Subgroups — #10610812

Order 640.

484 conjugacy classes of subgroups.

The subgroup lattice has 266 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	C₂×#4047	C₂	1	1	yes	maximal, characteristic
3	#65856	C₂	1	1	yes	maximal, characteristic
4	C₅×#424	C₂	1	1	yes	maximal, characteristic, complemented
5	#65407	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#3946	C₂	1	1	yes	maximal, characteristic, complemented
7	#65288	C₂	1	1	yes	maximal, characteristic, complemented
8	#65128	C₂	1	2	yes	maximal, complemented
9	#65128	C₂	1	2	yes	maximal, complemented
10	#65126	C₂	1	2	yes	maximal, complemented
11	D₅×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
12	C₂×#709	C₂²	1	1	yes	characteristic
13	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#117	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×#681	C₂²	1	1	yes	characteristic
18	C₂²×#168	C₂²	1	1	yes	characteristic
19	#4053	—	2	4	no	—
20	#4047	—	2	4	no	—
21	C₂×#676	C₂²	1	2	yes	—
22	C₂×#676	C₂²	1	2	yes	—
23	C₂×C₈×D₅	C₂²	1	2	yes	—
24	C₂×C₈×D₅	C₂²	1	2	yes	—
25	#4039	C₄	1	1	yes	characteristic
26	#4039	C₄	1	1	yes	characteristic
27	#4033	C₄	1	1	yes	characteristic
28	#4033	C₄	1	1	yes	characteristic
29	#4027	C₄	1	2	yes	—
30	#4027	C₄	1	2	yes	—
31	#4018	C₂²	1	2	yes	—
32	C₄²×D₅	C₂²	1	2	yes	—
33	C₅×C₄⋊C₈	C₂²	1	2	yes	—
34	C₅×C₄⋊C₈	C₂²	1	2	yes	—
35	#3946	C₂²	1	1	yes	characteristic
36	#3946	C₂²	1	2	yes	—
37	#3946	C₂²	1	1	yes	characteristic
38	#3946	C₂²	1	1	yes	characteristic
39	#3946	C₂²	1	1	yes	characteristic
40	#3946	C₂²	1	2	yes	—
41	C₄⋊#168	C₂²	1	2	yes	—
42	C₄⋊#168	C₂²	1	2	yes	—
43	#2906	—	5	1	no	maximal, Sylow
44	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₅	C₂×C₄	1	2	yes	—
46	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
48	#709	D₄	1	4	yes	—
49	#709	Q₈	1	4	yes	—
50	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₅	C₂³	1	2	yes	—
54	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
55	C₅×M₁₆	—	2	2	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂³	1	2	yes	—
60	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
61	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
63	C₄²×C₅	C₂³	1	2	yes	—
64	#685	C₂×C₄	1	2	yes	—
65	#685	C₂×C₄	1	2	yes	—
66	#683	C₂×C₄	1	2	yes	—
67	#683	C₂×C₄	1	2	yes	—
68	C₄×Dic₅	C₂×C₄	1	2	yes	—
69	C₄×Dic₅	C₂×C₄	1	2	yes	—
70	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₅	C₂³	1	1	yes	characteristic
73	C₄×Dic₅	C₂³	1	1	yes	characteristic
74	#681	—	2	2	no	—
75	C₂×#168	—	2	2	no	—
76	C₂×#168	C₂³	1	1	yes	characteristic
77	C₂×#168	C₂³	1	2	yes	—
78	C₂×#168	C₂³	1	1	yes	characteristic
79	#676	—	2	4	no	—
80	#676	—	2	4	no	—
81	C₈×D₅	—	2	4	no	—
82	C₈×D₅	—	2	4	no	—
83	C₂×#131	—	5	1	no	—
84	C₂×#131	—	5	1	no	—
85	C₄×#41	—	5	1	no	—
86	#424	—	5	1	no	—
87	#424	—	5	2	no	—
88	#424	—	5	2	no	—
89	#424	D₅	1	1	yes	characteristic, complemented
90	#423	—	5	1	no	—
91	#423	—	5	2	no	—
92	C₂×C₄⋊C₈	—	5	1	no	—
93	C₂×C₄⋊C₈	—	5	2	no	—
94	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₂²×D₅	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₅	C₄⋊C₄	1	8	yes	—
103	C₂×Dic₅	—	2	2	no	—
104	C₂×Dic₅	—	4	1	no	—
105	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₅	C₂²×C₄	1	2	yes	—
109	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
110	D₂₀	C₄⋊C₄	1	4	yes	—
111	C₄×D₅	—	2	4	no	—
112	C₄×D₅	—	4	2	no	—
113	C₄×D₅	C₂×D₄	1	4	yes	—
114	C₄×D₅	C₂×Q₈	1	4	yes	—
115	Dic₁₀	C₄⋊C₄	1	4	yes	—
116	C₅×C₈	—	2	1	no	—
117	C₅×C₈	—	2	2	no	—
118	C₅×C₈	—	2	1	no	—
119	#168	—	2	1	no	—
120	#168	—	2	2	no	—
121	#168	—	2	1	no	—
122	C₂×#41	—	5	1	no	—
123	#131	—	10	4	no	—
124	#131	—	10	4	no	—
125	C₂×M₁₆	—	5	1	no	—
126	C₂×M₁₆	—	5	2	no	—
127	C₂×M₁₆	—	5	2	no	—
128	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
129	C₂²×C₈	—	5	1	no	—
130	C₂²×C₈	—	5	2	no	—
131	C₂²×C₈	—	5	2	no	—
132	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
133	C₄×Q₈	—	5	1	no	—
134	C₄×Q₈	—	5	1	no	—
135	C₄×D₄	—	5	1	no	—
136	C₄×D₄	—	5	1	no	—
137	C₄×D₄	—	5	2	no	—
138	C₄×D₄	—	5	2	no	—
139	#117	—	5	2	no	—
140	#117	C₂×D₅	1	1	yes	characteristic
141	C₂×C₄²	—	5	1	no	—
142	C₂×C₄²	—	5	2	no	—
143	C₄⋊C₈	—	5	1	no	—
144	C₄⋊C₈	—	5	2	no	—
145	C₄⋊C₈	—	5	1	no	—
146	C₄⋊C₈	—	5	1	no	—
147	C₄⋊C₈	—	5	1	no	—
148	C₄⋊C₈	—	5	2	no	—
149	C₄⋊C₈	—	5	2	no	—
150	C₄⋊C₈	—	5	2	no	—
151	C₄⋊C₈	C₂×D₅	1	2	yes	—
152	C₄⋊C₈	C₂×D₅	1	2	yes	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	C₂×C₄⋊C₄	1	2	yes	—
156	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₂×D₅	—	2	2	no	—
158	C₂×D₅	—	2	2	no	—
159	C₂×D₅	C₂×C₄⋊C₄	1	4	yes	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
163	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
164	C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
165	Dic₅	—	2	1	no	—
166	Dic₅	—	2	1	no	—
167	Dic₅	—	2	1	no	—
168	Dic₅	C₂×C₄⋊C₄	1	4	yes	—
169	Dic₅	#131	1	2	yes	complemented
170	#41	—	5	4	no	—
171	#41	—	5	4	no	—
172	C₂×Q₈	—	5	1	no	—
173	C₂×D₄	—	5	2	no	—
174	C₂×D₄	—	5	1	no	—
175	C₂²×C₄	—	5	2	no	—
176	C₂²×C₄	—	5	1	no	—
177	C₂²×C₄	—	5	1	no	—
178	C₂²×C₄	—	5	2	no	—
179	C₂²×C₄	—	5	2	no	—
180	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
181	M₁₆	—	2	2	no	—
182	M₁₆	—	10	4	no	—
183	M₁₆	—	10	4	no	—
184	M₁₆	—	10	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	5	1	no	—
187	C₂×C₈	—	5	1	no	—
188	C₂×C₈	—	5	2	no	—
189	C₂×C₈	—	10	4	no	—
190	C₂×C₈	—	10	4	no	—
191	C₂×C₈	—	10	2	no	—
192	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₅	1	2	yes	—
194	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
195	C₄⋊C₄	—	5	2	no	—
196	C₄⋊C₄	—	5	2	no	—
197	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
198	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
199	C₂²⋊C₄	—	5	2	no	—
200	C₂²⋊C₄	—	5	2	no	—
201	C₂²⋊C₄	C₄×D₅	1	2	yes	—
202	C₄²	—	5	2	no	—
203	C₄²	—	5	1	no	—
204	C₄²	—	5	2	no	—
205	C₄²	—	5	1	no	—
206	C₄²	—	5	1	no	—
207	C₄²	—	5	1	no	—
208	C₄²	C₂²×D₅	1	2	yes	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₅	C₂×#131	1	1	yes	characteristic
212	C₂×C₅	#568	1	1	yes	characteristic
213	D₅	—	2	4	no	—
214	C₂³	—	5	2	no	—
215	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
216	Q₈	—	5	4	no	—
217	D₄	—	5	8	no	—
218	D₄	—	5	4	no	—
219	C₂×C₄	—	5	1	no	—
220	C₂×C₄	—	5	2	no	—
221	C₂×C₄	—	5	4	no	—
222	C₂×C₄	—	5	4	no	—
223	C₂×C₄	—	5	1	no	—
224	C₂×C₄	—	5	1	no	—
225	C₂×C₄	—	5	1	no	—
226	C₂×C₄	—	10	4	no	—
227	C₂×C₄	—	10	2	no	—
228	C₂×C₄	—	20	2	no	—
229	C₂×C₄	—	20	1	no	—
230	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
232	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
233	C₂×C₄	D₄×D₅	1	2	yes	—
234	C₂×C₄	Q₈×D₅	1	2	yes	—
235	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	2	no	—
239	C₈	—	10	1	no	—
240	C₈	—	10	2	no	—
241	C₈	—	10	1	no	—
242	C₅	#2906	1	1	yes	characteristic, complemented, Sylow
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	5	4	no	—
246	C₂²	—	10	2	no	—
247	C₂²	—	10	2	no	—
248	C₂²	D₅×C₄⋊C₄	1	2	yes	—
249	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	5	2	no	—
253	C₄	—	5	4	no	—
254	C₄	—	10	1	no	—
255	C₄	—	10	1	no	—
256	C₄	—	10	1	no	—
257	C₄	D₅×C₄⋊C₄	1	2	yes	—
258	C₄	C₂×D₄×D₅	1	1	yes	characteristic
259	C₄	C₂×Q₈×D₅	1	1	yes	characteristic
260	C₂	—	2	2	no	—
261	C₂	—	10	4	no	—
262	C₂	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
263	C₂	D₅×#131	1	1	yes	characteristic
264	C₂	#66084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#424	C₂
prime radical	C₂×#709	C₂²
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#424	D₅
hypocenter	C₅	#2906
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. C₅×#424	C₂	semidirect
5. #65407	C₂	semidirect
6. C₂×#3946	C₂	semidirect
7. #65288	C₂	semidirect
8. #65128	C₂	semidirect
9. #65128	C₂	semidirect
10. #65126	C₂	semidirect
11. D₅×C₄⋊C₈	C₂	semidirect
89. #424	D₅	semidirect
169. Dic₅	#131	semidirect
242. C₅	#2906	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
C₂×#4047	2
#65856	2
C₅×#424	2
#65407	2
C₂×#3946	2
#65288	2
#65128	2
#65126	2
D₅×C₄⋊C₈	2
#2906	5