Subgroups — C₆₁×#3381 #21443422

Order 7808.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#554	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#499	C₂	1	2	yes	maximal, complemented
4	C₆₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#479	C₂	1	2	yes	maximal, complemented
6	C₆₁×#478	C₂	1	2	yes	maximal
7	C₆₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#474	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
11	C₄×SD₁₆×C₆₁	C₂	1	2	yes	maximal
12	C₂×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₆₁	—	2	1	no	—
14	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	#125×C₆₁	—	2	2	no	—
17	#125×C₆₁	C₂²	1	2	yes	—
18	#125×C₆₁	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₆₁	—	2	1	no	—
20	C₂²⋊Q₈×C₆₁	—	2	2	no	—
21	C₄×Q₈×C₆₁	—	2	1	no	—
22	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
24	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₆₁	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
28	#107×C₆₁	C₂²	1	2	yes	—
29	#107×C₆₁	C₂²	1	2	yes	—
30	#106×C₆₁	C₂²	1	2	yes	—
31	#106×C₆₁	C₂²	1	1	yes	characteristic
32	#106×C₆₁	C₂²	1	2	yes	—
33	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
35	#103×C₆₁	C₂²	1	2	yes	—
36	#103×C₆₁	C₂²	1	1	yes	characteristic
37	#102×C₆₁	C₂²	1	2	yes	—
38	#102×C₆₁	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₆₁	C₂²	1	2	yes	—
41	C₂×Q₈×C₆₁	—	2	1	no	—
42	C₂×Q₈×C₆₁	—	2	1	no	—
43	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	—	2	1	no	—
47	C₂²×C₄×C₆₁	—	2	2	no	—
48	SD₁₆×C₆₁	Q₈	1	4	yes	complemented
49	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂³	1	2	yes	—
51	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₆₁	—	2	2	no	—
53	C₄⋊C₄×C₆₁	—	2	1	no	—
54	C₄⋊C₄×C₆₁	—	2	2	no	—
55	C₄⋊C₄×C₆₁	—	2	1	no	—
56	C₄⋊C₄×C₆₁	—	2	2	no	—
57	C₄⋊C₄×C₆₁	—	2	2	no	—
58	C₄⋊C₄×C₆₁	—	2	1	no	—
59	C₄⋊C₄×C₆₁	—	2	2	no	—
60	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
62	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
63	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
65	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
66	C₂²⋊C₄×C₆₁	—	2	1	no	—
67	C₂²⋊C₄×C₆₁	—	2	2	no	—
68	C₄²×C₆₁	—	2	2	no	—
69	C₄²×C₆₁	—	2	1	no	—
70	C₄²×C₆₁	C₂³	1	2	yes	—
71	C₄²×C₆₁	C₂³	1	1	yes	characteristic
72	C₂³×C₆₁	—	2	1	no	—
73	Q₈×C₆₁	—	2	2	no	—
74	Q₈×C₆₁	—	2	1	no	—
75	Q₈×C₆₁	—	4	2	no	—
76	Q₈×C₆₁	—	8	1	no	—
77	Q₈×C₆₁	C₂×Q₈	1	2	yes	—
78	D₄×C₆₁	—	2	1	no	—
79	D₄×C₆₁	C₂×Q₈	1	2	yes	—
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	2	1	no	—
82	C₂×C₄×C₆₁	—	2	2	no	—
83	C₂×C₄×C₆₁	—	2	1	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	4	2	no	—
86	C₂×C₄×C₆₁	—	8	2	no	—
87	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
92	C₈×C₆₁	—	2	1	no	—
93	C₈×C₆₁	—	2	2	no	—
94	C₈×C₆₁	C₂×Q₈	1	2	yes	—
95	C₂²×C₆₁	—	2	2	no	—
96	C₂²×C₆₁	—	4	1	no	—
97	C₂²×C₆₁	—	4	1	no	—
98	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₆₁	—	2	2	no	—
100	C₄×C₆₁	—	2	2	no	—
101	C₄×C₆₁	—	2	2	no	—
102	C₄×C₆₁	—	2	1	no	—
103	C₄×C₆₁	—	4	1	no	—
104	C₄×C₆₁	—	4	2	no	—
105	C₄×C₆₁	—	4	1	no	—
106	C₄×C₆₁	—	4	2	no	—
107	C₄×C₆₁	#135	1	2	yes	—
108	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
109	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
110	#3381	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₆₁	—	4	2	no	—
112	C₂×C₆₁	D₄×Q₈	1	1	yes	characteristic
113	C₂×C₆₁	C₂×#135	1	1	yes	characteristic
114	C₂×C₆₁	#577	1	1	yes	characteristic
115	#557	C₂×C₆₁	1	1	yes	characteristic, complemented
116	#554	C₂×C₆₁	1	1	yes	characteristic
117	#499	C₂×C₆₁	1	2	yes	complemented
118	#498	C₂×C₆₁	1	1	yes	characteristic, complemented
119	#479	C₂×C₆₁	1	2	yes	complemented
120	#478	C₂×C₆₁	1	2	yes	—
121	#477	C₂×C₆₁	1	1	yes	characteristic, complemented
122	#474	C₂×C₆₁	1	1	yes	characteristic
123	C₈×Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
124	C₄×SD₁₆	C₂×C₆₁	1	1	yes	characteristic
125	C₄×SD₁₆	C₂×C₆₁	1	2	yes	—
126	C₆₁	#3381	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
128	C₄⋊Q₈	—	2	1	no	—
129	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
130	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
131	#125	—	2	2	no	—
132	#125	C₂²×C₆₁	1	2	yes	—
133	#125	C₂²×C₆₁	1	2	yes	—
134	C₂²⋊Q₈	—	2	1	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
138	C₄×Q₈	C₂²×C₆₁	1	2	yes	—
139	C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
140	C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
141	C₄×D₄	C₂²×C₆₁	1	2	yes	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	#107	C₂²×C₆₁	1	2	yes	—
144	#107	C₂²×C₆₁	1	2	yes	—
145	#106	C₂²×C₆₁	1	2	yes	—
146	#106	C₂²×C₆₁	1	1	yes	characteristic
147	#106	C₂²×C₆₁	1	2	yes	—
148	C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
149	C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
150	#103	C₂²×C₆₁	1	2	yes	—
151	#103	C₂²×C₆₁	1	1	yes	characteristic
152	#102	C₂²×C₆₁	1	2	yes	—
153	#102	C₂²×C₆₁	1	1	yes	characteristic
154	C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₆₁	1	2	yes	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	D₄×C₆₁	1	1	yes	characteristic
159	C₂×Q₈	C₂³×C₆₁	1	1	yes	characteristic
160	C₂×D₄	C₂³×C₆₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	2	no	—
163	SD₁₆	Q₈×C₆₁	1	4	yes	complemented
164	C₂×C₈	C₂³×C₆₁	1	1	yes	characteristic
165	C₂×C₈	C₂³×C₆₁	1	2	yes	—
166	C₂×C₈	C₂³×C₆₁	1	1	yes	characteristic
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
176	C₄⋊C₄	D₄×C₆₁	1	2	yes	—
177	C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
178	C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
180	C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
181	C₂²⋊C₄	—	2	1	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	1	no	—
185	C₄²	C₂³×C₆₁	1	2	yes	—
186	C₄²	C₂³×C₆₁	1	1	yes	characteristic
187	C₂³	—	2	1	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	2	1	no	—
190	Q₈	—	4	2	no	—
191	Q₈	—	8	1	no	—
192	Q₈	C₂×Q₈×C₆₁	1	2	yes	—
193	D₄	—	2	1	no	—
194	D₄	C₂×Q₈×C₆₁	1	2	yes	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
203	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
204	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
206	C₂×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	2	no	—
209	C₈	C₂×Q₈×C₆₁	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	1	no	—
218	C₄	—	4	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	2	no	—
222	C₄	#135×C₆₁	1	2	yes	—
223	C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
224	C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
225	C₂	—	4	2	no	—
226	C₂	D₄×Q₈×C₆₁	1	1	yes	characteristic
227	C₂	C₂×#135×C₆₁	1	1	yes	characteristic
228	C₂	C₆₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₁
prime radical	C₂×D₄×C₆₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#557	C₂	semidirect
3. C₆₁×#499	C₂	semidirect
4. C₆₁×#498	C₂	semidirect
5. C₆₁×#479	C₂	semidirect
7. C₆₁×#477	C₂	semidirect
9. C₈×Q₈×C₆₁	C₂	semidirect
48. SD₁₆×C₆₁	Q₈	semidirect
110. #3381	C₆₁	direct (N × H)
115. #557	C₂×C₆₁	semidirect
117. #499	C₂×C₆₁	semidirect
118. #498	C₂×C₆₁	semidirect
119. #479	C₂×C₆₁	semidirect
121. #477	C₂×C₆₁	semidirect
123. C₈×Q₈	C₂×C₆₁	semidirect
126. C₆₁	#3381	direct (N × H)
163. SD₁₆	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#557	2
C₆₁×#554	2
C₆₁×#499	2
C₆₁×#498	2
C₆₁×#479	2
C₆₁×#478	2
C₆₁×#477	2
C₆₁×#474	2
C₈×Q₈×C₆₁	2
C₄×SD₁₆×C₆₁	2
#3381	61

Subgroups — C61×#3381 #21443422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#3381 #21443422