Subgroups — C₂×C₆₁×#2644 #29538050

Order 15616.

2222 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2644	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₆₁×#397	C₂	1	6	yes	maximal, complemented
3	C₂×C₆₁×#396	C₂	1	2	yes	maximal, complemented
4	C₂×C₆₁×#390	C₂	1	3	yes	maximal
5	C₂×C₆₁×#389	C₂	1	3	yes	maximal, complemented
6	C₂×C₆₁×#376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
9	C₂²×C₄⋊C₄×C₆₁	C₂²	1	6	yes	—
10	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	6	yes	—
11	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	6	yes	—
12	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
13	C₆₁×#397	C₂²	1	48	yes	complemented
14	C₆₁×#396	C₂²	1	16	yes	complemented
15	C₆₁×#390	C₂²	1	24	yes	—
16	C₆₁×#389	C₂²	1	24	yes	complemented
17	C₆₁×#376	C₂²	1	8	yes	complemented
18	C₂×#95×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×#95×C₆₁	C₂²	1	3	yes	—
20	C₂×#95×C₆₁	C₂²	1	6	yes	—
21	C₂⁵×C₆₁	C₂³	1	2	yes	—
22	C₂²×D₄×C₆₁	—	2	12	no	—
23	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
24	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
25	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
26	C₂³×C₄×C₆₁	C₂³	1	6	yes	—
27	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
29	C₂×C₄⋊C₄×C₆₁	—	4	12	no	—
30	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
31	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
32	C₂×C₂²⋊C₄×C₆₁	—	4	12	no	—
33	C₂×C₂²⋊C₄×C₆₁	—	4	12	no	—
34	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
35	C₂×C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
36	C₂×C₄²×C₆₁	—	2	12	no	—
37	C₂×C₄²×C₆₁	C₂³	1	12	yes	—
38	#95×C₆₁	C₂³	1	4	yes	—
39	#95×C₆₁	C₂³	1	12	yes	—
40	#95×C₆₁	C₂³	1	24	yes	—
41	C₂⁴×C₆₁	—	2	12	no	—
42	C₂⁴×C₆₁	—	2	12	no	—
43	C₂⁴×C₆₁	—	2	4	no	—
44	C₂⁴×C₆₁	C₂⁴	1	4	yes	—
45	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₂×D₄×C₆₁	—	2	24	no	—
47	C₂×D₄×C₆₁	—	4	16	no	—
48	C₂²×C₄×C₆₁	—	2	6	no	—
49	C₂²×C₄×C₆₁	—	2	6	no	—
50	C₂²×C₄×C₆₁	—	2	12	no	—
51	C₂²×C₄×C₆₁	—	2	6	no	—
52	C₂²×C₄×C₆₁	—	2	6	no	—
53	C₂²×C₄×C₆₁	—	2	12	no	—
54	C₂²×C₄×C₆₁	—	2	12	no	—
55	C₂²×C₄×C₆₁	—	2	6	no	—
56	C₂²×C₄×C₆₁	—	2	6	no	—
57	C₂²×C₄×C₆₁	#41	1	12	yes	—
58	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
59	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
60	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
61	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
62	C₄⋊C₄×C₆₁	—	2	24	no	—
63	C₄⋊C₄×C₆₁	—	4	24	no	—
64	C₂²⋊C₄×C₆₁	—	4	24	no	—
65	C₂²⋊C₄×C₆₁	—	4	24	no	—
66	C₄²×C₆₁	—	2	24	no	—
67	C₂³×C₆₁	—	2	12	no	—
68	C₂³×C₆₁	—	2	12	no	—
69	C₂³×C₆₁	—	2	4	no	—
70	C₂³×C₆₁	—	4	8	no	—
71	C₂³×C₆₁	—	4	24	no	—
72	C₂³×C₆₁	—	4	24	no	—
73	C₂³×C₆₁	C₂×#41	1	6	yes	—
74	C₂³×C₆₁	H(5,𝔽₂)	1	6	yes	—
75	C₂³×C₆₁	2₋¹⁺⁴	1	2	yes	—
76	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
77	D₄×C₆₁	—	4	16	no	—
78	C₂×C₄×C₆₁	—	2	6	no	—
79	C₂×C₄×C₆₁	—	2	6	no	—
80	C₂×C₄×C₆₁	—	2	12	no	—
81	C₂×C₄×C₆₁	—	2	6	no	—
82	C₂×C₄×C₆₁	—	2	6	no	—
83	C₂×C₄×C₆₁	—	2	12	no	—
84	C₂×C₄×C₆₁	—	2	12	no	—
85	C₂×C₄×C₆₁	—	2	6	no	—
86	C₂×C₄×C₆₁	—	2	6	no	—
87	C₂×C₄×C₆₁	—	4	12	no	—
88	C₂×C₄×C₆₁	—	4	24	no	—
89	C₂×C₄×C₆₁	—	4	12	no	—
90	C₂×C₄×C₆₁	#126	1	16	yes	—
91	C₂×C₄×C₆₁	C₂×#41	1	12	yes	—
92	C₂×#2644	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₆₁	—	4	4	no	—
94	C₂²×C₆₁	—	4	12	no	—
95	C₂²×C₆₁	—	4	12	no	—
96	C₂²×C₆₁	—	8	16	no	—
97	C₂²×C₆₁	C₂×#126	1	4	yes	—
98	C₂²×C₆₁	#540	1	12	yes	—
99	C₂²×C₆₁	#542	1	12	yes	—
100	C₂²×C₆₁	C₂²×#41	1	3	yes	—
101	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	3	yes	—
102	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₆₁	—	4	6	no	—
104	C₄×C₆₁	—	4	12	no	—
105	C₄×C₆₁	—	4	6	no	—
106	C₄×C₆₁	C₂×#126	1	8	yes	—
107	#2644	C₂×C₆₁	1	16	yes	direct product factor
108	C₂×#397	C₂×C₆₁	1	6	yes	complemented
109	C₂×#396	C₂×C₆₁	1	2	yes	complemented
110	C₂×#390	C₂×C₆₁	1	3	yes	—
111	C₂×#389	C₂×C₆₁	1	3	yes	complemented
112	C₂×#376	C₂×C₆₁	1	1	yes	characteristic, complemented
113	C₂×C₆₁	—	8	4	no	—
114	C₂×C₆₁	#2644	1	8	yes	direct product factor
115	C₂×C₆₁	C₂²×#126	1	1	yes	characteristic
116	C₂×C₆₁	C₂×#540	1	3	yes	—
117	C₂×C₆₁	C₂×#542	1	3	yes	—
118	C₂³×D₄	C₂²×C₆₁	1	1	yes	characteristic
119	C₂²×C₄⋊C₄	C₂²×C₆₁	1	3	yes	—
120	C₂²×C₄⋊C₄	C₂²×C₆₁	1	6	yes	—
121	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	6	yes	—
122	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	6	yes	—
123	C₂²×C₄²	C₂²×C₆₁	1	3	yes	—
124	#397	C₂²×C₆₁	1	48	yes	complemented
125	#396	C₂²×C₆₁	1	16	yes	complemented
126	#390	C₂²×C₆₁	1	24	yes	—
127	#389	C₂²×C₆₁	1	24	yes	complemented
128	#376	C₂²×C₆₁	1	8	yes	complemented
129	C₂×#95	C₂²×C₆₁	1	1	yes	characteristic
130	C₂×#95	C₂²×C₆₁	1	3	yes	—
131	C₂×#95	C₂²×C₆₁	1	6	yes	—
132	C₆₁	C₂×#2644	1	1	yes	characteristic, direct product factor, Sylow
133	C₂⁵	C₂³×C₆₁	1	2	yes	—
134	C₂²×D₄	—	2	12	no	—
135	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
136	C₂³×C₄	C₂³×C₆₁	1	6	yes	—
137	C₂³×C₄	C₂³×C₆₁	1	3	yes	—
138	C₂³×C₄	C₂³×C₆₁	1	3	yes	—
139	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	2	12	no	—
141	C₂×C₄⋊C₄	—	4	12	no	—
142	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
143	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
144	C₂×C₂²⋊C₄	—	4	12	no	—
145	C₂×C₂²⋊C₄	—	4	12	no	—
146	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
147	C₂×C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
148	C₂×C₄²	—	2	12	no	—
149	C₂×C₄²	C₂³×C₆₁	1	12	yes	—
150	#95	C₂³×C₆₁	1	4	yes	—
151	#95	C₂³×C₆₁	1	12	yes	—
152	#95	C₂³×C₆₁	1	24	yes	—
153	C₂⁴	—	2	12	no	—
154	C₂⁴	—	2	12	no	—
155	C₂⁴	—	2	4	no	—
156	C₂⁴	C₂⁴×C₆₁	1	4	yes	—
157	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
158	C₂×D₄	—	2	24	no	—
159	C₂×D₄	—	4	16	no	—
160	C₂²×C₄	—	2	6	no	—
161	C₂²×C₄	—	2	6	no	—
162	C₂²×C₄	—	2	12	no	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	12	no	—
166	C₂²×C₄	—	2	12	no	—
167	C₂²×C₄	—	2	6	no	—
168	C₂²×C₄	—	2	6	no	—
169	C₂²×C₄	#41×C₆₁	1	12	yes	—
170	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
171	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
172	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
173	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
174	C₄⋊C₄	—	2	24	no	—
175	C₄⋊C₄	—	4	24	no	—
176	C₂²⋊C₄	—	4	24	no	—
177	C₂²⋊C₄	—	4	24	no	—
178	C₄²	—	2	24	no	—
179	C₂³	—	2	12	no	—
180	C₂³	—	2	12	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	4	24	no	—
183	C₂³	—	4	24	no	—
184	C₂³	—	4	8	no	—
185	C₂³	C₂×#41×C₆₁	1	6	yes	—
186	C₂³	H(5,𝔽₂)×C₆₁	1	6	yes	—
187	C₂³	2₋¹⁺⁴×C₆₁	1	2	yes	—
188	C₂³	C₂⁵×C₆₁	1	1	yes	characteristic
189	D₄	—	4	16	no	—
190	C₂×C₄	—	2	6	no	—
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	6	no	—
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	2	12	no	—
196	C₂×C₄	—	2	12	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	4	12	no	—
200	C₂×C₄	—	4	24	no	—
201	C₂×C₄	—	4	12	no	—
202	C₂×C₄	#126×C₆₁	1	16	yes	—
203	C₂×C₄	C₂×#41×C₆₁	1	12	yes	—
204	C₂²	—	4	4	no	—
205	C₂²	—	4	12	no	—
206	C₂²	—	4	12	no	—
207	C₂²	—	8	16	no	—
208	C₂²	C₂×#126×C₆₁	1	4	yes	—
209	C₂²	C₆₁×#540	1	12	yes	—
210	C₂²	C₆₁×#542	1	12	yes	—
211	C₂²	C₂²×#41×C₆₁	1	3	yes	—
212	C₂²	C₂×H(5,𝔽₂)×C₆₁	1	3	yes	—
213	C₂²	C₂×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
214	C₄	—	4	6	no	—
215	C₄	—	4	12	no	—
216	C₄	—	4	6	no	—
217	C₄	C₂×#126×C₆₁	1	8	yes	—
218	C₂	—	8	4	no	—
219	C₂	C₆₁×#2644	1	8	yes	direct product factor
220	C₂	C₂²×#126×C₆₁	1	1	yes	characteristic
221	C₂	C₂×C₆₁×#540	1	3	yes	—
222	C₂	C₂×C₆₁×#542	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₁
prime radical	C₂³×D₄×C₆₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#2644	C₂	direct (N × H)
2. C₂×C₆₁×#397	C₂	semidirect
3. C₂×C₆₁×#396	C₂	semidirect
5. C₂×C₆₁×#389	C₂	semidirect
6. C₂×C₆₁×#376	C₂	semidirect
13. C₆₁×#397	C₂²	semidirect
14. C₆₁×#396	C₂²	semidirect
16. C₆₁×#389	C₂²	semidirect
17. C₆₁×#376	C₂²	semidirect
92. C₂×#2644	C₆₁	direct (N × H)
107. #2644	C₂×C₆₁	direct (N × H)
108. C₂×#397	C₂×C₆₁	semidirect
109. C₂×#396	C₂×C₆₁	semidirect
111. C₂×#389	C₂×C₆₁	semidirect
112. C₂×#376	C₂×C₆₁	semidirect
114. C₂×C₆₁	#2644	direct (N × H)
124. #397	C₂²×C₆₁	semidirect
125. #396	C₂²×C₆₁	semidirect
127. #389	C₂²×C₆₁	semidirect
128. #376	C₂²×C₆₁	semidirect
132. C₆₁	C₂×#2644	direct (N × H)
219. C₂	C₆₁×#2644	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×#2644	2
C₂×C₆₁×#397	2
C₂×C₆₁×#396	2
C₂×C₆₁×#390	2
C₂×C₆₁×#389	2
C₂×C₆₁×#376	2
C₂×#2644	61

Subgroups — C2×C61×#2644 #29538050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#2644 #29538050