Subgroups — #12014396

Order 1280.

410 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic
2	#10602804	C₂	1	1	yes	maximal, characteristic
3	#10602800	C₂	1	1	yes	maximal, characteristic
4	#10602192	C₂	1	2	yes	maximal, complemented
5	#10598304	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	C₅×#504	—	2	2	no	—
8	C₅×#504	—	2	2	no	—
9	C₂×C₅×M₃₂	C₄	1	2	yes	complemented
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₁₆	C₄	1	2	yes	complemented
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	#65529	C₄	1	1	yes	characteristic
14	#65526	C₄	1	1	yes	characteristic
15	#65390	C₂²	1	2	yes	—
16	C₂×#3944	C₂²	1	2	yes	—
17	#64776	C₄	1	2	yes	—
18	#64773	C₄	1	4	yes	complemented
19	#64722	—	2	2	no	—
20	C₁₆×Dic₅	—	2	2	no	—
21	#12613	—	5	1	no	maximal, Sylow
22	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
23	C₅×#131	—	2	2	no	—
24	C₅×#131	—	2	2	no	—
25	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	#4072	C₂×C₄	1	2	yes	—
30	C₂×#681	C₂×C₄	1	2	yes	—
31	C₅×M₃₂	—	2	2	no	—
32	C₅×M₃₂	—	2	4	no	—
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₂×C₅×C₁₆	—	2	2	no	—
35	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×C₁₆	D₄	1	2	yes	—
39	C₂×C₅×C₁₆	D₄	1	2	yes	—
40	#3968	C₈	1	2	yes	—
41	#3966	C₈	1	4	yes	—
42	#3965	C₈	1	2	yes	—
43	#3947	—	2	2	no	—
44	C₈×Dic₅	—	2	2	no	—
45	#3944	—	2	2	no	—
46	#3944	C₂×C₄	1	4	yes	—
47	C₂×#672	—	2	2	no	—
48	C₂×#504	D₅	1	1	yes	characteristic
49	#2115	—	5	1	no	—
50	#2108	—	5	2	no	—
51	#1757	—	5	1	no	—
52	#1378	—	5	2	no	—
53	C₅×#41	—	2	2	no	—
54	C₅×#41	—	2	2	no	—
55	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
56	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
57	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₄²	1	2	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₅×M₁₆	—	2	2	no	—
61	C₅×M₁₆	—	2	4	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	C₄²	1	2	yes	—
65	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
68	C₂×C₅×C₈	#41	1	2	yes	—
69	C₂²⋊Dic₅	C₂×C₈	1	4	yes	—
70	C₄⋊Dic₅	—	2	2	no	—
71	C₄×Dic₅	—	2	2	no	—
72	#681	—	2	2	no	—
73	#681	C₂×C₈	1	4	yes	—
74	C₂×#168	—	2	2	no	—
75	C₅×C₁₆	—	2	2	no	—
76	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
77	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
78	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
79	#672	—	2	4	no	—
80	C₂×#131	C₂×D₅	1	1	yes	characteristic
81	#504	—	2	2	no	—
82	#504	—	2	2	no	—
83	C₂×M₃₂	—	5	2	no	—
84	C₂×M₃₂	Dic₅	1	2	yes	complemented
85	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
86	C₂²×C₁₆	Dic₅	1	2	yes	complemented
87	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
88	#413	—	5	1	no	—
89	#410	—	5	1	no	—
90	#405	—	5	2	no	—
91	#349	—	5	4	no	—
92	#346	—	10	2	no	—
93	C₄×C₁₆	—	10	2	no	—
94	#344	—	5	2	no	—
95	C₂³×C₅	C₄×C₈	1	2	yes	—
96	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
97	C₅×Q₈	—	2	2	no	—
98	C₅×D₄	—	2	4	no	—
99	C₅×D₄	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅	#131	1	2	yes	—
107	C₂×Dic₅	—	2	4	no	—
108	C₅×C₈	—	2	2	no	—
109	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₅×C₈	#117	1	1	yes	characteristic
111	C₅×C₈	C₄×D₄	1	2	yes	—
112	#168	—	2	4	no	—
113	C₂×#41	C₄×D₅	1	1	yes	characteristic
114	#131	—	2	2	no	—
115	#131	—	2	2	no	—
116	C₂×M₁₆	—	5	2	no	—
117	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
118	C₂×M₁₆	C₄×D₅	1	2	yes	—
119	C₂²×C₈	C₂×Dic₅	1	2	yes	—
120	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
121	#117	—	5	2	no	—
122	M₃₂	—	2	2	no	—
123	M₃₂	—	2	4	no	—
124	M₃₂	—	5	4	no	—
125	M₃₂	—	10	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	10	2	no	—
129	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
130	C₂×C₁₆	C₂×Dic₅	1	1	yes	characteristic
131	C₂×C₁₆	C₂×Dic₅	1	1	yes	characteristic
132	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
133	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
134	#101	—	5	2	no	—
135	#100	—	5	2	no	—
136	#99	—	5	4	no	—
137	#97	—	10	2	no	—
138	C₄×C₈	—	10	2	no	—
139	C₂²×C₅	—	2	2	no	—
140	C₂²×C₅	—	4	1	no	—
141	C₂²×C₅	—	4	2	no	—
142	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
143	C₂²×C₅	C₈×D₄	1	2	yes	—
144	C₄×C₅	—	2	2	no	—
145	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₅	#405	1	1	yes	characteristic
147	C₄×C₅	C₈×D₄	1	2	yes	—
148	Dic₅	—	4	4	no	—
149	#41	—	2	2	no	—
150	#41	—	2	2	no	—
151	C₂×Q₈	C₈×D₅	1	1	yes	characteristic
152	C₂×D₄	C₈×D₅	1	2	yes	—
153	C₂×D₄	C₈×D₅	1	1	yes	characteristic
154	C₂²×C₄	C₄×Dic₅	1	2	yes	—
155	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
156	M₁₆	—	2	2	no	—
157	M₁₆	—	2	4	no	—
158	M₁₆	—	5	4	no	—
159	M₁₆	—	10	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	10	2	no	—
163	C₂×C₈	C₄×Dic₅	1	2	yes	—
164	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
165	C₂×C₈	#709	1	2	yes	—
166	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
168	C₄⋊C₄	—	10	2	no	—
169	C₂²⋊C₄	—	5	4	no	—
170	C₄²	—	10	2	no	—
171	C₁₆	—	2	2	no	—
172	C₁₆	—	10	4	no	—
173	C₁₆	C₂²⋊Dic₅	1	1	yes	characteristic
174	C₁₆	C₂²⋊Dic₅	1	2	yes	—
175	C₁₆	C₂²⋊Dic₅	1	1	yes	characteristic
176	C₂×C₅	—	2	2	no	—
177	C₂×C₅	—	2	1	no	—
178	C₂×C₅	—	4	2	no	—
179	C₂×C₅	C₈×C₂²⋊C₄	1	1	yes	characteristic
180	C₂³	C₈×Dic₅	1	2	yes	—
181	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
182	Q₈	—	2	2	no	—
183	D₄	—	2	4	no	—
184	D₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	10	4	no	—
188	C₂×C₄	C₈×Dic₅	1	2	yes	—
189	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
190	C₂×C₄	#4047	1	2	yes	—
191	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
192	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
193	C₈	—	2	2	no	—
194	C₈	—	10	4	no	—
195	C₈	#4072	1	1	yes	characteristic
196	C₈	C₄×C₂²⋊D₅	1	2	yes	—
197	C₈	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
198	C₅	#12613	1	1	yes	characteristic, complemented, Sylow
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
203	C₂²	C₈×C₂²⋊D₅	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	20	4	no	—
206	C₄	#65390	1	1	yes	characteristic
207	C₄	C₈×C₂²⋊D₅	1	2	yes	—
208	C₄	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	—	2	1	no	—
211	C₂	—	4	2	no	—
212	C₂	C₈×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊Dic₅
commutator	C₂²×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#504	C₂
socle	C₂×C₅	C₈×C₂²⋊C₄
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#504	D₅
hypocenter	C₅	#12613
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
4. #10602192	C₂	semidirect
5. #10598304	C₂	semidirect
9. C₂×C₅×M₃₂	C₄	semidirect
11. C₂²×C₅×C₁₆	C₄	semidirect
18. #64773	C₄	semidirect
84. C₂×M₃₂	Dic₅	semidirect
86. C₂²×C₁₆	Dic₅	semidirect
198. C₅	#12613	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#504	2
#10602804	2
#10602800	2
#10602192	2
#10598304	2
#12613	5