Subgroups — C₂²×#106×D₆₁ #30543653

Order 15616.

3614 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×D₆₁	C₂	1	2	yes	maximal
3	C₂²×#14519709	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14519708	C₂	1	2	yes	maximal, complemented
5	C₂×#106×D₆₁	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#106×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₆₁	—	2	2	no	—
8	C₂³×C₄×D₆₁	C₂²	1	1	yes	characteristic
9	C₂³×#92418	C₂²	1	1	yes	characteristic
10	C₂²×C₈×D₆₁	C₄	1	16	yes	complemented
11	C₂²×C₈×D₆₁	C₂²	1	12	yes	—
12	C₂²×C₄⋊Dic₆₁	C₂²	1	2	yes	—
13	C₂²×#11758196	—	2	2	no	—
14	C₂×C₄⋊C₄×D₆₁	—	2	12	no	—
15	C₂×C₄⋊C₄×D₆₁	C₂²	1	24	yes	—
16	C₂×#14519709	C₂²	1	12	yes	complemented
17	C₂×#14519708	C₂²	1	24	yes	complemented
18	#106×D₆₁	C₂²	1	64	yes	direct product factor
19	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
21	C₂×#106×C₆₁	C₂²	1	12	yes	complemented
22	C₂⁴×D₆₁	D₄	1	1	yes	characteristic
23	C₂³×Dic₆₁	—	2	2	no	—
24	C₂³×Dic₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×D₆₁	—	2	24	no	—
26	C₂²×C₄×D₆₁	—	4	16	no	—
27	C₂²×C₄×D₆₁	C₂×C₄	1	8	yes	—
28	C₂²×C₄×D₆₁	D₄	1	6	yes	—
29	C₂²×C₄×D₆₁	Q₈	1	8	yes	—
30	C₂²×C₄×D₆₁	C₂³	1	6	yes	—
31	C₂²×#92418	C₂×C₄	1	8	yes	complemented
32	C₂²×#92418	C₂³	1	6	yes	—
33	C₂×C₈×D₆₁	C₂×C₄	1	96	yes	complemented
34	C₂×C₈×D₆₁	C₂³	1	16	yes	—
35	C₂×C₄⋊Dic₆₁	—	2	6	no	—
36	C₂×C₄⋊Dic₆₁	C₂³	1	12	yes	—
37	C₂×#11758196	—	2	24	no	—
38	C₄⋊C₄×D₆₁	—	2	48	no	—
39	C₄⋊C₄×D₆₁	C₂³	1	32	yes	—
40	#14519709	C₂³	1	16	yes	complemented
41	#14519708	C₂³	1	32	yes	complemented
42	C₂³×C₄×C₆₁	—	2	2	no	—
43	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	complemented
45	C₂²×C₈×C₆₁	C₂³	1	6	yes	—
46	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
47	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
48	#106×C₆₁	C₂³	1	16	yes	complemented
49	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂³×D₆₁	C₄⋊C₄	1	8	yes	—
51	C₂³×D₆₁	SD₁₆	1	16	yes	—
52	C₂³×D₆₁	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₆₁	—	2	12	no	—
54	C₂²×C₄×C₆₁	—	4	8	no	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
59	C₂²×Dic₆₁	—	2	12	no	—
60	C₂²×Dic₆₁	—	4	8	no	—
61	C₂²×Dic₆₁	C₄⋊C₄	1	8	yes	—
62	C₂²×Dic₆₁	C₂×D₄	1	6	yes	—
63	C₂×C₄×D₆₁	—	2	32	no	—
64	C₂×C₄×D₆₁	—	4	96	no	—
65	C₂×C₄×D₆₁	C₄⋊C₄	1	48	yes	—
66	C₂×C₄×D₆₁	C₂²×C₄	1	24	yes	—
67	C₂×C₄×D₆₁	C₂×D₄	1	12	yes	—
68	C₂×C₄×D₆₁	C₂×Q₈	1	24	yes	—
69	C₂×C₄×D₆₁	C₂⁴	1	4	yes	—
70	C₂×C₈×C₆₁	C₂²×C₄	1	24	yes	complemented
71	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
72	C₄⋊C₄×C₆₁	—	2	12	no	—
73	C₄⋊C₄×C₆₁	C₂⁴	1	8	yes	—
74	C₄⋊Dic₆₁	—	2	12	no	—
75	C₄⋊Dic₆₁	C₂⁴	1	8	yes	—
76	#11758196	—	2	32	no	—
77	C₂×#92418	C₂²×C₄	1	24	yes	complemented
78	C₂×#92418	C₂⁴	1	4	yes	—
79	C₈×D₆₁	C₂²×C₄	1	64	yes	complemented
80	C₂³×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
81	C₂³×C₆₁	C₂×SD₁₆	1	8	yes	—
82	C₂³×C₆₁	C₂²×D₄	1	3	yes	—
83	C₂²×D₆₁	#106	1	64	yes	direct product factor
84	C₂²×D₆₁	C₂×C₄⋊C₄	1	24	yes	—
85	C₂²×D₆₁	C₂×SD₁₆	1	48	yes	—
86	C₂²×D₆₁	C₂²×D₄	1	4	yes	—
87	C₂×C₄×C₆₁	—	2	8	no	—
88	C₂×C₄×C₆₁	—	4	24	no	—
89	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	12	yes	—
90	C₂×C₄×C₆₁	C₂³×C₄	1	6	yes	—
91	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
92	C₂×C₄×C₆₁	C₂²×Q₈	1	6	yes	—
93	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
94	C₂×Dic₆₁	—	2	8	no	—
95	C₂×Dic₆₁	—	4	24	no	—
96	C₂×Dic₆₁	C₂×C₄⋊C₄	1	24	yes	—
97	C₂×Dic₆₁	C₂²×D₄	1	4	yes	—
98	C₄×D₆₁	—	4	64	no	—
99	C₄×D₆₁	C₂×C₄⋊C₄	1	48	yes	—
100	C₄×D₆₁	C₂³×C₄	1	8	yes	—
101	C₄×D₆₁	C₂²×Q₈	1	8	yes	—
102	C₈×C₆₁	C₂³×C₄	1	8	yes	complemented
103	#92418	C₂³×C₄	1	8	yes	complemented
104	C₂³×#106	—	61	1	no	maximal, Sylow
105	C₂²×C₆₁	C₂×#106	1	16	yes	complemented
106	C₂²×C₆₁	C₂²×C₄⋊C₄	1	6	yes	—
107	C₂²×C₆₁	C₂²×SD₁₆	1	12	yes	—
108	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
109	C₂×D₆₁	C₂×#106	1	96	yes	direct product factor
110	C₂×D₆₁	C₂²×C₄⋊C₄	1	8	yes	—
111	C₂×D₆₁	C₂²×SD₁₆	1	16	yes	—
112	C₄×C₆₁	—	4	8	no	—
113	C₄×C₆₁	C₂²×C₄⋊C₄	1	6	yes	—
114	C₄×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
115	C₄×C₆₁	C₂³×Q₈	1	1	yes	characteristic
116	Dic₆₁	—	4	8	no	—
117	Dic₆₁	C₂²×C₄⋊C₄	1	8	yes	—
118	C₂⁴×C₈	—	61	1	no	—
119	C₂³×C₄⋊C₄	—	61	2	no	—
120	C₂²×#106	—	61	24	no	—
121	C₂²×#106	—	61	1	no	—
122	C₂²×#106	—	61	2	no	—
123	C₂²×#106	D₆₁	1	1	yes	characteristic, direct product factor
124	C₂×C₆₁	C₂²×#106	1	12	yes	complemented
125	C₂×C₆₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₆₁	C₂³×SD₁₆	1	2	yes	—
127	D₆₁	C₂²×#106	1	16	yes	direct product factor
128	C₂⁴×C₄	—	61	1	no	—
129	C₂⁴×C₄	—	122	2	no	—
130	C₂³×C₈	—	61	1	no	—
131	C₂³×C₈	—	61	16	no	—
132	C₂³×C₈	—	61	12	no	—
133	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
134	C₂²×C₄⋊C₄	—	61	2	no	—
135	C₂²×C₄⋊C₄	—	61	24	no	—
136	C₂²×C₄⋊C₄	—	122	12	no	—
137	C₂²×C₄⋊C₄	—	122	2	no	—
138	C₂²×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
139	C₂×#106	—	61	12	no	—
140	C₂×#106	—	61	24	no	—
141	C₂×#106	—	61	64	no	—
142	C₂×#106	C₂×D₆₁	1	12	yes	direct product factor
143	C₆₁	C₂³×#106	1	1	yes	characteristic, complemented, Sylow
144	C₂⁵	—	61	1	no	—
145	C₂³×C₄	—	2	2	no	—
146	C₂³×C₄	—	61	1	no	—
147	C₂³×C₄	—	61	8	no	—
148	C₂³×C₄	—	61	6	no	—
149	C₂³×C₄	—	61	8	no	—
150	C₂³×C₄	—	61	6	no	—
151	C₂³×C₄	—	122	24	no	—
152	C₂³×C₄	—	122	2	no	—
153	C₂³×C₄	—	244	16	no	—
154	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
155	C₂²×C₈	—	61	8	no	—
156	C₂²×C₈	—	61	6	no	—
157	C₂²×C₈	—	61	96	no	—
158	C₂²×C₈	—	61	16	no	—
159	C₂²×C₈	C₄×D₆₁	1	8	yes	complemented
160	C₂²×C₈	C₂²×D₆₁	1	6	yes	—
161	C₂×C₄⋊C₄	—	2	6	no	—
162	C₂×C₄⋊C₄	—	61	12	no	—
163	C₂×C₄⋊C₄	—	61	32	no	—
164	C₂×C₄⋊C₄	—	122	48	no	—
165	C₂×C₄⋊C₄	—	122	24	no	—
166	C₂×C₄⋊C₄	—	122	6	no	—
167	C₂×C₄⋊C₄	C₂²×D₆₁	1	12	yes	—
168	#106	—	61	16	no	—
169	#106	—	61	32	no	—
170	#106	C₂²×D₆₁	1	16	yes	direct product factor
171	C₂⁴	—	61	6	no	—
172	C₂⁴	—	61	8	no	—
173	C₂⁴	—	61	16	no	—
174	C₂⁴	D₄×D₆₁	1	1	yes	characteristic
175	C₂²×C₄	—	2	12	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	—	61	6	no	—
178	C₂²×C₄	—	61	8	no	—
179	C₂²×C₄	—	61	24	no	—
180	C₂²×C₄	—	61	12	no	—
181	C₂²×C₄	—	61	24	no	—
182	C₂²×C₄	—	61	4	no	—
183	C₂²×C₄	—	61	48	no	—
184	C₂²×C₄	—	122	32	no	—
185	C₂²×C₄	—	122	12	no	—
186	C₂²×C₄	—	244	96	no	—
187	C₂²×C₄	—	244	8	no	—
188	C₂²×C₄	C₂×C₄×D₆₁	1	4	yes	—
189	C₂²×C₄	D₄×D₆₁	1	3	yes	—
190	C₂²×C₄	Q₈×D₆₁	1	4	yes	—
191	C₂²×C₄	C₂³×D₆₁	1	3	yes	—
192	C₂×C₈	—	61	24	no	—
193	C₂×C₈	—	61	4	no	—
194	C₂×C₈	—	61	64	no	—
195	C₂×C₈	C₂×C₄×D₆₁	1	24	yes	complemented
196	C₂×C₈	C₂³×D₆₁	1	4	yes	—
197	C₄⋊C₄	—	2	12	no	—
198	C₄⋊C₄	—	61	8	no	—
199	C₄⋊C₄	—	122	32	no	—
200	C₄⋊C₄	—	122	12	no	—
201	C₄⋊C₄	C₂³×D₆₁	1	8	yes	—
202	C₂³	—	61	24	no	—
203	C₂³	—	61	48	no	—
204	C₂³	—	61	64	no	—
205	C₂³	—	61	4	no	—
206	C₂³	C₄⋊C₄×D₆₁	1	4	yes	—
207	C₂³	SD₁₆×D₆₁	1	8	yes	—
208	C₂³	C₂×D₄×D₆₁	1	3	yes	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	4	24	no	—
211	C₂×C₄	—	61	24	no	—
212	C₂×C₄	—	61	4	no	—
213	C₂×C₄	—	61	48	no	—
214	C₂×C₄	—	61	8	no	—
215	C₂×C₄	—	61	8	no	—
216	C₂×C₄	—	122	8	no	—
217	C₂×C₄	—	244	64	no	—
218	C₂×C₄	—	244	24	no	—
219	C₂×C₄	C₄⋊C₄×D₆₁	1	12	yes	—
220	C₂×C₄	C₂²×C₄×D₆₁	1	6	yes	—
221	C₂×C₄	C₂×D₄×D₆₁	1	3	yes	—
222	C₂×C₄	C₂×Q₈×D₆₁	1	6	yes	—
223	C₂×C₄	C₂⁴×D₆₁	1	1	yes	characteristic
224	C₈	—	61	8	no	—
225	C₈	C₂²×C₄×D₆₁	1	8	yes	complemented
226	C₂²	—	61	8	no	—
227	C₂²	—	61	96	no	—
228	C₂²	—	61	16	no	—
229	C₂²	#106×D₆₁	1	16	yes	direct product factor
230	C₂²	C₂×C₄⋊C₄×D₆₁	1	6	yes	—
231	C₂²	C₂×SD₁₆×D₆₁	1	12	yes	—
232	C₂²	C₂²×D₄×D₆₁	1	1	yes	characteristic
233	C₄	—	4	8	no	—
234	C₄	—	61	8	no	—
235	C₄	—	244	8	no	—
236	C₄	C₂×C₄⋊C₄×D₆₁	1	6	yes	—
237	C₄	C₂³×C₄×D₆₁	1	1	yes	characteristic
238	C₄	C₂²×Q₈×D₆₁	1	1	yes	characteristic
239	C₂	—	61	16	no	—
240	C₂	C₂×#106×D₆₁	1	12	yes	direct product factor
241	C₂	C₂²×C₄⋊C₄×D₆₁	1	1	yes	characteristic
242	C₂	C₂²×SD₁₆×D₆₁	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₆₁
commutator	C₄×C₆₁	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₁
Fitting	C₂²×#106×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂×D₄
prime radical	C₂⁴×D₆₁	D₄
cosocle	C₂×C₄×C₆₁	C₂⁵
hypercenter	C₂²×#106	D₆₁
hypocenter	C₆₁	C₂³×#106
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
3. C₂²×#14519709	C₂	semidirect
4. C₂²×#14519708	C₂	semidirect
5. C₂×#106×D₆₁	C₂	direct (N × H)
6. C₂²×#106×C₆₁	C₂	semidirect
10. C₂²×C₈×D₆₁	C₄	semidirect
16. C₂×#14519709	C₂²	semidirect
17. C₂×#14519708	C₂²	semidirect
18. #106×D₆₁	C₂²	direct (N × H)
21. C₂×#106×C₆₁	C₂²	semidirect
31. C₂²×#92418	C₂×C₄	semidirect
33. C₂×C₈×D₆₁	C₂×C₄	semidirect
40. #14519709	C₂³	semidirect
41. #14519708	C₂³	semidirect
44. C₂²×C₈×C₆₁	C₂×C₄	semidirect
48. #106×C₆₁	C₂³	semidirect
70. C₂×C₈×C₆₁	C₂²×C₄	semidirect
77. C₂×#92418	C₂²×C₄	semidirect
79. C₈×D₆₁	C₂²×C₄	semidirect
83. C₂²×D₆₁	#106	direct (N × H)
102. C₈×C₆₁	C₂³×C₄	semidirect
103. #92418	C₂³×C₄	semidirect
105. C₂²×C₆₁	C₂×#106	semidirect
109. C₂×D₆₁	C₂×#106	direct (N × H)
123. C₂²×#106	D₆₁	direct (N × H)
124. C₂×C₆₁	C₂²×#106	semidirect
127. D₆₁	C₂²×#106	direct (N × H)
142. C₂×#106	C₂×D₆₁	direct (N × H)
143. C₆₁	C₂³×#106	semidirect
159. C₂²×C₈	C₄×D₆₁	semidirect
170. #106	C₂²×D₆₁	direct (N × H)
195. C₂×C₈	C₂×C₄×D₆₁	semidirect
225. C₈	C₂²×C₄×D₆₁	semidirect
229. C₂²	#106×D₆₁	direct (N × H)
240. C₂	C₂×#106×D₆₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×D₆₁	2
C₂²×C₄⋊C₄×D₆₁	2
C₂²×#14519709	2
C₂²×#14519708	2
C₂×#106×D₆₁	2
C₂²×#106×C₆₁	2
C₂³×#106	61

Subgroups — C22×#106×D61 #30543653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#106×D₆₁ #30543653