Subgroups — #12014305

Order 1280.

358 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1726	C₂	1	1	yes	maximal, characteristic
2	C₂×#64681	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic
4	#10598148	C₂	1	4	yes	maximal
5	C₂×C₅×#97	C₄	1	2	yes	—
6	C₂×C₅×#97	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#376	C₄	1	1	yes	characteristic
10	C₂²×#672	C₂²	1	2	yes	—
11	C₂²×#672	C₂²	1	2	yes	—
12	#64681	C₄	1	4	yes	complemented
13	C₄×#672	C₄	1	4	yes	complemented
14	#12592	—	5	1	no	maximal, Sylow
15	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₅×#97	—	2	4	no	—
21	C₅×#97	C₈	1	4	yes	—
22	C₄×C₅×C₈	—	2	4	no	—
23	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₅×#95	C₈	1	4	yes	—
26	C₂×#672	—	2	4	no	—
27	C₂×#672	—	2	4	no	—
28	C₂×#672	C₂×C₄	1	4	yes	—
29	C₂×#672	C₂×C₄	1	4	yes	—
30	C₂×#346	—	5	1	no	—
31	C₂×C₄×C₁₆	—	5	1	no	—
32	#1726	D₅	1	1	yes	characteristic
33	#1374	—	5	4	no	—
34	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₅×C₈	—	2	2	no	—
38	C₂×C₅×C₈	—	2	2	no	—
39	C₂×C₅×C₈	C₄²	1	2	yes	—
40	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
41	C₂×C₅×C₈	M₁₆	1	4	yes	—
42	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	#41	1	4	yes	—
45	C₄²×C₅	—	2	2	no	—
46	C₄²×C₅	—	2	2	no	—
47	C₄²×C₅	C₄²	1	2	yes	—
48	C₄²×C₅	C₂×C₈	1	2	yes	—
49	#672	—	2	4	no	—
50	#672	—	2	4	no	—
51	C₂²×C₁₆	—	5	2	no	—
52	C₂²×C₁₆	—	5	2	no	—
53	C₂×#97	Dic₅	1	2	yes	—
54	C₂×#97	Dic₅	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
56	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
57	#376	Dic₅	1	1	yes	characteristic
58	#346	—	5	4	no	—
59	C₄×C₁₆	—	5	4	no	—
60	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅	—	2	4	no	—
62	C₂×C₄×C₅	—	2	4	no	—
63	C₂×C₄×C₅	—	2	8	no	—
64	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
65	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅	#117	1	2	yes	—
67	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
69	C₅×C₈	—	2	4	no	—
70	C₅×C₈	#97	1	4	yes	—
71	C₅×C₈	#117	1	2	yes	—
72	C₅×C₈	#117	1	2	yes	—
73	C₂²×C₈	C₂×Dic₅	1	2	yes	—
74	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
75	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
76	C₂×C₄²	C₂×Dic₅	1	2	yes	—
77	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
78	C₂×C₁₆	—	5	4	no	—
79	C₂×C₁₆	—	5	4	no	—
80	C₂×C₁₆	—	10	4	no	—
81	C₂×C₁₆	—	10	4	no	—
82	#97	—	2	4	no	—
83	#97	#168	1	4	yes	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	C₄×D₅	1	2	yes	—
86	C₄×C₈	C₄×D₅	1	2	yes	—
87	#95	#168	1	4	yes	—
88	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
89	C₂²×C₅	#431	1	2	yes	—
90	C₂²×C₅	#504	1	4	yes	—
91	C₄×C₅	—	2	4	no	—
92	C₄×C₅	—	4	4	no	—
93	C₄×C₅	#376	1	1	yes	characteristic
94	C₄×C₅	C₂×#97	1	1	yes	characteristic
95	C₄×C₅	#431	1	2	yes	—
96	C₂²×C₄	C₂×#168	1	4	yes	—
97	C₂²×C₄	C₂×#168	1	2	yes	—
98	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂×#168	1	4	yes	—
102	C₂×C₈	#681	1	4	yes	—
103	C₂×C₈	C₄×Dic₅	1	2	yes	—
104	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
105	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
106	C₂×C₈	#709	1	4	yes	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂×#168	1	2	yes	—
110	C₄²	C₄×Dic₅	1	2	yes	—
111	C₁₆	—	10	4	no	—
112	C₁₆	—	10	4	no	—
113	C₂×C₅	#1744	1	1	yes	characteristic
114	C₂×C₅	#2108	1	2	yes	—
115	C₂×C₅	#2164	1	4	yes	—
116	C₂³	C₂²×#168	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	8	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	C₄×#168	1	4	yes	—
121	C₂×C₄	C₂²×#168	1	1	yes	characteristic
122	C₂×C₄	C₂×#681	1	2	yes	—
123	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
124	C₂×C₄	#4072	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	#3935	1	4	yes	—
127	C₈	#4018	1	2	yes	—
128	C₈	#4018	1	2	yes	—
129	C₅	#12592	1	1	yes	characteristic, complemented, Sylow
130	C₂²	C₂×C₄×#168	1	1	yes	characteristic
131	C₂²	#65214	1	2	yes	—
132	C₂²	#65429	1	4	yes	—
133	C₄	—	2	4	no	—
134	C₄	—	4	4	no	—
135	C₄	C₂×#3935	1	1	yes	characteristic
136	C₄	#65214	1	2	yes	—
137	C₄	#65221	1	1	yes	characteristic
138	C₂	#10600181	1	1	yes	characteristic
139	C₂	#10600674	1	2	yes	—
140	C₂	#10600676	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₅×#1726	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1726	D₅
hypocenter	C₅	#12592
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
12. #64681	C₄	semidirect
13. C₄×#672	C₄	semidirect
129. C₅	#12592	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1726	2
C₂×#64681	2
C₂×C₄×#672	2
#10598148	2
#12592	5