Subgroups — C₆₁×#33960 #29537393

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3393	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2249	C₂	1	2	yes	maximal
3	C₆₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2211	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2208	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₃₂×C₆₁	C₂	1	2	yes	maximal, complemented
8	C₄×D₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#556	—	2	2	no	—
10	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	—
11	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	—
12	C₂×SD₃₂×C₆₁	C₂²	1	4	yes	—
13	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#495	C₂²	1	4	yes	—
15	C₆₁×#494	C₂²	1	2	yes	—
16	C₆₁×#464	—	2	4	no	—
17	C₆₁×#461	—	2	2	no	—
18	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆×C₆₁	C₂²	1	4	yes	—
20	C₄×D₈×C₆₁	C₂²	1	2	yes	—
21	C₆₁×#366	C₂²	1	1	yes	characteristic
22	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₆₁×#358	C₂²	1	4	yes	—
25	C₆₁×#357	C₂²	1	2	yes	—
26	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
28	C₂×Q₁₆×C₆₁	—	2	2	no	—
29	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
30	C₂×SD₁₆×C₆₁	—	2	4	no	—
31	C₂×D₈×C₆₁	C₂³	1	2	yes	—
32	C₄⋊Q₈×C₆₁	—	2	2	no	—
33	#126×C₆₁	—	4	4	no	—
34	#124×C₆₁	—	2	4	no	—
35	C₂²⋊Q₈×C₆₁	—	4	2	no	—
36	C₄×Q₈×C₆₁	—	2	4	no	—
37	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
38	C₄×D₄×C₆₁	—	2	2	no	—
39	#117×C₆₁	—	4	2	no	—
40	Q₃₂×C₆₁	—	2	4	no	—
41	Q₃₂×C₆₁	—	4	2	no	—
42	SD₃₂×C₆₁	—	4	4	no	—
43	D₁₆×C₆₁	—	2	2	no	—
44	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
46	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
47	#107×C₆₁	C₂³	1	2	yes	—
48	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
49	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
50	#103×C₆₁	—	2	4	no	—
51	#102×C₆₁	—	2	2	no	—
52	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₆₁	C₂³	1	2	yes	—
54	C₂×Q₈×C₆₁	—	2	4	no	—
55	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
56	C₂×D₄×C₆₁	—	2	2	no	—
57	C₂²×C₄×C₆₁	—	4	2	no	—
58	Q₁₆×C₆₁	—	2	4	no	—
59	Q₁₆×C₆₁	—	2	4	no	—
60	Q₁₆×C₆₁	—	8	2	no	—
61	SD₁₆×C₆₁	—	8	4	no	—
62	D₈×C₆₁	—	2	2	no	—
63	D₈×C₆₁	—	2	2	no	—
64	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₆₁	—	2	2	no	—
68	C₄⋊C₄×C₆₁	—	4	2	no	—
69	C₄⋊C₄×C₆₁	—	4	4	no	—
70	C₄⋊C₄×C₆₁	—	4	2	no	—
71	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
72	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
73	C₂²⋊C₄×C₆₁	—	4	4	no	—
74	C₂²⋊C₄×C₆₁	—	4	2	no	—
75	C₂²⋊C₄×C₆₁	—	4	4	no	—
76	C₄²×C₆₁	—	4	4	no	—
77	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₁	C₂×D₄	1	2	yes	—
79	C₁₆×C₆₁	—	2	2	no	—
80	C₁₆×C₆₁	—	2	1	no	—
81	C₁₆×C₆₁	#41	1	2	yes	—
82	C₂³×C₆₁	—	4	2	no	—
83	Q₈×C₆₁	—	2	2	no	—
84	Q₈×C₆₁	—	4	4	no	—
85	Q₈×C₆₁	—	4	4	no	—
86	D₄×C₆₁	—	4	2	no	—
87	D₄×C₆₁	—	4	2	no	—
88	C₂×C₄×C₆₁	—	4	2	no	—
89	C₂×C₄×C₆₁	—	4	4	no	—
90	C₂×C₄×C₆₁	—	8	4	no	—
91	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₆₁	—	2	2	no	—
97	C₈×C₆₁	—	2	1	no	—
98	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
99	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
100	#33960	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₆₁	—	8	2	no	—
102	C₂²×C₆₁	—	8	2	no	—
103	C₂²×C₆₁	—	8	2	no	—
104	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
105	C₄×C₆₁	—	2	2	no	—
106	C₄×C₆₁	—	2	2	no	—
107	C₄×C₆₁	—	2	1	no	—
108	C₄×C₆₁	—	8	4	no	—
109	C₄×C₆₁	—	8	2	no	—
110	C₄×C₆₁	#508	1	2	yes	—
111	C₄×C₆₁	#550	1	1	yes	characteristic
112	C₄×C₆₁	#576	1	1	yes	characteristic
113	#3393	C₂×C₆₁	1	2	yes	complemented
114	#2249	C₂×C₆₁	1	2	yes	—
115	#2247	C₂×C₆₁	1	1	yes	characteristic, complemented
116	#2211	C₂×C₆₁	1	4	yes	complemented
117	#2208	C₂×C₆₁	1	2	yes	complemented
118	Q₈×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
119	C₄×Q₃₂	C₂×C₆₁	1	2	yes	complemented
120	C₄×D₁₆	C₂×C₆₁	1	1	yes	characteristic
121	C₂×C₆₁	—	16	2	no	—
122	C₂×C₆₁	#3391	1	1	yes	characteristic
123	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
124	C₂×C₆₁	#3417	1	1	yes	characteristic
125	#556	—	2	2	no	—
126	C₂×Q₃₂	C₂²×C₆₁	1	2	yes	—
127	C₂×Q₃₂	C₂²×C₆₁	1	2	yes	—
128	C₂×SD₃₂	C₂²×C₆₁	1	4	yes	—
129	C₂×D₁₆	C₂²×C₆₁	1	1	yes	characteristic
130	#495	C₂²×C₆₁	1	4	yes	—
131	#494	C₂²×C₆₁	1	2	yes	—
132	#464	—	2	4	no	—
133	#461	—	2	2	no	—
134	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
135	C₄×Q₁₆	C₂²×C₆₁	1	4	yes	—
136	C₄×D₈	C₂²×C₆₁	1	2	yes	—
137	#366	C₂²×C₆₁	1	1	yes	characteristic
138	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
139	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
140	#358	C₂²×C₆₁	1	4	yes	—
141	#357	C₂²×C₆₁	1	2	yes	—
142	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
143	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
144	C₆₁	#33960	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
147	C₂×SD₁₆	—	2	4	no	—
148	C₂×D₈	C₂³×C₆₁	1	2	yes	—
149	C₄⋊Q₈	—	2	2	no	—
150	#126	—	4	4	no	—
151	#124	—	2	4	no	—
152	C₂²⋊Q₈	—	4	2	no	—
153	C₄×Q₈	—	2	4	no	—
154	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
155	C₄×D₄	—	2	2	no	—
156	#117	—	4	2	no	—
157	Q₃₂	—	2	4	no	—
158	Q₃₂	—	4	2	no	—
159	SD₃₂	—	4	4	no	—
160	D₁₆	—	2	2	no	—
161	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
162	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
163	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
164	#107	C₂³×C₆₁	1	2	yes	—
165	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
166	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
167	#103	—	2	4	no	—
168	#102	—	2	2	no	—
169	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
170	C₄×C₈	C₂³×C₆₁	1	2	yes	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
173	C₂×D₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	Q₁₆	—	2	4	no	—
176	Q₁₆	—	2	4	no	—
177	Q₁₆	—	8	2	no	—
178	SD₁₆	—	8	4	no	—
179	D₈	—	2	2	no	—
180	D₈	—	2	2	no	—
181	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
183	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
189	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
190	C₂²⋊C₄	—	4	4	no	—
191	C₂²⋊C₄	—	4	2	no	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₄²	—	4	4	no	—
194	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₆₁	1	2	yes	—
196	C₁₆	—	2	2	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	#41×C₆₁	1	2	yes	—
199	C₂³	—	4	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	4	4	no	—
202	Q₈	—	4	4	no	—
203	D₄	—	4	2	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	4	no	—
208	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
210	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
211	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
212	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	2	1	no	—
215	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
216	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
217	C₂²	—	8	2	no	—
218	C₂²	—	8	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	4	no	—
225	C₄	—	8	2	no	—
226	C₄	C₆₁×#508	1	2	yes	—
227	C₄	C₆₁×#550	1	1	yes	characteristic
228	C₄	C₆₁×#576	1	1	yes	characteristic
229	C₂	—	16	2	no	—
230	C₂	C₆₁×#3391	1	1	yes	characteristic
231	C₂	C₂×C₆₁×#508	1	1	yes	characteristic
232	C₂	C₆₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₂×D₁₆×C₆₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×Q₈×C₆₁ ◁ C₈×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3393	C₂	semidirect
3. C₆₁×#2247	C₂	semidirect
4. C₆₁×#2211	C₂	semidirect
5. C₆₁×#2208	C₂	semidirect
6. Q₈×C₁₆×C₆₁	C₂	semidirect
7. C₄×Q₃₂×C₆₁	C₂	semidirect
100. #33960	C₆₁	direct (N × H)
113. #3393	C₂×C₆₁	semidirect
115. #2247	C₂×C₆₁	semidirect
116. #2211	C₂×C₆₁	semidirect
117. #2208	C₂×C₆₁	semidirect
118. Q₈×C₁₆	C₂×C₆₁	semidirect
119. C₄×Q₃₂	C₂×C₆₁	semidirect
144. C₆₁	#33960	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×#3393	2
C₆₁×#2249	2
C₆₁×#2247	2
C₆₁×#2211	2
C₆₁×#2208	2
Q₈×C₁₆×C₆₁	2
C₄×Q₃₂×C₆₁	2
C₄×D₁₆×C₆₁	2
#33960	61

Subgroups — C61×#33960 #29537393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33960 #29537393