Subgroups — C₆₁×#33748 #29537181

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3169	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2240	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2218	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2177	C₂	1	4	yes	maximal, complemented
7	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#532	—	2	2	no	—
9	C₂×D₁₆×C₆₁	C₂²	1	4	yes	—
10	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#500	C₂²	1	2	yes	—
12	C₆₁×#493	C₂²	1	2	yes	—
13	C₆₁×#486	—	2	2	no	—
14	C₆₁×#466	—	2	2	no	—
15	C₆₁×#466	C₂²	1	4	yes	—
16	C₄×D₈×C₆₁	—	2	2	no	—
17	C₄×D₈×C₆₁	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#367	C₂²	1	4	yes	—
20	C₆₁×#357	C₂²	1	8	yes	—
21	C₆₁×#346	C₂²	1	2	yes	—
22	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×#41×C₆₁	—	4	2	no	—
25	C₂×D₈×C₆₁	—	2	2	no	—
26	C₂×D₈×C₆₁	C₂³	1	4	yes	—
27	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₆₁	—	2	2	no	—
30	#127×C₆₁	—	2	2	no	—
31	#124×C₆₁	—	4	2	no	—
32	#121×C₆₁	—	2	4	no	—
33	#121×C₆₁	—	4	2	no	—
34	C₄×D₄×C₆₁	—	2	4	no	—
35	C₄×D₄×C₆₁	—	4	2	no	—
36	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
37	D₁₆×C₆₁	—	2	8	no	—
38	M₃₂×C₆₁	—	2	4	no	—
39	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
40	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
41	#107×C₆₁	—	2	2	no	—
42	#107×C₆₁	C₂³	1	4	yes	—
43	#102×C₆₁	—	2	4	no	—
44	#102×C₆₁	—	2	4	no	—
45	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
49	#41×C₆₁	—	8	8	no	—
50	C₂×Q₈×C₆₁	—	4	2	no	—
51	C₂×D₄×C₆₁	—	2	4	no	—
52	C₂×D₄×C₆₁	—	4	4	no	—
53	C₂×D₄×C₆₁	—	4	2	no	—
54	C₂×D₄×C₆₁	—	4	2	no	—
55	C₂²×C₄×C₆₁	—	4	4	no	—
56	C₂²×C₄×C₆₁	D₈	1	2	yes	—
57	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	D₈×C₆₁	—	2	8	no	—
59	D₈×C₆₁	—	4	4	no	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₆₁	—	2	4	no	—
67	C₄⋊C₄×C₆₁	—	4	2	no	—
68	C₂²⋊C₄×C₆₁	—	4	4	no	—
69	C₂²⋊C₄×C₆₁	—	4	4	no	—
70	C₄²×C₆₁	D₈	1	2	yes	—
71	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₆₁	—	2	2	no	—
74	C₁₆×C₆₁	#41	1	4	yes	complemented
75	C₂³×C₆₁	—	4	4	no	—
76	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
77	Q₈×C₆₁	—	8	4	no	—
78	D₄×C₆₁	—	4	4	no	—
79	D₄×C₆₁	—	4	4	no	—
80	D₄×C₆₁	—	8	8	no	—
81	D₄×C₆₁	—	8	4	no	—
82	D₄×C₆₁	—	8	4	no	—
83	C₂×C₄×C₆₁	—	2	1	no	—
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	4	no	—
86	C₂×C₄×C₆₁	—	4	4	no	—
87	C₂×C₄×C₆₁	—	8	8	no	—
88	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
92	C₈×C₆₁	—	2	1	no	—
93	C₈×C₆₁	—	2	1	no	—
94	C₈×C₆₁	#135	1	2	yes	—
95	C₈×C₆₁	C₂×#41	1	2	yes	—
96	#33748	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₆₁	—	2	2	no	—
98	C₂²×C₆₁	—	2	1	no	—
99	C₂²×C₆₁	—	8	4	no	—
100	C₂²×C₆₁	—	8	4	no	—
101	C₂²×C₆₁	—	8	4	no	—
102	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
103	C₄×C₆₁	—	2	2	no	—
104	C₄×C₆₁	—	2	1	no	—
105	C₄×C₆₁	—	8	4	no	—
106	C₄×C₆₁	#509	1	4	yes	—
107	C₄×C₆₁	#532	1	1	yes	characteristic
108	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
109	#3169	C₂×C₆₁	1	2	yes	complemented
110	#2254	C₂×C₆₁	1	1	yes	characteristic, complemented
111	#2246	C₂×C₆₁	1	1	yes	characteristic, complemented
112	#2240	C₂×C₆₁	1	2	yes	complemented
113	#2218	C₂×C₆₁	1	4	yes	complemented
114	#2177	C₂×C₆₁	1	4	yes	complemented
115	C₄×M₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
116	C₂×C₆₁	—	4	1	no	—
117	C₂×C₆₁	—	16	4	no	—
118	C₂×C₆₁	#3169	1	1	yes	characteristic
119	C₂×C₆₁	C₂×#509	1	2	yes	—
120	#532	—	2	2	no	—
121	C₂×D₁₆	C₂²×C₆₁	1	4	yes	—
122	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
123	#500	C₂²×C₆₁	1	2	yes	—
124	#493	C₂²×C₆₁	1	2	yes	—
125	#486	—	2	2	no	—
126	#466	—	2	2	no	—
127	#466	C₂²×C₆₁	1	4	yes	—
128	C₄×D₈	—	2	2	no	—
129	C₄×D₈	C₂²×C₆₁	1	4	yes	—
130	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
131	#367	C₂²×C₆₁	1	4	yes	—
132	#357	C₂²×C₆₁	1	8	yes	—
133	#346	C₂²×C₆₁	1	2	yes	—
134	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
135	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
136	C₆₁	#33748	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#41	—	4	2	no	—
138	C₂×D₈	—	2	2	no	—
139	C₂×D₈	C₂³×C₆₁	1	4	yes	—
140	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
141	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
142	C₄⋊Q₈	—	2	2	no	—
143	#127	—	2	2	no	—
144	#124	—	4	2	no	—
145	#121	—	2	4	no	—
146	#121	—	4	2	no	—
147	C₄×D₄	—	2	4	no	—
148	C₄×D₄	—	4	2	no	—
149	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
150	D₁₆	—	2	8	no	—
151	M₃₂	—	2	4	no	—
152	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
153	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
154	#107	—	2	2	no	—
155	#107	C₂³×C₆₁	1	4	yes	—
156	#102	—	2	4	no	—
157	#102	—	2	4	no	—
158	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
159	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
160	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
161	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
162	#41	—	8	8	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	4	4	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	D₈×C₆₁	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
171	D₈	—	2	8	no	—
172	D₈	—	4	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
178	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	4	no	—
182	C₂²⋊C₄	—	4	4	no	—
183	C₄²	D₈×C₆₁	1	2	yes	—
184	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
185	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
186	C₁₆	—	2	2	no	—
187	C₁₆	#41×C₆₁	1	4	yes	complemented
188	C₂³	—	4	4	no	—
189	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
190	Q₈	—	8	4	no	—
191	D₄	—	4	4	no	—
192	D₄	—	4	4	no	—
193	D₄	—	8	8	no	—
194	D₄	—	8	4	no	—
195	D₄	—	8	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	8	no	—
201	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
203	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
204	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	#135×C₆₁	1	2	yes	—
208	C₈	C₂×#41×C₆₁	1	2	yes	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	8	4	no	—
212	C₂²	—	8	4	no	—
213	C₂²	—	8	4	no	—
214	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	4	no	—
218	C₄	C₆₁×#509	1	4	yes	—
219	C₄	C₆₁×#532	1	1	yes	characteristic
220	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	—	16	4	no	—
223	C₂	C₆₁×#3169	1	1	yes	characteristic
224	C₂	C₂×C₆₁×#509	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 ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3169	C₂	semidirect
2. C₆₁×#2254	C₂	semidirect
3. C₆₁×#2246	C₂	semidirect
4. C₆₁×#2240	C₂	semidirect
5. C₆₁×#2218	C₂	semidirect
6. C₆₁×#2177	C₂	semidirect
7. C₄×M₃₂×C₆₁	C₂	semidirect
74. C₁₆×C₆₁	#41	semidirect
96. #33748	C₆₁	direct (N × H)
109. #3169	C₂×C₆₁	semidirect
110. #2254	C₂×C₆₁	semidirect
111. #2246	C₂×C₆₁	semidirect
112. #2240	C₂×C₆₁	semidirect
113. #2218	C₂×C₆₁	semidirect
114. #2177	C₂×C₆₁	semidirect
115. C₄×M₃₂	C₂×C₆₁	semidirect
136. C₆₁	#33748	direct (N × H)
187. C₁₆	#41×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3169	2
C₆₁×#2254	2
C₆₁×#2246	2
C₆₁×#2240	2
C₆₁×#2218	2
C₆₁×#2177	2
C₄×M₃₂×C₆₁	2
#33748	61

Subgroups — C61×#33748 #29537181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33748 #29537181