Subgroups — C₂×#131×D₆₁ #21460522

Order 7808.

1504 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519785	C₂	1	3	yes	maximal, complemented
3	C₂×#14519784	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14519783	C₂	1	3	yes	maximal, complemented
5	#131×D₆₁	C₂	1	16	yes	maximal, direct product factor
6	C₂×M₁₆×D₆₁	C₂	1	3	yes	maximal, complemented
7	C₂²×C₈×D₆₁	C₂	1	3	yes	maximal, complemented
8	C₂×#131×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11758226	C₄	1	1	yes	characteristic
10	C₂×#11758222	C₂²	1	3	yes	—
11	C₂×#11758224	C₄	1	3	yes	—
12	#41×D₆₁	C₄	1	8	yes	—
13	#41×D₆₁	C₂²	1	8	yes	—
14	C₂×Q₈×D₆₁	C₄	1	1	yes	characteristic
15	C₂×D₄×D₆₁	C₄	1	3	yes	—
16	C₂²×C₄×D₆₁	C₂²	1	3	yes	—
17	#14519785	C₂²	1	24	yes	complemented
18	#14519784	C₂²	1	8	yes	complemented
19	#14519783	C₂²	1	24	yes	complemented
20	C₂×#11758194	C₂²	1	3	yes	complemented
21	C₂×#11758189	C₂²	1	6	yes	complemented
22	M₁₆×D₆₁	C₂²	1	24	yes	complemented
23	C₂²×#92418	C₂²	1	3	yes	complemented
24	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₆₁	C₂²	1	3	yes	—
26	C₂×C₈×D₆₁	C₂²	1	24	yes	complemented
27	C₂×C₈×D₆₁	C₂²	1	3	yes	complemented
28	C₂×C₈×D₆₁	C₂²	1	3	yes	complemented
29	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
30	#131×C₆₁	C₂²	1	8	yes	complemented
31	C₂×M₁₆×C₆₁	C₂²	1	3	yes	complemented
32	C₂²×C₈×C₆₁	C₂²	1	3	yes	complemented
33	C₂³×D₆₁	C₂×C₄	1	3	yes	—
34	#41×C₆₁	C₂×C₄	1	4	yes	—
35	#41×C₆₁	C₂³	1	4	yes	—
36	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₆₁	C₂×C₄	1	3	yes	—
38	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
39	C₂×C₂²⋊D₆₁	C₂×C₄	1	6	yes	—
40	C₂²×Dic₆₁	C₂×C₄	1	3	yes	—
41	#11758226	C₂×C₄	1	8	yes	—
42	Q₈×D₆₁	C₂×C₄	1	8	yes	—
43	#11758224	C₂×C₄	1	24	yes	—
44	D₄×D₆₁	C₂×C₄	1	24	yes	—
45	#11758222	C₂×C₄	1	12	yes	—
46	#11758222	C₂³	1	12	yes	—
47	C₂×D₂₄₄	C₂×C₄	1	3	yes	—
48	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
49	C₂×C₄×D₆₁	C₂×C₄	1	12	yes	—
50	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
51	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₆₁	C₂³	1	12	yes	—
53	C₂×C₄×D₆₁	C₂³	1	3	yes	—
54	C₂×Dic₁₂₂	C₂×C₄	1	3	yes	—
55	M₁₆×C₆₁	C₂³	1	12	yes	complemented
56	C₂×C₈×C₆₁	C₂³	1	3	yes	—
57	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂³	1	12	yes	complemented
59	#11758194	C₂³	1	12	yes	complemented
60	C₂×#92418	C₂³	1	1	yes	characteristic
61	C₂×#92418	C₂³	1	3	yes	—
62	C₂×#92418	C₂³	1	12	yes	complemented
63	#11758189	C₂³	1	24	yes	complemented
64	C₈×D₆₁	C₂³	1	4	yes	—
65	C₈×D₆₁	C₂³	1	12	yes	—
66	C₈×D₆₁	C₂³	1	12	yes	complemented
67	C₈×D₆₁	C₂³	1	12	yes	complemented
68	C₂³×C₆₁	C₂²×C₄	1	3	yes	—
69	C₂²×D₆₁	—	2	12	no	—
70	C₂²×D₆₁	C₂²×C₄	1	3	yes	—
71	C₂²×D₆₁	C₂²×C₄	1	12	yes	—
72	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
73	Q₈×C₆₁	C₂²×C₄	1	4	yes	—
74	D₄×C₆₁	C₂²×C₄	1	12	yes	—
75	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
76	C₂×C₄×C₆₁	C₂²×C₄	1	6	yes	—
77	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂⁴	1	6	yes	—
79	C₂²⋊D₆₁	C₂²×C₄	1	24	yes	—
80	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
81	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₆₁	C₂²×C₄	1	12	yes	—
83	D₂₄₄	C₂²×C₄	1	12	yes	—
84	C₄×D₆₁	C₂²×C₄	1	2	yes	—
85	C₄×D₆₁	C₂²×C₄	1	6	yes	—
86	C₄×D₆₁	C₂²×C₄	1	12	yes	—
87	C₄×D₆₁	C₂²×C₄	1	12	yes	—
88	C₄×D₆₁	C₂⁴	1	2	yes	—
89	C₄×D₆₁	C₂⁴	1	6	yes	—
90	Dic₁₂₂	C₂²×C₄	1	12	yes	—
91	C₈×C₆₁	C₂⁴	1	2	yes	—
92	C₈×C₆₁	C₂⁴	1	6	yes	—
93	#92418	C₂⁴	1	2	yes	—
94	#92418	C₂⁴	1	6	yes	—
95	C₂²×C₆₁	—	2	6	no	—
96	C₂²×C₆₁	C₂³×C₄	1	6	yes	—
97	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₆₁	—	2	6	no	—
99	C₂×D₆₁	—	2	24	no	—
100	C₂×D₆₁	#131	1	4	yes	direct product factor
101	C₂×D₆₁	C₂³×C₄	1	6	yes	—
102	C₂×D₆₁	C₂³×C₄	1	2	yes	—
103	C₄×C₆₁	C₂³×C₄	1	6	yes	—
104	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
105	C₄×C₆₁	C₂⁵	1	1	yes	characteristic
106	Dic₆₁	C₂³×C₄	1	6	yes	—
107	Dic₆₁	C₂³×C₄	1	2	yes	—
108	C₂²×#131	—	61	1	no	maximal, Sylow
109	C₂×C₆₁	—	2	6	no	—
110	C₂×C₆₁	C₂×#131	1	2	yes	complemented
111	C₂×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
112	D₆₁	—	2	6	no	—
113	D₆₁	C₂×#131	1	4	yes	direct product factor
114	C₂²×#41	—	61	1	no	—
115	C₂×#131	—	61	1	no	—
116	C₂×#131	—	61	16	no	—
117	C₂×#131	—	61	3	no	—
118	C₂×#131	—	61	3	no	—
119	C₂×#131	D₆₁	1	1	yes	characteristic, direct product factor
120	C₂²×M₁₆	—	61	3	no	—
121	C₂³×C₈	—	61	3	no	—
122	C₆₁	C₂²×#131	1	1	yes	characteristic, complemented, Sylow
123	C₂×#41	—	61	8	no	—
124	C₂×#41	—	61	8	no	—
125	C₂×#41	—	61	1	no	—
126	C₂×#41	—	61	3	no	—
127	C₂×#41	—	61	3	no	—
128	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
129	C₂²×Q₈	—	61	1	no	—
130	C₂²×D₄	—	61	3	no	—
131	C₂³×C₄	—	61	3	no	—
132	#131	—	61	8	no	—
133	#131	—	61	24	no	—
134	#131	—	61	24	no	—
135	#131	C₂×D₆₁	1	8	yes	direct product factor
136	C₂×M₁₆	—	61	3	no	—
137	C₂×M₁₆	—	61	24	no	—
138	C₂×M₁₆	—	61	6	no	—
139	C₂×M₁₆	C₂×D₆₁	1	3	yes	complemented
140	C₂²×C₈	—	61	3	no	—
141	C₂²×C₈	—	61	3	no	—
142	C₂²×C₈	—	61	3	no	—
143	C₂²×C₈	—	61	1	no	—
144	C₂²×C₈	—	61	3	no	—
145	C₂²×C₈	—	61	24	no	—
146	C₂²×C₈	C₂×D₆₁	1	3	yes	complemented
147	C₂⁴	—	61	3	no	—
148	#41	—	61	8	no	—
149	#41	—	61	24	no	—
150	#41	—	61	12	no	—
151	#41	—	61	12	no	—
152	#41	C₄×D₆₁	1	4	yes	—
153	#41	C₂²×D₆₁	1	4	yes	—
154	C₂×Q₈	—	61	8	no	—
155	C₂×Q₈	—	61	3	no	—
156	C₂×Q₈	C₄×D₆₁	1	1	yes	characteristic
157	C₂×D₄	—	61	6	no	—
158	C₂×D₄	—	61	24	no	—
159	C₂×D₄	—	61	3	no	—
160	C₂×D₄	C₄×D₆₁	1	3	yes	—
161	C₂²×C₄	—	61	3	no	—
162	C₂²×C₄	—	61	3	no	—
163	C₂²×C₄	—	61	3	no	—
164	C₂²×C₄	—	61	12	no	—
165	C₂²×C₄	—	61	1	no	—
166	C₂²×C₄	—	61	3	no	—
167	C₂²×C₄	—	61	12	no	—
168	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
169	M₁₆	—	61	12	no	—
170	M₁₆	—	61	24	no	—
171	M₁₆	C₂²×D₆₁	1	12	yes	complemented
172	C₂×C₈	—	61	1	no	—
173	C₂×C₈	—	61	3	no	—
174	C₂×C₈	—	61	12	no	—
175	C₂×C₈	—	61	4	no	—
176	C₂×C₈	—	61	12	no	—
177	C₂×C₈	—	61	12	no	—
178	C₂×C₈	—	61	12	no	—
179	C₂×C₈	C₂²×D₆₁	1	3	yes	—
180	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₆₁	1	12	yes	complemented
182	C₂³	—	61	1	no	—
183	C₂³	—	61	12	no	—
184	C₂³	—	61	3	no	—
185	C₂³	—	122	12	no	—
186	C₂³	C₂×C₄×D₆₁	1	3	yes	—
187	Q₈	—	61	12	no	—
188	Q₈	C₂×C₄×D₆₁	1	4	yes	—
189	D₄	—	61	24	no	—
190	D₄	—	61	12	no	—
191	D₄	C₂×C₄×D₆₁	1	12	yes	—
192	C₂×C₄	—	61	1	no	—
193	C₂×C₄	—	61	3	no	—
194	C₂×C₄	—	61	12	no	—
195	C₂×C₄	—	61	2	no	—
196	C₂×C₄	—	61	12	no	—
197	C₂×C₄	—	61	12	no	—
198	C₂×C₄	—	61	6	no	—
199	C₂×C₄	—	61	2	no	—
200	C₂×C₄	—	61	6	no	—
201	C₂×C₄	C₂×C₄×D₆₁	1	3	yes	—
202	C₂×C₄	C₂×C₄×D₆₁	1	6	yes	—
203	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
204	C₂×C₄	C₂³×D₆₁	1	6	yes	—
205	C₈	—	61	2	no	—
206	C₈	—	61	6	no	—
207	C₈	C₂³×D₆₁	1	2	yes	—
208	C₈	C₂³×D₆₁	1	6	yes	—
209	C₂²	—	2	6	no	—
210	C₂²	—	61	4	no	—
211	C₂²	—	61	2	no	—
212	C₂²	—	61	6	no	—
213	C₂²	—	122	24	no	—
214	C₂²	—	122	6	no	—
215	C₂²	C₂²×C₄×D₆₁	1	6	yes	—
216	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
217	C₄	—	61	2	no	—
218	C₄	—	61	6	no	—
219	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
220	C₄	C₂²×C₄×D₆₁	1	6	yes	—
221	C₄	C₂⁴×D₆₁	1	1	yes	characteristic
222	C₂	—	2	6	no	—
223	C₂	—	61	4	no	—
224	C₂	—	122	6	no	—
225	C₂	#131×D₆₁	1	2	yes	direct product factor
226	C₂	C₂³×C₄×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₆₁
commutator	C₂×C₆₁	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₆₁
Fitting	C₂×#131×C₆₁	C₂
socle	C₂²×C₆₁	C₂³×C₄
hypercenter	C₂×#131	D₆₁
hypocenter	C₆₁	C₂²×#131
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#14519785	C₂	semidirect
3. C₂×#14519784	C₂	semidirect
4. C₂×#14519783	C₂	semidirect
5. #131×D₆₁	C₂	direct (N × H)
6. C₂×M₁₆×D₆₁	C₂	semidirect
7. C₂²×C₈×D₆₁	C₂	semidirect
8. C₂×#131×C₆₁	C₂	semidirect
17. #14519785	C₂²	semidirect
18. #14519784	C₂²	semidirect
19. #14519783	C₂²	semidirect
20. C₂×#11758194	C₂²	semidirect
21. C₂×#11758189	C₂²	semidirect
22. M₁₆×D₆₁	C₂²	semidirect
23. C₂²×#92418	C₂²	semidirect
26. C₂×C₈×D₆₁	C₂²	semidirect
27. C₂×C₈×D₆₁	C₂²	semidirect
28. C₂×C₈×D₆₁	C₂²	semidirect
30. #131×C₆₁	C₂²	semidirect
31. C₂×M₁₆×C₆₁	C₂²	semidirect
32. C₂²×C₈×C₆₁	C₂²	semidirect
55. M₁₆×C₆₁	C₂³	semidirect
58. C₂×C₈×C₆₁	C₂³	semidirect
59. #11758194	C₂³	semidirect
62. C₂×#92418	C₂³	semidirect
63. #11758189	C₂³	semidirect
66. C₈×D₆₁	C₂³	semidirect
67. C₈×D₆₁	C₂³	semidirect
100. C₂×D₆₁	#131	direct (N × H)
110. C₂×C₆₁	C₂×#131	semidirect
113. D₆₁	C₂×#131	direct (N × H)
119. C₂×#131	D₆₁	direct (N × H)
122. C₆₁	C₂²×#131	semidirect
135. #131	C₂×D₆₁	direct (N × H)
139. C₂×M₁₆	C₂×D₆₁	semidirect
146. C₂²×C₈	C₂×D₆₁	semidirect
171. M₁₆	C₂²×D₆₁	semidirect
181. C₂×C₈	C₂²×D₆₁	semidirect
225. C₂	#131×D₆₁	direct (N × H)

Subgroups — C2×#131×D61 #21460522

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#131×D₆₁ #21460522