Subgroups — C₆₁×#22703 #29526136

Order 15616.

1054 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
2	D₄×C₄⋊C₄×C₆₁	C₂	1	2	yes	maximal
3	C₆₁×#1948	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1853	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1807	C₂	1	2	yes	maximal, complemented
7	C₄×#106×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₆₁×#340	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₆₁	—	2	2	no	—
11	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
13	C₈×D₄×C₆₁	C₄	1	8	yes	complemented
14	C₂×#106×C₆₁	C₂²	1	2	yes	—
15	C₂×#106×C₆₁	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#389	—	2	2	no	—
20	C₆₁×#385	—	2	4	no	—
21	C₆₁×#384	C₂²	1	2	yes	—
22	C₆₁×#380	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
24	C₆₁×#340	C₂²	1	1	yes	characteristic
25	C₆₁×#340	C₂²	1	1	yes	characteristic
26	C₆₁×#340	C₂²	1	2	yes	—
27	C₆₁×#340	C₂²	1	2	yes	—
28	C₆₁×#340	C₂²	1	4	yes	—
29	C₂²×D₄×C₆₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₁	—	2	4	no	—
31	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₆₁	—	4	4	no	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	complemented
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	C₄×D₄×C₆₁	—	4	8	no	—
38	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₆₁	Q₈	1	4	yes	—
40	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
41	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₁	—	4	4	no	—
45	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
51	C₂×C₄²×C₆₁	—	2	2	no	—
52	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
53	#106×C₆₁	—	2	4	no	—
54	#106×C₆₁	—	4	4	no	—
55	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	complemented
56	C₂²⋊C₈×C₆₁	C₂×C₄	1	8	yes	complemented
57	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	complemented
58	#95×C₆₁	—	2	2	no	—
59	#95×C₆₁	—	2	4	no	—
60	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
61	C₂×D₄×C₆₁	C₄⋊C₄	1	4	yes	—
62	C₂×D₄×C₆₁	SD₁₆	1	8	yes	—
63	C₂²×C₄×C₆₁	—	2	4	no	—
64	C₂²×C₄×C₆₁	—	2	4	no	—
65	C₂²×C₄×C₆₁	—	2	2	no	—
66	C₂²×C₄×C₆₁	—	2	2	no	—
67	C₂²×C₄×C₆₁	—	2	4	no	—
68	C₂²×C₄×C₆₁	—	4	8	no	—
69	C₂²×C₄×C₆₁	—	4	8	no	—
70	C₂²×C₄×C₆₁	—	8	4	no	—
71	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₆₁	—	2	2	no	—
78	C₂×C₈×C₆₁	—	2	4	no	—
79	C₂×C₈×C₆₁	—	2	2	no	—
80	C₂×C₈×C₆₁	—	4	8	no	—
81	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
84	C₄⋊C₄×C₆₁	—	2	2	no	—
85	C₄⋊C₄×C₆₁	—	2	2	no	—
86	C₄⋊C₄×C₆₁	—	2	4	no	—
87	C₄⋊C₄×C₆₁	—	2	2	no	—
88	C₄⋊C₄×C₆₁	—	2	4	no	—
89	C₄⋊C₄×C₆₁	—	4	8	no	—
90	C₄⋊C₄×C₆₁	—	4	4	no	—
91	C₄⋊C₄×C₆₁	—	8	2	no	—
92	C₄⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
93	C₄⋊C₄×C₆₁	#41	1	4	yes	—
94	C₂²⋊C₄×C₆₁	—	4	8	no	—
95	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	8	yes	—
96	C₄²×C₆₁	—	4	4	no	—
97	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
98	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
99	C₂³×C₆₁	—	2	4	no	—
100	C₂³×C₆₁	—	2	2	no	—
101	C₂³×C₆₁	—	2	2	no	—
102	C₂³×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
103	C₂³×C₆₁	C₂×SD₁₆	1	8	yes	—
104	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
105	D₄×C₆₁	#106	1	16	yes	complemented
106	C₂×C₄×C₆₁	—	2	4	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	4	no	—
110	C₂×C₄×C₆₁	—	2	4	no	—
111	C₂×C₄×C₆₁	—	2	8	no	—
112	C₂×C₄×C₆₁	—	2	4	no	—
113	C₂×C₄×C₆₁	—	4	2	no	—
114	C₂×C₄×C₆₁	—	4	2	no	—
115	C₂×C₄×C₆₁	—	4	4	no	—
116	C₂×C₄×C₆₁	—	4	2	no	—
117	C₂×C₄×C₆₁	—	4	2	no	—
118	C₂×C₄×C₆₁	—	4	4	no	—
119	C₂×C₄×C₆₁	—	4	4	no	—
120	C₂×C₄×C₆₁	—	4	2	no	—
121	C₂×C₄×C₆₁	—	8	16	no	—
122	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
124	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
126	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
129	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
130	C₈×C₆₁	—	2	2	no	—
131	C₈×C₆₁	—	2	2	no	—
132	C₈×C₆₁	—	2	4	no	—
133	#22703	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₆₁	—	2	4	no	—
135	C₂²×C₆₁	—	2	4	no	—
136	C₂²×C₆₁	—	2	8	no	—
137	C₂²×C₆₁	—	2	8	no	—
138	C₂²×C₆₁	C₂×#106	1	8	yes	—
139	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₆₁	#546	1	2	yes	—
141	C₂²×C₆₁	C₂²×SD₁₆	1	2	yes	—
142	C₂²×C₆₁	#576	1	1	yes	characteristic
143	C₂²×C₆₁	#578	1	1	yes	characteristic
144	C₄×C₆₁	—	2	4	no	—
145	C₄×C₆₁	—	2	2	no	—
146	C₄×C₆₁	—	4	4	no	—
147	C₄×C₆₁	—	8	4	no	—
148	C₄×C₆₁	—	8	2	no	—
149	C₄×C₆₁	C₂×#106	1	4	yes	—
150	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
151	C₄×C₆₁	#518	1	1	yes	characteristic
152	C₄×C₆₁	#554	1	2	yes	—
153	C₂×C₈×D₄	C₂×C₆₁	1	1	yes	characteristic
154	D₄×C₄⋊C₄	C₂×C₆₁	1	2	yes	—
155	#1948	C₂×C₆₁	1	2	yes	complemented
156	#1853	C₂×C₆₁	1	4	yes	complemented
157	#1834	C₂×C₆₁	1	1	yes	characteristic, complemented
158	#1807	C₂×C₆₁	1	2	yes	complemented
159	C₄×#106	C₂×C₆₁	1	1	yes	characteristic, complemented
160	C₂×#340	C₂×C₆₁	1	2	yes	complemented
161	C₂×C₆₁	—	2	8	no	—
162	C₂×C₆₁	#2349	1	1	yes	characteristic
163	C₂×C₆₁	C₂²×#106	1	1	yes	characteristic
164	C₂×C₆₁	#2913	1	1	yes	characteristic
165	C₂×C₆₁	#3295	1	2	yes	—
166	C₂×C₆₁	#3298	1	2	yes	—
167	C₂³×C₈	C₂²×C₆₁	1	2	yes	—
168	C₂×C₄×D₄	—	2	2	no	—
169	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
170	C₂²×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
171	C₈×D₄	C₄×C₆₁	1	8	yes	complemented
172	C₂×#106	C₂²×C₆₁	1	2	yes	—
173	C₂×#106	C₂²×C₆₁	1	4	yes	—
174	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
175	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
177	#389	—	2	2	no	—
178	#385	—	2	4	no	—
179	#384	C₂²×C₆₁	1	2	yes	—
180	#380	C₂²×C₆₁	1	4	yes	—
181	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
182	#340	C₂²×C₆₁	1	1	yes	characteristic
183	#340	C₂²×C₆₁	1	2	yes	—
184	#340	C₂²×C₆₁	1	2	yes	—
185	#340	C₂²×C₆₁	1	4	yes	—
186	#340	C₂²×C₆₁	1	1	yes	characteristic
187	C₆₁	#22703	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×D₄	D₄×C₆₁	1	1	yes	characteristic
189	C₂³×C₄	—	2	4	no	—
190	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
191	C₂²×C₈	—	4	4	no	—
192	C₂²×C₈	C₂×C₄×C₆₁	1	8	yes	complemented
193	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
195	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
196	C₄×D₄	—	4	8	no	—
197	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
198	C₄×D₄	Q₈×C₆₁	1	4	yes	—
199	C₂×C₄⋊C₄	—	2	8	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	4	4	no	—
204	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
206	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	4	no	—
209	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
212	#106	—	2	4	no	—
213	#106	—	4	4	no	—
214	C₄⋊C₈	C₂×C₄×C₆₁	1	4	yes	complemented
215	C₂²⋊C₈	C₂×C₄×C₆₁	1	8	yes	complemented
216	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	complemented
217	#95	—	2	2	no	—
218	#95	—	2	4	no	—
219	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
220	C₂×D₄	C₄⋊C₄×C₆₁	1	4	yes	—
221	C₂×D₄	SD₁₆×C₆₁	1	8	yes	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	8	no	—
228	C₂²×C₄	—	4	8	no	—
229	C₂²×C₄	—	8	4	no	—
230	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
231	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×Q₈×C₆₁	1	4	yes	—
235	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	8	no	—
240	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	8	2	no	—
251	C₄⋊C₄	C₄⋊C₄×C₆₁	1	4	yes	—
252	C₄⋊C₄	#41×C₆₁	1	4	yes	—
253	C₂²⋊C₄	—	4	8	no	—
254	C₂²⋊C₄	C₄⋊C₄×C₆₁	1	8	yes	—
255	C₄²	—	4	4	no	—
256	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
257	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
262	C₂³	C₂×SD₁₆×C₆₁	1	8	yes	—
263	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
264	D₄	#106×C₆₁	1	16	yes	complemented
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	16	no	—
281	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
282	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
283	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
284	C₂×C₄	C₂×SD₁₆×C₆₁	1	4	yes	—
285	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
286	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
287	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
288	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
289	C₈	—	2	2	no	—
290	C₈	—	2	2	no	—
291	C₈	—	2	4	no	—
292	C₂²	—	2	8	no	—
293	C₂²	—	2	8	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	C₂×#106×C₆₁	1	8	yes	—
297	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
298	C₂²	C₆₁×#546	1	2	yes	—
299	C₂²	C₂²×SD₁₆×C₆₁	1	2	yes	—
300	C₂²	C₆₁×#576	1	1	yes	characteristic
301	C₂²	C₆₁×#578	1	1	yes	characteristic
302	C₄	—	2	4	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	4	no	—
306	C₄	—	8	2	no	—
307	C₄	C₂×#106×C₆₁	1	4	yes	—
308	C₄	C₄×#41×C₆₁	1	1	yes	characteristic
309	C₄	C₆₁×#518	1	1	yes	characteristic
310	C₄	C₆₁×#554	1	2	yes	—
311	C₂	—	2	8	no	—
312	C₂	C₆₁×#2349	1	1	yes	characteristic
313	C₂	C₂²×#106×C₆₁	1	1	yes	characteristic
314	C₂	C₆₁×#2913	1	1	yes	characteristic
315	C₂	C₆₁×#3295	1	2	yes	—
316	C₂	C₆₁×#3298	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1948	C₂	semidirect
4. C₆₁×#1853	C₂	semidirect
5. C₆₁×#1834	C₂	semidirect
6. C₆₁×#1807	C₂	semidirect
7. C₄×#106×C₆₁	C₂	semidirect
8. C₂×C₆₁×#340	C₂	semidirect
13. C₈×D₄×C₆₁	C₄	semidirect
33. C₂²×C₈×C₆₁	C₂×C₄	semidirect
55. C₄⋊C₈×C₆₁	C₂×C₄	semidirect
56. C₂²⋊C₈×C₆₁	C₂×C₄	semidirect
57. C₄×C₈×C₆₁	C₂×C₄	semidirect
105. D₄×C₆₁	#106	semidirect
133. #22703	C₆₁	direct (N × H)
155. #1948	C₂×C₆₁	semidirect
156. #1853	C₂×C₆₁	semidirect
157. #1834	C₂×C₆₁	semidirect
158. #1807	C₂×C₆₁	semidirect
159. C₄×#106	C₂×C₆₁	semidirect
160. C₂×#340	C₂×C₆₁	semidirect
171. C₈×D₄	C₄×C₆₁	semidirect
187. C₆₁	#22703	direct (N × H)
192. C₂²×C₈	C₂×C₄×C₆₁	semidirect
214. C₄⋊C₈	C₂×C₄×C₆₁	semidirect
215. C₂²⋊C₈	C₂×C₄×C₆₁	semidirect
216. C₄×C₈	C₂×C₄×C₆₁	semidirect
264. D₄	#106×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₆₁	2
D₄×C₄⋊C₄×C₆₁	2
C₆₁×#1948	2
C₆₁×#1853	2
C₆₁×#1834	2
C₆₁×#1807	2
C₄×#106×C₆₁	2
C₂×C₆₁×#340	2
#22703	61

Subgroups — C61×#22703 #29526136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#22703 #29526136