Subgroups — #12254654

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#64984	C₂	1	1	yes	maximal, characteristic
3	#10602174	C₂	1	4	yes	maximal, complemented
4	#10602145	C₂	1	4	yes	maximal, complemented
5	#10601445	C₂	1	4	yes	maximal, complemented
6	C₄×#3944	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#533	—	2	1	no	—
8	C₂²×D₄₀	—	2	1	no	—
9	C₂²×D₄₀	C₂²	1	2	yes	—
10	C₂×#4020	—	2	1	no	—
11	C₅×#500	C₂²	1	2	yes	complemented
12	C₅×#499	—	2	1	no	—
13	C₅×#499	C₂²	1	2	yes	complemented
14	C₅×#498	—	2	1	no	—
15	C₅×#427	—	2	1	no	—
16	C₅×#427	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×#3944	C₂²	1	2	yes	—
19	#64984	—	2	1	no	—
20	#64984	—	2	1	no	—
21	#64984	C₂²	1	2	yes	—
22	#64984	C₂²	1	2	yes	—
23	#64711	C₂²	1	16	yes	complemented
24	#64681	C₂²	1	2	yes	complemented
25	C₄×#672	C₂²	1	2	yes	complemented
26	#33704	—	5	1	no	maximal, Sylow
27	C₂²×D₂₀	—	2	2	no	—
28	C₂²×D₂₀	—	4	2	no	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₅×C₄⋊Q₈	—	2	2	no	—
32	C₅×#125	—	2	2	no	—
33	C₅×C₂²⋊Q₈	—	4	2	no	—
34	C₄×C₅×Q₈	—	4	2	no	—
35	C₅×#117	—	2	2	no	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×D₄₀	—	2	8	no	—
38	C₂×D₄₀	—	2	2	no	—
39	C₂×D₄₀	—	2	2	no	—
40	C₂×D₄₀	—	2	2	no	—
41	C₂×D₄₀	—	4	4	no	—
42	C₂×D₄₀	C₂³	1	4	yes	—
43	#4020	—	2	2	no	—
44	#4020	—	2	2	no	—
45	#4020	—	4	2	no	—
46	C₅×#107	—	2	2	no	—
47	C₅×#107	C₂³	1	4	yes	—
48	C₅×#106	—	2	2	no	—
49	C₅×#106	—	2	2	no	—
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
54	#3944	—	2	4	no	—
55	C₂×#672	C₂³	1	4	yes	—
56	#3160	D₅	1	1	yes	characteristic, complemented
57	C₂×#493	—	5	1	no	—
58	#2244	—	5	4	no	—
59	#2240	—	5	4	no	—
60	#2140	—	5	4	no	—
61	C₄×M₃₂	—	5	1	no	—
62	C₂³×D₅	—	4	2	no	—
63	C₂×C₅×Q₈	—	4	2	no	—
64	C₂²×C₄×C₅	D₈	1	2	yes	—
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×D₂₀	—	2	4	no	—
67	C₂×D₂₀	—	4	4	no	—
68	C₂×D₂₀	—	4	4	no	—
69	C₂×D₂₀	—	4	2	no	—
70	C₂×D₂₀	—	4	4	no	—
71	C₂×D₂₀	—	4	4	no	—
72	C₂×D₂₀	—	8	8	no	—
73	C₂×C₅×C₈	SD₁₆	1	4	yes	—
74	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	#41	1	4	yes	—
78	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
79	C₅×C₄⋊C₄	—	2	4	no	—
80	C₅×C₄⋊C₄	—	4	4	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₂²⋊C₄	—	4	2	no	—
83	C₄²×C₅	—	4	2	no	—
84	C₄²×C₅	D₈	1	2	yes	—
85	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
87	D₄₀	—	2	8	no	—
88	D₄₀	—	4	4	no	—
89	D₄₀	—	4	4	no	—
90	D₄₀	—	4	4	no	—
91	#672	—	2	4	no	—
92	C₂²×D₈	—	5	2	no	—
93	C₂²×D₈	—	10	1	no	—
94	#533	—	2	1	no	—
95	C₂×#127	—	10	1	no	—
96	C₂×M₃₂	—	5	2	no	—
97	#500	C₂×D₅	1	2	yes	complemented
98	#499	—	2	1	no	—
99	#499	C₂×D₅	1	2	yes	complemented
100	#498	—	2	1	no	—
101	#493	—	5	2	no	—
102	#493	—	5	2	no	—
103	#493	—	10	1	no	—
104	#493	—	10	1	no	—
105	#427	—	2	1	no	—
106	#427	C₂×D₅	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
108	#357	—	5	16	no	—
109	#346	—	5	2	no	—
110	C₄×C₁₆	—	5	2	no	—
111	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
112	C₂²×D₅	—	4	4	no	—
113	C₂²×D₅	—	8	8	no	—
114	C₂²×D₅	—	8	4	no	—
115	C₅×Q₈	—	8	4	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	4	4	no	—
118	C₂×C₄×C₅	#124	1	2	yes	—
119	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
124	D₂₀	—	4	4	no	—
125	D₂₀	—	4	4	no	—
126	D₂₀	—	8	8	no	—
127	D₂₀	—	8	4	no	—
128	D₂₀	—	8	8	no	—
129	C₅×C₈	#124	1	2	yes	—
130	C₅×C₈	C₂×SD₁₆	1	2	yes	—
131	C₅×C₈	C₂×SD₁₆	1	2	yes	—
132	C₅×C₈	C₂×#41	1	2	yes	—
133	C₂²×D₄	—	10	2	no	—
134	C₂²×D₄	—	20	2	no	—
135	C₂×D₈	—	5	4	no	—
136	C₂×D₈	—	10	8	no	—
137	C₂×D₈	—	10	2	no	—
138	C₂×D₈	—	10	2	no	—
139	C₂×D₈	—	10	2	no	—
140	C₂×D₈	—	20	4	no	—
141	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	2	no	—
144	#127	—	10	2	no	—
145	#127	—	10	2	no	—
146	#127	—	20	2	no	—
147	#125	—	2	2	no	—
148	C₂²⋊Q₈	—	4	2	no	—
149	C₄×Q₈	—	4	2	no	—
150	#117	—	2	2	no	—
151	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
152	M₃₂	—	10	4	no	—
153	C₂×C₁₆	—	5	4	no	—
154	#107	—	2	2	no	—
155	#107	C₂²×D₅	1	4	yes	—
156	#106	—	2	2	no	—
157	#106	—	2	2	no	—
158	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	#486	1	2	yes	—
164	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
165	C₂×D₅	—	8	4	no	—
166	C₂×D₅	—	8	8	no	—
167	C₂×D₅	—	16	8	no	—
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	8	4	no	—
170	C₄×C₅	#486	1	2	yes	—
171	C₄×C₅	#509	1	4	yes	—
172	C₄×C₅	C₂×#124	1	1	yes	characteristic
173	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
174	C₂⁴	—	20	2	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×D₄	—	10	4	no	—
177	C₂×D₄	—	20	4	no	—
178	C₂×D₄	—	20	4	no	—
179	C₂×D₄	—	20	2	no	—
180	C₂×D₄	—	20	4	no	—
181	C₂×D₄	—	20	4	no	—
182	C₂×D₄	—	40	8	no	—
183	C₂²×C₄	#686	1	2	yes	—
184	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
185	D₈	—	10	8	no	—
186	D₈	—	20	4	no	—
187	D₈	—	20	4	no	—
188	D₈	—	20	4	no	—
189	C₂×C₈	#688	1	4	yes	—
190	C₂×C₈	#713	1	4	yes	—
191	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
193	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
194	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₄²	—	4	2	no	—
200	C₄²	#686	1	2	yes	—
201	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
203	C₁₆	—	10	4	no	—
204	C₂×C₅	—	2	2	no	—
205	C₂×C₅	C₂×#486	1	1	yes	characteristic
206	C₂×C₅	C₂×#509	1	2	yes	—
207	D₅	—	16	4	no	—
208	C₂³	—	20	4	no	—
209	C₂³	—	40	8	no	—
210	C₂³	—	40	4	no	—
211	C₂³	C₂×#686	1	1	yes	characteristic
212	Q₈	—	8	4	no	—
213	D₄	—	20	4	no	—
214	D₄	—	20	4	no	—
215	D₄	—	40	8	no	—
216	D₄	—	40	4	no	—
217	D₄	—	40	8	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	C₂×#686	1	2	yes	—
221	C₂×C₄	C₂×#686	1	1	yes	characteristic
222	C₂×C₄	C₂×#686	1	2	yes	—
223	C₂×C₄	C₂×#688	1	2	yes	—
224	C₂×C₄	#4093	1	2	yes	—
225	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
226	C₈	C₂×#688	1	2	yes	—
227	C₈	C₂×#688	1	2	yes	—
228	C₈	#4093	1	2	yes	—
229	C₈	C₂×#713	1	2	yes	—
230	C₅	#33704	1	1	yes	characteristic, complemented, Sylow
231	C₂²	—	2	2	no	—
232	C₂²	—	40	8	no	—
233	C₂²	—	40	4	no	—
234	C₂²	—	80	8	no	—
235	C₂²	#65377	1	2	yes	—
236	C₂²	C₂²×#686	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	8	4	no	—
239	C₄	#65377	1	2	yes	—
240	C₄	#65482	1	4	yes	—
241	C₄	C₂²×#688	1	1	yes	characteristic
242	C₄	C₂×#4093	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	80	4	no	—
245	C₂	C₂×#65377	1	1	yes	characteristic
246	C₂	C₂×#65482	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#686
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#3160	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂×#64984	C₂
hypercenter	#3160	D₅
hypocenter	C₅	#33704
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
1. C₅×#3160	C₂	semidirect
3. #10602174	C₂	semidirect
4. #10602145	C₂	semidirect
5. #10601445	C₂	semidirect
6. C₄×#3944	C₂	semidirect
11. C₅×#500	C₂²	semidirect
13. C₅×#499	C₂²	semidirect
23. #64711	C₂²	semidirect
24. #64681	C₂²	semidirect
25. C₄×#672	C₂²	semidirect
56. #3160	D₅	semidirect
97. #500	C₂×D₅	semidirect
99. #499	C₂×D₅	semidirect
230. C₅	#33704	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3160	2
C₂×#64984	2
#10602174	2
#10602145	2
#10601445	2
C₄×#3944	2
#33704	5