Subgroups — C₆₁×#8294 #29511727

Order 15616.

358 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#108×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₆₁	C₂	1	2	yes	maximal
4	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1278	C₂	1	2	yes	maximal
6	C₆₁×#431	C₂²	1	2	yes	—
7	C₂×#108×C₆₁	C₄	1	2	yes	—
8	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#341	C₄	1	2	yes	—
13	C₆₁×#335	C₄	1	2	yes	—
14	C₆₁×#334	C₄	1	2	yes	—
15	C₆₁×#322	—	2	2	no	—
16	C₆₁×#322	C₄	1	2	yes	—
17	C₆₁×#322	C₄	1	2	yes	—
18	C₈²×C₆₁	—	2	2	no	—
19	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₆₁	—	2	1	no	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
24	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
25	#108×C₆₁	C₈	1	8	yes	complemented
26	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
27	C₂²⋊C₈×C₆₁	—	2	2	no	—
28	#97×C₆₁	—	2	2	no	—
29	C₄×C₈×C₆₁	—	2	2	no	—
30	C₄×C₈×C₆₁	—	2	2	no	—
31	C₄×C₈×C₆₁	—	2	1	no	—
32	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₆₁	D₄	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
42	M₁₆×C₆₁	—	2	4	no	—
43	M₁₆×C₆₁	C₂×C₈	1	8	yes	—
44	C₂×C₈×C₆₁	—	2	4	no	—
45	C₂×C₈×C₆₁	—	2	2	no	—
46	C₂×C₈×C₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	2	4	no	—
48	C₂×C₈×C₆₁	—	4	2	no	—
49	C₂×C₈×C₆₁	—	4	2	no	—
50	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
56	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₆₁	#41	1	1	yes	characteristic
60	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₆₁	#131	1	2	yes	—
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	—	2	1	no	—
73	C₈×C₆₁	—	2	4	no	—
74	C₈×C₆₁	—	2	8	no	—
75	C₈×C₆₁	—	4	2	no	—
76	C₈×C₆₁	—	4	4	no	—
77	C₈×C₆₁	C₄×C₈	1	2	yes	—
78	C₈×C₆₁	C₄×C₈	1	2	yes	—
79	#8294	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₆₁	—	2	1	no	—
81	C₂²×C₆₁	—	2	1	no	—
82	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₆₁	C₈×Q₈	1	2	yes	—
84	C₄×C₆₁	—	2	1	no	—
85	C₄×C₆₁	—	2	1	no	—
86	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
87	C₄×C₆₁	#405	1	1	yes	characteristic
88	C₄×C₆₁	#430	1	2	yes	—
89	C₄×C₆₁	C₈×D₄	1	2	yes	—
90	C₄×C₆₁	#444	1	2	yes	—
91	C₄×#108	C₂×C₆₁	1	1	yes	characteristic
92	#1564	C₂×C₆₁	1	1	yes	characteristic
93	C₈×M₁₆	C₂×C₆₁	1	2	yes	—
94	C₂×#322	C₂×C₆₁	1	1	yes	characteristic
95	#1278	C₂×C₆₁	1	2	yes	—
96	C₂×C₆₁	—	2	2	no	—
97	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₆₁	#1773	1	1	yes	characteristic
99	C₂×C₆₁	#1778	1	1	yes	characteristic
100	#431	C₂²×C₆₁	1	2	yes	—
101	C₂×#108	C₄×C₆₁	1	2	yes	—
102	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
103	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
106	#341	C₄×C₆₁	1	2	yes	—
107	#335	C₄×C₆₁	1	2	yes	—
108	#334	C₄×C₆₁	1	2	yes	—
109	#322	—	2	2	no	—
110	#322	C₄×C₆₁	1	2	yes	—
111	#322	C₄×C₆₁	1	2	yes	—
112	C₈²	—	2	2	no	—
113	C₆₁	#8294	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
118	C₂²×C₈	Q₈×C₆₁	1	2	yes	—
119	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
120	#108	C₈×C₆₁	1	8	yes	complemented
121	C₄⋊C₈	C₂×C₄×C₆₁	1	4	yes	—
122	C₂²⋊C₈	—	2	2	no	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
128	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
129	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
130	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
131	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
132	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
133	C₄×C₈	D₄×C₆₁	1	2	yes	—
134	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
135	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
136	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	C₂×C₈×C₆₁	1	8	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	C₄²×C₆₁	1	1	yes	characteristic
146	C₂×C₈	C₄²×C₆₁	1	1	yes	characteristic
147	C₂×C₈	C₄²×C₆₁	1	1	yes	characteristic
148	C₂×C₈	C₄²×C₆₁	1	1	yes	characteristic
149	C₂×C₈	C₄⋊C₄×C₆₁	1	4	yes	—
150	C₂×C₈	C₂×C₈×C₆₁	1	4	yes	—
151	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
152	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
153	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
154	C₄²	#41×C₆₁	1	1	yes	characteristic
155	C₂³	C₄×Q₈×C₆₁	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
160	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
161	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
162	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
163	C₂×C₄	C₄×Q₈×C₆₁	1	1	yes	characteristic
164	C₂×C₄	C₂²×C₈×C₆₁	1	2	yes	—
165	C₂×C₄	#131×C₆₁	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	4	no	—
169	C₈	—	2	8	no	—
170	C₈	—	4	2	no	—
171	C₈	—	4	4	no	—
172	C₈	C₄×C₈×C₆₁	1	2	yes	—
173	C₈	C₄×C₈×C₆₁	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×C₄⋊C₄×C₆₁	1	1	yes	characteristic
177	C₂²	C₈×Q₈×C₆₁	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×C₄×C₈×C₆₁	1	1	yes	characteristic
181	C₄	C₆₁×#405	1	1	yes	characteristic
182	C₄	C₆₁×#430	1	2	yes	—
183	C₄	C₈×D₄×C₆₁	1	2	yes	—
184	C₄	C₆₁×#444	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₈×C₄⋊C₄×C₆₁	1	1	yes	characteristic
187	C₂	C₆₁×#1773	1	1	yes	characteristic
188	C₂	C₆₁×#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₆₁	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	C₄×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
25. #108×C₆₁	C₈	semidirect
79. #8294	C₆₁	direct (N × H)
113. C₆₁	#8294	direct (N × H)
120. #108	C₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#108×C₆₁	2
C₆₁×#1564	2
C₈×M₁₆×C₆₁	2
C₂×C₆₁×#322	2
C₆₁×#1278	2
#8294	61

Subgroups — C61×#8294 #29511727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8294 #29511727