Subgroups — C₂×C₆₁×#2253 #29537157

Order 15616.

494 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#499	C₂	1	2	yes	maximal
2	C₆₁×#2253	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₆₁×#367	C₂	1	2	yes	maximal
4	C₂×C₆₁×#366	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₆₁	—	2	2	no	—
7	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#499	—	2	4	no	—
10	C₆₁×#499	C₂²	1	8	yes	—
11	C₂×#107×C₆₁	C₂²	1	4	yes	—
12	C₂×#106×C₆₁	—	2	2	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#367	C₂²	1	8	yes	—
15	C₆₁×#366	C₂²	1	8	yes	—
16	C₄×C₁₆×C₆₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
19	#125×C₆₁	—	2	8	no	—
20	#125×C₆₁	—	4	4	no	—
21	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
22	C₂×C₄⋊C₄×C₆₁	—	4	4	no	—
23	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₆₁	Q₈	1	4	yes	—
25	C₂×C₁₆×C₆₁	Q₈	1	4	yes	—
26	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
27	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
28	#107×C₆₁	—	2	4	no	—
29	#107×C₆₁	C₂³	1	8	yes	—
30	#106×C₆₁	—	2	8	no	—
31	C₄×C₈×C₆₁	D₄	1	2	yes	—
32	C₄×C₈×C₆₁	C₂³	1	2	yes	—
33	C₂²×C₄×C₆₁	—	4	4	no	—
34	C₂²×C₄×C₆₁	D₈	1	2	yes	—
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
37	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
40	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
41	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
42	C₄⋊C₄×C₆₁	—	2	8	no	—
43	C₄⋊C₄×C₆₁	—	4	4	no	—
44	C₄⋊C₄×C₆₁	—	4	16	no	—
45	C₄²×C₆₁	D₈	1	2	yes	—
46	C₄²×C₆₁	C₂×D₄	1	2	yes	—
47	C₁₆×C₆₁	C₂×Q₈	1	4	yes	—
48	C₁₆×C₆₁	C₂×Q₈	1	4	yes	—
49	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	—	2	4	no	—
51	C₂×C₄×C₆₁	—	4	8	no	—
52	C₂×C₄×C₆₁	—	8	8	no	—
53	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
55	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
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	1	yes	characteristic
61	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
62	C₂×#2253	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₆₁	#500	1	2	yes	—
64	C₂²×C₆₁	#508	1	4	yes	—
65	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
66	C₄×C₆₁	—	2	4	no	—
67	C₄×C₆₁	—	8	8	no	—
68	C₄×C₆₁	#500	1	2	yes	—
69	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
70	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
71	C₂×#499	C₂×C₆₁	1	2	yes	—
72	#2253	C₂×C₆₁	1	8	yes	direct product factor
73	C₂×#367	C₂×C₆₁	1	2	yes	—
74	C₂×#366	C₂×C₆₁	1	2	yes	—
75	C₂×C₄×C₁₆	C₂×C₆₁	1	1	yes	characteristic
76	C₂×C₆₁	#2253	1	4	yes	direct product factor
77	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
78	C₂×C₆₁	C₂×#508	1	2	yes	—
79	C₂×#125	—	2	2	no	—
80	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
81	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
82	#499	—	2	4	no	—
83	#499	C₂²×C₆₁	1	8	yes	—
84	C₂×#107	C₂²×C₆₁	1	4	yes	—
85	C₂×#106	—	2	2	no	—
86	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
87	#367	C₂²×C₆₁	1	8	yes	—
88	#366	C₂²×C₆₁	1	8	yes	—
89	C₄×C₁₆	C₂²×C₆₁	1	4	yes	—
90	C₆₁	C₂×#2253	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
92	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
93	#125	—	2	8	no	—
94	#125	—	4	4	no	—
95	C₂×C₄⋊C₄	—	2	4	no	—
96	C₂×C₄⋊C₄	—	4	4	no	—
97	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
98	C₂×C₁₆	Q₈×C₆₁	1	4	yes	—
99	C₂×C₁₆	Q₈×C₆₁	1	4	yes	—
100	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
101	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
102	#107	—	2	4	no	—
103	#107	C₂³×C₆₁	1	8	yes	—
104	#106	—	2	8	no	—
105	C₄×C₈	D₄×C₆₁	1	2	yes	—
106	C₄×C₈	C₂³×C₆₁	1	2	yes	—
107	C₂²×C₄	—	4	4	no	—
108	C₂²×C₄	D₈×C₆₁	1	2	yes	—
109	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
110	C₂×C₈	Q₁₆×C₆₁	1	4	yes	—
111	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
113	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
114	C₂×C₈	C₂×Q₈×C₆₁	1	2	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	16	no	—
119	C₄²	D₈×C₆₁	1	2	yes	—
120	C₄²	C₂×D₄×C₆₁	1	2	yes	—
121	C₁₆	C₂×Q₈×C₆₁	1	4	yes	—
122	C₁₆	C₂×Q₈×C₆₁	1	4	yes	—
123	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	8	no	—
126	C₂×C₄	—	8	8	no	—
127	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
128	C₂×C₄	C₂×D₈×C₆₁	1	4	yes	—
129	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
130	C₂×C₄	C₂×Q₁₆×C₆₁	1	2	yes	—
131	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
132	C₈	C₄⋊Q₈×C₆₁	1	2	yes	—
133	C₈	C₂×Q₁₆×C₆₁	1	4	yes	—
134	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
135	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
136	C₂²	C₆₁×#500	1	2	yes	—
137	C₂²	C₆₁×#508	1	4	yes	—
138	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	8	8	no	—
141	C₄	C₆₁×#500	1	2	yes	—
142	C₄	C₂×C₄⋊Q₈×C₆₁	1	1	yes	characteristic
143	C₄	C₂²×Q₁₆×C₆₁	1	1	yes	characteristic
144	C₂	C₆₁×#2253	1	4	yes	direct product factor
145	C₂	C₂×C₆₁×#500	1	1	yes	characteristic
146	C₂	C₂×C₆₁×#508	1	2	yes	—

	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×#2253	C₂	direct (N × H)
62. C₂×#2253	C₆₁	direct (N × H)
72. #2253	C₂×C₆₁	direct (N × H)
76. C₂×C₆₁	#2253	direct (N × H)
90. C₆₁	C₂×#2253	direct (N × H)
144. C₂	C₆₁×#2253	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×#499	2
C₆₁×#2253	2
C₂×C₆₁×#367	2
C₂×C₆₁×#366	2
C₂×C₄×C₁₆×C₆₁	2
C₂×#2253	61

Subgroups — C2×C61×#2253 #29537157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#2253 #29537157