Subgroups — #10610941

Order 640.

476 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4053	C₂	1	1	yes	maximal, characteristic
2	#65857	C₂	1	1	yes	maximal, characteristic
3	#65838	C₂	1	1	yes	maximal, characteristic
4	C₅×#428	C₂	1	1	yes	maximal, characteristic, complemented
5	#65410	C₂	1	1	yes	maximal, characteristic, complemented
6	#65287	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#3939	C₂	1	1	yes	maximal, characteristic, complemented
8	#65170	C₂	1	2	yes	maximal, complemented
9	D₅×#107	C₂	1	2	yes	maximal, complemented
10	#65150	C₂	1	2	yes	maximal, complemented
11	#65149	C₂	1	2	yes	maximal, complemented
12	C₂×#709	C₂²	1	1	yes	characteristic
13	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₅×#117	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊D₅	—	2	2	no	—
17	#4072	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
19	C₂²×#168	C₂²	1	1	yes	characteristic
20	#4053	C₄	1	8	yes	complemented
21	C₂×#676	C₂²	1	2	yes	—
22	C₂×C₈×D₅	C₂²	1	2	yes	—
23	#4039	—	2	1	no	—
24	#4038	C₂²	1	2	yes	—
25	#4038	C₂²	1	2	yes	—
26	D₅×C₄⋊C₄	C₂²	1	2	yes	—
27	D₅×C₄⋊C₄	C₂²	1	2	yes	—
28	#4033	—	2	1	no	—
29	#4027	—	2	2	no	—
30	C₄×D₂₀	—	2	1	no	—
31	C₄×Dic₁₀	—	2	1	no	—
32	C₅×#107	C₂²	1	2	yes	—
33	C₅×#106	C₂²	1	2	yes	—
34	#3950	C₂²	1	2	yes	—
35	#3949	C₂²	1	2	yes	—
36	#3940	C₂²	1	1	yes	characteristic
37	#3940	C₂²	1	1	yes	characteristic
38	#3939	C₂²	1	1	yes	characteristic
39	#3939	C₂²	1	1	yes	characteristic
40	#3939	C₂²	1	2	yes	—
41	#3939	C₂²	1	2	yes	—
42	#2913	—	5	1	no	maximal, Sylow
43	C₂²×C₄×C₅	—	2	1	no	—
44	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₅	D₄	1	2	yes	—
46	C₂²×Dic₅	—	2	1	no	—
47	#709	C₂×C₄	1	4	yes	—
48	#709	Q₈	1	4	yes	—
49	C₂×D₂₀	D₄	1	1	yes	characteristic
50	C₂×C₄×D₅	—	2	2	no	—
51	C₂×C₄×D₅	—	2	2	no	—
52	C₂×C₄×D₅	C₂³	1	2	yes	—
53	C₂×Dic₁₀	D₄	1	1	yes	characteristic
54	C₅×M₁₆	C₂×C₄	1	4	yes	complemented
55	C₂×C₅×C₈	C₂³	1	2	yes	—
56	C₅×C₄⋊C₄	—	2	1	no	—
57	C₅×C₄⋊C₄	C₂³	1	2	yes	—
58	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
59	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
60	C₅×C₂²⋊C₄	—	2	1	no	—
61	C₄²×C₅	—	2	1	no	—
62	C₂²⋊Dic₅	—	2	1	no	—
63	#685	—	2	2	no	—
64	#685	—	2	2	no	—
65	C₄⋊Dic₅	—	2	1	no	—
66	C₄⋊Dic₅	C₂³	1	2	yes	—
67	C₄⋊Dic₅	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₅	C₂³	1	1	yes	characteristic
69	#683	—	2	2	no	—
70	#683	—	2	2	no	—
71	C₄×Dic₅	—	2	1	no	—
72	C₄×Dic₅	—	2	2	no	—
73	C₄×Dic₅	—	2	2	no	—
74	C₂×#168	C₂×C₄	1	4	yes	complemented
75	C₂×#168	C₂³	1	1	yes	characteristic
76	C₂×#168	C₂³	1	1	yes	characteristic
77	#676	C₂×C₄	1	8	yes	complemented
78	C₈×D₅	C₂×C₄	1	8	yes	complemented
79	C₂×#131	—	5	1	no	—
80	#520	—	5	1	no	—
81	#520	—	5	1	no	—
82	#428	—	5	1	no	—
83	#428	—	5	2	no	—
84	#428	—	5	2	no	—
85	#428	D₅	1	1	yes	characteristic, complemented
86	#427	—	5	1	no	—
87	#427	—	5	2	no	—
88	C₂×#107	—	5	1	no	—
89	C₂×#107	—	5	2	no	—
90	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×D₅	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₅	—	2	1	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	8	1	no	—
96	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	C₂²⋊D₅	C₄⋊C₄	1	8	yes	—
101	C₂×Dic₅	—	2	1	no	—
102	C₂×Dic₅	—	2	1	no	—
103	C₂×Dic₅	—	2	2	no	—
104	C₂×Dic₅	—	8	1	no	—
105	C₂×Dic₅	C₂×D₄	1	2	yes	—
106	D₂₀	C₄⋊C₄	1	4	yes	—
107	C₄×D₅	—	8	2	no	—
108	C₄×D₅	—	8	2	no	—
109	C₄×D₅	C₂²×C₄	1	4	yes	—
110	C₄×D₅	C₂×Q₈	1	4	yes	—
111	Dic₁₀	C₄⋊C₄	1	4	yes	—
112	C₅×C₈	C₂²×C₄	1	4	yes	—
113	#168	C₂²×C₄	1	2	yes	—
114	#168	C₂²×C₄	1	2	yes	—
115	C₂×#41	—	5	1	no	—
116	#131	—	5	8	no	—
117	C₂×M₁₆	—	5	2	no	—
118	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
119	C₂²×C₈	—	5	1	no	—
120	C₂²×C₈	—	5	2	no	—
121	C₄×Q₈	—	10	1	no	—
122	C₄×Q₈	—	10	1	no	—
123	C₄×D₄	—	10	1	no	—
124	C₄×D₄	—	10	2	no	—
125	C₄×D₄	—	10	1	no	—
126	C₄×D₄	—	10	2	no	—
127	#117	—	5	1	no	—
128	#117	—	5	2	no	—
129	#117	—	5	2	no	—
130	#117	C₂×D₅	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	5	1	no	—
132	C₂×C₄⋊C₄	—	5	2	no	—
133	C₂×C₄⋊C₄	—	5	2	no	—
134	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
135	#107	—	5	2	no	—
136	#107	—	5	1	no	—
137	#107	—	5	2	no	—
138	#107	—	5	2	no	—
139	#107	—	5	1	no	—
140	#107	C₂×D₅	1	2	yes	—
141	#106	—	5	2	no	—
142	#106	—	5	1	no	—
143	#106	—	5	1	no	—
144	#106	C₂×D₅	1	2	yes	—
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	—	2	1	no	—
147	C₂²×C₅	C₂×C₄⋊C₄	1	2	yes	—
148	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₅	—	2	2	no	—
150	C₂×D₅	—	2	2	no	—
151	C₂×D₅	C₂×C₄⋊C₄	1	4	yes	—
152	C₄×C₅	—	4	1	no	—
153	C₄×C₅	—	4	2	no	—
154	C₄×C₅	—	4	1	no	—
155	C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
156	C₄×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
158	Dic₅	—	4	1	no	—
159	Dic₅	—	4	2	no	—
160	Dic₅	—	4	1	no	—
161	Dic₅	C₂×C₄⋊C₄	1	4	yes	—
162	#41	—	5	4	no	—
163	#41	—	5	4	no	—
164	C₂×Q₈	—	5	1	no	—
165	C₂×D₄	—	5	2	no	—
166	C₂×D₄	—	5	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	5	2	no	—
169	C₂²×C₄	—	10	2	no	—
170	C₂²×C₄	—	10	2	no	—
171	C₂²×C₄	—	10	1	no	—
172	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
173	M₁₆	—	5	8	no	—
174	M₁₆	C₄×D₅	1	4	yes	complemented
175	C₂×C₈	—	5	4	no	—
176	C₂×C₈	—	5	1	no	—
177	C₂×C₈	—	5	1	no	—
178	C₂×C₈	—	5	8	no	—
179	C₂×C₈	C₂²×D₅	1	2	yes	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	5	1	no	—
182	C₄⋊C₄	—	5	2	no	—
183	C₄⋊C₄	—	5	1	no	—
184	C₄⋊C₄	—	10	2	no	—
185	C₄⋊C₄	—	10	2	no	—
186	C₄⋊C₄	—	10	1	no	—
187	C₄⋊C₄	C₂²×D₅	1	1	yes	characteristic
188	C₄⋊C₄	C₂²×D₅	1	1	yes	characteristic
189	C₄⋊C₄	C₂²×D₅	1	2	yes	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	10	2	no	—
192	C₂²⋊C₄	—	10	2	no	—
193	C₂²⋊C₄	—	10	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	10	2	no	—
196	C₄²	—	10	2	no	—
197	C₄²	—	10	1	no	—
198	C₂×C₅	—	2	2	no	—
199	C₂×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₅	#576	1	1	yes	characteristic
201	C₂×C₅	#578	1	1	yes	characteristic
202	D₅	—	2	4	no	—
203	C₂³	—	5	2	no	—
204	C₂³	D₄×D₅	1	1	yes	characteristic
205	Q₈	—	5	4	no	—
206	D₄	—	5	8	no	—
207	D₄	—	5	4	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	5	2	no	—
212	C₂×C₄	—	5	4	no	—
213	C₂×C₄	—	5	4	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	10	1	no	—
216	C₂×C₄	—	10	2	no	—
217	C₂×C₄	—	10	1	no	—
218	C₂×C₄	—	40	2	no	—
219	C₂×C₄	—	40	2	no	—
220	C₂×C₄	—	40	1	no	—
221	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
222	C₂×C₄	D₄×D₅	1	1	yes	characteristic
223	C₂×C₄	Q₈×D₅	1	2	yes	—
224	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
225	C₈	—	5	2	no	—
226	C₈	—	5	2	no	—
227	C₈	C₂×C₄×D₅	1	4	yes	—
228	C₅	#2913	1	1	yes	characteristic, complemented, Sylow
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	5	4	no	—
232	C₂²	—	10	2	no	—
233	C₂²	—	10	2	no	—
234	C₂²	D₅×C₄⋊C₄	1	2	yes	—
235	C₂²	C₂×D₄×D₅	1	1	yes	characteristic
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	5	4	no	—
240	C₄	—	20	1	no	—
241	C₄	—	20	2	no	—
242	C₄	—	20	1	no	—
243	C₄	D₅×C₄⋊C₄	1	2	yes	—
244	C₄	C₂²×C₄×D₅	1	1	yes	characteristic
245	C₄	C₂×Q₈×D₅	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	10	4	no	—
248	C₂	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
249	C₂	#66108	1	1	yes	characteristic
250	C₂	#66113	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅
commutator	C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅
Fitting	C₅×#428	C₂
socle	C₂²×C₅	C₂²×D₄
prime radical	C₂×#709	C₂²
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#428	D₅
hypocenter	C₅	#2913
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. C₅×#428	C₂	semidirect
5. #65410	C₂	semidirect
6. #65287	C₂	semidirect
7. C₂×#3939	C₂	semidirect
8. #65170	C₂	semidirect
9. D₅×#107	C₂	semidirect
10. #65150	C₂	semidirect
11. #65149	C₂	semidirect
20. #4053	C₄	semidirect
54. C₅×M₁₆	C₂×C₄	semidirect
74. C₂×#168	C₂×C₄	semidirect
77. #676	C₂×C₄	semidirect
78. C₈×D₅	C₂×C₄	semidirect
85. #428	D₅	semidirect
174. M₁₆	C₄×D₅	semidirect
228. C₅	#2913	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4053	2
#65857	2
#65838	2
C₅×#428	2
#65410	2
#65287	2
C₂×#3939	2
#65170	2
D₅×#107	2
#65150	2
#65149	2
#2913	5