Subgroups — C₆₁×#2370 #21442411

Order 7808.

464 conjugacy classes of subgroups.

The subgroup lattice has 246 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
2	C₆₁×#389	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#389	C₂	1	2	yes	maximal
4	C₆₁×#389	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#387	C₂	1	2	yes	maximal
6	C₆₁×#387	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₆₁	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
12	#125×C₆₁	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₆₁	C₂²	1	2	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	2	yes	—
27	C₂×C₄²×C₆₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
29	#95×C₆₁	C₂²	1	1	yes	characteristic
30	#95×C₆₁	C₂²	1	1	yes	characteristic
31	#95×C₆₁	C₂²	1	2	yes	—
32	#95×C₆₁	C₂²	1	2	yes	—
33	#95×C₆₁	C₂²	1	2	yes	—
34	#95×C₆₁	C₂²	1	2	yes	—
35	#95×C₆₁	C₂²	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₆₁	—	2	2	no	—
48	C₄⋊C₄×C₆₁	—	2	4	no	—
49	C₄⋊C₄×C₆₁	—	2	2	no	—
50	C₄⋊C₄×C₆₁	—	2	2	no	—
51	C₄⋊C₄×C₆₁	—	4	2	no	—
52	C₄⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
53	C₄⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
54	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
55	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
56	C₄⋊C₄×C₆₁	Q₈	1	4	yes	—
57	C₄²×C₆₁	—	2	4	no	—
58	C₄²×C₆₁	—	2	2	no	—
59	C₄²×C₆₁	—	2	2	no	—
60	C₄²×C₆₁	—	4	2	no	—
61	C₄²×C₆₁	—	4	2	no	—
62	C₄²×C₆₁	C₂×C₄	1	4	yes	—
63	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	1	no	—
66	C₂×C₄×C₆₁	—	2	1	no	—
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	2	1	no	—
72	C₂×C₄×C₆₁	—	2	1	no	—
73	C₂×C₄×C₆₁	—	2	2	no	—
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	—	2	2	no	—
77	C₂×C₄×C₆₁	—	2	1	no	—
78	C₂×C₄×C₆₁	—	2	1	no	—
79	C₂×C₄×C₆₁	—	2	1	no	—
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	2	2	no	—
82	C₂×C₄×C₆₁	—	2	1	no	—
83	C₂×C₄×C₆₁	—	2	2	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
91	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
93	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
94	C₂×C₄×C₆₁	#41	1	2	yes	—
95	C₂×C₄×C₆₁	#41	1	2	yes	—
96	C₂×C₄×C₆₁	#41	1	2	yes	—
97	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
98	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
100	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
101	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
102	C₂²×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂²×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₆₁	—	2	4	no	—
105	C₄×C₆₁	—	2	2	no	—
106	C₄×C₆₁	—	2	2	no	—
107	C₄×C₆₁	—	2	4	no	—
108	C₄×C₆₁	—	2	2	no	—
109	C₄×C₆₁	—	2	2	no	—
110	C₄×C₆₁	—	2	4	no	—
111	C₄×C₆₁	—	4	1	no	—
112	C₄×C₆₁	—	4	1	no	—
113	C₄×C₆₁	—	4	2	no	—
114	C₄×C₆₁	C₄×Q₈	1	4	yes	—
115	#2370	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
117	C₂×C₆₁	C₄×#41	1	1	yes	characteristic
118	C₂×C₆₁	#520	1	1	yes	characteristic
119	C₂×C₆₁	#552	1	1	yes	characteristic
120	C₂×C₆₁	#553	1	1	yes	characteristic
121	C₂×C₆₁	#554	1	1	yes	characteristic
122	C₂×C₆₁	#557	1	1	yes	characteristic
123	C₂×#125	C₂×C₆₁	1	1	yes	characteristic
124	#389	C₂×C₆₁	1	1	yes	characteristic
125	#389	C₂×C₆₁	1	2	yes	—
126	#389	C₂×C₆₁	1	1	yes	characteristic
127	#387	C₂×C₆₁	1	2	yes	—
128	#387	C₂×C₆₁	1	2	yes	—
129	C₄×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic
130	C₄×C₄⋊C₄	C₂×C₆₁	1	2	yes	—
131	C₄×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic
132	C₄×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic
133	C₄×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic
134	C₆₁	#2370	1	1	yes	characteristic, direct product factor, Sylow
135	#125	C₄×C₆₁	1	8	yes	complemented
136	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
141	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
142	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
146	C₂×C₄²	C₂²×C₆₁	1	2	yes	—
147	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
148	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
149	C₂×C₄²	C₂²×C₆₁	1	2	yes	—
150	C₂×C₄²	C₂²×C₆₁	1	2	yes	—
151	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
152	#95	C₂²×C₆₁	1	1	yes	characteristic
153	#95	C₂²×C₆₁	1	1	yes	characteristic
154	#95	C₂²×C₆₁	1	2	yes	—
155	#95	C₂²×C₆₁	1	2	yes	—
156	#95	C₂²×C₆₁	1	2	yes	—
157	#95	C₂²×C₆₁	1	2	yes	—
158	#95	C₂²×C₆₁	1	2	yes	—
159	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
162	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
164	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
166	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
167	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	C₂×C₄×C₆₁	1	8	yes	—
176	C₄⋊C₄	C₂×C₄×C₆₁	1	8	yes	—
177	C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
178	C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
179	C₄⋊C₄	Q₈×C₆₁	1	4	yes	—
180	C₄²	—	2	4	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₂×C₄×C₆₁	1	4	yes	—
186	C₂³	C₂⁴×C₆₁	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	C₂²×C₄×C₆₁	1	2	yes	—
211	C₂×C₄	C₂²×C₄×C₆₁	1	2	yes	—
212	C₂×C₄	C₂²×C₄×C₆₁	1	4	yes	—
213	C₂×C₄	C₂²×C₄×C₆₁	1	4	yes	—
214	C₂×C₄	C₂²×C₄×C₆₁	1	2	yes	—
215	C₂×C₄	C₂×Q₈×C₆₁	1	4	yes	—
216	C₂×C₄	C₂×Q₈×C₆₁	1	2	yes	—
217	C₂×C₄	#41×C₆₁	1	2	yes	—
218	C₂×C₄	#41×C₆₁	1	2	yes	—
219	C₂×C₄	#41×C₆₁	1	2	yes	—
220	C₂²	C₂³×C₄×C₆₁	1	1	yes	characteristic
221	C₂²	C₂²×Q₈×C₆₁	1	1	yes	characteristic
222	C₂²	C₂×#41×C₆₁	1	1	yes	characteristic
223	C₂²	C₂×#41×C₆₁	1	1	yes	characteristic
224	C₂²	C₂×#41×C₆₁	1	1	yes	characteristic
225	C₂²	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
226	C₂²	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	4	no	—
230	C₄	—	2	4	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	C₄×Q₈×C₆₁	1	4	yes	—
238	C₂	C₂×C₄×Q₈×C₆₁	1	1	yes	characteristic
239	C₂	C₄×#41×C₆₁	1	1	yes	characteristic
240	C₂	C₆₁×#520	1	1	yes	characteristic
241	C₂	C₆₁×#552	1	1	yes	characteristic
242	C₂	C₆₁×#553	1	1	yes	characteristic
243	C₂	C₆₁×#554	1	1	yes	characteristic
244	C₂	C₆₁×#557	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
12. #125×C₆₁	C₄	semidirect
115. #2370	C₆₁	direct (N × H)
134. C₆₁	#2370	direct (N × H)
135. #125	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×#125×C₆₁	2
C₆₁×#389	2
C₆₁×#387	2
C₄×C₄⋊C₄×C₆₁	2
#2370	61

Subgroups — C61×#2370 #21442411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#2370 #21442411