Subgroups — C₆₁×#33397 #29536830

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₁	C₂	1	2	yes	maximal
2	C₂×C₆₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2151	C₂	1	8	yes	maximal, complemented
4	C₆₁×#2150	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₆₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₆₁	—	2	8	no	—
9	C₂×M₃₂×C₆₁	C₂²	1	4	yes	—
10	C₂×M₃₂×C₆₁	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₆₁	C₂²	1	4	yes	—
12	C₆₁×#405	C₄	1	4	yes	—
13	C₆₁×#405	C₂²	1	4	yes	—
14	C₂×#97×C₆₁	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
16	C₆₁×C₄⋊C₁₆	C₂²	1	16	yes	complemented
17	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₆₁	—	2	4	no	—
19	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₁	D₄	1	4	yes	—
24	C₂²×C₈×C₆₁	Q₈	1	4	yes	—
25	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
27	#117×C₆₁	C₈	1	4	yes	—
28	#117×C₆₁	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
30	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
31	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
32	M₃₂×C₆₁	—	2	8	no	—
33	M₃₂×C₆₁	—	2	8	no	—
34	C₂×C₁₆×C₆₁	—	2	8	no	—
35	C₂×C₁₆×C₆₁	C₂³	1	8	yes	—
36	#97×C₆₁	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₆₁	C₂³	1	4	yes	—
39	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
45	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
47	M₁₆×C₆₁	—	2	8	no	—
48	C₂×C₈×C₆₁	—	2	4	no	—
49	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
50	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	—
51	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
55	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
60	C₂²⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
61	C₄²×C₆₁	C₂×C₈	1	4	yes	—
62	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
63	C₁₆×C₆₁	—	2	8	no	—
64	C₂³×C₆₁	—	4	2	no	—
65	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
66	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
67	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₆₁	—	4	4	no	—
69	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
70	C₂×C₄×C₆₁	C₄⋊C₈	1	8	yes	—
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
74	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
78	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₆₁	—	2	4	no	—
80	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
81	C₈×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
82	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
83	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
84	#33397	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₆₁	—	2	2	no	—
86	C₂²×C₆₁	—	2	4	no	—
87	C₂²×C₆₁	—	4	4	no	—
88	C₂²×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
89	C₂²×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
90	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₆₁	—	2	8	no	—
92	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
93	C₄×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
94	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
96	C₂²×M₃₂	C₂×C₆₁	1	2	yes	—
97	C₂×#405	C₂×C₆₁	1	1	yes	characteristic
98	#2151	C₂×C₆₁	1	8	yes	complemented
99	#2150	C₂×C₆₁	1	4	yes	complemented
100	C₂×C₆₁	—	2	2	no	—
101	C₂×C₆₁	—	2	4	no	—
102	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₆₁	#3407	1	2	yes	—
104	C₂²×M₁₆	C₄×C₆₁	1	1	yes	characteristic
105	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
106	C₂×#117	C₄×C₆₁	1	1	yes	characteristic
107	C₂×M₃₂	—	2	8	no	—
108	C₂×M₃₂	C₂²×C₆₁	1	4	yes	—
109	C₂×M₃₂	C₂²×C₆₁	1	4	yes	—
110	C₂²×C₁₆	C₂²×C₆₁	1	4	yes	—
111	#405	C₄×C₆₁	1	4	yes	—
112	#405	C₂²×C₆₁	1	4	yes	—
113	C₂×#97	C₄×C₆₁	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
115	C₄⋊C₁₆	C₂²×C₆₁	1	16	yes	complemented
116	C₆₁	#33397	1	1	yes	characteristic, direct product factor, Sylow
117	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
118	C₂×M₁₆	—	2	4	no	—
119	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
120	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
121	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
122	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
123	C₂²×C₈	D₄×C₆₁	1	4	yes	—
124	C₂²×C₈	Q₈×C₆₁	1	4	yes	—
125	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
127	#117	C₈×C₆₁	1	4	yes	—
128	#117	C₂×C₄×C₆₁	1	4	yes	—
129	C₂×C₄⋊C₄	C₈×C₆₁	1	2	yes	—
130	C₂×C₂²⋊C₄	C₈×C₆₁	1	2	yes	—
131	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
132	M₃₂	—	2	8	no	—
133	M₃₂	—	2	8	no	—
134	C₂×C₁₆	—	2	8	no	—
135	C₂×C₁₆	C₂³×C₆₁	1	8	yes	—
136	#97	C₂×C₄×C₆₁	1	8	yes	—
137	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
138	C₄×C₈	C₂³×C₆₁	1	4	yes	—
139	C₂⁴	C₂×C₈×C₆₁	1	1	yes	characteristic
140	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
141	C₂²×C₄	C₂×C₈×C₆₁	1	4	yes	—
142	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
144	C₂²×C₄	M₁₆×C₆₁	1	4	yes	—
145	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
147	M₁₆	—	2	8	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₄⋊C₄×C₆₁	1	4	yes	—
150	C₂×C₈	C₄⋊C₄×C₆₁	1	8	yes	—
151	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
152	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
153	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
154	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
155	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
156	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
157	C₂×C₈	C₂×Q₈×C₆₁	1	4	yes	—
158	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
159	C₄⋊C₄	C₂×C₈×C₆₁	1	8	yes	—
160	C₂²⋊C₄	C₂×C₈×C₆₁	1	8	yes	—
161	C₄²	C₂×C₈×C₆₁	1	4	yes	—
162	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
163	C₁₆	—	2	8	no	—
164	C₂³	—	4	2	no	—
165	C₂³	C₄⋊C₈×C₆₁	1	4	yes	—
166	C₂³	C₂²×C₈×C₆₁	1	2	yes	—
167	C₂³	C₂²×C₈×C₆₁	1	1	yes	characteristic
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₄⋊C₈×C₆₁	1	4	yes	—
170	C₂×C₄	C₄⋊C₈×C₆₁	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
172	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
173	C₂×C₄	C₂²×C₈×C₆₁	1	8	yes	—
174	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₈×C₆₁	1	2	yes	—
176	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
177	C₂×C₄	C₂×M₁₆×C₆₁	1	4	yes	—
178	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
181	C₈	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
182	C₈	C₂²×D₄×C₆₁	1	1	yes	characteristic
183	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂×C₄⋊C₈×C₆₁	1	2	yes	—
188	C₂²	C₂×C₄⋊C₈×C₆₁	1	4	yes	—
189	C₂²	C₂³×C₈×C₆₁	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	C₂×C₄⋊C₈×C₆₁	1	2	yes	—
192	C₄	C₂×C₄⋊C₈×C₆₁	1	4	yes	—
193	C₄	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
194	C₄	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂²×C₄⋊C₈×C₆₁	1	1	yes	characteristic
198	C₂	C₆₁×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂³×C₈
prime radical	C₂⁴×C₆₁	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#2151	C₂	semidirect
4. C₆₁×#2150	C₂	semidirect
16. C₆₁×C₄⋊C₁₆	C₂²	semidirect
84. #33397	C₆₁	direct (N × H)
98. #2151	C₂×C₆₁	semidirect
99. #2150	C₂×C₆₁	semidirect
115. C₄⋊C₁₆	C₂²×C₆₁	semidirect
116. C₆₁	#33397	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₆₁	2
C₂×C₆₁×#405	2
C₆₁×#2151	2
C₆₁×#2150	2
#33397	61

Subgroups — C61×#33397 #29536830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33397 #29536830