Subgroups — C₆₁×#60191 #29563624

Order 15616.

1810 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3458	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2961	C₂	1	12	yes	maximal, complemented
3	C₆₁×#2960	C₂	1	12	yes	maximal, complemented
4	C₆₁×#2905	C₂	1	3	yes	maximal
5	C₂×C₆₁×#405	C₂	1	3	yes	maximal
6	C₂²×M₁₆×C₆₁	C₂²	1	3	yes	—
7	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#544	C₄	1	12	yes	—
9	C₆₁×#543	C₄	1	4	yes	—
10	C₆₁×#533	C₂²	1	12	yes	—
11	C₂×#117×C₆₁	C₂²	1	3	yes	—
12	C₆₁×#446	C₂²	1	48	yes	complemented
13	C₈×Q₈×C₆₁	C₂²	1	16	yes	complemented
14	C₆₁×#424	C₂²	1	24	yes	—
15	C₆₁×#423	C₂²	1	12	yes	—
16	C₆₁×#405	C₂²	1	12	yes	—
17	C₆₁×#405	C₂²	1	12	yes	—
18	C₂×#97×C₆₁	C₂²	1	6	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	6	yes	—
20	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₆₁	—	2	12	no	—
22	C₂×M₁₆×C₆₁	C₂³	1	12	yes	—
23	C₂²×C₈×C₆₁	Q₈	1	8	yes	complemented
24	C₂²×C₈×C₆₁	C₂³	1	6	yes	—
25	C₂²×C₈×C₆₁	C₂³	1	3	yes	—
26	C₂²×C₈×C₆₁	C₂³	1	3	yes	—
27	C₄⋊Q₈×C₆₁	C₂×C₄	1	24	yes	—
28	#125×C₆₁	C₂×C₄	1	24	yes	—
29	C₂²⋊Q₈×C₆₁	C₂×C₄	1	48	yes	—
30	C₄×Q₈×C₆₁	C₂³	1	16	yes	—
31	#117×C₆₁	C₂×C₄	1	12	yes	—
32	#117×C₆₁	C₂³	1	12	yes	—
33	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	6	yes	—
34	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	6	yes	—
35	C₂×C₄²×C₆₁	C₂³	1	6	yes	—
36	C₄⋊C₈×C₆₁	C₂³	1	48	yes	—
37	#97×C₆₁	C₂³	1	24	yes	—
38	C₄×C₈×C₆₁	C₂³	1	12	yes	—
39	C₄×C₈×C₆₁	C₂³	1	12	yes	—
40	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×Q₈×C₆₁	C₂²×C₄	1	16	yes	—
42	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
43	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₆₁	#41	1	4	yes	—
46	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
47	M₁₆×C₆₁	—	2	24	no	—
48	C₂×C₈×C₆₁	—	2	12	no	—
49	C₂×C₈×C₆₁	C₂×Q₈	1	12	yes	—
50	C₂×C₈×C₆₁	C₂×Q₈	1	12	yes	—
51	C₂×C₈×C₆₁	C₂⁴	1	12	yes	—
52	C₂×C₈×C₆₁	C₂⁴	1	3	yes	—
53	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₆₁	C₂²×C₄	1	24	yes	—
55	C₄⋊C₄×C₆₁	C₂²×C₄	1	48	yes	—
56	C₂²⋊C₄×C₆₁	C₂²×C₄	1	24	yes	—
57	C₄²×C₆₁	C₂²×C₄	1	12	yes	—
58	C₄²×C₆₁	C₂⁴	1	12	yes	—
59	C₂³×C₆₁	—	4	2	no	—
60	C₂³×C₆₁	C₄×Q₈	1	4	yes	—
61	C₂³×C₆₁	C₂³×C₄	1	3	yes	—
62	Q₈×C₆₁	—	4	16	no	—
63	C₂×C₄×C₆₁	—	4	12	no	—
64	C₂×C₄×C₆₁	C₄×Q₈	1	12	yes	—
65	C₂×C₄×C₆₁	C₂³×C₄	1	24	yes	—
66	C₂×C₄×C₆₁	C₂³×C₄	1	3	yes	—
67	C₂×C₄×C₆₁	C₂²×Q₈	1	6	yes	—
68	C₂×C₄×C₆₁	C₂×#41	1	6	yes	—
69	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
70	C₈×C₆₁	—	2	12	no	—
71	C₈×C₆₁	C₂²×Q₈	1	6	yes	—
72	C₈×C₆₁	C₂²×Q₈	1	2	yes	—
73	#60191	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₆₁	—	2	6	no	—
75	C₂²×C₆₁	—	4	4	no	—
76	C₂²×C₆₁	C₂×C₄×Q₈	1	6	yes	—
77	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
78	C₄×C₆₁	—	2	24	no	—
79	C₄×C₆₁	C₂×C₄×Q₈	1	6	yes	—
80	C₄×C₆₁	C₂³×Q₈	1	1	yes	characteristic
81	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
82	#3458	C₂×C₆₁	1	1	yes	characteristic
83	#2961	C₂×C₆₁	1	12	yes	complemented
84	#2960	C₂×C₆₁	1	12	yes	complemented
85	#2905	C₂×C₆₁	1	3	yes	—
86	C₂×#405	C₂×C₆₁	1	3	yes	—
87	C₂×C₆₁	—	2	6	no	—
88	C₂×C₆₁	C₂²×C₄×Q₈	1	1	yes	characteristic
89	C₂×C₆₁	#3573	1	2	yes	—
90	C₂²×M₁₆	C₂²×C₆₁	1	3	yes	—
91	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
92	#544	C₄×C₆₁	1	12	yes	—
93	#543	C₄×C₆₁	1	4	yes	—
94	#533	C₂²×C₆₁	1	12	yes	—
95	C₂×#117	C₂²×C₆₁	1	3	yes	—
96	#446	C₂²×C₆₁	1	48	yes	complemented
97	C₈×Q₈	C₂²×C₆₁	1	16	yes	complemented
98	#424	C₂²×C₆₁	1	24	yes	—
99	#423	C₂²×C₆₁	1	12	yes	—
100	#405	C₂²×C₆₁	1	12	yes	—
101	#405	C₂²×C₆₁	1	12	yes	—
102	C₂×#97	C₂²×C₆₁	1	6	yes	—
103	C₂×C₄×C₈	C₂²×C₆₁	1	6	yes	—
104	C₆₁	#60191	1	1	yes	characteristic, direct product factor, Sylow
105	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
106	C₂×M₁₆	—	2	12	no	—
107	C₂×M₁₆	C₂³×C₆₁	1	12	yes	—
108	C₂²×C₈	Q₈×C₆₁	1	8	yes	complemented
109	C₂²×C₈	C₂³×C₆₁	1	6	yes	—
110	C₂²×C₈	C₂³×C₆₁	1	3	yes	—
111	C₂²×C₈	C₂³×C₆₁	1	3	yes	—
112	C₄⋊Q₈	C₂×C₄×C₆₁	1	24	yes	—
113	#125	C₂×C₄×C₆₁	1	24	yes	—
114	C₂²⋊Q₈	C₂×C₄×C₆₁	1	48	yes	—
115	C₄×Q₈	C₂³×C₆₁	1	16	yes	—
116	#117	C₂×C₄×C₆₁	1	12	yes	—
117	#117	C₂³×C₆₁	1	12	yes	—
118	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	6	yes	—
119	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	6	yes	—
120	C₂×C₄²	C₂³×C₆₁	1	6	yes	—
121	C₄⋊C₈	C₂³×C₆₁	1	48	yes	—
122	#97	C₂³×C₆₁	1	24	yes	—
123	C₄×C₈	C₂³×C₆₁	1	12	yes	—
124	C₄×C₈	C₂³×C₆₁	1	12	yes	—
125	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
126	C₂×Q₈	C₂²×C₄×C₆₁	1	16	yes	—
127	C₂²×C₄	C₂²×C₄×C₆₁	1	12	yes	—
128	C₂²×C₄	C₂²×C₄×C₆₁	1	3	yes	—
129	C₂²×C₄	C₂×Q₈×C₆₁	1	4	yes	—
130	C₂²×C₄	#41×C₆₁	1	4	yes	—
131	C₂²×C₄	C₂⁴×C₆₁	1	3	yes	—
132	M₁₆	—	2	24	no	—
133	C₂×C₈	—	2	12	no	—
134	C₂×C₈	C₂×Q₈×C₆₁	1	12	yes	—
135	C₂×C₈	C₂×Q₈×C₆₁	1	12	yes	—
136	C₂×C₈	C₂⁴×C₆₁	1	12	yes	—
137	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
138	C₂×C₈	C₂⁴×C₆₁	1	3	yes	—
139	C₄⋊C₄	C₂²×C₄×C₆₁	1	24	yes	—
140	C₄⋊C₄	C₂²×C₄×C₆₁	1	48	yes	—
141	C₂²⋊C₄	C₂²×C₄×C₆₁	1	24	yes	—
142	C₄²	C₂²×C₄×C₆₁	1	12	yes	—
143	C₄²	C₂⁴×C₆₁	1	12	yes	—
144	C₂³	—	4	2	no	—
145	C₂³	C₄×Q₈×C₆₁	1	4	yes	—
146	C₂³	C₂³×C₄×C₆₁	1	3	yes	—
147	Q₈	—	4	16	no	—
148	C₂×C₄	—	4	12	no	—
149	C₂×C₄	C₄×Q₈×C₆₁	1	12	yes	—
150	C₂×C₄	C₂³×C₄×C₆₁	1	24	yes	—
151	C₂×C₄	C₂³×C₄×C₆₁	1	3	yes	—
152	C₂×C₄	C₂²×Q₈×C₆₁	1	6	yes	—
153	C₂×C₄	C₂×#41×C₆₁	1	6	yes	—
154	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
155	C₈	—	2	12	no	—
156	C₈	C₂²×Q₈×C₆₁	1	6	yes	—
157	C₈	C₂²×Q₈×C₆₁	1	2	yes	—
158	C₂²	—	2	6	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂×C₄×Q₈×C₆₁	1	6	yes	—
161	C₂²	C₂⁴×C₄×C₆₁	1	1	yes	characteristic
162	C₄	—	2	24	no	—
163	C₄	C₂×C₄×Q₈×C₆₁	1	6	yes	—
164	C₄	C₂³×Q₈×C₆₁	1	1	yes	characteristic
165	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
166	C₂	—	2	6	no	—
167	C₂	C₂²×C₄×Q₈×C₆₁	1	1	yes	characteristic
168	C₂	C₆₁×#3573	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
socle	C₂²×C₆₁	C₂⁴×C₄
prime radical	C₂⁴×C₆₁	C₂²×C₄
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
2. C₆₁×#2961	C₂	semidirect
3. C₆₁×#2960	C₂	semidirect
12. C₆₁×#446	C₂²	semidirect
13. C₈×Q₈×C₆₁	C₂²	semidirect
23. C₂²×C₈×C₆₁	Q₈	semidirect
73. #60191	C₆₁	direct (N × H)
83. #2961	C₂×C₆₁	semidirect
84. #2960	C₂×C₆₁	semidirect
96. #446	C₂²×C₆₁	semidirect
97. C₈×Q₈	C₂²×C₆₁	semidirect
104. C₆₁	#60191	direct (N × H)
108. C₂²×C₈	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3458	2
C₆₁×#2961	2
C₆₁×#2960	2
C₆₁×#2905	2
C₂×C₆₁×#405	2
#60191	61

Subgroups — C61×#60191 #29563624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#60191 #29563624