Subgroups — C₄×C₆₁×#346 #29515971

Order 15616.

510 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#346	C₂	1	4	yes	maximal
2	C₂×C₄×C₁₆×C₆₁	C₂	1	2	yes	maximal
3	C₄²×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₆₁	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₄³×C₆₁	C₄	1	1	yes	characteristic
10	C₆₁×#346	C₄	1	16	yes	direct product factor
11	C₄×C₁₆×C₆₁	C₄	1	8	yes	complemented
12	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₆₁	C₈	1	4	yes	—
16	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₆₁	C₂×C₄	1	8	yes	—
19	C₂×C₁₆×C₆₁	C₂×C₄	1	16	yes	—
20	C₄×C₈×C₆₁	C₈	1	8	yes	—
21	C₄×C₈×C₆₁	C₈	1	4	yes	—
22	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₆₁	C₂×C₄	1	8	yes	—
24	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
25	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
26	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₆₁	C₄²	1	2	yes	—
28	C₂×C₈×C₆₁	C₄²	1	4	yes	—
29	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
30	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
33	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
34	C₄²×C₆₁	—	2	8	no	—
35	C₄²×C₆₁	C₄²	1	4	yes	—
36	C₄²×C₆₁	C₄²	1	2	yes	—
37	C₄²×C₆₁	C₂×C₈	1	4	yes	—
38	C₄²×C₆₁	C₂×C₈	1	2	yes	—
39	C₁₆×C₆₁	C₄²	1	16	yes	complemented
40	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₄×C₆₁	—	2	8	no	—
42	C₂×C₄×C₆₁	C₄×C₈	1	8	yes	—
43	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
44	C₂×C₄×C₆₁	M₃₂	1	8	yes	—
45	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
47	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
49	C₈×C₆₁	C₄×C₈	1	4	yes	—
50	C₈×C₆₁	C₄×C₈	1	8	yes	—
51	C₈×C₆₁	C₂×C₄²	1	4	yes	—
52	C₄×#346	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
54	C₂²×C₆₁	C₂×C₄×C₈	1	2	yes	—
55	C₂²×C₆₁	C₂×M₃₂	1	4	yes	—
56	C₄×C₆₁	—	2	8	no	—
57	C₄×C₆₁	#346	1	8	yes	direct product factor
58	C₄×C₆₁	C₄³	1	1	yes	characteristic
59	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
61	C₂×#346	C₂×C₆₁	1	4	yes	—
62	C₂×C₄×C₁₆	C₂×C₆₁	1	2	yes	—
63	C₄²×C₈	C₂×C₆₁	1	1	yes	characteristic
64	C₂×C₆₁	C₄²×C₈	1	1	yes	characteristic
65	C₂×C₆₁	C₂×#346	1	2	yes	—
66	C₂×C₆₁	C₄×M₃₂	1	4	yes	—
67	C₂²×C₁₆	C₂²×C₆₁	1	4	yes	—
68	C₂×C₄×C₈	C₄×C₆₁	1	2	yes	—
69	C₂×C₄×C₈	C₄×C₆₁	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
72	C₄³	C₄×C₆₁	1	1	yes	characteristic
73	#346	C₄×C₆₁	1	16	yes	direct product factor
74	C₄×C₁₆	C₄×C₆₁	1	8	yes	complemented
75	C₆₁	C₄×#346	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
77	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
78	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
79	C₂×C₄²	C₈×C₆₁	1	4	yes	—
80	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
81	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
82	C₂×C₁₆	C₂×C₄×C₆₁	1	8	yes	—
83	C₂×C₁₆	C₂×C₄×C₆₁	1	16	yes	—
84	C₄×C₈	C₈×C₆₁	1	8	yes	—
85	C₄×C₈	C₈×C₆₁	1	4	yes	—
86	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
87	C₄×C₈	C₂×C₄×C₆₁	1	8	yes	—
88	C₂²×C₄	C₂×C₈×C₆₁	1	4	yes	—
89	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
90	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
91	C₂×C₈	C₄²×C₆₁	1	2	yes	—
92	C₂×C₈	C₄²×C₆₁	1	4	yes	—
93	C₂×C₈	C₂×C₈×C₆₁	1	4	yes	—
94	C₂×C₈	C₂×C₈×C₆₁	1	4	yes	—
95	C₂×C₈	C₂×C₈×C₆₁	1	4	yes	—
96	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
97	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
98	C₄²	—	2	8	no	—
99	C₄²	C₄²×C₆₁	1	4	yes	—
100	C₄²	C₄²×C₆₁	1	2	yes	—
101	C₄²	C₂×C₈×C₆₁	1	4	yes	—
102	C₄²	C₂×C₈×C₆₁	1	2	yes	—
103	C₁₆	C₄²×C₆₁	1	16	yes	complemented
104	C₂³	C₂²×C₈×C₆₁	1	1	yes	characteristic
105	C₂×C₄	—	2	8	no	—
106	C₂×C₄	C₄×C₈×C₆₁	1	8	yes	—
107	C₂×C₄	C₄×C₈×C₆₁	1	4	yes	—
108	C₂×C₄	M₃₂×C₆₁	1	8	yes	—
109	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
111	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
112	C₂×C₄	C₂²×C₈×C₆₁	1	2	yes	—
113	C₈	C₄×C₈×C₆₁	1	4	yes	—
114	C₈	C₄×C₈×C₆₁	1	8	yes	—
115	C₈	C₂×C₄²×C₆₁	1	4	yes	—
116	C₂²	C₂×C₄×C₈×C₆₁	1	1	yes	characteristic
117	C₂²	C₂×C₄×C₈×C₆₁	1	2	yes	—
118	C₂²	C₂×M₃₂×C₆₁	1	4	yes	—
119	C₄	—	2	8	no	—
120	C₄	C₆₁×#346	1	8	yes	direct product factor
121	C₄	C₄³×C₆₁	1	1	yes	characteristic
122	C₄	C₂×C₄×C₈×C₆₁	1	1	yes	characteristic
123	C₄	C₂×C₄×C₈×C₆₁	1	2	yes	—
124	C₂	C₄²×C₈×C₆₁	1	1	yes	characteristic
125	C₂	C₂×C₆₁×#346	1	2	yes	—
126	C₂	C₄×M₃₂×C₆₁	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
10. C₆₁×#346	C₄	direct (N × H)
11. C₄×C₁₆×C₆₁	C₄	semidirect
39. C₁₆×C₆₁	C₄²	semidirect
52. C₄×#346	C₆₁	direct (N × H)
57. C₄×C₆₁	#346	direct (N × H)
73. #346	C₄×C₆₁	direct (N × H)
74. C₄×C₁₆	C₄×C₆₁	semidirect
75. C₆₁	C₄×#346	direct (N × H)
103. C₁₆	C₄²×C₆₁	semidirect
120. C₄	C₆₁×#346	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#346	2
C₂×C₄×C₁₆×C₆₁	2
C₄²×C₈×C₆₁	2
C₄×#346	61

Subgroups — C4×C61×#346 #29515971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₆₁×#346 #29515971