Subgroups — C₂×C₆₁×#1426 #29517070

Order 15616.

190 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×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#368	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#367	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1426	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
6	C₂×#108×C₆₁	—	2	1	no	—
7	C₂×#107×C₆₁	—	2	1	no	—
8	C₆₁×M₆₄	C₄	1	2	yes	complemented
9	C₆₁×M₆₄	C₂²	1	2	yes	—
10	C₂×C₃₂×C₆₁	C₄	1	2	yes	complemented
11	C₆₁×#368	—	2	1	no	—
12	C₆₁×#368	C₂²	1	2	yes	—
13	C₆₁×#367	—	2	1	no	—
14	C₆₁×#367	C₂²	1	2	yes	—
15	C₂×M₁₆×C₆₁	—	4	1	no	—
16	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
18	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
22	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
23	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
24	#108×C₆₁	—	2	2	no	—
25	#108×C₆₁	—	4	1	no	—
26	#107×C₆₁	—	2	2	no	—
27	#107×C₆₁	—	4	1	no	—
28	C₃₂×C₆₁	C₂×C₄	1	4	yes	complemented
29	C₂²×C₄×C₆₁	—	8	1	no	—
30	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
31	M₁₆×C₆₁	—	4	2	no	—
32	M₁₆×C₆₁	—	8	1	no	—
33	C₂×C₈×C₆₁	—	8	1	no	—
34	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
37	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
38	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₆₁	—	4	2	no	—
41	C₄⋊C₄×C₆₁	—	8	1	no	—
42	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₂³×C₆₁	SD₃₂	1	1	yes	characteristic
47	C₂×C₄×C₆₁	—	8	2	no	—
48	C₂×C₄×C₆₁	—	16	2	no	—
49	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
51	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
52	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
53	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
55	C₈×C₆₁	—	8	2	no	—
56	C₈×C₆₁	#107	1	1	yes	characteristic
57	C₈×C₆₁	#107	1	1	yes	characteristic
58	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×#1426	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₆₁	—	2	2	no	—
62	C₂²×C₆₁	#367	1	1	yes	characteristic
63	C₂²×C₆₁	#367	1	1	yes	characteristic
64	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
65	C₄×C₆₁	—	16	2	no	—
66	C₄×C₆₁	#367	1	1	yes	characteristic
67	C₄×C₆₁	#367	1	1	yes	characteristic
68	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
69	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
70	C₂×M₆₄	C₂×C₆₁	1	1	yes	characteristic
71	C₂×#368	C₂×C₆₁	1	1	yes	characteristic
72	C₂×#367	C₂×C₆₁	1	1	yes	characteristic
73	#1426	C₂×C₆₁	1	4	yes	direct product factor
74	C₂×C₆₁	—	2	1	no	—
75	C₂×C₆₁	—	2	1	no	—
76	C₂×C₆₁	#1426	1	2	yes	direct product factor
77	C₂×C₆₁	C₂×#367	1	1	yes	characteristic
78	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
79	C₂×#108	—	2	1	no	—
80	C₂×#107	—	2	1	no	—
81	M₆₄	C₄×C₆₁	1	2	yes	complemented
82	M₆₄	C₂²×C₆₁	1	2	yes	—
83	C₂×C₃₂	C₄×C₆₁	1	2	yes	complemented
84	#368	—	2	1	no	—
85	#368	C₂²×C₆₁	1	2	yes	—
86	#367	—	2	1	no	—
87	#367	C₂²×C₆₁	1	2	yes	—
88	C₆₁	C₂×#1426	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×M₁₆	—	4	1	no	—
90	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	4	1	no	—
92	C₂×C₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
93	C₂×C₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
94	C₂×C₁₆	D₄×C₆₁	1	1	yes	characteristic
95	C₂×C₁₆	Q₈×C₆₁	1	1	yes	characteristic
96	C₂×C₁₆	Q₈×C₆₁	1	1	yes	characteristic
97	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
98	#108	—	2	2	no	—
99	#108	—	4	1	no	—
100	#107	—	2	2	no	—
101	#107	—	4	1	no	—
102	C₃₂	C₂×C₄×C₆₁	1	4	yes	complemented
103	C₂²×C₄	—	8	1	no	—
104	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
105	M₁₆	—	4	2	no	—
106	M₁₆	—	8	1	no	—
107	C₂×C₈	—	8	1	no	—
108	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
109	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
110	C₂×C₈	D₈×C₆₁	1	1	yes	characteristic
111	C₂×C₈	Q₁₆×C₆₁	1	1	yes	characteristic
112	C₂×C₈	Q₁₆×C₆₁	1	1	yes	characteristic
113	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	8	1	no	—
116	C₁₆	C₄⋊C₄×C₆₁	1	1	yes	characteristic
117	C₁₆	C₄⋊C₄×C₆₁	1	1	yes	characteristic
118	C₁₆	C₂²×C₄×C₆₁	1	1	yes	characteristic
119	C₁₆	C₂×Q₈×C₆₁	1	1	yes	characteristic
120	C₂³	SD₃₂×C₆₁	1	1	yes	characteristic
121	C₂×C₄	—	8	2	no	—
122	C₂×C₄	—	16	2	no	—
123	C₂×C₄	#107×C₆₁	1	1	yes	characteristic
124	C₂×C₄	#107×C₆₁	1	1	yes	characteristic
125	C₂×C₄	SD₃₂×C₆₁	1	1	yes	characteristic
126	C₂×C₄	SD₃₂×C₆₁	1	1	yes	characteristic
127	C₂×C₄	SD₃₂×C₆₁	1	1	yes	characteristic
128	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
129	C₈	—	8	2	no	—
130	C₈	#107×C₆₁	1	1	yes	characteristic
131	C₈	#107×C₆₁	1	1	yes	characteristic
132	C₈	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
133	C₈	C₂×Q₁₆×C₆₁	1	1	yes	characteristic
134	C₂²	—	2	2	no	—
135	C₂²	C₆₁×#367	1	1	yes	characteristic
136	C₂²	C₆₁×#367	1	1	yes	characteristic
137	C₂²	C₂×SD₃₂×C₆₁	1	1	yes	characteristic
138	C₄	—	16	2	no	—
139	C₄	C₆₁×#367	1	1	yes	characteristic
140	C₄	C₆₁×#367	1	1	yes	characteristic
141	C₄	C₂×#107×C₆₁	1	1	yes	characteristic
142	C₄	C₂×SD₃₂×C₆₁	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	C₆₁×#1426	1	2	yes	direct product factor
146	C₂	C₂×C₆₁×#367	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	#367
commutator	C₁₆	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₁₆	C₂³×C₆₁
prime radical	C₂³×C₆₁	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ C₂²×C₁₆×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₁×#1426	C₂	direct (N × H)
8. C₆₁×M₆₄	C₄	semidirect
10. C₂×C₃₂×C₆₁	C₄	semidirect
28. C₃₂×C₆₁	C₂×C₄	semidirect
60. C₂×#1426	C₆₁	direct (N × H)
73. #1426	C₂×C₆₁	direct (N × H)
76. C₂×C₆₁	#1426	direct (N × H)
81. M₆₄	C₄×C₆₁	semidirect
83. C₂×C₃₂	C₄×C₆₁	semidirect
88. C₆₁	C₂×#1426	direct (N × H)
102. C₃₂	C₂×C₄×C₆₁	semidirect
145. C₂	C₆₁×#1426	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×M₆₄	2
C₂×C₆₁×#368	2
C₂×C₆₁×#367	2
C₆₁×#1426	2
C₂×#1426	61

Subgroups — C2×C61×#1426 #29517070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1426 #29517070