Subgroups — #12097412

Order 1280.

994 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#65414	C₂	1	2	yes	maximal
3	#10613933	C₂	1	2	yes	maximal
4	C₂×#64772	C₂	1	2	yes	maximal
5	#10600633	C₂	1	4	yes	maximal, complemented
6	C₄×#3951	C₂	1	4	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₂²	1	1	yes	characteristic
10	C₂²×#681	C₂²	1	4	yes	—
11	C₅×#405	C₄	1	8	yes	—
12	C₂×C₅×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
14	#65414	C₄	1	16	yes	complemented
15	C₂×#3951	C₂²	1	8	yes	—
16	#65286	—	2	8	no	—
17	C₄×#681	C₂²	1	8	yes	—
18	#64772	C₂²	1	4	yes	—
19	#64772	C₂²	1	4	yes	—
20	#20459	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
23	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
24	C₂²×C₅×C₈	—	2	4	no	—
25	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₅×#117	C₂×C₄	1	4	yes	—
30	C₅×#117	D₄	1	4	yes	—
31	C₂×C₅×C₄⋊C₄	Q₈	1	2	yes	—
32	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
33	C₂×C₄²×C₅	C₂³	1	2	yes	—
34	C₂×#681	—	2	4	no	—
35	C₂×#681	—	2	8	no	—
36	C₂×#681	C₂×C₄	1	16	yes	—
37	C₂×#681	C₂³	1	8	yes	—
38	C₂²×#168	—	2	4	no	—
39	C₅×#97	C₂×C₄	1	8	yes	—
40	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
41	#3951	C₂×C₄	1	32	yes	—
42	#3935	—	2	8	no	—
43	C₄×#168	—	2	8	no	—
44	C₂×#430	—	5	2	no	—
45	#2873	—	5	2	no	—
46	C₂×#405	D₅	1	1	yes	characteristic
47	#1778	—	5	4	no	—
48	C₄×#108	—	5	4	no	—
49	C₂×#341	—	5	2	no	—
50	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	#41	1	4	yes	—
57	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
58	C₅×M₁₆	C₄²	1	16	yes	—
59	C₂×C₅×C₈	—	2	8	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
63	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
64	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
65	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
66	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
67	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
68	C₄²×C₅	C₂²×C₄	1	4	yes	—
69	C₄²×C₅	C₂×D₄	1	4	yes	—
70	#681	—	2	16	no	—
71	#681	—	2	8	no	—
72	#681	C₂²×C₄	1	16	yes	—
73	C₂×#168	—	2	16	no	—
74	C₂×#168	—	2	8	no	—
75	C₂²×M₁₆	—	5	4	no	—
76	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
77	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
78	C₂×#117	C₂×D₅	1	1	yes	characteristic
79	#430	—	5	16	no	—
80	C₂×#108	—	5	8	no	—
81	#405	—	10	8	no	—
82	#405	Dic₅	1	8	yes	—
83	C₄×M₁₆	—	5	8	no	—
84	C₂×#97	C₂×D₅	1	2	yes	—
85	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
86	#341	—	5	4	no	—
87	#341	—	5	4	no	—
88	C₂³×C₅	—	4	2	no	—
89	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₅	C₄×Q₈	1	4	yes	—
91	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅	—	4	4	no	—
93	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
94	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₅	#117	1	4	yes	—
97	C₂×C₄×C₅	C₄×D₄	1	8	yes	—
98	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
99	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
100	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×#41	1	2	yes	—
103	C₅×C₈	—	2	2	no	—
104	C₅×C₈	—	2	2	no	—
105	C₅×C₈	C₂×C₄²	1	8	yes	—
106	#168	—	2	16	no	—
107	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
108	C₂×M₁₆	—	5	16	no	—
109	C₂×M₁₆	—	5	8	no	—
110	C₂×M₁₆	—	10	4	no	—
111	C₂×M₁₆	—	10	8	no	—
112	C₂×M₁₆	C₄×D₅	1	8	yes	—
113	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
114	C₂²×C₈	—	2	4	no	—
115	C₂²×C₈	—	10	4	no	—
116	C₂²×C₈	C₂×Dic₅	1	4	yes	—
117	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅	1	2	yes	—
120	#117	D₂₀	1	4	yes	—
121	#117	C₂×Dic₅	1	4	yes	—
122	C₂×C₄⋊C₄	Dic₁₀	1	2	yes	—
123	C₂×C₂²⋊C₄	Dic₁₀	1	2	yes	—
124	C₂×C₄²	C₂²×D₅	1	2	yes	—
125	#108	—	5	32	no	—
126	#97	—	10	8	no	—
127	#97	C₂×Dic₅	1	8	yes	—
128	C₄×C₈	—	10	8	no	—
129	C₄×C₈	C₂×Dic₅	1	8	yes	—
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	—	2	4	no	—
132	C₂²×C₅	—	4	4	no	—
133	C₂²×C₅	C₄×C₄⋊C₄	1	4	yes	—
134	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₅	C₂×C₄×Q₈	1	2	yes	—
136	C₄×C₅	—	2	8	no	—
137	C₄×C₅	C₄×C₄⋊C₄	1	4	yes	—
138	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
139	C₄×C₅	C₂×#117	1	1	yes	characteristic
140	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
141	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
142	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
143	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₄×D₅	1	4	yes	—
145	C₂²×C₄	C₂×D₂₀	1	2	yes	—
146	C₂²×C₄	#709	1	4	yes	—
147	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
148	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
149	M₁₆	—	5	16	no	—
150	M₁₆	—	10	8	no	—
151	M₁₆	—	10	16	no	—
152	M₁₆	C₄×Dic₅	1	16	yes	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	10	8	no	—
157	C₂×C₈	—	10	16	no	—
158	C₂×C₈	C₂×C₄×D₅	1	8	yes	—
159	C₂×C₈	C₂²×Dic₅	1	2	yes	—
160	C₂×C₈	C₂²×Dic₅	1	2	yes	—
161	C₂×C₈	C₂²×Dic₅	1	4	yes	—
162	C₄⋊C₄	C₄⋊Dic₅	1	8	yes	—
163	C₂²⋊C₄	C₄⋊Dic₅	1	8	yes	—
164	C₄²	C₂×D₂₀	1	4	yes	—
165	C₄²	C₂²×Dic₅	1	4	yes	—
166	C₂×C₅	—	2	2	no	—
167	C₂×C₅	—	2	4	no	—
168	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₅	#2916	1	2	yes	—
170	C₂³	—	4	2	no	—
171	C₂³	C₄×Dic₁₀	1	4	yes	—
172	C₂³	C₂×C₄⋊Dic₅	1	2	yes	—
173	C₂³	C₂²×Dic₁₀	1	1	yes	characteristic
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	C₄×Dic₁₀	1	4	yes	—
176	C₂×C₄	C₄×D₂₀	1	8	yes	—
177	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₅	1	8	yes	—
179	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
180	C₂×C₄	#4072	1	4	yes	—
181	C₂×C₄	C₂²×C₄×D₅	1	2	yes	—
182	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
183	C₂×C₄	C₂×#709	1	2	yes	—
184	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	2	no	—
187	C₈	—	10	16	no	—
188	C₈	C₂×C₄×Dic₅	1	8	yes	—
189	C₅	#20459	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	C₄×C₄⋊Dic₅	1	4	yes	—
194	C₂²	C₂×C₄×Dic₁₀	1	2	yes	—
195	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
196	C₄	—	2	8	no	—
197	C₄	C₄×C₄⋊Dic₅	1	4	yes	—
198	C₄	C₂×C₄×D₂₀	1	2	yes	—
199	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
200	C₄	C₂×#4072	1	1	yes	characteristic
201	C₂	—	2	2	no	—
202	C₂	—	2	4	no	—
203	C₂	C₂×C₄×C₄⋊Dic₅	1	1	yes	characteristic
204	C₂	#10615520	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₅×#405	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×#405	D₅
hypocenter	C₅	#20459
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
5. #10600633	C₂	semidirect
6. C₄×#3951	C₂	semidirect
14. #65414	C₄	semidirect
189. C₅	#20459	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#405	2
C₂×#65414	2
#10613933	2
C₂×#64772	2
#10600633	2
C₄×#3951	2
#20459	5