Subgroups — C₆₁×#13264 #29516697

Order 15616.

386 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1374	C₂	1	4	yes	maximal
5	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
8	C₈×Q₈×C₆₁	—	2	2	no	—
9	C₈×Q₈×C₆₁	—	2	2	no	—
10	C₂×C₄⋊C₈×C₆₁	C₄	1	2	yes	—
11	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×C₄⋊C₁₆	—	2	2	no	—
15	C₆₁×#346	—	2	2	no	—
16	C₂²×Q₈×C₆₁	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
20	C₄×Q₈×C₆₁	—	2	2	no	—
21	C₄×Q₈×C₆₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₆₁	C₈	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
24	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₁	—	2	2	no	—
27	C₂×C₁₆×C₆₁	—	2	2	no	—
28	C₂×C₁₆×C₆₁	—	2	2	no	—
29	C₂×C₁₆×C₆₁	—	2	4	no	—
30	C₂×C₁₆×C₆₁	Q₈	1	4	yes	complemented
31	C₄⋊C₈×C₆₁	—	2	4	no	—
32	C₄⋊C₈×C₆₁	—	2	2	no	—
33	C₄×C₈×C₆₁	—	2	2	no	—
34	C₄×C₈×C₆₁	—	2	2	no	—
35	C₄×C₈×C₆₁	D₄	1	2	yes	—
36	C₄×C₈×C₆₁	D₄	1	2	yes	—
37	C₂×Q₈×C₆₁	—	2	2	no	—
38	C₂×Q₈×C₆₁	—	2	4	no	—
39	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	—	2	2	no	—
45	C₂×C₈×C₆₁	—	2	2	no	—
46	C₂×C₈×C₆₁	—	2	4	no	—
47	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₆₁	#41	1	2	yes	—
52	C₄⋊C₄×C₆₁	—	2	4	no	—
53	C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₄²×C₆₁	—	2	2	no	—
55	C₄²×C₆₁	—	2	2	no	—
56	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₆₁	M₁₆	1	2	yes	—
58	C₁₆×C₆₁	—	2	4	no	—
59	C₁₆×C₆₁	—	4	2	no	—
60	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
61	Q₈×C₆₁	—	2	8	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	—	2	4	no	—
68	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
69	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄×C₆₁	M₃₂	1	4	yes	—
71	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#131	1	2	yes	—
75	C₈×C₆₁	—	2	2	no	—
76	C₈×C₆₁	—	4	2	no	—
77	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₆₁	#124	1	1	yes	characteristic
79	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
80	#13264	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	C₂²×C₆₁	C₈×Q₈	1	2	yes	—
83	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
84	C₂²×C₆₁	#504	1	2	yes	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	—	2	4	no	—
87	C₄×C₆₁	—	2	2	no	—
88	C₄×C₆₁	—	4	2	no	—
89	C₄×C₆₁	#391	1	1	yes	characteristic
90	C₄×C₆₁	#408	1	1	yes	characteristic
91	C₄×C₆₁	#446	1	2	yes	—
92	C₂×C₈×Q₈	C₂×C₆₁	1	1	yes	characteristic
93	C₂×C₄⋊C₁₆	C₂×C₆₁	1	1	yes	characteristic
94	C₂×#346	C₂×C₆₁	1	1	yes	characteristic
95	#1374	C₂×C₆₁	1	4	yes	—
96	C₂×C₆₁	#1984	1	1	yes	characteristic
97	C₂×C₆₁	#2112	1	1	yes	characteristic
98	C₂×C₆₁	#2113	1	1	yes	characteristic
99	C₂×C₆₁	#2183	1	4	yes	—
100	C₂×C₄×Q₈	C₄×C₆₁	1	1	yes	characteristic
101	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
102	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
103	C₈×Q₈	—	2	2	no	—
104	C₈×Q₈	—	2	2	no	—
105	C₂×C₄⋊C₈	C₄×C₆₁	1	2	yes	—
106	C₂×C₄⋊C₈	C₄×C₆₁	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
109	C₄⋊C₁₆	—	2	2	no	—
110	#346	—	2	2	no	—
111	C₆₁	#13264	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×Q₈	C₈×C₆₁	1	1	yes	characteristic
113	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
114	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
116	C₄×Q₈	—	2	2	no	—
117	C₄×Q₈	—	2	2	no	—
118	C₂×C₄⋊C₄	C₈×C₆₁	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₈×C₆₁	1	2	yes	—
120	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
121	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	2	4	no	—
126	C₂×C₁₆	Q₈×C₆₁	1	4	yes	complemented
127	C₄⋊C₈	—	2	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	D₄×C₆₁	1	2	yes	—
132	C₄×C₈	D₄×C₆₁	1	2	yes	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
136	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
139	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
144	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
145	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
146	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
147	C₂×C₈	#41×C₆₁	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
153	C₄²	M₁₆×C₆₁	1	2	yes	—
154	C₁₆	—	2	4	no	—
155	C₁₆	—	4	2	no	—
156	C₂³	C₂²×C₈×C₆₁	1	1	yes	characteristic
157	Q₈	—	2	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
165	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
166	C₂×C₄	M₃₂×C₆₁	1	4	yes	—
167	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
169	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
170	C₂×C₄	#131×C₆₁	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	4	2	no	—
173	C₈	C₂²⋊Q₈×C₆₁	1	2	yes	—
174	C₈	#124×C₆₁	1	1	yes	characteristic
175	C₈	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
176	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
177	C₂²	C₈×Q₈×C₆₁	1	2	yes	—
178	C₂²	C₂×M₃₂×C₆₁	1	2	yes	—
179	C₂²	C₆₁×#504	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	2	2	no	—
183	C₄	—	4	2	no	—
184	C₄	C₆₁×#391	1	1	yes	characteristic
185	C₄	C₆₁×#408	1	1	yes	characteristic
186	C₄	C₆₁×#446	1	2	yes	—
187	C₂	C₆₁×#1984	1	1	yes	characteristic
188	C₂	C₆₁×#2112	1	1	yes	characteristic
189	C₂	C₆₁×#2113	1	1	yes	characteristic
190	C₂	C₆₁×#2183	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
30. C₂×C₁₆×C₆₁	Q₈	semidirect
80. #13264	C₆₁	direct (N × H)
111. C₆₁	#13264	direct (N × H)
126. C₂×C₁₆	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₆₁	2
C₂×C₆₁×C₄⋊C₁₆	2
C₂×C₆₁×#346	2
C₆₁×#1374	2
#13264	61

Subgroups — C61×#13264 #29516697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13264 #29516697