Subgroups — C₆₁×#22815 #29526248

Order 15616.

754 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2356	C₂	1	2	yes	maximal
3	C₆₁×#1971	C₂	1	2	yes	maximal
4	C₆₁×#1928	C₂	1	4	yes	maximal
5	C₆₁×#1846	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#1805	C₂	1	2	yes	maximal
7	C₆₁×#1776	C₂	1	2	yes	maximal
8	C₆₁×#1770	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₆₁	C₄	1	8	yes	complemented
12	C₂×#107×C₆₁	C₂²	1	2	yes	—
13	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#106×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#103×C₆₁	C₂²	1	4	yes	—
17	C₂×#103×C₆₁	C₂²	1	2	yes	—
18	C₂×#97×C₆₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#391	C₂²	1	4	yes	—
21	C₆₁×#387	—	2	4	no	—
22	C₆₁×#383	—	2	2	no	—
23	C₆₁×#382	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
25	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#376	—	2	2	no	—
27	C₆₁×#340	C₂²	1	4	yes	—
28	C₆₁×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
30	C₂×Q₁₆×C₆₁	—	2	2	no	—
31	C₂×Q₁₆×C₆₁	—	2	2	no	—
32	C₂×Q₁₆×C₆₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₆₁	—	2	4	no	—
37	C₄×Q₈×C₆₁	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₆₁	—	2	2	no	—
43	C₂×C₄²×C₆₁	—	2	4	no	—
44	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
46	#107×C₆₁	—	4	2	no	—
47	#107×C₆₁	C₂×C₄	1	4	yes	—
48	#106×C₆₁	—	2	4	no	—
49	C₄⋊C₈×C₆₁	—	2	2	no	—
50	#103×C₆₁	—	4	4	no	—
51	#103×C₆₁	C₂×C₄	1	8	yes	—
52	#97×C₆₁	—	2	4	no	—
53	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
54	#95×C₆₁	—	2	2	no	—
55	#95×C₆₁	—	2	4	no	—
56	#95×C₆₁	—	2	4	no	—
57	#95×C₆₁	—	2	2	no	—
58	#95×C₆₁	—	2	4	no	—
59	#95×C₆₁	D₄	1	2	yes	—
60	C₂×Q₈×C₆₁	—	2	2	no	—
61	C₂×Q₈×C₆₁	—	2	2	no	—
62	C₂×Q₈×C₆₁	—	2	2	no	—
63	C₂×Q₈×C₆₁	—	2	2	no	—
64	C₂×Q₈×C₆₁	—	2	2	no	—
65	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₆₁	—	2	2	no	—
67	C₂²×C₄×C₆₁	—	2	2	no	—
68	C₂²×C₄×C₆₁	—	2	4	no	—
69	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
73	Q₁₆×C₆₁	—	2	8	no	—
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	1	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	2	2	no	—
78	C₂×C₈×C₆₁	—	2	2	no	—
79	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₁	#41	1	2	yes	—
82	C₂×C₈×C₆₁	#41	1	2	yes	—
83	C₄⋊C₄×C₆₁	—	2	8	no	—
84	C₄⋊C₄×C₆₁	—	2	2	no	—
85	C₄⋊C₄×C₆₁	—	2	4	no	—
86	C₄⋊C₄×C₆₁	—	2	4	no	—
87	C₄⋊C₄×C₆₁	—	4	2	no	—
88	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
89	C₄²×C₆₁	—	2	8	no	—
90	C₄²×C₆₁	—	2	2	no	—
91	C₄²×C₆₁	—	2	2	no	—
92	C₄²×C₆₁	—	4	8	no	—
93	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
94	C₄²×C₆₁	#41	1	2	yes	—
95	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₆₁	—	2	4	no	—
97	Q₈×C₆₁	—	2	4	no	—
98	Q₈×C₆₁	—	4	2	no	—
99	Q₈×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	2	no	—
102	C₂×C₄×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	2	4	no	—
105	C₂×C₄×C₆₁	—	2	4	no	—
106	C₂×C₄×C₆₁	—	2	4	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	—	4	4	no	—
110	C₂×C₄×C₆₁	—	4	4	no	—
111	C₂×C₄×C₆₁	—	4	4	no	—
112	C₂×C₄×C₆₁	—	4	4	no	—
113	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
121	C₈×C₆₁	—	2	2	no	—
122	C₈×C₆₁	—	4	2	no	—
123	C₈×C₆₁	#117	1	4	yes	—
124	#22815	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂²×C₆₁	#547	1	1	yes	characteristic
127	C₂²×C₆₁	#550	1	1	yes	characteristic
128	C₂²×C₆₁	#575	1	2	yes	—
129	C₂²×C₆₁	#578	1	2	yes	—
130	C₄×C₆₁	—	2	2	no	—
131	C₄×C₆₁	—	2	2	no	—
132	C₄×C₆₁	—	4	4	no	—
133	C₄×C₆₁	—	4	4	no	—
134	C₄×C₆₁	—	4	4	no	—
135	C₄×C₆₁	—	8	4	no	—
136	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
137	C₄×C₆₁	#520	1	1	yes	characteristic
138	C₄×C₆₁	#555	1	1	yes	characteristic
139	C₄×C₆₁	#556	1	1	yes	characteristic
140	C₂×C₄×Q₁₆	C₂×C₆₁	1	1	yes	characteristic
141	#2356	C₂×C₆₁	1	2	yes	—
142	#1971	C₂×C₆₁	1	2	yes	—
143	#1928	C₂×C₆₁	1	4	yes	—
144	#1846	C₂×C₆₁	1	1	yes	characteristic
145	#1805	C₂×C₆₁	1	2	yes	—
146	#1776	C₂×C₆₁	1	2	yes	—
147	#1770	C₂×C₆₁	1	1	yes	characteristic
148	C₂×C₆₁	#2354	1	1	yes	characteristic
149	C₂×C₆₁	#2943	1	1	yes	characteristic
150	C₂×C₆₁	#2950	1	1	yes	characteristic
151	C₂×C₆₁	#3352	1	2	yes	—
152	C₂×C₆₁	#3402	1	2	yes	—
153	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
154	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
155	C₄×Q₁₆	C₄×C₆₁	1	8	yes	complemented
156	C₂×#107	C₂²×C₆₁	1	2	yes	—
157	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
158	C₂×#106	C₂²×C₆₁	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
160	C₂×#103	C₂²×C₆₁	1	4	yes	—
161	C₂×#103	C₂²×C₆₁	1	2	yes	—
162	C₂×#97	C₂²×C₆₁	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
164	#391	C₂²×C₆₁	1	4	yes	—
165	#387	—	2	4	no	—
166	#383	—	2	2	no	—
167	#382	C₂²×C₆₁	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
169	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
170	#376	—	2	2	no	—
171	#340	C₂²×C₆₁	1	4	yes	—
172	#336	C₂²×C₆₁	1	2	yes	—
173	C₆₁	#22815	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×Q₁₆	—	2	2	no	—
177	C₂×Q₁₆	C₂×C₄×C₆₁	1	4	yes	—
178	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
179	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
181	C₄×Q₈	—	2	4	no	—
182	C₄×Q₈	C₂×C₄×C₆₁	1	8	yes	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
186	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	4	no	—
189	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
190	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
191	#107	—	4	2	no	—
192	#107	C₂×C₄×C₆₁	1	4	yes	—
193	#106	—	2	4	no	—
194	C₄⋊C₈	—	2	2	no	—
195	#103	—	4	4	no	—
196	#103	C₂×C₄×C₆₁	1	8	yes	—
197	#97	—	2	4	no	—
198	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
199	#95	—	2	2	no	—
200	#95	—	2	4	no	—
201	#95	—	2	4	no	—
202	#95	—	2	2	no	—
203	#95	—	2	4	no	—
204	#95	D₄×C₆₁	1	2	yes	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	C₂²×C₄×C₆₁	1	4	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
215	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
217	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
218	Q₁₆	—	2	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
225	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
226	C₂×C₈	#41×C₆₁	1	2	yes	—
227	C₂×C₈	#41×C₆₁	1	2	yes	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
234	C₄²	—	2	8	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	8	no	—
238	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
239	C₄²	#41×C₆₁	1	2	yes	—
240	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
259	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
260	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
261	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
264	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
265	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	4	2	no	—
268	C₈	#117×C₆₁	1	4	yes	—
269	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
270	C₂²	C₆₁×#547	1	1	yes	characteristic
271	C₂²	C₆₁×#550	1	1	yes	characteristic
272	C₂²	C₆₁×#575	1	2	yes	—
273	C₂²	C₆₁×#578	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	—	4	4	no	—
279	C₄	—	8	4	no	—
280	C₄	C₂×#117×C₆₁	1	1	yes	characteristic
281	C₄	C₆₁×#520	1	1	yes	characteristic
282	C₄	C₆₁×#555	1	1	yes	characteristic
283	C₄	C₆₁×#556	1	1	yes	characteristic
284	C₂	C₆₁×#2354	1	1	yes	characteristic
285	C₂	C₆₁×#2943	1	1	yes	characteristic
286	C₂	C₆₁×#2950	1	1	yes	characteristic
287	C₂	C₆₁×#3352	1	2	yes	—
288	C₂	C₆₁×#3402	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 ◁ C₄×C₄⋊C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄×Q₁₆×C₆₁	C₄	semidirect
124. #22815	C₆₁	direct (N × H)
155. C₄×Q₁₆	C₄×C₆₁	semidirect
173. C₆₁	#22815	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×C₄×Q₁₆×C₆₁	2
C₆₁×#2356	2
C₆₁×#1971	2
C₆₁×#1928	2
C₆₁×#1846	2
C₆₁×#1805	2
C₆₁×#1776	2
C₆₁×#1770	2
#22815	61

Subgroups — C61×#22815 #29526248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#22815 #29526248