Subgroups — #12008281

Order 1280.

330 conjugacy classes of subgroups.

The subgroup lattice has 144 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	C₄×#3944	C₂	1	1	yes	maximal, characteristic
3	#10600517	C₂	1	1	yes	maximal, characteristic
4	#10597989	C₂	1	2	yes	maximal, complemented
5	C₁₆×#168	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×C₁₆	C₄	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3944	C₄	1	2	yes	—
9	#65214	C₄	1	2	yes	—
10	C₄×C₅×C₁₆	C₄	1	2	yes	—
11	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	#64681	C₄	1	2	yes	—
14	C₄×#672	C₂²	1	2	yes	—
15	#64675	C₄	1	2	yes	—
16	C₈×#168	C₂²	1	2	yes	—
17	#11399	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₅×C₁₆	C₈	1	8	yes	complemented
22	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
24	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂²⋊#168	C₈	1	4	yes	—
29	#3944	C₈	1	8	yes	complemented
30	C₂×#672	C₂×C₄	1	4	yes	—
31	C₄⋊#168	C₈	1	4	yes	—
32	C₄×#168	C₂×C₄	1	4	yes	—
33	C₄×M₃₂	—	5	1	no	—
34	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic
35	#1450	—	5	1	no	—
36	#1311	—	5	2	no	—
37	C₈×C₁₆	—	5	2	no	—
38	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
40	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
41	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
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	C₄²×C₅	C₄²	1	1	yes	characteristic
47	C₄²×C₅	C₂×C₈	1	2	yes	—
48	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×#168	C₂×C₈	1	8	yes	—
50	C₅×C₁₆	C₂×C₈	1	4	yes	—
51	C₅×C₁₆	C₂×C₈	1	4	yes	—
52	#672	C₂×C₈	1	8	yes	—
53	C₂×M₃₂	—	5	2	no	—
54	C₂²×C₁₆	Dic₅	1	2	yes	—
55	#431	—	5	2	no	—
56	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
57	#346	—	5	2	no	—
58	C₄×C₁₆	—	5	2	no	—
59	C₄×C₁₆	Dic₅	1	2	yes	—
60	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
61	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
62	#322	—	5	2	no	—
63	C₈²	—	5	2	no	—
64	C₂³×C₅	C₄×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅	—	2	4	no	—
66	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
67	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
70	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
72	C₅×C₈	C₄×C₈	1	2	yes	—
73	C₅×C₈	C₄×C₈	1	2	yes	—
74	C₅×C₈	C₂²×C₈	1	2	yes	—
75	C₅×C₈	C₂²×C₈	1	2	yes	—
76	#168	—	2	8	no	—
77	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
78	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
79	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
80	M₃₂	—	5	8	no	—
81	C₂×C₁₆	—	5	4	no	—
82	C₂×C₁₆	#168	1	8	yes	complemented
83	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
84	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
85	C₄⋊C₈	—	5	4	no	—
86	C₂²⋊C₈	—	5	4	no	—
87	C₄×C₈	—	5	4	no	—
88	C₄×C₈	C₄×D₅	1	1	yes	characteristic
89	C₄×C₈	C₄×D₅	1	1	yes	characteristic
90	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
92	C₂²×C₅	—	2	2	no	—
93	C₂²×C₅	C₈²	1	2	yes	—
94	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
95	C₄×C₅	—	2	2	no	—
96	C₄×C₅	C₈²	1	2	yes	—
97	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
98	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
99	C₄×C₅	#504	1	4	yes	—
100	C₂²×C₄	C₈×D₅	1	2	yes	—
101	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
102	C₂×C₈	—	5	8	no	—
103	C₂×C₈	C₈×D₅	1	4	yes	—
104	C₂×C₈	C₂×#168	1	4	yes	—
105	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
106	C₂×C₈	C₄×Dic₅	1	1	yes	characteristic
107	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
108	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
109	C₄²	C₈×D₅	1	2	yes	—
110	C₄²	C₄×Dic₅	1	1	yes	characteristic
111	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
112	C₁₆	—	5	8	no	—
113	C₁₆	C₂×#168	1	4	yes	—
114	C₁₆	C₂×#168	1	4	yes	—
115	C₂×C₅	—	2	2	no	—
116	C₂×C₅	C₂×C₈²	1	1	yes	characteristic
117	C₂×C₅	#2108	1	2	yes	—
118	C₂³	C₈×Dic₅	1	1	yes	characteristic
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	C₄×#168	1	2	yes	—
121	C₂×C₄	C₈×Dic₅	1	2	yes	—
122	C₂×C₄	C₈×Dic₅	1	2	yes	—
123	C₂×C₄	C₈×Dic₅	1	1	yes	characteristic
124	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
125	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
126	C₈	—	10	8	no	—
127	C₈	C₄×#168	1	2	yes	—
128	C₈	C₈×Dic₅	1	2	yes	—
129	C₈	C₂×C₈×D₅	1	2	yes	—
130	C₈	C₂²×#168	1	2	yes	—
131	C₅	#11399	1	1	yes	characteristic, complemented, Sylow
132	C₂²	—	2	2	no	—
133	C₂²	C₈×#168	1	2	yes	—
134	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	C₈×#168	1	2	yes	—
137	C₄	#65190	1	4	yes	—
138	C₄	C₂×C₄×#168	1	1	yes	characteristic
139	C₄	C₂×C₈×Dic₅	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	C₂×C₈×#168	1	1	yes	characteristic
142	C₂	#10602192	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₂×D₅
commutator	C₂×C₅	C₂×C₈²
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₂×C₄×C₅×C₁₆	C₂
socle	C₂²×C₅	C₂×C₄×C₈
prime radical	C₂³×C₅	C₄×C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₅
hypocenter	C₅	#11399
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
4. #10597989	C₂	semidirect
5. C₁₆×#168	C₂	semidirect
21. C₂×C₅×C₁₆	C₈	semidirect
29. #3944	C₈	semidirect
82. C₂×C₁₆	#168	semidirect
131. C₅	#11399	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
C₄×#3944	2
#10600517	2
#10597989	2
C₁₆×#168	2
#11399	5