	—
189	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
190	#335	C₂²×C₆₁	1	4	yes	—
191	#333	C₂²×C₆₁	1	8	yes	—
192	C₆₁	#19272	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×#41	D₄×C₆₁	1	1	yes	characteristic
194	C₂²×Q₈	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	D₄×C₆₁	1	2	yes	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
202	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
203	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
207	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
208	#117	—	2	1	no	—
209	#117	D₄×C₆₁	1	1	yes	characteristic
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
214	#107	—	2	4	no	—
215	#107	—	2	4	no	—
216	#107	—	2	4	no	—
217	#106	—	2	4	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
221	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
222	#103	—	2	4	no	—
223	#102	—	2	4	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₄×C₈	—	2	2	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
229	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₆₁	1	2	yes	—
231	#41	—	2	2	no	—
232	#41	Q₁₆×C₆₁	1	4	yes	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	8	4	no	—
236	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
245	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
246	M₁₆	—	4	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	4	no	—
251	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
260	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
269	C₄²	#41×C₆₁	1	1	yes	characteristic
270	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
271	C₂³	—	2	1	no	—
272	C₂³	#121×C₆₁	1	1	yes	characteristic
273	Q₈	—	2	1	no	—
274	Q₈	—	8	4	no	—
275	Q₈	—	8	4	no	—
276	Q₈	C₂×Q₁₆×C₆₁	1	2	yes	—
277	D₄	—	2	4	no	—
278	D₄	—	2	1	no	—
279	D₄	C₂×Q₁₆×C₆₁	1	2	yes	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
291	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
292	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
293	C₂×C₄	C₂×Q₁₆×C₆₁	1	2	yes	—
294	C₂×C₄	#135×C₆₁	1	2	yes	—
295	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
297	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	4	4	no	—
300	C₈	—	4	2	no	—
301	C₈	#137×C₆₁	1	4	yes	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	C₆₁×#467	1	2	yes	—
308	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	8	4	no	—
315	C₄	C₆₁×#467	1	2	yes	—
316	C₄	C₆₁×#472	1	2	yes	—
317	C₄	C₂²×Q₁₆×C₆₁	1	1	yes	characteristic
318	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
319	C₄	C₂×#137×C₆₁	1	2	yes	—
320	C₂	—	2	2	no	—
321	C₂	—	4	2	no	—
322	C₂	C₆₁×#3039	1	1	yes	characteristic
323	C₂	C₂×C₆₁×#467	1	1	yes	characteristic
324	C₂	C₂×C₆₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₂×#41×C₆₁	D₄
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. C₂×C₆₁×#500	C₂	semidirect
4. C₆₁×#1676	C₂	semidirect
5. C₆₁×#1675	C₂	semidirect
6. C₆₁×#1567	C₂	semidirect
7. C₆₁×#1548	C₂	semidirect
139. #19272	C₆₁	direct (N × H)
158. C₂×#500	C₂×C₆₁	semidirect
161. #1676	C₂×C₆₁	semidirect
162. #1675	C₂×C₆₁	semidirect
163. #1567	C₂×C₆₁	semidirect
164. #1548	C₂×C₆₁	semidirect
192. C₆₁	#19272	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×#500	2
C₆₁×#3079	2
C₈×#41×C₆₁	2
C₆₁×#1676	2
C₆₁×#1675	2
C₆₁×#1567	2
C₆₁×#1548	2
#19272	61

Subgroups — C61×#19272 #29522705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19272 #29522705