Subgroups — C₂×C₆₁×#2252 #29537156

Order 15616.

558 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#500	C₂	1	2	yes	maximal
2	C₆₁×#2252	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₆₁×#366	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₁	—	2	2	no	—
6	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
7	C₆₁×#500	—	2	4	no	—
8	C₆₁×#500	C₂²	1	8	yes	—
9	C₂×#107×C₆₁	—	2	2	no	—
10	C₂×#107×C₆₁	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#366	C₂²	1	16	yes	—
13	C₄×C₁₆×C₆₁	C₂²	1	4	yes	—
14	C₂²×Q₈×C₆₁	—	4	2	no	—
15	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
17	C₄⋊Q₈×C₆₁	—	2	8	no	—
18	C₄⋊Q₈×C₆₁	—	4	4	no	—
19	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
20	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
21	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₁	Q₈	1	8	yes	complemented
23	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
24	#107×C₆₁	—	2	8	no	—
25	#107×C₆₁	—	2	4	no	—
26	#107×C₆₁	C₂³	1	8	yes	—
27	C₄×C₈×C₆₁	D₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂³	1	2	yes	—
29	C₂×Q₈×C₆₁	—	4	8	no	—
30	C₂×Q₈×C₆₁	—	8	8	no	—
31	C₂²×C₄×C₆₁	—	4	4	no	—
32	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
36	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
39	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
40	C₄⋊C₄×C₆₁	—	2	8	no	—
41	C₄⋊C₄×C₆₁	—	4	4	no	—
42	C₄⋊C₄×C₆₁	—	4	8	no	—
43	C₄²×C₆₁	D₈	1	2	yes	—
44	C₄²×C₆₁	C₂×D₄	1	2	yes	—
45	C₁₆×C₆₁	C₂×Q₈	1	8	yes	complemented
46	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
47	Q₈×C₆₁	—	8	16	no	—
48	C₂×C₄×C₆₁	—	4	8	no	—
49	C₂×C₄×C₆₁	—	8	8	no	—
50	C₂×C₄×C₆₁	D₁₆	1	4	yes	—
51	C₂×C₄×C₆₁	Q₃₂	1	4	yes	—
52	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
53	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
56	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
57	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
58	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
59	C₈×C₆₁	C₂×Q₁₆	1	4	yes	—
60	C₈×C₆₁	C₂²×Q₈	1	2	yes	—
61	C₂×#2252	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂²×C₆₁	#500	1	2	yes	—
63	C₂²×C₆₁	C₂×D₁₆	1	2	yes	—
64	C₂²×C₆₁	C₂×Q₃₂	1	2	yes	—
65	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
66	C₄×C₆₁	—	8	8	no	—
67	C₄×C₆₁	#500	1	2	yes	—
68	C₄×C₆₁	C₂×D₁₆	1	4	yes	—
69	C₄×C₆₁	C₂×Q₃₂	1	4	yes	—
70	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
72	C₂×#500	C₂×C₆₁	1	2	yes	—
73	#2252	C₂×C₆₁	1	8	yes	direct product factor
74	C₂×#366	C₂×C₆₁	1	4	yes	—
75	C₂×C₄×C₁₆	C₂×C₆₁	1	1	yes	characteristic
76	C₂×C₆₁	#2252	1	4	yes	direct product factor
77	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
78	C₂×C₆₁	C₂²×D₁₆	1	1	yes	characteristic
79	C₂×C₆₁	C₂²×Q₃₂	1	1	yes	characteristic
80	C₂×C₄⋊Q₈	—	2	2	no	—
81	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
82	#500	—	2	4	no	—
83	#500	C₂²×C₆₁	1	8	yes	—
84	C₂×#107	—	2	2	no	—
85	C₂×#107	C₂²×C₆₁	1	4	yes	—
86	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
87	#366	C₂²×C₆₁	1	16	yes	—
88	C₄×C₁₆	C₂²×C₆₁	1	4	yes	—
89	C₆₁	C₂×#2252	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×Q₈	—	4	2	no	—
91	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
92	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
93	C₄⋊Q₈	—	2	8	no	—
94	C₄⋊Q₈	—	4	4	no	—
95	C₂×C₄⋊C₄	—	2	4	no	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
98	C₂×C₁₆	Q₈×C₆₁	1	8	yes	complemented
99	C₂×C₁₆	C₂³×C₆₁	1	4	yes	—
100	#107	—	2	8	no	—
101	#107	—	2	4	no	—
102	#107	C₂³×C₆₁	1	8	yes	—
103	C₄×C₈	D₄×C₆₁	1	2	yes	—
104	C₄×C₈	C₂³×C₆₁	1	2	yes	—
105	C₂×Q₈	—	4	8	no	—
106	C₂×Q₈	—	8	8	no	—
107	C₂²×C₄	—	4	4	no	—
108	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
109	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
110	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
111	C₂×C₈	Q₁₆×C₆₁	1	4	yes	—
112	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
113	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
114	C₂×C₈	C₂×Q₈×C₆₁	1	4	yes	—
115	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
116	C₄⋊C₄	—	2	8	no	—
117	C₄⋊C₄	—	4	4	no	—
118	C₄⋊C₄	—	4	8	no	—
119	C₄²	D₈×C₆₁	1	2	yes	—
120	C₄²	C₂×D₄×C₆₁	1	2	yes	—
121	C₁₆	C₂×Q₈×C₆₁	1	8	yes	complemented
122	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
123	Q₈	—	8	16	no	—
124	C₂×C₄	—	4	8	no	—
125	C₂×C₄	—	8	8	no	—
126	C₂×C₄	D₁₆×C₆₁	1	4	yes	—
127	C₂×C₄	Q₃₂×C₆₁	1	4	yes	—
128	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
129	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
130	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
131	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
132	C₂×C₄	C₂×Q₁₆×C₆₁	1	2	yes	—
133	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
134	C₈	C₄⋊Q₈×C₆₁	1	2	yes	—
135	C₈	C₂×Q₁₆×C₆₁	1	4	yes	—
136	C₈	C₂²×Q₈×C₆₁	1	2	yes	—
137	C₂²	C₆₁×#500	1	2	yes	—
138	C₂²	C₂×D₁₆×C₆₁	1	2	yes	—
139	C₂²	C₂×Q₃₂×C₆₁	1	2	yes	—
140	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
141	C₄	—	8	8	no	—
142	C₄	C₆₁×#500	1	2	yes	—
143	C₄	C₂×D₁₆×C₆₁	1	4	yes	—
144	C₄	C₂×Q₃₂×C₆₁	1	4	yes	—
145	C₄	C₂×C₄⋊Q₈×C₆₁	1	1	yes	characteristic
146	C₄	C₂²×Q₁₆×C₆₁	1	1	yes	characteristic
147	C₂	C₆₁×#2252	1	4	yes	direct product factor
148	C₂	C₂×C₆₁×#500	1	1	yes	characteristic
149	C₂	C₂²×D₁₆×C₆₁	1	1	yes	characteristic
150	C₂	C₂²×Q₃₂×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2252	C₂	direct (N × H)
22. C₂×C₁₆×C₆₁	Q₈	semidirect
45. C₁₆×C₆₁	C₂×Q₈	semidirect
61. C₂×#2252	C₆₁	direct (N × H)
73. #2252	C₂×C₆₁	direct (N × H)
76. C₂×C₆₁	#2252	direct (N × H)
89. C₆₁	C₂×#2252	direct (N × H)
98. C₂×C₁₆	Q₈×C₆₁	semidirect
121. C₁₆	C₂×Q₈×C₆₁	semidirect
147. C₂	C₆₁×#2252	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×#500	2
C₆₁×#2252	2
C₂×C₆₁×#366	2
C₂×C₄×C₁₆×C₆₁	2
C₂×#2252	61

Subgroups — C2×C61×#2252 #29537156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#2252 #29537156