Subgroups — #12010908

Order 1280.

220 conjugacy classes of subgroups.

The subgroup lattice has 180 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	#10600538	C₂	1	1	yes	maximal, characteristic
3	#10600537	C₂	1	1	yes	maximal, characteristic
4	#10597931	C₂	1	2	yes	maximal, complemented
5	#10597930	C₂	1	2	yes	maximal, complemented
6	C₅×#431	C₄	1	1	yes	characteristic
7	C₅×#431	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#322	C₄	1	1	yes	characteristic
10	C₅×#322	C₄	1	1	yes	characteristic
11	C₅×C₈²	C₂²	1	1	yes	characteristic
12	C₅×C₈²	C₂²	1	1	yes	characteristic
13	C₂²⋊#672	—	2	1	no	—
14	C₂²⋊#672	—	2	1	no	—
15	C₄⋊#672	C₂²	1	2	yes	—
16	C₄⋊#672	C₂²	1	2	yes	—
17	#64681	—	2	1	no	—
18	C₄×#672	—	2	1	no	—
19	#12014	—	5	1	no	maximal, Sylow
20	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₅×C₄⋊C₈	—	2	1	no	—
24	C₅×C₄⋊C₈	C₈	1	2	yes	—
25	C₅×C₂²⋊C₈	—	2	1	no	—
26	C₅×C₂²⋊C₈	C₈	1	2	yes	—
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₅×C₈	D₄	1	1	yes	characteristic
33	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
34	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂×#672	—	2	2	no	—
36	C₂×#672	—	2	2	no	—
37	#2163	—	5	1	no	—
38	#2162	—	5	1	no	—
39	#1450	D₅	1	1	yes	characteristic
40	#1370	—	5	2	no	—
41	#1367	—	5	2	no	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	—	2	1	no	—
48	C₂×C₅×C₈	—	2	1	no	—
49	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
52	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
53	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₅	M₁₆	1	1	yes	characteristic
58	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
59	#672	—	4	2	no	—
60	#672	—	4	2	no	—
61	#431	Dic₅	1	1	yes	characteristic
62	#431	Dic₅	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
64	C₄⋊C₁₆	—	5	2	no	—
65	C₄⋊C₁₆	—	5	2	no	—
66	C₂²⋊C₁₆	—	10	1	no	—
67	C₂²⋊C₁₆	—	10	1	no	—
68	#346	—	10	1	no	—
69	C₄×C₁₆	—	10	1	no	—
70	#322	Dic₅	1	1	yes	characteristic
71	#322	Dic₅	1	1	yes	characteristic
72	C₈²	C₂×D₅	1	1	yes	characteristic
73	C₈²	C₂×D₅	1	1	yes	characteristic
74	C₂³×C₅	C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	2	1	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	4	1	no	—
79	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
85	C₅×C₈	—	2	1	no	—
86	C₅×C₈	—	2	1	no	—
87	C₅×C₈	—	2	2	no	—
88	C₅×C₈	—	2	2	no	—
89	C₅×C₈	—	4	1	no	—
90	C₅×C₈	—	4	1	no	—
91	C₅×C₈	#108	1	2	yes	—
92	C₅×C₈	#135	1	2	yes	—
93	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
94	C₂²×C₈	D₂₀	1	1	yes	characteristic
95	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
96	C₂×C₁₆	—	10	2	no	—
97	C₂×C₁₆	—	10	2	no	—
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	#168	1	2	yes	—
100	C₂²⋊C₈	—	2	1	no	—
101	C₂²⋊C₈	#168	1	2	yes	—
102	C₄×C₈	Dic₁₀	1	1	yes	characteristic
103	C₄×C₈	D₂₀	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
105	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
106	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
108	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
109	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
110	C₂²×C₅	—	2	1	no	—
111	C₂²×C₅	—	2	1	no	—
112	C₂²×C₅	—	2	1	no	—
113	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×C₅	—	2	1	no	—
115	C₄×C₅	—	2	1	no	—
116	C₄×C₅	—	2	1	no	—
117	C₄×C₅	—	2	1	no	—
118	C₄×C₅	#427	1	1	yes	characteristic
119	C₄×C₅	C₂×#108	1	1	yes	characteristic
120	C₄×C₅	#504	1	2	yes	—
121	C₂²×C₄	C₂×#168	1	1	yes	characteristic
122	C₂²×C₄	#681	1	1	yes	characteristic
123	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	C₂×#168	1	2	yes	—
129	C₂×C₈	C₂×#168	1	2	yes	—
130	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
131	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
132	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
133	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
134	C₄²	C₂×#168	1	1	yes	characteristic
135	C₄²	#681	1	1	yes	characteristic
136	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
137	C₄²	C₂²×Dic₅	1	1	yes	characteristic
138	C₁₆	—	20	2	no	—
139	C₁₆	—	20	2	no	—
140	C₂×C₅	—	4	1	no	—
141	C₂×C₅	#1564	1	1	yes	characteristic
142	C₂×C₅	#2151	1	1	yes	characteristic
143	C₂×C₅	#2153	1	1	yes	characteristic
144	C₂³	C₄⋊#168	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
150	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
151	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
152	C₂×C₄	C₂²×#168	1	1	yes	characteristic
153	C₂×C₄	C₂×#681	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	2	no	—
158	C₈	—	2	2	no	—
159	C₈	—	4	1	no	—
160	C₈	—	4	1	no	—
161	C₈	#3951	1	2	yes	—
162	C₈	#4050	1	2	yes	—
163	C₅	#12014	1	1	yes	characteristic, complemented, Sylow
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	#65395	1	1	yes	characteristic
173	C₄	C₂×#3951	1	1	yes	characteristic
174	C₄	#65429	1	2	yes	—
175	C₂	—	4	1	no	—
176	C₂	#10600522	1	1	yes	characteristic
177	C₂	#10600675	1	1	yes	characteristic
178	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	#12014
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
5. #10597930	C₂	semidirect
163. C₅	#12014	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
#10600538	2
#10600537	2
#10597931	2
#10597930	2
#12014	5