Subgroups — C₆₁×#29149 #29532582

Order 15616.

878 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#127×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2025	C₂	1	4	yes	maximal, complemented
4	C₆₁×#1961	C₂	1	2	yes	maximal
5	C₆₁×#1836	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1802	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×#125×C₆₁	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₆₁	—	2	2	no	—
11	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×#102×C₆₁	C₂²	1	8	yes	—
14	C₂×#97×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#390	—	2	2	no	—
16	C₆₁×#390	C₂²	1	2	yes	—
17	C₆₁×#389	—	2	2	no	—
18	C₆₁×#389	—	2	2	no	—
19	C₆₁×#384	C₂²	1	1	yes	characteristic
20	C₆₁×#384	C₂²	1	2	yes	—
21	C₆₁×#382	C₂²	1	2	yes	—
22	C₄³×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#340	C₂²	1	8	yes	—
24	C₆₁×#336	C₂²	1	2	yes	—
25	C₂²×D₄×C₆₁	—	2	2	no	—
26	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
27	C₂³×C₄×C₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
29	#127×C₆₁	—	2	2	no	—
30	#127×C₆₁	Q₈	1	4	yes	—
31	#125×C₆₁	—	4	4	no	—
32	C₄×D₄×C₆₁	—	2	8	no	—
33	C₄×D₄×C₆₁	—	4	8	no	—
34	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
42	C₂×C₄²×C₆₁	D₄	1	2	yes	—
43	C₂×C₄²×C₆₁	D₄	1	2	yes	—
44	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₆₁	—	2	4	no	—
48	#102×C₆₁	—	4	8	no	—
49	#97×C₆₁	—	2	4	no	—
50	#95×C₆₁	—	2	4	no	—
51	C₂⁴×C₆₁	—	2	2	no	—
52	C₂×D₄×C₆₁	—	2	8	no	—
53	C₂×D₄×C₆₁	—	2	4	no	—
54	C₂×D₄×C₆₁	—	2	2	no	—
55	C₂×D₄×C₆₁	—	2	4	no	—
56	C₂×D₄×C₆₁	—	4	4	no	—
57	C₂×D₄×C₆₁	—	4	4	no	—
58	C₂×D₄×C₆₁	C₂×Q₈	1	4	yes	—
59	C₂²×C₄×C₆₁	—	2	4	no	—
60	C₂²×C₄×C₆₁	—	2	2	no	—
61	C₂²×C₄×C₆₁	—	4	4	no	—
62	C₂²×C₄×C₆₁	—	4	8	no	—
63	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₆₁	—	2	4	no	—
69	C₂×C₈×C₆₁	—	2	4	no	—
70	C₂×C₈×C₆₁	—	2	4	no	—
71	C₄⋊C₄×C₆₁	—	2	4	no	—
72	C₄⋊C₄×C₆₁	—	2	8	no	—
73	C₄⋊C₄×C₆₁	—	4	4	no	—
74	C₄⋊C₄×C₆₁	—	4	4	no	—
75	C₄⋊C₄×C₆₁	—	4	8	no	—
76	C₄⋊C₄×C₆₁	—	4	4	no	—
77	C₂²⋊C₄×C₆₁	—	4	8	no	—
78	C₂²⋊C₄×C₆₁	—	4	4	no	—
79	C₄²×C₆₁	—	2	4	no	—
80	C₄²×C₆₁	—	2	2	no	—
81	C₄²×C₆₁	—	4	2	no	—
82	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
83	C₄²×C₆₁	#41	1	2	yes	—
84	C₄²×C₆₁	#41	1	4	yes	—
85	C₂³×C₆₁	—	2	4	no	—
86	C₂³×C₆₁	—	4	2	no	—
87	C₂³×C₆₁	—	4	2	no	—
88	C₂³×C₆₁	—	4	4	no	—
89	C₂³×C₆₁	—	4	4	no	—
90	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
91	D₄×C₆₁	—	2	8	no	—
92	D₄×C₆₁	—	4	16	no	—
93	D₄×C₆₁	—	4	4	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	2	2	no	—
96	C₂×C₄×C₆₁	—	2	2	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	—	2	2	no	—
99	C₂×C₄×C₆₁	—	2	2	no	—
100	C₂×C₄×C₆₁	—	4	2	no	—
101	C₂×C₄×C₆₁	—	4	4	no	—
102	C₂×C₄×C₆₁	—	4	4	no	—
103	C₂×C₄×C₆₁	—	4	4	no	—
104	C₂×C₄×C₆₁	—	4	4	no	—
105	C₂×C₄×C₆₁	—	8	8	no	—
106	C₂×C₄×C₆₁	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₆₁	#124	1	2	yes	—
108	C₂×C₄×C₆₁	#124	1	2	yes	—
109	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
110	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
112	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
114	C₈×C₆₁	—	4	4	no	—
115	#29149	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₆₁	—	4	4	no	—
117	C₂²×C₆₁	—	4	8	no	—
118	C₂²×C₆₁	—	8	2	no	—
119	C₂²×C₆₁	—	8	2	no	—
120	C₂²×C₆₁	—	8	4	no	—
121	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₆₁	C₂×#124	1	1	yes	characteristic
123	C₂²×C₆₁	#535	1	1	yes	characteristic
124	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
125	C₂²×C₆₁	#575	1	2	yes	—
126	C₄×C₆₁	—	2	4	no	—
127	C₄×C₆₁	—	2	2	no	—
128	C₄×C₆₁	—	4	2	no	—
129	C₄×C₆₁	—	8	2	no	—
130	C₄×C₆₁	—	8	4	no	—
131	C₄×C₆₁	#490	1	4	yes	—
132	C₄×C₆₁	#554	1	2	yes	—
133	C₄×C₆₁	#555	1	2	yes	—
134	#2859	C₂×C₆₁	1	1	yes	characteristic, complemented
135	C₄×#127	C₂×C₆₁	1	1	yes	characteristic
136	#2025	C₂×C₆₁	1	4	yes	complemented
137	#1961	C₂×C₆₁	1	2	yes	—
138	#1836	C₂×C₆₁	1	2	yes	complemented
139	#1802	C₂×C₆₁	1	4	yes	complemented
140	#1767	C₂×C₆₁	1	1	yes	characteristic, complemented
141	C₂×C₆₁	—	8	4	no	—
142	C₂×C₆₁	#2563	1	1	yes	characteristic
143	C₂×C₆₁	#3077	1	2	yes	—
144	C₂×C₆₁	C₂×#490	1	1	yes	characteristic
145	C₂×C₆₁	#3142	1	1	yes	characteristic
146	C₂×C₆₁	#3214	1	2	yes	—
147	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
148	C₂×#125	C₂²×C₆₁	1	2	yes	—
149	C₂×C₄×D₄	—	2	2	no	—
150	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
152	C₂×#102	C₂²×C₆₁	1	8	yes	—
153	C₂×#97	C₂²×C₆₁	1	2	yes	—
154	#390	—	2	2	no	—
155	#390	C₂²×C₆₁	1	2	yes	—
156	#389	—	2	2	no	—
157	#389	—	2	2	no	—
158	#384	C₂²×C₆₁	1	1	yes	characteristic
159	#384	C₂²×C₆₁	1	2	yes	—
160	#382	C₂²×C₆₁	1	2	yes	—
161	C₄³	C₂²×C₆₁	1	1	yes	characteristic
162	#340	C₂²×C₆₁	1	8	yes	—
163	#336	C₂²×C₆₁	1	2	yes	—
164	C₆₁	#29149	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×D₄	—	2	2	no	—
166	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
169	#127	—	2	2	no	—
170	#127	Q₈×C₆₁	1	4	yes	—
171	#125	—	4	4	no	—
172	C₄×D₄	—	2	8	no	—
173	C₄×D₄	—	4	8	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
179	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
180	C₂×C₂²⋊C₄	—	2	4	no	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₄²	D₄×C₆₁	1	2	yes	—
183	C₂×C₄²	D₄×C₆₁	1	2	yes	—
184	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
187	C₄⋊C₈	—	2	4	no	—
188	#102	—	4	8	no	—
189	#97	—	2	4	no	—
190	#95	—	2	4	no	—
191	C₂⁴	—	2	2	no	—
192	C₂×D₄	—	2	8	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	4	4	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	C₂×Q₈×C₆₁	1	4	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	8	no	—
203	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	8	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	4	8	no	—
218	C₂²⋊C₄	—	4	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
223	C₄²	#41×C₆₁	1	2	yes	—
224	C₄²	#41×C₆₁	1	4	yes	—
225	C₂³	—	2	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	4	4	no	—
229	C₂³	—	4	2	no	—
230	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
231	D₄	—	2	8	no	—
232	D₄	—	4	16	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	8	8	no	—
246	C₂×C₄	C₂²⋊Q₈×C₆₁	1	4	yes	—
247	C₂×C₄	#124×C₆₁	1	2	yes	—
248	C₂×C₄	#124×C₆₁	1	2	yes	—
249	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	4	yes	—
250	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
252	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
253	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
254	C₈	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	8	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
261	C₂²	C₂×#124×C₆₁	1	1	yes	characteristic
262	C₂²	C₆₁×#535	1	1	yes	characteristic
263	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	2	yes	—
264	C₂²	C₆₁×#575	1	2	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	4	no	—
270	C₄	C₆₁×#490	1	4	yes	—
271	C₄	C₆₁×#554	1	2	yes	—
272	C₄	C₆₁×#555	1	2	yes	—
273	C₂	—	8	4	no	—
274	C₂	C₆₁×#2563	1	1	yes	characteristic
275	C₂	C₆₁×#3077	1	2	yes	—
276	C₂	C₂×C₆₁×#490	1	1	yes	characteristic
277	C₂	C₆₁×#3142	1	1	yes	characteristic
278	C₂	C₆₁×#3214	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×#127×C₆₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#2859	C₂	semidirect
3. C₆₁×#2025	C₂	semidirect
5. C₆₁×#1836	C₂	semidirect
6. C₆₁×#1802	C₂	semidirect
7. C₆₁×#1767	C₂	semidirect
115. #29149	C₆₁	direct (N × H)
134. #2859	C₂×C₆₁	semidirect
136. #2025	C₂×C₆₁	semidirect
138. #1836	C₂×C₆₁	semidirect
139. #1802	C₂×C₆₁	semidirect
140. #1767	C₂×C₆₁	semidirect
164. C₆₁	#29149	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#2859	2
C₄×#127×C₆₁	2
C₆₁×#2025	2
C₆₁×#1961	2
C₆₁×#1836	2
C₆₁×#1802	2
C₆₁×#1767	2
#29149	61

Subgroups — C61×#29149 #29532582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29149 #29532582