Subgroups — #12010874

Order 1280.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1450	C₂	1	1	yes	maximal, characteristic
2	#10600536	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic
4	#10597931	C₂	1	4	yes	maximal, complemented
5	C₅×#431	C₄	1	1	yes	characteristic
6	C₅×#431	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3944	—	2	1	no	—
9	C₂²×#672	—	2	1	no	—
10	C₅×#322	C₄	1	2	yes	—
11	C₅×C₈²	C₂²	1	2	yes	—
12	C₄⋊#672	—	2	1	no	—
13	C₄⋊#672	—	2	1	no	—
14	C₄⋊#672	C₂²	1	2	yes	—
15	C₄⋊#672	C₂²	1	2	yes	—
16	#12005	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₅×C₄⋊C₈	—	2	1	no	—
21	C₅×C₄⋊C₈	C₈	1	2	yes	—
22	C₅×C₂²⋊C₈	—	2	1	no	—
23	C₅×C₂²⋊C₈	C₈	1	2	yes	—
24	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₄×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
30	#3944	—	4	2	no	—
31	C₂×#672	—	2	2	no	—
32	C₂×#672	—	2	2	no	—
33	C₂×#672	—	4	2	no	—
34	#2150	—	5	1	no	—
35	C₂×C₄⋊C₁₆	—	5	1	no	—
36	#1450	D₅	1	1	yes	characteristic
37	#1370	—	5	4	no	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
41	C₂×C₅×C₈	—	2	2	no	—
42	C₂×C₅×C₈	—	2	2	no	—
43	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
46	C₂×C₅×C₈	D₈	1	2	yes	—
47	C₂×C₅×C₈	Q₁₆	1	2	yes	—
48	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₅	M₁₆	1	1	yes	characteristic
53	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
54	#672	—	4	2	no	—
55	#672	—	4	2	no	—
56	C₂×M₃₂	—	10	1	no	—
57	C₂²×C₁₆	—	10	1	no	—
58	#431	Dic₅	1	1	yes	characteristic
59	#431	Dic₅	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
61	C₄⋊C₁₆	—	5	2	no	—
62	C₄⋊C₁₆	—	5	2	no	—
63	C₄⋊C₁₆	—	10	1	no	—
64	C₄⋊C₁₆	—	10	1	no	—
65	#322	Dic₅	1	2	yes	—
66	C₈²	C₂×D₅	1	2	yes	—
67	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	4	1	no	—
70	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₅	#107	1	2	yes	—
74	C₂×C₄×C₅	#108	1	2	yes	—
75	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
78	C₅×C₈	—	2	1	no	—
79	C₅×C₈	—	2	1	no	—
80	C₅×C₈	—	2	4	no	—
81	C₅×C₈	—	4	2	no	—
82	C₅×C₈	#107	1	2	yes	—
83	C₅×C₈	C₂×D₈	1	1	yes	characteristic
84	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
85	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
86	C₂²×C₈	D₂₀	1	1	yes	characteristic
87	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
88	M₃₂	—	20	2	no	—
89	C₂×C₁₆	—	10	2	no	—
90	C₂×C₁₆	—	10	2	no	—
91	C₂×C₁₆	—	20	2	no	—
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	#168	1	2	yes	—
94	C₂²⋊C₈	—	2	1	no	—
95	C₂²⋊C₈	#168	1	2	yes	—
96	C₄×C₈	Dic₁₀	1	1	yes	characteristic
97	C₄×C₈	D₂₀	1	1	yes	characteristic
98	C₄×C₈	C₂×Dic₅	1	2	yes	—
99	C₄×C₈	C₂×Dic₅	1	2	yes	—
100	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
102	C₂²×C₅	—	2	1	no	—
103	C₂²×C₅	—	2	1	no	—
104	C₂²×C₅	#335	1	2	yes	—
105	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₅	—	2	1	no	—
107	C₄×C₅	—	2	1	no	—
108	C₄×C₅	—	2	1	no	—
109	C₄×C₅	#335	1	2	yes	—
110	C₄×C₅	C₂×#107	1	1	yes	characteristic
111	C₄×C₅	C₂×#108	1	1	yes	characteristic
112	C₄×C₅	#504	1	2	yes	—
113	C₂²×C₄	C₂×#168	1	1	yes	characteristic
114	C₂²×C₄	#681	1	1	yes	characteristic
115	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	D₄₀	1	2	yes	—
119	C₂×C₈	Dic₂₀	1	2	yes	—
120	C₂×C₈	C₂×#168	1	4	yes	—
121	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
122	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
124	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
125	C₄²	C₂×#168	1	1	yes	characteristic
126	C₄²	#681	1	1	yes	characteristic
127	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
128	C₄²	C₂²×Dic₅	1	1	yes	characteristic
129	C₁₆	—	20	2	no	—
130	C₁₆	—	20	2	no	—
131	C₂×C₅	—	2	2	no	—
132	C₂×C₅	C₂×#335	1	1	yes	characteristic
133	C₂×C₅	#2151	1	1	yes	characteristic
134	C₂×C₅	#2153	1	1	yes	characteristic
135	C₂³	C₄⋊#168	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
139	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
140	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
141	C₂×C₄	#3950	1	2	yes	—
142	C₂×C₄	#3951	1	2	yes	—
143	C₂×C₄	C₂²×#168	1	1	yes	characteristic
144	C₂×C₄	C₂×#681	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	4	no	—
149	C₈	—	4	2	no	—
150	C₈	#3950	1	2	yes	—
151	C₈	C₂×D₄₀	1	1	yes	characteristic
152	C₈	C₂×Dic₂₀	1	1	yes	characteristic
153	C₅	#12005	1	1	yes	characteristic, complemented, Sylow
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#64678	1	2	yes	—
157	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#64678	1	2	yes	—
162	C₄	C₂×#3950	1	1	yes	characteristic
163	C₄	C₂×#3951	1	1	yes	characteristic
164	C₄	#65429	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×#64678	1	1	yes	characteristic
167	C₂	#10600675	1	1	yes	characteristic
168	C₂	#10600678	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₄⋊C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	#1450	D₅
hypocenter	C₅	#12005
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
4. #10597931	C₂	semidirect
153. C₅	#12005	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1450	2
#10600536	2
C₂×C₄⋊#672	2
#10597931	2
#12005	5