Subgroups — #12014311

Order 1280.

354 conjugacy classes of subgroups.

The subgroup lattice has 162 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×#3947	C₂	1	1	yes	maximal, characteristic
3	C₄×#3944	C₂	1	1	yes	maximal, characteristic
4	#10598302	C₂	1	4	yes	maximal
5	C₂×C₅×M₃₂	C₂²	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#3947	C₄	1	2	yes	—
8	C₂×#3947	C₂²	1	2	yes	—
9	C₂×#3944	C₂²	1	2	yes	—
10	C₄²×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₄×#168	C₄	1	1	yes	characteristic
12	C₅×#346	C₄	1	2	yes	complemented
13	C₄×C₅×C₁₆	C₄	1	2	yes	complemented
14	#64681	C₄	1	2	yes	complemented
15	C₄×#672	C₄	1	2	yes	complemented
16	#12594	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅	—	2	2	no	—
21	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
22	C₂²×#168	C₂×C₄	1	2	yes	—
23	C₅×M₃₂	—	2	4	no	—
24	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	#3947	—	2	4	no	—
30	#3947	C₈	1	8	yes	—
31	#3944	—	2	4	no	—
32	C₂×#672	C₂×C₄	1	4	yes	—
33	C₄×#168	—	2	2	no	—
34	C₄×M₃₂	—	5	1	no	—
35	C₄×M₃₂	D₅	1	1	yes	characteristic
36	C₄×#97	—	5	1	no	—
37	#1375	—	5	4	no	—
38	C₂²×C₄×C₅	M₁₆	1	2	yes	—
39	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅	—	2	2	no	—
41	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
42	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
43	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
44	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	#41	1	4	yes	—
47	C₄²×C₅	C₄²	1	1	yes	characteristic
48	C₄²×C₅	C₄²	1	1	yes	characteristic
49	C₄²×C₅	M₁₆	1	2	yes	—
50	C₄×Dic₅	—	2	2	no	—
51	C₄×Dic₅	—	4	4	no	—
52	C₄×Dic₅	C₂×C₈	1	4	yes	—
53	C₂×#168	—	2	2	no	—
54	C₂×#168	—	2	2	no	—
55	C₂×#168	C₂×C₈	1	4	yes	—
56	C₅×C₁₆	—	2	4	no	—
57	#672	—	2	4	no	—
58	C₂×M₃₂	—	5	2	no	—
59	C₂×M₃₂	C₂×D₅	1	2	yes	—
60	C₂×#97	—	5	2	no	—
61	C₂×#97	—	5	2	no	—
62	C₂×C₄×C₈	—	5	1	no	—
63	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
64	C₄³	—	5	1	no	—
65	#346	—	5	2	no	—
66	#346	Dic₅	1	2	yes	complemented
67	C₄×C₁₆	—	5	2	no	—
68	C₄×C₁₆	Dic₅	1	2	yes	complemented
69	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅	—	2	4	no	—
71	C₂×C₄×C₅	#97	1	2	yes	—
72	C₂×C₄×C₅	#97	1	2	yes	—
73	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₅	#117	1	2	yes	—
75	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₅	—	2	4	no	—
78	C₂×Dic₅	—	4	4	no	—
79	C₅×C₈	C₄×C₈	1	2	yes	—
80	C₅×C₈	C₄×C₈	1	2	yes	—
81	C₅×C₈	#117	1	2	yes	—
82	C₅×C₈	#117	1	2	yes	—
83	#168	—	2	4	no	—
84	C₂²×C₈	—	5	2	no	—
85	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
86	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
87	C₂×C₄²	—	5	2	no	—
88	C₂×C₄²	—	10	2	no	—
89	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
90	M₃₂	—	2	4	no	—
91	M₃₂	—	10	4	no	—
92	C₂×C₁₆	—	5	4	no	—
93	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
94	#97	—	5	8	no	—
95	#97	—	10	4	no	—
96	C₄×C₈	—	10	2	no	—
97	C₄×C₈	C₄×D₅	1	1	yes	characteristic
98	C₄×C₈	C₄×D₅	1	1	yes	characteristic
99	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
100	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
101	C₂²×C₅	—	2	2	no	—
102	C₂²×C₅	C₂×#97	1	1	yes	characteristic
103	C₂²×C₅	#431	1	2	yes	—
104	C₄×C₅	—	2	2	no	—
105	C₄×C₅	#347	1	4	yes	—
106	C₄×C₅	#376	1	1	yes	characteristic
107	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
108	C₄×C₅	#431	1	2	yes	—
109	Dic₅	—	4	4	no	—
110	C₂²×C₄	—	10	2	no	—
111	C₂²×C₄	#676	1	2	yes	—
112	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
113	C₂×C₈	—	5	4	no	—
114	C₂×C₈	—	10	2	no	—
115	C₂×C₈	—	10	2	no	—
116	C₂×C₈	C₈×D₅	1	4	yes	—
117	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
118	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
119	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
120	C₂×C₈	#709	1	4	yes	—
121	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
122	C₄²	—	5	4	no	—
123	C₄²	—	10	2	no	—
124	C₄²	—	20	4	no	—
125	C₄²	#676	1	2	yes	—
126	C₄²	C₄×Dic₅	1	1	yes	characteristic
127	C₄²	C₄×Dic₅	1	1	yes	characteristic
128	C₁₆	—	2	4	no	—
129	C₁₆	—	10	4	no	—
130	C₂×C₅	—	2	2	no	—
131	C₂×C₅	#1744	1	1	yes	characteristic
132	C₂×C₅	C₂×#347	1	2	yes	—
133	C₂³	C₂×#676	1	1	yes	characteristic
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	10	4	no	—
136	C₂×C₄	—	20	4	no	—
137	C₂×C₄	#3947	1	2	yes	—
138	C₂×C₄	#3947	1	2	yes	—
139	C₂×C₄	#4018	1	2	yes	—
140	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
141	C₂×C₄	C₂×#676	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
143	C₈	—	10	4	no	—
144	C₈	C₈×Dic₅	1	2	yes	—
145	C₈	C₈×Dic₅	1	2	yes	—
146	C₈	#4018	1	2	yes	—
147	C₈	#4072	1	2	yes	—
148	C₅	#12594	1	1	yes	characteristic, complemented, Sylow
149	C₂²	—	2	2	no	—
150	C₂²	#64974	1	2	yes	—
151	C₂²	C₂×#3947	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	20	4	no	—
154	C₄	#64732	1	4	yes	—
155	C₄	#64974	1	2	yes	—
156	C₄	#65221	1	1	yes	characteristic
157	C₄	C₂×C₈×Dic₅	1	1	yes	characteristic
158	C₂	—	2	2	no	—
159	C₂	#10600543	1	1	yes	characteristic
160	C₂	C₂×#64732	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₄×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₄×M₃₂	D₅
hypocenter	C₅	#12594
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
12. C₅×#346	C₄	semidirect
13. C₄×C₅×C₁₆	C₄	semidirect
14. #64681	C₄	semidirect
15. C₄×#672	C₄	semidirect
66. #346	Dic₅	semidirect
68. C₄×C₁₆	Dic₅	semidirect
148. C₅	#12594	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×#3947	2
C₄×#3944	2
#10598302	2
#12594	5