Subgroups — #12015717

Order 1280.

382 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#64683	C₂	1	4	yes	maximal
3	#10600536	C₂	1	2	yes	maximal
4	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
5	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₄³×C₅	C₄	1	1	yes	characteristic
9	C₂×#3944	C₂²	1	4	yes	—
10	#64683	—	2	8	no	—
11	C₄⋊#672	—	2	4	no	—
12	#12900	—	5	1	no	maximal, Sylow
13	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
15	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅	—	2	2	no	—
17	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅×C₈	—	2	4	no	—
20	C₄×C₅×C₈	C₈	1	2	yes	—
21	C₄×C₅×C₈	C₈	1	2	yes	—
22	C₄×C₅×C₈	D₄	1	2	yes	—
23	C₄×C₅×C₈	D₄	1	4	yes	—
24	C₄×C₅×C₈	Q₈	1	4	yes	—
25	C₄×C₅×C₈	Q₈	1	2	yes	—
26	#3944	—	2	8	no	—
27	C₂×#672	—	2	4	no	—
28	C₂×#364	—	5	4	no	—
29	#2150	—	5	2	no	—
30	C₄²×C₈	D₅	1	1	yes	characteristic
31	C₂²×C₄×C₅	—	2	2	no	—
32	C₂²×C₄×C₅	M₁₆	1	2	yes	—
33	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₅×C₈	—	2	2	no	—
35	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
36	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
37	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
38	C₂×C₅×C₈	M₁₆	1	4	yes	—
39	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
40	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
42	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₅	—	2	8	no	—
44	C₄²×C₅	—	2	2	no	—
45	C₄²×C₅	C₄⋊C₄	1	4	yes	—
46	C₄²×C₅	C₄⋊C₄	1	2	yes	—
47	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
49	#672	—	4	4	no	—
50	C₂×M₃₂	—	5	4	no	—
51	C₂×C₄×C₈	Dic₅	1	2	yes	—
52	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
54	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
55	C₄³	Dic₅	1	1	yes	characteristic
56	#364	—	10	8	no	—
57	C₄⋊C₁₆	—	10	4	no	—
58	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₅	—	2	8	no	—
60	C₂×C₄×C₅	—	2	4	no	—
61	C₂×C₄×C₅	—	2	4	no	—
62	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
63	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
64	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
68	C₅×C₈	—	2	2	no	—
69	C₅×C₈	C₄⋊C₈	1	8	yes	—
70	C₅×C₈	#127	1	1	yes	characteristic
71	C₅×C₈	C₄⋊Q₈	1	2	yes	—
72	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
73	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
74	C₂²×C₈	C₂×Dic₅	1	2	yes	—
75	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
76	C₂×C₄²	—	2	2	no	—
77	C₂×C₄²	C₂×Dic₅	1	2	yes	—
78	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
79	M₃₂	—	10	8	no	—
80	C₂×C₁₆	—	10	4	no	—
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	#168	1	2	yes	—
83	C₄×C₈	#168	1	2	yes	—
84	C₄×C₈	Dic₁₀	1	4	yes	—
85	C₄×C₈	Dic₁₀	1	2	yes	—
86	C₄×C₈	D₂₀	1	4	yes	—
87	C₄×C₈	D₂₀	1	2	yes	—
88	C₂²×C₅	—	2	2	no	—
89	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
90	C₂²×C₅	#423	1	2	yes	—
91	C₄×C₅	—	2	8	no	—
92	C₄×C₅	—	2	2	no	—
93	C₄×C₅	#364	1	4	yes	complemented
94	C₄×C₅	#384	1	1	yes	characteristic
95	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
96	C₄×C₅	#423	1	1	yes	characteristic
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	#681	1	2	yes	—
99	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
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	C₄⋊Dic₅	1	4	yes	—
105	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
106	C₂×C₈	C₂×Dic₁₀	1	2	yes	—
107	C₂×C₈	C₂×D₂₀	1	2	yes	—
108	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
109	C₄²	—	2	8	no	—
110	C₄²	—	2	2	no	—
111	C₄²	C₂×#168	1	1	yes	characteristic
112	C₄²	C₂×#168	1	1	yes	characteristic
113	C₄²	C₄⋊Dic₅	1	4	yes	—
114	C₄²	C₄⋊Dic₅	1	2	yes	—
115	C₁₆	—	20	4	no	—
116	C₂×C₅	—	2	2	no	—
117	C₂×C₅	#1842	1	1	yes	characteristic
118	C₂×C₅	C₂×#364	1	2	yes	—
119	C₂³	C₂×#681	1	1	yes	characteristic
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	C₄⋊#168	1	2	yes	—
124	C₂×C₄	C₄⋊#168	1	2	yes	—
125	C₂×C₄	C₂²×#168	1	1	yes	characteristic
126	C₂×C₄	C₂×#681	1	2	yes	—
127	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
128	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	C₄⋊#168	1	8	yes	—
131	C₈	#4015	1	2	yes	—
132	C₈	#4015	1	1	yes	characteristic
133	C₈	#4020	1	1	yes	characteristic
134	C₅	#12900	1	1	yes	characteristic, complemented, Sylow
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
137	C₂²	#65213	1	2	yes	—
138	C₄	—	2	8	no	—
139	C₄	—	2	2	no	—
140	C₄	#64683	1	4	yes	complemented
141	C₄	C₂×C₄⋊#168	1	2	yes	—
142	C₄	#65213	1	1	yes	characteristic
143	C₄	#65224	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	#10600187	1	1	yes	characteristic
146	C₂	C₂×#64683	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₄²×C₅×C₈	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₄²×C₈	D₅
hypocenter	C₅	#12900
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
93. C₄×C₅	#364	semidirect
134. C₅	#12900	semidirect
140. C₄	#64683	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₅×C₈	2
C₂×#64683	2
#10600536	2
#12900	5