Subgroups — C₆₁×#22690 #29526123

Order 15616.

846 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#443	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2951	C₂	1	2	yes	maximal
3	C₆₁×#2912	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1952	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1860	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1815	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×#131×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#520	C₂²	1	4	yes	—
12	C₆₁×#443	C₄	1	8	yes	complemented
13	C₆₁×#440	—	2	4	no	—
14	C₄×Q₁₆×C₆₁	—	2	2	no	—
15	C₄×D₈×C₆₁	—	2	2	no	—
16	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#428	—	2	4	no	—
19	C₆₁×#428	C₂²	1	4	yes	—
20	C₆₁×#427	—	2	2	no	—
21	C₆₁×#427	C₂²	1	1	yes	characteristic
22	C₂×#107×C₆₁	C₂²	1	2	yes	—
23	C₂×#106×C₆₁	—	2	2	no	—
24	C₂×#106×C₆₁	C₂²	1	2	yes	—
25	C₂×#104×C₆₁	C₂²	1	2	yes	—
26	C₆₁×#417	C₂²	1	4	yes	—
27	C₆₁×#405	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#382	C₂²	1	2	yes	—
30	C₆₁×#343	C₂²	1	4	yes	—
31	C₆₁×#339	C₂²	1	2	yes	—
32	C₂×#41×C₆₁	C₂³	1	2	yes	—
33	#135×C₆₁	C₂×C₄	1	4	yes	—
34	#135×C₆₁	Q₈	1	4	yes	—
35	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic
36	C₂×SD₁₆×C₆₁	D₄	1	2	yes	—
37	C₂×D₈×C₆₁	D₄	1	1	yes	characteristic
38	#131×C₆₁	—	2	4	no	—
39	#131×C₆₁	C₂×C₄	1	8	yes	complemented
40	C₂×M₁₆×C₆₁	—	2	2	no	—
41	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
42	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₁	—	2	2	no	—
44	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₆₁	—	2	4	no	—
47	C₄×D₄×C₆₁	—	2	4	no	—
48	C₄×D₄×C₆₁	—	4	4	no	—
49	#117×C₆₁	—	2	4	no	—
50	#117×C₆₁	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
52	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
53	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
54	#108×C₆₁	—	2	2	no	—
55	#108×C₆₁	C₂×C₄	1	4	yes	complemented
56	#107×C₆₁	—	2	2	no	—
57	#107×C₆₁	—	2	2	no	—
58	#107×C₆₁	—	2	4	no	—
59	#107×C₆₁	D₄	1	2	yes	—
60	#106×C₆₁	—	2	2	no	—
61	#106×C₆₁	—	2	2	no	—
62	#106×C₆₁	—	2	4	no	—
63	#106×C₆₁	D₄	1	1	yes	characteristic
64	#106×C₆₁	D₄	1	1	yes	characteristic
65	#104×C₆₁	C₂×C₄	1	8	yes	complemented
66	#103×C₆₁	—	2	4	no	—
67	#102×C₆₁	—	2	4	no	—
68	#97×C₆₁	—	2	2	no	—
69	C₄×C₈×C₆₁	—	2	2	no	—
70	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	complemented
71	#95×C₆₁	—	4	2	no	—
72	#41×C₆₁	—	2	2	no	—
73	#41×C₆₁	—	2	2	no	—
74	#41×C₆₁	C₂²×C₄	1	4	yes	—
75	#41×C₆₁	C₂×Q₈	1	4	yes	—
76	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
77	C₂×D₄×C₆₁	—	2	2	no	—
78	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₁	—	2	2	no	—
80	C₂²×C₄×C₆₁	—	2	1	no	—
81	C₂²×C₄×C₆₁	—	2	4	no	—
82	C₂²×C₄×C₆₁	—	4	4	no	—
83	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₆₁	C₄⋊C₄	1	4	yes	—
85	SD₁₆×C₆₁	C₄⋊C₄	1	8	yes	—
86	D₈×C₆₁	C₄⋊C₄	1	4	yes	—
87	M₁₆×C₆₁	—	2	2	no	—
88	M₁₆×C₆₁	—	2	8	no	—
89	M₁₆×C₆₁	—	2	2	no	—
90	M₁₆×C₆₁	—	2	2	no	—
91	M₁₆×C₆₁	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₆₁	—	2	2	no	—
93	C₂×C₈×C₆₁	—	2	8	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
102	C₂×C₈×C₆₁	#41	1	2	yes	—
103	C₄⋊C₄×C₆₁	—	2	4	no	—
104	C₄⋊C₄×C₆₁	—	2	4	no	—
105	C₄⋊C₄×C₆₁	—	4	4	no	—
106	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
108	C₂²⋊C₄×C₆₁	—	2	2	no	—
109	C₂²⋊C₄×C₆₁	—	4	4	no	—
110	C₄²×C₆₁	—	2	2	no	—
111	C₄²×C₆₁	—	2	1	no	—
112	C₄²×C₆₁	—	4	4	no	—
113	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
114	C₂³×C₆₁	—	2	2	no	—
115	C₂³×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
116	Q₈×C₆₁	—	2	2	no	—
117	Q₈×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
118	D₄×C₆₁	—	2	4	no	—
119	D₄×C₆₁	—	2	4	no	—
120	D₄×C₆₁	—	2	2	no	—
121	D₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	—	2	4	no	—
124	C₂×C₄×C₆₁	—	2	4	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	—	2	2	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	4	no	—
130	C₂×C₄×C₆₁	—	8	4	no	—
131	C₂×C₄×C₆₁	—	16	4	no	—
132	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
138	C₈×C₆₁	—	2	4	no	—
139	C₈×C₆₁	—	4	2	no	—
140	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
141	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₆₁	C₄×D₄	1	2	yes	—
143	C₈×C₆₁	C₄×D₄	1	2	yes	—
144	#22690	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₆₁	—	2	2	no	—
146	C₂²×C₆₁	—	2	4	no	—
147	C₂²×C₆₁	—	4	4	no	—
148	C₂²×C₆₁	#520	1	1	yes	characteristic
149	C₂²×C₆₁	D₄²	1	1	yes	characteristic
150	C₂²×C₆₁	#554	1	1	yes	characteristic
151	C₄×C₆₁	—	2	4	no	—
152	C₄×C₆₁	—	4	2	no	—
153	C₄×C₆₁	—	4	2	no	—
154	C₄×C₆₁	—	4	2	no	—
155	C₄×C₆₁	—	8	4	no	—
156	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₆₁	#547	1	1	yes	characteristic
159	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
160	C₂×#443	C₂×C₆₁	1	1	yes	characteristic
161	#2951	C₂×C₆₁	1	2	yes	—
162	#2912	C₂×C₆₁	1	2	yes	complemented
163	#1952	C₂×C₆₁	1	2	yes	complemented
164	#1860	C₂×C₆₁	1	4	yes	complemented
165	#1846	C₂×C₆₁	1	1	yes	characteristic, complemented
166	#1815	C₂×C₆₁	1	2	yes	complemented
167	#1772	C₂×C₆₁	1	1	yes	characteristic, complemented
168	C₂×C₆₁	—	2	1	no	—
169	C₂×C₆₁	—	2	1	no	—
170	C₂×C₆₁	—	4	4	no	—
171	C₂×C₆₁	D₄×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₆₁	#3290	1	2	yes	—
173	C₂×#135	C₂²×C₆₁	1	1	yes	characteristic
174	C₂×#131	C₂²×C₆₁	1	2	yes	—
175	#520	C₂²×C₆₁	1	4	yes	—
176	#443	C₄×C₆₁	1	8	yes	complemented
177	#440	—	2	4	no	—
178	C₄×Q₁₆	—	2	2	no	—
179	C₄×D₈	—	2	2	no	—
180	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
181	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
182	#428	—	2	4	no	—
183	#428	C₂²×C₆₁	1	4	yes	—
184	#427	—	2	2	no	—
185	#427	C₂²×C₆₁	1	1	yes	characteristic
186	C₂×#107	C₂²×C₆₁	1	2	yes	—
187	C₂×#106	—	2	2	no	—
188	C₂×#106	C₂²×C₆₁	1	2	yes	—
189	C₂×#104	C₂²×C₆₁	1	2	yes	—
190	#417	C₂²×C₆₁	1	4	yes	—
191	#405	C₂²×C₆₁	1	2	yes	—
192	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
193	#382	C₂²×C₆₁	1	2	yes	—
194	#343	C₂²×C₆₁	1	4	yes	—
195	#339	C₂²×C₆₁	1	2	yes	—
196	C₆₁	#22690	1	1	yes	characteristic, direct product factor, Sylow
197	C₂×#41	C₂³×C₆₁	1	2	yes	—
198	#135	C₂×C₄×C₆₁	1	4	yes	—
199	#135	Q₈×C₆₁	1	4	yes	—
200	C₂×Q₁₆	D₄×C₆₁	1	1	yes	characteristic
201	C₂×SD₁₆	D₄×C₆₁	1	2	yes	—
202	C₂×D₈	D₄×C₆₁	1	1	yes	characteristic
203	#131	—	2	4	no	—
204	#131	C₂×C₄×C₆₁	1	8	yes	complemented
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
207	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
211	C₄×Q₈	—	2	4	no	—
212	C₄×D₄	—	2	4	no	—
213	C₄×D₄	—	4	4	no	—
214	#117	—	2	4	no	—
215	#117	C₂³×C₆₁	1	2	yes	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
218	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
219	#108	—	2	2	no	—
220	#108	C₂×C₄×C₆₁	1	4	yes	complemented
221	#107	—	2	2	no	—
222	#107	—	2	2	no	—
223	#107	—	2	4	no	—
224	#107	D₄×C₆₁	1	2	yes	—
225	#106	—	2	2	no	—
226	#106	—	2	2	no	—
227	#106	—	2	4	no	—
228	#106	D₄×C₆₁	1	1	yes	characteristic
229	#106	D₄×C₆₁	1	1	yes	characteristic
230	#104	C₂×C₄×C₆₁	1	8	yes	complemented
231	#103	—	2	4	no	—
232	#102	—	2	4	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	complemented
236	#95	—	4	2	no	—
237	#41	—	2	2	no	—
238	#41	—	2	2	no	—
239	#41	C₂²×C₄×C₆₁	1	4	yes	—
240	#41	C₂×Q₈×C₆₁	1	4	yes	—
241	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
249	Q₁₆	C₄⋊C₄×C₆₁	1	4	yes	—
250	SD₁₆	C₄⋊C₄×C₆₁	1	8	yes	—
251	D₈	C₄⋊C₄×C₆₁	1	4	yes	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	8	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	C₂²×C₄×C₆₁	1	4	yes	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	8	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
261	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
262	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
264	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
266	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
267	C₂×C₈	#41×C₆₁	1	2	yes	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
272	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	4	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	4	4	no	—
278	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
279	C₂³	—	2	2	no	—
280	C₂³	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
283	D₄	—	2	4	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	—	16	4	no	—
297	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
300	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
301	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
302	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
306	C₈	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
307	C₈	C₄×D₄×C₆₁	1	2	yes	—
308	C₈	C₄×D₄×C₆₁	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₆₁×#520	1	1	yes	characteristic
313	C₂²	D₄²×C₆₁	1	1	yes	characteristic
314	C₂²	C₆₁×#554	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	8	4	no	—
320	C₄	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
321	C₄	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
322	C₄	C₆₁×#547	1	1	yes	characteristic
323	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	4	4	no	—
327	C₂	D₄×C₄⋊C₄×C₆₁	1	1	yes	characteristic
328	C₂	C₆₁×#3290	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×#135×C₆₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#2912	C₂	semidirect
4. C₆₁×#1952	C₂	semidirect
5. C₆₁×#1860	C₂	semidirect
6. C₆₁×#1846	C₂	semidirect
7. C₆₁×#1815	C₂	semidirect
8. C₆₁×#1772	C₂	semidirect
12. C₆₁×#443	C₄	semidirect
39. #131×C₆₁	C₂×C₄	semidirect
55. #108×C₆₁	C₂×C₄	semidirect
65. #104×C₆₁	C₂×C₄	semidirect
70. C₄×C₈×C₆₁	C₂×C₄	semidirect
144. #22690	C₆₁	direct (N × H)
162. #2912	C₂×C₆₁	semidirect
163. #1952	C₂×C₆₁	semidirect
164. #1860	C₂×C₆₁	semidirect
165. #1846	C₂×C₆₁	semidirect
166. #1815	C₂×C₆₁	semidirect
167. #1772	C₂×C₆₁	semidirect
176. #443	C₄×C₆₁	semidirect
196. C₆₁	#22690	direct (N × H)
204. #131	C₂×C₄×C₆₁	semidirect
220. #108	C₂×C₄×C₆₁	semidirect
230. #104	C₂×C₄×C₆₁	semidirect
235. C₄×C₈	C₂×C₄×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×#443	2
C₆₁×#2951	2
C₆₁×#2912	2
C₆₁×#1952	2
C₆₁×#1860	2
C₆₁×#1846	2
C₆₁×#1815	2
C₆₁×#1772	2
#22690	61

Subgroups — C61×#22690 #29526123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#22690 #29526123