Subgroups — #12014129

Order 1280.

402 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×#347	C₂	1	1	yes	maximal, characteristic
2	C₂×#64732	C₂	1	2	yes	maximal
3	Dic₅×M₃₂	C₂	1	2	yes	maximal
4	#10600597	C₂	1	1	yes	maximal, characteristic
5	C₂×#64686	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₅×#97	C₂²	1	1	yes	characteristic
8	C₂×#3947	C₄	1	1	yes	characteristic
9	C₂×#3947	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×#3944	C₂²	1	2	yes	—
13	#65221	C₄	1	1	yes	characteristic
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	C₅×#347	C₄	1	4	yes	complemented
16	#64732	C₄	1	8	yes	complemented
17	#64722	C₄	1	4	yes	complemented
18	C₁₆×Dic₅	C₄	1	4	yes	complemented
19	#64686	C₄	1	4	yes	complemented
20	#12541	—	5	1	no	maximal, Sylow
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×#168	C₂×C₄	1	1	yes	characteristic
27	C₂²×#168	C₂×C₄	1	1	yes	characteristic
28	C₅×M₃₂	C₂×C₄	1	4	yes	—
29	C₅×M₃₂	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
31	C₅×#97	C₂×C₄	1	2	yes	—
32	C₅×#97	C₂×C₄	1	2	yes	—
33	#3963	—	2	2	no	—
34	#3947	—	2	2	no	—
35	#3947	C₂×C₄	1	4	yes	—
36	C₈×Dic₅	—	2	2	no	—
37	C₈×Dic₅	C₂×C₄	1	2	yes	—
38	C₈×Dic₅	C₂×C₄	1	2	yes	—
39	#3944	C₂×C₄	1	4	yes	—
40	#3944	C₂×C₄	1	4	yes	—
41	C₂×#672	C₂×C₄	1	4	yes	—
42	C₄×#168	—	2	2	no	—
43	C₂×#347	—	5	2	no	—
44	C₂×#347	—	5	1	no	—
45	C₂×#347	D₅	1	1	yes	characteristic
46	C₄×M₃₂	—	5	2	no	—
47	#1726	—	5	1	no	—
48	C₂²×C₄×C₅	—	2	1	no	—
49	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₅	—	2	1	no	—
51	C₂²×Dic₅	M₁₆	1	2	yes	—
52	C₂×C₅×C₈	—	2	1	no	—
53	C₂×C₅×C₈	—	2	1	no	—
54	C₂×C₅×C₈	C₄²	1	2	yes	—
55	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
56	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
58	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₅	—	2	1	no	—
60	C₄²×C₅	C₄²	1	2	yes	—
61	C₄×Dic₅	—	2	1	no	—
62	C₄×Dic₅	C₄²	1	2	yes	—
63	C₄×Dic₅	C₄²	1	2	yes	—
64	C₄×Dic₅	M₁₆	1	2	yes	—
65	C₂×#168	—	2	1	no	—
66	C₂×#168	—	2	1	no	—
67	C₂×#168	C₄²	1	2	yes	—
68	C₂×#168	C₄²	1	1	yes	characteristic
69	C₂×#168	C₄²	1	1	yes	characteristic
70	C₂×#168	M₁₆	1	2	yes	—
71	C₂×#168	M₁₆	1	2	yes	—
72	C₅×C₁₆	C₄²	1	8	yes	—
73	#672	C₄²	1	8	yes	—
74	C₂×M₃₂	—	5	2	no	—
75	C₂×M₃₂	C₂×D₅	1	2	yes	—
76	C₂×#97	—	5	1	no	—
77	C₂×#97	—	5	1	no	—
78	C₂×#97	C₂×D₅	1	1	yes	characteristic
79	C₂×C₄×C₈	—	5	1	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	C₂×C₄×C₈	—	5	1	no	—
82	#376	—	5	1	no	—
83	#347	—	5	8	no	—
84	#347	—	5	4	no	—
85	#347	Dic₅	1	4	yes	complemented
86	#346	—	5	4	no	—
87	C₄×C₁₆	—	5	4	no	—
88	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	4	2	no	—
91	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
97	C₂×Dic₅	—	2	2	no	—
98	C₂×Dic₅	—	4	2	no	—
99	C₂×Dic₅	—	4	2	no	—
100	C₂×Dic₅	#97	1	4	yes	—
101	C₅×C₈	—	2	2	no	—
102	C₅×C₈	C₂×C₄²	1	2	yes	—
103	C₅×C₈	C₂×C₄²	1	2	yes	—
104	#168	—	2	2	no	—
105	#168	#97	1	2	yes	—
106	#168	#97	1	2	yes	—
107	C₂²×C₈	—	5	1	no	—
108	C₂²×C₈	—	5	1	no	—
109	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
111	C₂×C₄²	—	5	1	no	—
112	C₂×C₄²	—	5	1	no	—
113	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
114	M₃₂	—	5	4	no	—
115	M₃₂	—	5	4	no	—
116	M₃₂	C₄×D₅	1	4	yes	—
117	M₃₂	C₂×Dic₅	1	4	yes	—
118	C₂×C₁₆	—	5	4	no	—
119	C₂×C₁₆	C₄×D₅	1	4	yes	—
120	#97	—	5	4	no	—
121	#97	—	10	2	no	—
122	#97	C₄×D₅	1	2	yes	—
123	#97	C₂×Dic₅	1	2	yes	—
124	C₄×C₈	—	5	2	no	—
125	C₄×C₈	—	5	2	no	—
126	C₄×C₈	—	10	2	no	—
127	C₄×C₈	—	10	2	no	—
128	#95	—	10	2	no	—
129	C₂²×C₅	—	2	2	no	—
130	C₂²×C₅	C₂×#97	1	1	yes	characteristic
131	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
132	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
133	C₄×C₅	—	4	2	no	—
134	C₄×C₅	C₄³	1	1	yes	characteristic
135	C₄×C₅	C₂×#97	1	1	yes	characteristic
136	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
137	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
138	Dic₅	—	2	4	no	—
139	Dic₅	—	4	2	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	5	2	no	—
142	C₂²×C₄	—	10	1	no	—
143	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	5	1	no	—
147	C₂×C₈	—	5	1	no	—
148	C₂×C₈	—	5	2	no	—
149	C₂×C₈	—	5	2	no	—
150	C₂×C₈	—	5	2	no	—
151	C₂×C₈	—	10	1	no	—
152	C₂×C₈	—	10	1	no	—
153	C₂×C₈	C₄×Dic₅	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
155	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
156	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
157	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
158	C₄²	—	2	1	no	—
159	C₄²	—	5	2	no	—
160	C₄²	—	5	2	no	—
161	C₄²	—	5	2	no	—
162	C₄²	—	10	1	no	—
163	C₄²	C₄×Dic₅	1	2	yes	—
164	C₁₆	—	5	8	no	—
165	C₁₆	C₄×Dic₅	1	8	yes	—
166	C₂×C₅	—	2	1	no	—
167	C₂×C₅	—	2	1	no	—
168	C₂×C₅	C₄×#97	1	1	yes	characteristic
169	C₂×C₅	#2110	1	2	yes	—
170	C₂³	D₅×M₁₆	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	5	4	no	—
174	C₂×C₄	—	10	2	no	—
175	C₂×C₄	—	20	2	no	—
176	C₂×C₄	—	20	2	no	—
177	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
178	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
179	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
180	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
181	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	5	2	no	—
185	C₈	—	5	2	no	—
186	C₈	—	10	2	no	—
187	C₈	C₄²×D₅	1	2	yes	—
188	C₈	C₂×C₄×Dic₅	1	2	yes	—
189	C₅	#12541	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	2	2	no	—
191	C₂²	D₅×#97	1	1	yes	characteristic
192	C₂²	#64996	1	1	yes	characteristic
193	C₂²	M₁₆×Dic₅	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	10	4	no	—
196	C₄	—	20	2	no	—
197	C₄	D₅×#97	1	1	yes	characteristic
198	C₄	#64996	1	1	yes	characteristic
199	C₄	C₄²×Dic₅	1	1	yes	characteristic
200	C₄	M₁₆×Dic₅	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#10599880	1	2	yes	—
204	C₂	Dic₅×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₅
commutator	C₄×C₅	C₄³
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#347	C₂
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#347	D₅
hypocenter	C₅	#12541
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
15. C₅×#347	C₄	semidirect
16. #64732	C₄	semidirect
17. #64722	C₄	semidirect
18. C₁₆×Dic₅	C₄	semidirect
19. #64686	C₄	semidirect
85. #347	Dic₅	semidirect
189. C₅	#12541	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#347	2
C₂×#64732	2
Dic₅×M₃₂	2
#10600597	2
C₂×#64686	2
#12541	5