Subgroups — #12253804

Order 1280.

650 conjugacy classes of subgroups.

The subgroup lattice has 250 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	#10615538	C₂	1	2	yes	maximal
3	C₂²×Dic₄₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602240	C₂	1	4	yes	maximal, complemented
5	#10602238	C₂	1	2	yes	maximal, complemented
6	#10602209	C₂	1	4	yes	maximal, complemented
7	#10602197	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#131	C₂²	1	1	yes	characteristic
9	#66159	—	2	2	no	—
10	C₂²×Dic₂₀	C₂²	1	2	yes	—
11	C₅×#504	—	2	4	no	—
12	C₂×C₅×M₃₂	C₂²	1	2	yes	—
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	#65426	—	2	4	no	—
17	#65409	—	2	2	no	—
18	#65405	C₂²	1	8	yes	—
19	#65395	C₂²	1	2	yes	—
20	C₂×Dic₄₀	—	2	4	no	—
21	C₂×Dic₄₀	C₂²	1	2	yes	—
22	C₂×Dic₄₀	C₂²	1	2	yes	—
23	#64725	C₂²	1	2	yes	—
24	#64724	C₂²	1	2	yes	—
25	#64723	C₂²	1	8	yes	—
26	#33608	—	5	1	no	maximal, Sylow
27	C₂×C₅×#41	D₄	1	1	yes	characteristic
28	C₂²×Dic₁₀	—	2	2	no	—
29	C₅×#131	—	2	2	no	—
30	C₅×#131	—	2	2	no	—
31	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	D₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	#4090	—	4	2	no	—
36	#4082	—	4	2	no	—
37	#4081	—	4	4	no	—
38	#4072	—	2	2	no	—
39	#4070	—	2	8	no	—
40	C₂×Dic₂₀	—	2	4	no	—
41	C₂×Dic₂₀	—	2	4	no	—
42	C₂×Dic₂₀	—	2	2	no	—
43	C₂×Dic₂₀	C₂³	1	4	yes	—
44	C₅×M₃₂	—	2	2	no	—
45	C₅×M₃₂	—	2	4	no	—
46	C₂×C₅×C₁₆	—	2	4	no	—
47	C₂×C₅×C₁₆	D₄	1	4	yes	—
48	C₂×C₅×C₁₆	C₂³	1	2	yes	—
49	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
51	#3950	C₂³	1	4	yes	—
52	#3949	—	2	2	no	—
53	#3948	—	2	8	no	—
54	Dic₄₀	—	2	4	no	—
55	Dic₄₀	—	2	4	no	—
56	C₂²×Q₃₂	—	5	1	no	—
57	C₂×#504	D₅	1	1	yes	characteristic
58	#3059	—	5	2	no	—
59	#2219	—	5	4	no	—
60	#2216	—	5	4	no	—
61	#2158	—	5	1	no	—
62	#2141	—	5	2	no	—
63	C₅×#41	—	2	2	no	—
64	C₅×#41	—	2	2	no	—
65	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
66	C₂×C₅×D₄	D₈	1	2	yes	—
67	C₂×C₅×D₄	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₅	—	4	2	no	—
71	C₂×Dic₁₀	—	2	4	no	—
72	C₂×Dic₁₀	—	4	4	no	—
73	C₂×Dic₁₀	—	4	4	no	—
74	C₂×Dic₁₀	—	4	2	no	—
75	C₅×M₁₆	—	2	2	no	—
76	C₅×M₁₆	—	2	4	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
80	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₅×C₈	#41	1	2	yes	—
83	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
84	C₂²⋊Dic₅	—	4	8	no	—
85	C₂²⋊Dic₅	—	4	2	no	—
86	C₄⋊Dic₅	—	2	4	no	—
87	#683	—	4	8	no	—
88	C₄×Dic₅	—	4	2	no	—
89	Dic₂₀	—	2	4	no	—
90	Dic₂₀	—	2	4	no	—
91	Dic₂₀	—	4	4	no	—
92	C₅×C₁₆	—	2	2	no	—
93	C₅×C₁₆	C₂×D₄	1	2	yes	—
94	C₅×C₁₆	C₂×D₄	1	2	yes	—
95	C₂²×Q₁₆	—	5	2	no	—
96	C₂×#131	C₂×D₅	1	1	yes	characteristic
97	#536	—	10	2	no	—
98	C₂×Q₃₂	—	5	2	no	—
99	C₂×Q₃₂	—	5	2	no	—
100	C₂×Q₃₂	—	10	4	no	—
101	#504	—	2	4	no	—
102	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
103	C₂×M₃₂	C₂×D₅	1	2	yes	—
104	C₂²×C₁₆	C₂×D₅	1	2	yes	—
105	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
106	#470	—	10	4	no	—
107	#467	—	5	8	no	—
108	#427	—	5	2	no	—
109	#419	—	10	2	no	—
110	#367	—	5	2	no	—
111	#366	—	5	2	no	—
112	#358	—	5	8	no	—
113	C₂³×C₅	C₂×D₈	1	2	yes	—
114	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
115	C₅×Q₈	—	2	2	no	—
116	C₅×D₄	—	2	4	no	—
117	C₅×D₄	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	#121	1	2	yes	—
121	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅	#135	1	2	yes	—
124	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₅	—	4	4	no	—
126	C₂×Dic₅	—	4	4	no	—
127	C₂×Dic₅	—	8	4	no	—
128	Dic₁₀	—	4	4	no	—
129	Dic₁₀	—	4	4	no	—
130	Dic₁₀	—	8	4	no	—
131	C₅×C₈	—	2	2	no	—
132	C₅×C₈	#121	1	2	yes	—
133	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
134	C₅×C₈	C₂×#41	1	1	yes	characteristic
135	C₂×#41	D₂₀	1	1	yes	characteristic
136	C₂²×Q₈	—	10	2	no	—
137	C₂×Q₁₆	—	5	4	no	—
138	C₂×Q₁₆	—	10	2	no	—
139	C₂×Q₁₆	—	10	4	no	—
140	C₂×Q₁₆	—	10	4	no	—
141	#131	—	2	2	no	—
142	#131	—	2	2	no	—
143	C₂×M₁₆	D₂₀	1	1	yes	characteristic
144	C₂×M₁₆	D₂₀	1	2	yes	—
145	C₂²×C₈	C₂²×D₅	1	2	yes	—
146	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
147	C₄⋊Q₈	—	20	2	no	—
148	#124	—	20	2	no	—
149	#123	—	20	4	no	—
150	C₂²⋊Q₈	—	10	8	no	—
151	#117	—	10	2	no	—
152	Q₃₂	—	10	4	no	—
153	Q₃₂	—	10	4	no	—
154	M₃₂	—	2	2	no	—
155	M₃₂	—	2	4	no	—
156	C₂×C₁₆	—	2	4	no	—
157	C₂×C₁₆	C₂²⋊D₅	1	4	yes	—
158	C₂×C₁₆	C₂²×D₅	1	2	yes	—
159	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
160	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
161	#107	—	5	4	no	—
162	#106	—	10	2	no	—
163	#103	—	10	8	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	4	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	#466	1	2	yes	—
168	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	#466	1	2	yes	—
171	C₄×C₅	C₂×#121	1	1	yes	characteristic
172	C₄×C₅	C₂×#135	1	1	yes	characteristic
173	Dic₅	—	8	4	no	—
174	Dic₅	—	8	4	no	—
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	C₂×Q₈	—	10	4	no	—
178	C₂×Q₈	—	20	2	no	—
179	C₂×Q₈	—	20	4	no	—
180	C₂×Q₈	—	20	4	no	—
181	C₂×Q₈	D₄₀	1	1	yes	characteristic
182	C₂×D₄	D₄₀	1	2	yes	—
183	C₂×D₄	D₄₀	1	1	yes	characteristic
184	C₂²×C₄	—	20	2	no	—
185	C₂²×C₄	C₂×D₂₀	1	2	yes	—
186	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
187	Q₁₆	—	10	4	no	—
188	Q₁₆	—	10	4	no	—
189	Q₁₆	—	20	4	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₂×D₂₀	1	2	yes	—
195	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
196	C₂×C₈	#709	1	2	yes	—
197	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
198	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
199	C₄⋊C₄	—	10	4	no	—
200	C₄⋊C₄	—	20	8	no	—
201	C₂²⋊C₄	—	20	2	no	—
202	C₂²⋊C₄	—	20	8	no	—
203	C₄²	—	20	2	no	—
204	C₁₆	—	2	2	no	—
205	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
206	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
207	C₂×C₅	—	2	2	no	—
208	C₂×C₅	—	4	2	no	—
209	C₂×C₅	C₂×#466	1	1	yes	characteristic
210	C₂×C₅	#3417	1	2	yes	—
211	C₂³	C₂×D₄₀	1	2	yes	—
212	C₂³	C₂×D₄₀	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	20	4	no	—
215	Q₈	—	20	4	no	—
216	Q₈	—	40	4	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	20	4	no	—
222	C₂×C₄	—	20	4	no	—
223	C₂×C₄	—	40	4	no	—
224	C₂×C₄	C₂×D₄₀	1	2	yes	—
225	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
226	C₂×C₄	#4050	1	2	yes	—
227	C₂×C₄	#4076	1	2	yes	—
228	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	#4076	1	2	yes	—
231	C₈	C₂×#709	1	1	yes	characteristic
232	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
233	C₅	#33608	1	1	yes	characteristic, complemented, Sylow
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	#65404	1	2	yes	—
238	C₂²	C₂²×D₄₀	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	40	4	no	—
241	C₄	—	40	4	no	—
242	C₄	#65404	1	2	yes	—
243	C₄	C₂×#4050	1	1	yes	characteristic
244	C₄	C₂×#4076	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	4	2	no	—
247	C₂	#10613746	1	2	yes	—
248	C₂	C₂×#65404	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₀
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#504	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂×C₅×#41	D₄
hypercenter	C₂×#504	D₅
hypocenter	C₅	#33608
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
3. C₂²×Dic₄₀	C₂	semidirect
4. #10602240	C₂	semidirect
5. #10602238	C₂	semidirect
6. #10602209	C₂	semidirect
7. #10602197	C₂	semidirect
233. C₅	#33608	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
#10615538	2
C₂²×Dic₄₀	2
#10602240	2
#10602238	2
#10602209	2
#10602197	2
#33608	5