Subgroups — C₆₁×#22730 #29526163

Order 15616.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#446	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2355	C₂	1	2	yes	maximal
3	C₆₁×#1949	C₂	1	2	yes	maximal
4	C₆₁×#1949	C₂	1	2	yes	maximal
5	C₆₁×#1948	C₂	1	2	yes	maximal
6	C₆₁×#1847	C₂	1	2	yes	maximal
7	C₆₁×#1846	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#1776	C₂	1	2	yes	maximal
9	C₄×#106×C₆₁	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#446	C₄	1	8	yes	complemented
12	C₂×#107×C₆₁	C₂²	1	2	yes	—
13	C₂×#107×C₆₁	C₂²	1	2	yes	—
14	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#106×C₆₁	C₂²	1	2	yes	—
18	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#97×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#391	—	2	2	no	—
24	C₆₁×#387	—	2	2	no	—
25	C₆₁×#387	—	2	2	no	—
26	C₆₁×#384	C₂²	1	2	yes	—
27	C₆₁×#382	C₂²	1	2	yes	—
28	C₆₁×#382	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
33	C₆₁×#340	C₂²	1	2	yes	—
34	C₆₁×#340	C₂²	1	2	yes	—
35	C₆₁×#340	C₂²	1	2	yes	—
36	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₆₁	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₆₁	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₁	—	2	2	no	—
51	C₂×C₄²×C₆₁	—	2	2	no	—
52	C₂×C₄²×C₆₁	—	2	2	no	—
53	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
55	#107×C₆₁	—	2	4	no	—
56	#107×C₆₁	—	4	2	no	—
57	#106×C₆₁	—	2	2	no	—
58	#106×C₆₁	—	4	1	no	—
59	#106×C₆₁	—	4	2	no	—
60	#106×C₆₁	—	4	1	no	—
61	#106×C₆₁	Q₈	1	4	yes	complemented
62	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
63	C₄⋊C₈×C₆₁	C₂×C₄	1	8	yes	—
64	#97×C₆₁	C₂×C₄	1	8	yes	—
65	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
66	#95×C₆₁	—	2	2	no	—
67	#95×C₆₁	—	2	2	no	—
68	#95×C₆₁	—	2	2	no	—
69	#95×C₆₁	—	2	2	no	—
70	#95×C₆₁	—	2	2	no	—
71	#95×C₆₁	—	2	2	no	—
72	#95×C₆₁	—	2	2	no	—
73	C₂×Q₈×C₆₁	—	4	2	no	—
74	C₂×Q₈×C₆₁	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₆₁	—	2	2	no	—
76	C₂²×C₄×C₆₁	—	2	2	no	—
77	C₂²×C₄×C₆₁	—	2	2	no	—
78	C₂²×C₄×C₆₁	—	2	2	no	—
79	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₆₁	—	2	1	no	—
85	C₂×C₈×C₆₁	—	2	2	no	—
86	C₂×C₈×C₆₁	—	2	1	no	—
87	C₂×C₈×C₆₁	—	2	2	no	—
88	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₆₁	#41	1	2	yes	—
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	—	2	2	no	—
95	C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₄⋊C₄×C₆₁	—	2	2	no	—
100	C₄⋊C₄×C₆₁	—	4	4	no	—
101	C₄⋊C₄×C₆₁	—	8	2	no	—
102	C₄⋊C₄×C₆₁	C₄⋊C₄	1	8	yes	—
103	C₄⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
104	C₄⋊C₄×C₆₁	C₂×Q₈	1	4	yes	—
105	C₄²×C₆₁	—	4	4	no	—
106	C₄²×C₆₁	—	8	2	no	—
107	C₄²×C₆₁	—	8	2	no	—
108	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
109	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
110	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
111	C₄²×C₆₁	C₂×Q₈	1	4	yes	—
112	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₆₁	—	4	4	no	—
114	C₂×C₄×C₆₁	—	2	2	no	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	—	2	2	no	—
118	C₂×C₄×C₆₁	—	2	2	no	—
119	C₂×C₄×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	—	2	4	no	—
121	C₂×C₄×C₆₁	—	4	2	no	—
122	C₂×C₄×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	—	4	2	no	—
124	C₂×C₄×C₆₁	—	4	2	no	—
125	C₂×C₄×C₆₁	—	4	2	no	—
126	C₂×C₄×C₆₁	—	4	2	no	—
127	C₂×C₄×C₆₁	—	4	2	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	—	4	2	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
136	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
142	C₈×C₆₁	—	2	2	no	—
143	C₈×C₆₁	—	2	4	no	—
144	C₈×C₆₁	C₄×Q₈	1	4	yes	—
145	#22730	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
148	C₂²×C₆₁	#550	1	1	yes	characteristic
149	C₂²×C₆₁	#575	1	2	yes	—
150	C₂²×C₆₁	#577	1	2	yes	—
151	C₄×C₆₁	—	2	4	no	—
152	C₄×C₆₁	—	2	2	no	—
153	C₄×C₆₁	—	2	2	no	—
154	C₄×C₆₁	—	2	4	no	—
155	C₄×C₆₁	—	4	4	no	—
156	C₄×C₆₁	—	8	2	no	—
157	C₄×C₆₁	—	8	2	no	—
158	C₄×C₆₁	—	8	2	no	—
159	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
160	C₄×C₆₁	#520	1	1	yes	characteristic
161	C₄×C₆₁	#557	1	1	yes	characteristic
162	C₄×C₆₁	Q₈²	1	1	yes	characteristic
163	C₂×#446	C₂×C₆₁	1	1	yes	characteristic
164	#2355	C₂×C₆₁	1	2	yes	—
165	#1949	C₂×C₆₁	1	2	yes	—
166	#1949	C₂×C₆₁	1	2	yes	—
167	#1948	C₂×C₆₁	1	2	yes	—
168	#1847	C₂×C₆₁	1	2	yes	—
169	#1846	C₂×C₆₁	1	1	yes	characteristic
170	#1776	C₂×C₆₁	1	2	yes	—
171	C₄×#106	C₂×C₆₁	1	1	yes	characteristic
172	C₂×C₆₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₆₁	#2911	1	1	yes	characteristic
174	C₂×C₆₁	#2912	1	1	yes	characteristic
175	C₂×C₆₁	#3388	1	2	yes	—
176	C₂×C₆₁	#3401	1	2	yes	—
177	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
178	#446	C₄×C₆₁	1	8	yes	complemented
179	C₂×#107	C₂²×C₆₁	1	2	yes	—
180	C₂×#107	C₂²×C₆₁	1	2	yes	—
181	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
182	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
183	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
184	C₂×#106	C₂²×C₆₁	1	2	yes	—
185	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
187	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
188	C₂×#97	C₂²×C₆₁	1	2	yes	—
189	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
190	#391	—	2	2	no	—
191	#387	—	2	2	no	—
192	#387	—	2	2	no	—
193	#384	C₂²×C₆₁	1	2	yes	—
194	#382	C₂²×C₆₁	1	2	yes	—
195	#382	C₂²×C₆₁	1	2	yes	—
196	C₄×C₄⋊C₄	—	2	2	no	—
197	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
198	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
199	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
200	#340	C₂²×C₆₁	1	2	yes	—
201	#340	C₂²×C₆₁	1	2	yes	—
202	#340	C₂²×C₆₁	1	2	yes	—
203	C₆₁	#22730	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
207	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
208	C₄×Q₈	C₂×C₄×C₆₁	1	4	yes	—
209	C₄×Q₈	Q₈×C₆₁	1	4	yes	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
213	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
223	#107	—	2	4	no	—
224	#107	—	4	2	no	—
225	#106	—	2	2	no	—
226	#106	—	4	1	no	—
227	#106	—	4	2	no	—
228	#106	—	4	1	no	—
229	#106	Q₈×C₆₁	1	4	yes	complemented
230	C₄⋊C₈	C₂×C₄×C₆₁	1	4	yes	—
231	C₄⋊C₈	C₂×C₄×C₆₁	1	8	yes	—
232	#97	C₂×C₄×C₆₁	1	8	yes	—
233	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	C₄⋊C₄×C₆₁	1	4	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
251	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
257	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
258	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
259	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
260	C₂×C₈	#41×C₆₁	1	2	yes	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	8	2	no	—
270	C₄⋊C₄	C₄⋊C₄×C₆₁	1	8	yes	—
271	C₄⋊C₄	C₄⋊C₄×C₆₁	1	4	yes	—
272	C₄⋊C₄	C₂×Q₈×C₆₁	1	4	yes	—
273	C₄²	—	4	4	no	—
274	C₄²	—	8	2	no	—
275	C₄²	—	8	2	no	—
276	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
277	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
278	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
279	C₄²	C₂×Q₈×C₆₁	1	4	yes	—
280	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
281	Q₈	—	4	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
301	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
302	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
304	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
308	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
309	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	2	4	no	—
312	C₈	C₄×Q₈×C₆₁	1	4	yes	—
313	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
314	C₂²	D₄×Q₈×C₆₁	1	1	yes	characteristic
315	C₂²	C₆₁×#550	1	1	yes	characteristic
316	C₂²	C₆₁×#575	1	2	yes	—
317	C₂²	C₆₁×#577	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₂×C₄×Q₈×C₆₁	1	1	yes	characteristic
327	C₄	C₆₁×#520	1	1	yes	characteristic
328	C₄	C₆₁×#557	1	1	yes	characteristic
329	C₄	Q₈²×C₆₁	1	1	yes	characteristic
330	C₂	Q₈×C₄⋊C₄×C₆₁	1	1	yes	characteristic
331	C₂	C₆₁×#2911	1	1	yes	characteristic
332	C₂	C₆₁×#2912	1	1	yes	characteristic
333	C₂	C₆₁×#3388	1	2	yes	—
334	C₂	C₆₁×#3401	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₆₁×#446	C₄	semidirect
61. #106×C₆₁	Q₈	semidirect
145. #22730	C₆₁	direct (N × H)
178. #446	C₄×C₆₁	semidirect
203. C₆₁	#22730	direct (N × H)
229. #106	Q₈×C₆₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₆₁×#446	2
C₆₁×#2355	2
C₆₁×#1949	2
C₆₁×#1948	2
C₆₁×#1847	2
C₆₁×#1846	2
C₆₁×#1776	2
C₄×#106×C₆₁	2
#22730	61

Subgroups — C61×#22730 #29526163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#22730 #29526163