Subgroups — C₆₁×#17316 #29520749

Order 15616.

890 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₆₁	C₂	1	2	yes	maximal
2	C₄×#131×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1450	C₂	1	3	yes	maximal, complemented
4	C₈×M₁₆×C₆₁	C₂	1	6	yes	maximal, complemented
5	C₂×C₆₁×#322	C₂	1	3	yes	maximal, complemented
6	C₂×#131×C₆₁	C₄	1	2	yes	—
7	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₆₁	C₄	1	4	yes	—
9	C₈×D₄×C₆₁	C₄	1	12	yes	—
10	C₆₁×#431	C₂²	1	6	yes	—
11	C₆₁×#405	C₄	1	6	yes	—
12	C₄×M₁₆×C₆₁	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	6	yes	—
15	C₆₁×#322	C₂²	1	1	yes	characteristic
16	C₆₁×#322	C₂²	1	6	yes	—
17	C₆₁×#322	C₂²	1	3	yes	—
18	C₈²×C₆₁	C₂²	1	6	yes	—
19	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
20	#131×C₆₁	C₈	1	16	yes	complemented
21	C₂×M₁₆×C₆₁	C₂×C₄	1	6	yes	—
22	C₂²×C₈×C₆₁	C₂×C₄	1	12	yes	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
24	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₄×D₄×C₆₁	C₂×C₄	1	6	yes	—
26	#117×C₆₁	C₂×C₄	1	3	yes	—
27	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
28	C₄⋊C₈×C₆₁	C₂×C₄	1	12	yes	—
29	C₂²⋊C₈×C₆₁	C₂×C₄	1	12	yes	—
30	#97×C₆₁	C₂×C₄	1	6	yes	—
31	C₄×C₈×C₆₁	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₆₁	C₂×C₄	1	12	yes	—
33	C₄×C₈×C₆₁	C₂³	1	3	yes	—
34	C₄×C₈×C₆₁	C₂³	1	6	yes	—
35	C₄×C₈×C₆₁	C₂³	1	2	yes	—
36	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	#41×C₆₁	C₂×C₈	1	8	yes	—
38	C₂×Q₈×C₆₁	C₄²	1	1	yes	characteristic
39	C₂×D₄×C₆₁	C₄²	1	3	yes	—
40	C₂²×C₄×C₆₁	C₂²×C₄	1	6	yes	—
41	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
42	M₁₆×C₆₁	C₂×C₈	1	24	yes	complemented
43	C₂×C₈×C₆₁	—	4	12	no	—
44	C₂×C₈×C₆₁	C₂×C₈	1	24	yes	complemented
45	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₆₁	C₂²×C₄	1	12	yes	—
47	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₆₁	C₂²×C₄	1	6	yes	—
49	C₄⋊C₄×C₆₁	C₄²	1	6	yes	—
50	C₂²⋊C₄×C₆₁	C₄²	1	6	yes	—
51	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
52	C₄²×C₆₁	C₂²×C₄	1	6	yes	—
53	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
54	C₂³×C₆₁	C₂×C₄²	1	3	yes	—
55	Q₈×C₆₁	C₄×C₈	1	4	yes	—
56	D₄×C₆₁	C₄×C₈	1	12	yes	—
57	C₂×C₄×C₆₁	—	2	12	no	—
58	C₂×C₄×C₆₁	C₂×C₄²	1	6	yes	—
59	C₂×C₄×C₆₁	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₆₁	C₂²×C₈	1	12	yes	—
61	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
63	C₈×C₆₁	—	2	4	no	—
64	C₈×C₆₁	—	2	12	no	—
65	C₈×C₆₁	C₂²×C₈	1	4	yes	—
66	C₈×C₆₁	C₂²×C₈	1	12	yes	—
67	#17316	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₆₁	—	2	6	no	—
69	C₂²×C₆₁	C₂×C₄×C₈	1	6	yes	—
70	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
71	C₄×C₆₁	—	2	6	no	—
72	C₄×C₆₁	C₂×C₄×C₈	1	6	yes	—
73	C₄×C₆₁	C₂³×C₈	1	2	yes	—
74	C₄×C₆₁	#568	1	4	yes	—
75	C₈×#41	C₂×C₆₁	1	2	yes	—
76	C₄×#131	C₂×C₆₁	1	1	yes	characteristic
77	#1450	C₂×C₆₁	1	3	yes	complemented
78	C₈×M₁₆	C₂×C₆₁	1	6	yes	complemented
79	C₂×#322	C₂×C₆₁	1	3	yes	complemented
80	C₂×C₆₁	—	2	6	no	—
81	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
82	C₂×C₆₁	#2875	1	2	yes	—
83	C₂×#131	C₄×C₆₁	1	2	yes	—
84	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
85	C₈×Q₈	C₄×C₆₁	1	4	yes	—
86	C₈×D₄	C₄×C₆₁	1	12	yes	—
87	#431	C₂²×C₆₁	1	6	yes	—
88	#405	C₄×C₆₁	1	6	yes	—
89	C₄×M₁₆	C₂²×C₆₁	1	3	yes	—
90	C₂×C₄×C₈	C₂²×C₆₁	1	3	yes	—
91	C₂×C₄×C₈	C₂²×C₆₁	1	6	yes	—
92	#322	C₂²×C₆₁	1	1	yes	characteristic
93	#322	C₂²×C₆₁	1	6	yes	—
94	#322	C₂²×C₆₁	1	3	yes	—
95	C₈²	C₂²×C₆₁	1	6	yes	—
96	C₆₁	#17316	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
98	#131	C₈×C₆₁	1	16	yes	complemented
99	C₂×M₁₆	C₂×C₄×C₆₁	1	6	yes	—
100	C₂²×C₈	C₂×C₄×C₆₁	1	12	yes	—
101	C₂²×C₈	C₂×C₄×C₆₁	1	6	yes	—
102	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
103	C₄×D₄	C₂×C₄×C₆₁	1	6	yes	—
104	#117	C₂×C₄×C₆₁	1	3	yes	—
105	C₂×C₄²	C₂³×C₆₁	1	3	yes	—
106	C₄⋊C₈	C₂×C₄×C₆₁	1	12	yes	—
107	C₂²⋊C₈	C₂×C₄×C₆₁	1	12	yes	—
108	#97	C₂×C₄×C₆₁	1	6	yes	—
109	C₄×C₈	C₂×C₄×C₆₁	1	6	yes	—
110	C₄×C₈	C₂×C₄×C₆₁	1	12	yes	—
111	C₄×C₈	C₂³×C₆₁	1	3	yes	—
112	C₄×C₈	C₂³×C₆₁	1	6	yes	—
113	C₄×C₈	C₂³×C₆₁	1	2	yes	—
114	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
115	#41	C₂×C₈×C₆₁	1	8	yes	—
116	C₂×Q₈	C₄²×C₆₁	1	1	yes	characteristic
117	C₂×D₄	C₄²×C₆₁	1	3	yes	—
118	C₂²×C₄	C₂²×C₄×C₆₁	1	6	yes	—
119	C₂²×C₄	C₂²×C₄×C₆₁	1	3	yes	—
120	M₁₆	C₂×C₈×C₆₁	1	24	yes	complemented
121	C₂×C₈	—	4	12	no	—
122	C₂×C₈	C₂×C₈×C₆₁	1	24	yes	complemented
123	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
124	C₂×C₈	C₂²×C₄×C₆₁	1	12	yes	—
125	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
126	C₂×C₈	C₂²×C₄×C₆₁	1	6	yes	—
127	C₄⋊C₄	C₄²×C₆₁	1	6	yes	—
128	C₂²⋊C₄	C₄²×C₆₁	1	6	yes	—
129	C₄²	C₂²×C₄×C₆₁	1	3	yes	—
130	C₄²	C₂²×C₄×C₆₁	1	6	yes	—
131	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
132	C₂³	C₂×C₄²×C₆₁	1	3	yes	—
133	Q₈	C₄×C₈×C₆₁	1	4	yes	—
134	D₄	C₄×C₈×C₆₁	1	12	yes	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	C₂×C₄²×C₆₁	1	6	yes	—
137	C₂×C₄	C₂×C₄²×C₆₁	1	3	yes	—
138	C₂×C₄	C₂²×C₈×C₆₁	1	12	yes	—
139	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
140	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
141	C₈	—	2	4	no	—
142	C₈	—	2	12	no	—
143	C₈	C₂²×C₈×C₆₁	1	4	yes	—
144	C₈	C₂²×C₈×C₆₁	1	12	yes	—
145	C₂²	—	2	6	no	—
146	C₂²	C₂×C₄×C₈×C₆₁	1	6	yes	—
147	C₂²	C₂²×C₄²×C₆₁	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	C₂×C₄×C₈×C₆₁	1	6	yes	—
150	C₄	C₂³×C₈×C₆₁	1	2	yes	—
151	C₄	C₆₁×#568	1	4	yes	—
152	C₂	—	2	6	no	—
153	C₂	C₂²×C₄×C₈×C₆₁	1	1	yes	characteristic
154	C₂	C₆₁×#2875	1	2	yes	—

	subgroup	G/H
center	C₄²×C₆₁	C₂⁴
commutator	C₂	C₂²×C₄×C₈×C₆₁
Frattini, cosocle	C₄²	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂²×C₄²
prime radical	C₂×#41×C₆₁	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1450	C₂	semidirect
4. C₈×M₁₆×C₆₁	C₂	semidirect
5. C₂×C₆₁×#322	C₂	semidirect
20. #131×C₆₁	C₈	semidirect
42. M₁₆×C₆₁	C₂×C₈	semidirect
44. C₂×C₈×C₆₁	C₂×C₈	semidirect
67. #17316	C₆₁	direct (N × H)
77. #1450	C₂×C₆₁	semidirect
78. C₈×M₁₆	C₂×C₆₁	semidirect
79. C₂×#322	C₂×C₆₁	semidirect
96. C₆₁	#17316	direct (N × H)
98. #131	C₈×C₆₁	semidirect
120. M₁₆	C₂×C₈×C₆₁	semidirect
122. C₂×C₈	C₂×C₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₆₁	2
C₄×#131×C₆₁	2
C₆₁×#1450	2
C₈×M₁₆×C₆₁	2
C₂×C₆₁×#322	2
#17316	61

Subgroups — C61×#17316 #29520749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#17316 #29520749