Subgroups — C₆₁×#33740 #29537173

Order 15616.

532 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3176	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2244	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2218	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2215	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2177	C₂	1	2	yes	maximal, complemented
8	C₄×D₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
9	C₆₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#540	—	2	2	no	—
11	C₂×D₁₆×C₆₁	C₂²	1	2	yes	—
12	C₂×D₁₆×C₆₁	C₂²	1	2	yes	—
13	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#499	C₂²	1	2	yes	—
16	C₆₁×#493	C₂²	1	2	yes	—
17	C₆₁×#490	—	2	2	no	—
18	C₆₁×#469	—	2	2	no	—
19	C₆₁×#466	C₂²	1	4	yes	—
20	C₆₁×#442	—	2	2	no	—
21	C₄×D₈×C₆₁	C₂²	1	4	yes	—
22	C₆₁×#405	C₂²	1	1	yes	characteristic
23	C₆₁×#367	C₂²	1	2	yes	—
24	C₆₁×#366	C₂²	1	2	yes	—
25	C₆₁×#357	C₂²	1	4	yes	—
26	C₆₁×#357	C₂²	1	4	yes	—
27	C₆₁×#346	C₂²	1	1	yes	characteristic
28	C₆₁×#346	C₂²	1	1	yes	characteristic
29	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×D₈×C₆₁	—	2	2	no	—
32	C₂×D₈×C₆₁	C₂³	1	4	yes	—
33	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	#127×C₆₁	—	2	2	no	—
36	#125×C₆₁	—	2	2	no	—
37	#123×C₆₁	—	4	4	no	—
38	#121×C₆₁	—	2	4	no	—
39	#121×C₆₁	—	4	4	no	—
40	C₄×D₄×C₆₁	—	2	4	no	—
41	#117×C₆₁	D₄	1	1	yes	characteristic
42	D₁₆×C₆₁	—	2	4	no	—
43	D₁₆×C₆₁	—	2	4	no	—
44	M₃₂×C₆₁	—	2	2	no	—
45	C₂×C₁₆×C₆₁	—	2	2	no	—
46	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
50	#107×C₆₁	C₂³	1	4	yes	—
51	#106×C₆₁	—	2	2	no	—
52	#102×C₆₁	—	2	4	no	—
53	#102×C₆₁	—	2	4	no	—
54	#97×C₆₁	D₄	1	1	yes	characteristic
55	#97×C₆₁	D₄	1	1	yes	characteristic
56	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₆₁	—	2	4	no	—
59	C₂×D₄×C₆₁	—	4	4	no	—
60	C₂×D₄×C₆₁	—	4	4	no	—
61	C₂²×C₄×C₆₁	—	4	4	no	—
62	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
63	D₈×C₆₁	—	2	4	no	—
64	D₈×C₆₁	—	2	4	no	—
65	D₈×C₆₁	—	4	4	no	—
66	M₁₆×C₆₁	—	2	2	no	—
67	C₂×C₈×C₆₁	—	2	1	no	—
68	C₂×C₈×C₆₁	—	2	1	no	—
69	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₆₁	—	2	4	no	—
74	C₄⋊C₄×C₆₁	—	4	4	no	—
75	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
76	C₂²⋊C₄×C₆₁	—	4	4	no	—
77	C₂²⋊C₄×C₆₁	—	4	4	no	—
78	C₂²⋊C₄×C₆₁	D₈	1	2	yes	—
79	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₁₆×C₆₁	—	2	1	no	—
82	C₁₆×C₆₁	—	2	1	no	—
83	C₁₆×C₆₁	#41	1	2	yes	—
84	C₁₆×C₆₁	#41	1	2	yes	—
85	C₂³×C₆₁	—	4	4	no	—
86	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
87	D₄×C₆₁	—	4	4	no	—
88	D₄×C₆₁	—	4	4	no	—
89	D₄×C₆₁	—	16	4	no	—
90	D₄×C₆₁	—	16	4	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	2	1	no	—
93	C₂×C₄×C₆₁	—	4	4	no	—
94	C₂×C₄×C₆₁	—	16	4	no	—
95	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₆₁	—	2	1	no	—
100	C₈×C₆₁	—	2	1	no	—
101	C₈×C₆₁	#135	1	2	yes	—
102	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
103	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
104	#33740	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₆₁	—	2	2	no	—
106	C₂²×C₆₁	—	2	1	no	—
107	C₂²×C₆₁	—	8	4	no	—
108	C₂²×C₆₁	—	8	4	no	—
109	C₂²×C₆₁	—	8	4	no	—
110	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
111	C₄×C₆₁	—	2	2	no	—
112	C₄×C₆₁	—	2	2	no	—
113	C₄×C₆₁	—	2	1	no	—
114	C₄×C₆₁	—	8	4	no	—
115	C₄×C₆₁	#532	1	1	yes	characteristic
116	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
117	#3176	C₂×C₆₁	1	2	yes	complemented
118	#2253	C₂×C₆₁	1	1	yes	characteristic, complemented
119	#2246	C₂×C₆₁	1	1	yes	characteristic, complemented
120	#2244	C₂×C₆₁	1	2	yes	complemented
121	#2218	C₂×C₆₁	1	2	yes	complemented
122	#2215	C₂×C₆₁	1	2	yes	complemented
123	#2177	C₂×C₆₁	1	2	yes	complemented
124	C₄×D₁₆	C₂×C₆₁	1	2	yes	complemented
125	#2108	C₂×C₆₁	1	1	yes	characteristic, complemented
126	C₂×C₆₁	—	4	1	no	—
127	C₂×C₆₁	—	16	4	no	—
128	C₂×C₆₁	#3169	1	1	yes	characteristic
129	C₂×C₆₁	#3415	1	2	yes	—
130	#540	—	2	2	no	—
131	C₂×D₁₆	C₂²×C₆₁	1	2	yes	—
132	C₂×D₁₆	C₂²×C₆₁	1	2	yes	—
133	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
134	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
135	#499	C₂²×C₆₁	1	2	yes	—
136	#493	C₂²×C₆₁	1	2	yes	—
137	#490	—	2	2	no	—
138	#469	—	2	2	no	—
139	#466	C₂²×C₆₁	1	4	yes	—
140	#442	—	2	2	no	—
141	C₄×D₈	C₂²×C₆₁	1	4	yes	—
142	#405	C₂²×C₆₁	1	1	yes	characteristic
143	#367	C₂²×C₆₁	1	2	yes	—
144	#366	C₂²×C₆₁	1	2	yes	—
145	#357	C₂²×C₆₁	1	4	yes	—
146	#357	C₂²×C₆₁	1	4	yes	—
147	#346	C₂²×C₆₁	1	1	yes	characteristic
148	#346	C₂²×C₆₁	1	1	yes	characteristic
149	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
150	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
151	C₆₁	#33740	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	C₂³×C₆₁	1	4	yes	—
154	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
156	#127	—	2	2	no	—
157	#125	—	2	2	no	—
158	#123	—	4	4	no	—
159	#121	—	2	4	no	—
160	#121	—	4	4	no	—
161	C₄×D₄	—	2	4	no	—
162	#117	D₄×C₆₁	1	1	yes	characteristic
163	D₁₆	—	2	4	no	—
164	D₁₆	—	2	4	no	—
165	M₃₂	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
168	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
169	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
170	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
171	#107	C₂³×C₆₁	1	4	yes	—
172	#106	—	2	2	no	—
173	#102	—	2	4	no	—
174	#102	—	2	4	no	—
175	#97	D₄×C₆₁	1	1	yes	characteristic
176	#97	D₄×C₆₁	1	1	yes	characteristic
177	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
178	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
184	D₈	—	2	4	no	—
185	D₈	—	2	4	no	—
186	D₈	—	4	4	no	—
187	M₁₆	—	2	2	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
191	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
192	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
193	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₂²⋊C₄	—	4	4	no	—
199	C₂²⋊C₄	D₈×C₆₁	1	2	yes	—
200	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
201	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
202	C₁₆	—	2	1	no	—
203	C₁₆	—	2	1	no	—
204	C₁₆	#41×C₆₁	1	2	yes	—
205	C₁₆	#41×C₆₁	1	2	yes	—
206	C₂³	—	4	4	no	—
207	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	16	4	no	—
211	D₄	—	16	4	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	16	4	no	—
216	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
217	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
218	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
219	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	#135×C₆₁	1	2	yes	—
223	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
224	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
225	C₂²	—	2	2	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	4	no	—
229	C₂²	—	8	4	no	—
230	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	8	4	no	—
235	C₄	C₆₁×#532	1	1	yes	characteristic
236	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
237	C₂	—	4	1	no	—
238	C₂	—	16	4	no	—
239	C₂	C₆₁×#3169	1	1	yes	characteristic
240	C₂	C₆₁×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ #117×C₆₁ ◁ C₆₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3176	C₂	semidirect
2. C₆₁×#2253	C₂	semidirect
3. C₆₁×#2246	C₂	semidirect
4. C₆₁×#2244	C₂	semidirect
5. C₆₁×#2218	C₂	semidirect
6. C₆₁×#2215	C₂	semidirect
7. C₆₁×#2177	C₂	semidirect
8. C₄×D₁₆×C₆₁	C₂	semidirect
9. C₆₁×#2108	C₂	semidirect
104. #33740	C₆₁	direct (N × H)
117. #3176	C₂×C₆₁	semidirect
118. #2253	C₂×C₆₁	semidirect
119. #2246	C₂×C₆₁	semidirect
120. #2244	C₂×C₆₁	semidirect
121. #2218	C₂×C₆₁	semidirect
122. #2215	C₂×C₆₁	semidirect
123. #2177	C₂×C₆₁	semidirect
124. C₄×D₁₆	C₂×C₆₁	semidirect
125. #2108	C₂×C₆₁	semidirect
151. C₆₁	#33740	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×#3176	2
C₆₁×#2253	2
C₆₁×#2246	2
C₆₁×#2244	2
C₆₁×#2218	2
C₆₁×#2215	2
C₆₁×#2177	2
C₄×D₁₆×C₆₁	2
C₆₁×#2108	2
#33740	61

Subgroups — C61×#33740 #29537173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33740 #29537173