Subgroups — #12014123

Order 1280.

478 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#64722	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₅	C₂	1	2	yes	maximal
4	#10600542	C₂	1	1	yes	maximal, characteristic
5	C₂×#64681	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3947	C₄	1	2	yes	—
10	C₂×C₈×Dic₅	C₂²	1	2	yes	—
11	C₂²×#672	C₂²	1	2	yes	—
12	#65221	C₄	1	1	yes	characteristic
13	C₂×#3935	C₄	1	1	yes	characteristic
14	C₄×C₅×C₁₆	C₄	1	4	yes	complemented
15	#64722	C₄	1	8	yes	complemented
16	C₁₆×Dic₅	C₄	1	8	yes	complemented
17	#64681	C₄	1	4	yes	complemented
18	#12539	—	5	1	no	maximal, Sylow
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
23	C₂²×#168	C₂×C₄	1	2	yes	—
24	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
25	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
30	#3963	C₈	1	4	yes	—
31	#3947	C₈	1	8	yes	—
32	C₈×Dic₅	C₂×C₄	1	4	yes	—
33	C₈×Dic₅	C₂×C₄	1	4	yes	—
34	C₂×#672	C₂×C₄	1	4	yes	—
35	C₂×#672	C₂×C₄	1	4	yes	—
36	C₂×#672	C₂×C₄	1	4	yes	—
37	#3935	C₈	1	4	yes	—
38	C₂×#346	—	5	2	no	—
39	C₂×#346	—	5	1	no	—
40	C₂×C₄×C₁₆	—	5	2	no	—
41	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic
42	#1726	—	5	1	no	—
43	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅	C₂×C₈	1	4	yes	—
46	C₂×C₅×C₈	C₄²	1	2	yes	—
47	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
48	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
50	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅	C₄²	1	2	yes	—
53	C₄²×C₅	C₂×C₈	1	2	yes	—
54	C₄×Dic₅	C₄²	1	4	yes	—
55	C₄×Dic₅	C₂×C₈	1	4	yes	—
56	C₂×#168	C₄²	1	4	yes	—
57	C₂×#168	C₂×C₈	1	4	yes	—
58	C₂×#168	C₂×C₈	1	4	yes	—
59	C₅×C₁₆	C₄²	1	4	yes	—
60	C₅×C₁₆	C₄²	1	4	yes	—
61	#672	C₄²	1	8	yes	—
62	C₂²×C₁₆	—	5	2	no	—
63	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
64	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
65	C₂×#97	—	5	2	no	—
66	C₂×#97	—	5	1	no	—
67	C₂×C₄×C₈	—	5	2	no	—
68	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
69	#376	—	5	1	no	—
70	#346	—	5	8	no	—
71	#346	—	5	4	no	—
72	C₄×C₁₆	—	5	8	no	—
73	C₄×C₁₆	Dic₅	1	4	yes	complemented
74	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅	—	2	4	no	—
76	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
77	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
78	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
81	C₂×Dic₅	—	2	8	no	—
82	C₂×Dic₅	C₄×C₈	1	8	yes	—
83	C₅×C₈	C₄×C₈	1	4	yes	—
84	C₅×C₈	C₂×C₄²	1	2	yes	—
85	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
86	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
87	#168	C₄×C₈	1	8	yes	—
88	C₂²×C₈	—	5	2	no	—
89	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
91	C₂×C₄²	—	5	2	no	—
92	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
93	C₂×C₁₆	—	5	4	no	—
94	C₂×C₁₆	—	5	4	no	—
95	C₂×C₁₆	—	5	4	no	—
96	C₂×C₁₆	C₄×D₅	1	4	yes	—
97	C₂×C₁₆	C₄×D₅	1	4	yes	—
98	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
99	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
100	#97	—	5	8	no	—
101	#97	—	5	4	no	—
102	C₄×C₈	—	5	4	no	—
103	C₄×C₈	—	5	4	no	—
104	C₄×C₈	C₄×D₅	1	2	yes	—
105	C₄×C₈	C₂×Dic₅	1	2	yes	—
106	#95	—	5	4	no	—
107	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
109	C₂²×C₅	#504	1	4	yes	—
110	C₄×C₅	—	2	4	no	—
111	C₄×C₅	C₄³	1	1	yes	characteristic
112	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
113	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
114	Dic₅	—	2	8	no	—
115	C₂²×C₄	—	5	4	no	—
116	C₂²×C₄	C₈×D₅	1	2	yes	—
117	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
118	C₂×C₈	—	5	4	no	—
119	C₂×C₈	—	5	4	no	—
120	C₂×C₈	—	5	4	no	—
121	C₂×C₈	C₈×D₅	1	4	yes	—
122	C₂×C₈	C₄×Dic₅	1	2	yes	—
123	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
124	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
125	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
126	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
127	C₄²	—	5	4	no	—
128	C₄²	—	5	4	no	—
129	C₄²	C₈×D₅	1	2	yes	—
130	C₄²	C₄×Dic₅	1	2	yes	—
131	C₁₆	—	5	8	no	—
132	C₁₆	C₄×Dic₅	1	4	yes	—
133	C₁₆	C₄×Dic₅	1	4	yes	—
134	C₂×C₅	C₄²×C₈	1	1	yes	characteristic
135	C₂×C₅	#2108	1	4	yes	—
136	C₂×C₅	#2108	1	2	yes	—
137	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	5	8	no	—
140	C₂×C₄	—	10	8	no	—
141	C₂×C₄	C₈×Dic₅	1	4	yes	—
142	C₂×C₄	C₄²×D₅	1	2	yes	—
143	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
144	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
146	C₈	—	5	8	no	—
147	C₈	C₈×Dic₅	1	4	yes	—
148	C₈	C₄²×D₅	1	2	yes	—
149	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
150	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
151	C₅	#12539	1	1	yes	characteristic, complemented, Sylow
152	C₂²	C₄×C₈×D₅	1	2	yes	—
153	C₂²	#65190	1	4	yes	—
154	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
155	C₄	—	2	4	no	—
156	C₄	—	10	8	no	—
157	C₄	C₄×C₈×D₅	1	2	yes	—
158	C₄	C₄²×Dic₅	1	1	yes	characteristic
159	C₄	C₂×C₈×Dic₅	1	1	yes	characteristic
160	C₂	#10599844	1	4	yes	—
161	C₂	C₄×C₈×Dic₅	1	1	yes	characteristic
162	C₂	#10602192	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₅
commutator	C₂×C₅	C₄²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₄×C₅×C₁₆	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₅
hypocenter	C₅	#12539
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
14. C₄×C₅×C₁₆	C₄	semidirect
15. #64722	C₄	semidirect
16. C₁₆×Dic₅	C₄	semidirect
17. #64681	C₄	semidirect
73. C₄×C₁₆	Dic₅	semidirect
151. C₅	#12539	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₁₆	2
C₂×#64722	2
C₂×C₁₆×Dic₅	2
#10600542	2
C₂×#64681	2
#12539	5