Subgroups — #12008302

Order 1280.

298 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₄×#3944	C₂	1	1	yes	maximal, characteristic
3	C₂×#64675	C₂	1	1	yes	maximal, characteristic
4	#10598003	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×M₃₂	C₄	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#3944	C₄	1	2	yes	—
8	C₂×C₄×#168	C₄	1	2	yes	—
9	C₅×#346	C₂²	1	2	yes	—
10	C₄×C₅×C₁₆	C₄	1	2	yes	—
11	#64681	C₂²	1	2	yes	—
12	C₄×#672	C₄	1	2	yes	—
13	#64675	C₄	1	2	yes	—
14	#64675	C₂²	1	2	yes	—
15	#11404	—	5	1	no	maximal, Sylow
16	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂²×#168	—	2	2	no	—
20	C₅×M₃₂	—	2	4	no	—
21	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
26	#3944	C₈	1	8	yes	complemented
27	C₂×#672	C₂×C₄	1	4	yes	—
28	C₄×#168	—	2	2	no	—
29	C₄×#168	C₂×C₄	1	4	yes	—
30	C₄×M₃₂	—	5	1	no	—
31	C₄×M₃₂	D₅	1	1	yes	characteristic
32	C₂×#322	—	5	1	no	—
33	#1313	—	5	4	no	—
34	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₅	M₁₆	1	2	yes	—
36	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
37	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
38	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
39	C₂×C₅×C₈	M₁₆	1	4	yes	—
40	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₅	C₄²	1	1	yes	characteristic
43	C₄²×C₅	M₁₆	1	2	yes	—
44	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×#168	—	2	4	no	—
46	C₂×#168	—	4	4	no	—
47	C₅×C₁₆	—	2	4	no	—
48	#672	C₂×C₈	1	8	yes	complemented
49	C₂×M₃₂	—	5	2	no	—
50	C₂×M₃₂	Dic₅	1	2	yes	—
51	C₂×C₄×C₈	—	5	2	no	—
52	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
53	#346	—	5	2	no	—
54	#346	C₂×D₅	1	2	yes	—
55	C₄×C₁₆	—	5	2	no	—
56	C₄×C₁₆	Dic₅	1	2	yes	—
57	#322	—	5	2	no	—
58	#322	—	5	2	no	—
59	C₂³×C₅	#97	1	1	yes	characteristic
60	C₂×C₄×C₅	—	2	4	no	—
61	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
62	C₂×C₄×C₅	#97	1	2	yes	—
63	C₂×C₄×C₅	#97	1	1	yes	characteristic
64	C₂×C₄×C₅	#97	1	2	yes	—
65	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
67	C₅×C₈	C₄×C₈	1	2	yes	—
68	C₅×C₈	#97	1	2	yes	—
69	C₅×C₈	C₂²×C₈	1	2	yes	—
70	C₅×C₈	C₂×M₁₆	1	2	yes	—
71	#168	—	4	4	no	—
72	C₂²×C₈	—	10	2	no	—
73	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
74	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
75	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
76	M₃₂	—	2	4	no	—
77	M₃₂	—	5	8	no	—
78	C₂×C₁₆	—	5	4	no	—
79	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
80	C₄×C₈	—	5	4	no	—
81	C₄×C₈	—	10	2	no	—
82	C₄×C₈	C₄×D₅	1	1	yes	characteristic
83	C₄×C₈	C₄×D₅	1	1	yes	characteristic
84	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
85	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
86	C₂²×C₅	—	2	2	no	—
87	C₂²×C₅	#322	1	2	yes	—
88	C₂²×C₅	C₂×#97	1	1	yes	characteristic
89	C₄×C₅	—	2	2	no	—
90	C₄×C₅	#322	1	2	yes	—
91	C₄×C₅	#347	1	4	yes	—
92	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₅	C₂×#97	1	1	yes	characteristic
94	C₂²×C₄	#676	1	2	yes	—
95	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
96	C₂×C₈	—	10	4	no	—
97	C₂×C₈	—	20	4	no	—
98	C₂×C₈	C₈×D₅	1	4	yes	—
99	C₂×C₈	#681	1	4	yes	—
100	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
101	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
102	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
103	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
104	C₄²	#676	1	2	yes	—
105	C₄²	C₄×Dic₅	1	1	yes	characteristic
106	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
107	C₁₆	—	2	4	no	—
108	C₁₆	—	5	8	no	—
109	C₂×C₅	—	2	2	no	—
110	C₂×C₅	C₂×#322	1	1	yes	characteristic
111	C₂×C₅	C₂×#347	1	2	yes	—
112	C₂³	#3947	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	#3935	1	2	yes	—
115	C₂×C₄	C₈×Dic₅	1	2	yes	—
116	C₂×C₄	#3947	1	1	yes	characteristic
117	C₂×C₄	#3947	1	2	yes	—
118	C₂×C₄	C₂×#676	1	2	yes	—
119	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
120	C₈	—	20	4	no	—
121	C₈	#3935	1	2	yes	—
122	C₈	C₈×Dic₅	1	2	yes	—
123	C₈	C₂×C₈×D₅	1	2	yes	—
124	C₈	C₂×#681	1	2	yes	—
125	C₅	#11404	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	2	2	no	—
127	C₂²	#64674	1	2	yes	—
128	C₂²	C₂×#3947	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	#64674	1	2	yes	—
131	C₄	#64732	1	4	yes	—
132	C₄	C₂×#3935	1	1	yes	characteristic
133	C₄	C₂×C₈×Dic₅	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×#64674	1	1	yes	characteristic
136	C₂	C₂×#64732	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₄×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	#97
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₄×M₃₂	D₅
hypocenter	C₅	#11404
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
4. #10598003	C₂	semidirect
26. #3944	C₈	semidirect
48. #672	C₂×C₈	semidirect
125. C₅	#11404	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×M₃₂	2
C₄×#3944	2
C₂×#64675	2
#10598003	2
#11404	5