Subgroups — C₆₁×#2566 #21442607

Order 7808.

548 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#396	C₂	1	4	yes	maximal, complemented
4	C₆₁×#390	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#390	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#385	C₂	1	4	yes	maximal, complemented
7	C₆₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₆₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₆₁	C₂²	1	2	yes	—
12	#125×C₆₁	—	2	4	no	—
13	C₄×D₄×C₆₁	—	2	4	no	—
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
22	C₂×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
24	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
27	#95×C₆₁	C₂²	1	2	yes	—
28	#95×C₆₁	C₂²	1	4	yes	—
29	C₂⁴×C₆₁	C₂³	1	2	yes	—
30	C₂×D₄×C₆₁	—	2	2	no	—
31	C₂×D₄×C₆₁	—	2	2	no	—
32	C₂×D₄×C₆₁	Q₈	1	4	yes	—
33	C₂²×C₄×C₆₁	—	2	4	no	—
34	C₂²×C₄×C₆₁	—	4	4	no	—
35	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₆₁	—	2	2	no	—
44	C₄⋊C₄×C₆₁	—	2	4	no	—
45	C₄⋊C₄×C₆₁	—	2	4	no	—
46	C₄⋊C₄×C₆₁	—	2	2	no	—
47	C₄⋊C₄×C₆₁	—	2	2	no	—
48	C₄⋊C₄×C₆₁	—	2	4	no	—
49	C₄⋊C₄×C₆₁	—	4	4	no	—
50	C₂²⋊C₄×C₆₁	—	2	4	no	—
51	C₂²⋊C₄×C₆₁	—	2	8	no	—
52	C₂²⋊C₄×C₆₁	—	4	4	no	—
53	C₄²×C₆₁	—	2	2	no	—
54	C₄²×C₆₁	—	2	2	no	—
55	C₄²×C₆₁	D₄	1	4	yes	—
56	C₂³×C₆₁	—	2	2	no	—
57	C₂³×C₆₁	—	2	4	no	—
58	C₂³×C₆₁	—	2	2	no	—
59	C₂³×C₆₁	—	2	4	no	—
60	C₂³×C₆₁	C₂×Q₈	1	4	yes	—
61	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
62	D₄×C₆₁	—	2	8	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	—	2	4	no	—
72	C₂×C₄×C₆₁	—	2	2	no	—
73	C₂×C₄×C₆₁	—	2	2	no	—
74	C₂×C₄×C₆₁	—	2	4	no	—
75	C₂×C₄×C₆₁	—	2	4	no	—
76	C₂×C₄×C₆₁	—	4	8	no	—
77	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₆₁	#41	1	4	yes	—
82	C₂×C₄×C₆₁	#41	1	2	yes	—
83	C₂²×C₆₁	—	2	4	no	—
84	C₂²×C₆₁	—	2	8	no	—
85	C₂²×C₆₁	—	4	2	no	—
86	C₂²×C₆₁	—	4	2	no	—
87	C₂²×C₆₁	—	4	4	no	—
88	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
89	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
90	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
91	C₂²×C₆₁	C₂×#41	1	2	yes	—
92	C₂²×C₆₁	H(5,𝔽₂)	1	2	yes	—
93	C₄×C₆₁	—	2	2	no	—
94	C₄×C₆₁	—	2	2	no	—
95	C₄×C₆₁	—	2	4	no	—
96	C₄×C₆₁	—	4	2	no	—
97	C₄×C₆₁	—	4	2	no	—
98	C₄×C₆₁	—	4	2	no	—
99	C₄×C₆₁	—	4	2	no	—
100	C₄×C₆₁	C₂²⋊Q₈	1	4	yes	—
101	#2566	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₆₁	—	4	4	no	—
103	C₂×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₆₁	#532	1	1	yes	characteristic
105	C₂×C₆₁	#535	1	1	yes	characteristic
106	C₂×C₆₁	#553	1	2	yes	—
107	C₂×C₆₁	#554	1	2	yes	—
108	C₂×#125	C₂×C₆₁	1	1	yes	characteristic, complemented
109	C₂×C₄×D₄	C₂×C₆₁	1	1	yes	characteristic
110	#396	C₂×C₆₁	1	4	yes	complemented
111	#390	C₂×C₆₁	1	1	yes	characteristic
112	#390	C₂×C₆₁	1	1	yes	characteristic
113	#385	C₂×C₆₁	1	4	yes	complemented
114	#384	C₂×C₆₁	1	1	yes	characteristic, complemented
115	#382	C₂×C₆₁	1	1	yes	characteristic, complemented
116	C₄×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
117	C₆₁	#2566	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×D₄	C₂²×C₆₁	1	1	yes	characteristic
119	C₂³×C₄	C₂²×C₆₁	1	2	yes	—
120	#125	—	2	4	no	—
121	C₄×D₄	—	2	4	no	—
122	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
128	C₂×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
132	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
133	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
134	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
135	#95	C₂²×C₆₁	1	2	yes	—
136	#95	C₂²×C₆₁	1	4	yes	—
137	C₂⁴	C₂³×C₆₁	1	2	yes	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	Q₈×C₆₁	1	4	yes	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
144	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
145	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
146	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
147	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
150	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	8	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	D₄×C₆₁	1	4	yes	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	4	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	4	no	—
168	C₂³	C₂×Q₈×C₆₁	1	4	yes	—
169	C₂³	C₂⁴×C₆₁	1	1	yes	characteristic
170	D₄	—	2	8	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
186	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
187	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
188	C₂×C₄	C₂×Q₈×C₆₁	1	2	yes	—
189	C₂×C₄	#41×C₆₁	1	4	yes	—
190	C₂×C₄	#41×C₆₁	1	2	yes	—
191	C₂²	—	2	8	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	4	2	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	4	no	—
196	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
197	C₂²	C₂²×Q₈×C₆₁	1	1	yes	characteristic
198	C₂²	C₂×#41×C₆₁	1	2	yes	—
199	C₂²	C₂×#41×C₆₁	1	1	yes	characteristic
200	C₂²	H(5,𝔽₂)×C₆₁	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	4	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₄	C₂²⋊Q₈×C₆₁	1	4	yes	—
209	C₂	—	4	4	no	—
210	C₂	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
211	C₂	C₆₁×#532	1	1	yes	characteristic
212	C₂	C₆₁×#535	1	1	yes	characteristic
213	C₂	C₆₁×#553	1	2	yes	—
214	C₂	C₆₁×#554	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#125×C₆₁	C₂	semidirect
3. C₆₁×#396	C₂	semidirect
6. C₆₁×#385	C₂	semidirect
7. C₆₁×#384	C₂	semidirect
8. C₆₁×#382	C₂	semidirect
9. C₄×C₄⋊C₄×C₆₁	C₂	semidirect
101. #2566	C₆₁	direct (N × H)
108. C₂×#125	C₂×C₆₁	semidirect
110. #396	C₂×C₆₁	semidirect
113. #385	C₂×C₆₁	semidirect
114. #384	C₂×C₆₁	semidirect
115. #382	C₂×C₆₁	semidirect
116. C₄×C₄⋊C₄	C₂×C₆₁	semidirect
117. C₆₁	#2566	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×#125×C₆₁	2
C₂×C₄×D₄×C₆₁	2
C₆₁×#396	2
C₆₁×#390	2
C₆₁×#385	2
C₆₁×#384	2
C₆₁×#382	2
C₄×C₄⋊C₄×C₆₁	2
#2566	61

Subgroups — C61×#2566 #21442607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#2566 #21442607