Subgroups — C₂×C₅×#452 #10616596

Order 640.

760 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#452	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#102	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#41	—	2	4	no	—
10	C₂×C₅×#41	—	2	4	no	—
11	C₂×C₅×#41	—	2	2	no	—
12	C₂×C₅×#41	C₂²	1	4	yes	—
13	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₅×D₄	—	2	1	no	—
16	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₅	—	2	1	no	—
18	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
19	C₂×C₅×Q₁₆	—	2	4	no	—
20	C₂×C₅×Q₁₆	C₂²	1	4	yes	complemented
21	C₂×C₅×SD₁₆	—	2	4	no	—
22	C₂×C₅×SD₁₆	C₂²	1	4	yes	complemented
23	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×C₂²⋊Q₈	—	2	2	no	—
26	C₅×C₂²⋊Q₈	C₂²	1	4	yes	complemented
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
30	C₅×#103	C₂²	1	4	yes	complemented
31	C₅×#102	C₂²	1	4	yes	complemented
32	C₅×C₂²⋊C₈	C₂²	1	4	yes	complemented
33	C₂×#452	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂⁴×C₅	—	2	1	no	—
35	C₂⁴×C₅	D₄	1	1	yes	characteristic
36	C₅×#41	—	2	16	no	—
37	C₅×#41	—	2	8	no	—
38	C₅×#41	—	2	8	no	—
39	C₂×C₅×Q₈	—	2	2	no	—
40	C₂×C₅×Q₈	—	2	1	no	—
41	C₂×C₅×Q₈	—	2	1	no	—
42	C₂×C₅×Q₈	—	2	2	no	—
43	C₂×C₅×Q₈	—	2	2	no	—
44	C₂×C₅×Q₈	D₄	1	4	yes	—
45	C₂×C₅×Q₈	C₂³	1	2	yes	—
46	C₂×C₅×Q₈	C₂³	1	2	yes	—
47	C₂×C₅×D₄	—	2	8	no	—
48	C₂×C₅×D₄	—	2	4	no	—
49	C₂×C₅×D₄	—	2	1	no	—
50	C₂×C₅×D₄	—	2	2	no	—
51	C₂×C₅×D₄	D₄	1	4	yes	—
52	C₂×C₅×D₄	C₂³	1	2	yes	—
53	C₂²×C₄×C₅	—	2	8	no	—
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	2	4	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	—	2	1	no	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	D₄	1	2	yes	—
61	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂³	1	2	yes	—
63	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₅×Q₁₆	—	2	8	no	—
65	C₅×SD₁₆	—	2	8	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	C₂³	1	2	yes	—
69	C₂×C₅×C₈	C₂³	1	2	yes	—
70	C₅×C₄⋊C₄	—	2	4	no	—
71	C₅×C₄⋊C₄	—	2	1	no	—
72	C₅×C₄⋊C₄	C₂³	1	2	yes	—
73	C₅×C₂²⋊C₄	—	2	4	no	—
74	C₂²×#41	C₂×C₅	1	1	yes	characteristic
75	C₂²×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
76	C₂²×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
77	C₂×C₂²⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
78	#452	C₂×C₅	1	8	yes	direct product factor
79	C₂×#103	C₂×C₅	1	1	yes	characteristic, complemented
80	C₂×#102	C₂×C₅	1	1	yes	characteristic, complemented
81	C₂×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
82	C₂³×C₅	—	2	2	no	—
83	C₂³×C₅	—	2	4	no	—
84	C₂³×C₅	—	2	2	no	—
85	C₂³×C₅	—	2	2	no	—
86	C₂³×C₅	—	2	2	no	—
87	C₂³×C₅	—	4	2	no	—
88	C₂³×C₅	—	4	2	no	—
89	C₂³×C₅	C₂×D₄	1	2	yes	—
90	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
91	C₅×Q₈	—	2	4	no	—
92	C₅×Q₈	—	2	2	no	—
93	C₅×Q₈	—	2	2	no	—
94	C₅×Q₈	—	2	2	no	—
95	C₅×Q₈	—	4	2	no	—
96	C₅×Q₈	C₂×D₄	1	4	yes	—
97	C₅×D₄	—	2	16	no	—
98	C₅×D₄	—	2	4	no	—
99	C₅×D₄	—	2	2	no	—
100	C₅×D₄	C₂×D₄	1	4	yes	—
101	C₂×C₄×C₅	—	2	16	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
117	C₅×C₈	—	2	2	no	—
118	C₅×C₈	—	2	2	no	—
119	C₂×#41	—	2	4	no	—
120	C₂×#41	—	2	4	no	—
121	C₂×#41	—	2	2	no	—
122	C₂×#41	C₂²×C₅	1	4	yes	—
123	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
124	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
125	C₂²×D₄	—	2	1	no	—
126	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
127	C₂³×C₄	—	2	1	no	—
128	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
129	C₂×Q₁₆	—	2	4	no	—
130	C₂×Q₁₆	C₂²×C₅	1	4	yes	complemented
131	C₂×SD₁₆	—	2	4	no	—
132	C₂×SD₁₆	C₂²×C₅	1	4	yes	complemented
133	C₂²×C₈	C₂²×C₅	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₅	1	1	yes	characteristic
135	C₂²⋊Q₈	—	2	2	no	—
136	C₂²⋊Q₈	C₂²×C₅	1	4	yes	complemented
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
139	C₂×C₂²⋊C₄	—	2	1	no	—
140	#103	C₂²×C₅	1	4	yes	complemented
141	#102	C₂²×C₅	1	4	yes	complemented
142	C₂²⋊C₈	C₂²×C₅	1	4	yes	complemented
143	C₂²×C₅	—	2	2	no	—
144	C₂²×C₅	—	2	4	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	—	4	2	no	—
148	C₂²×C₅	—	4	2	no	—
149	C₂²×C₅	—	4	4	no	—
150	C₂²×C₅	—	4	8	no	—
151	C₂²×C₅	#120	1	2	yes	—
152	C₂²×C₅	#135	1	2	yes	—
153	C₂²×C₅	#137	1	2	yes	—
154	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	2	4	no	—
157	C₄×C₅	—	4	2	no	—
158	C₄×C₅	—	4	2	no	—
159	C₄×C₅	#120	1	2	yes	—
160	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
162	C₂⁴	—	2	1	no	—
163	C₂⁴	C₅×D₄	1	1	yes	characteristic
164	#41	—	2	16	no	—
165	#41	—	2	8	no	—
166	#41	—	2	8	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	1	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	C₅×D₄	1	4	yes	—
173	C₂×Q₈	C₂³×C₅	1	2	yes	—
174	C₂×Q₈	C₂³×C₅	1	2	yes	—
175	C₂×D₄	—	2	8	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	C₅×D₄	1	4	yes	—
180	C₂×D₄	C₂³×C₅	1	2	yes	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₅×D₄	1	1	yes	characteristic
189	C₂²×C₄	C₅×D₄	1	2	yes	—
190	C₂²×C₄	C₂³×C₅	1	2	yes	—
191	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
192	Q₁₆	—	2	8	no	—
193	SD₁₆	—	2	8	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	C₂³×C₅	1	2	yes	—
197	C₂×C₈	C₂³×C₅	1	2	yes	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₂³×C₅	1	2	yes	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂×C₅	—	4	2	no	—
203	C₂×C₅	—	4	4	no	—
204	C₂×C₅	#452	1	4	yes	direct product factor
205	C₂×C₅	C₂×#120	1	1	yes	characteristic
206	C₂×C₅	C₂×#135	1	1	yes	characteristic
207	C₂×C₅	C₂×#137	1	1	yes	characteristic
208	C₂³	—	2	2	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₂×C₅×D₄	1	2	yes	—
216	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
217	Q₈	—	2	4	no	—
218	Q₈	—	2	2	no	—
219	Q₈	—	2	2	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	4	2	no	—
222	Q₈	C₂×C₅×D₄	1	4	yes	—
223	D₄	—	2	16	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	D₄	C₂×C₅×D₄	1	4	yes	—
227	C₂×C₄	—	2	16	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
239	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
240	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
241	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
242	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₅	C₂×#452	1	1	yes	characteristic, direct product factor, Sylow
246	C₂²	—	2	2	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	8	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₅×#120	1	2	yes	—
255	C₂²	C₅×#135	1	2	yes	—
256	C₂²	C₅×#137	1	2	yes	—
257	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	C₅×#120	1	2	yes	—
263	C₄	C₂²×C₅×D₄	1	1	yes	characteristic
264	C₄	C₂²×C₅×D₄	1	1	yes	characteristic
265	C₂	—	4	2	no	—
266	C₂	—	4	4	no	—
267	C₂	C₅×#452	1	4	yes	direct product factor
268	C₂	C₂×C₅×#120	1	1	yes	characteristic
269	C₂	C₂×C₅×#135	1	1	yes	characteristic
270	C₂	C₂×C₅×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂²×C₅×#41	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×Q₁₆	C₂	semidirect
3. C₂²×C₅×SD₁₆	C₂	semidirect
4. C₂×C₅×C₂²⋊Q₈	C₂	semidirect
5. C₅×#452	C₂	direct (N × H)
6. C₂×C₅×#103	C₂	semidirect
7. C₂×C₅×#102	C₂	semidirect
8. C₂×C₅×C₂²⋊C₈	C₂	semidirect
20. C₂×C₅×Q₁₆	C₂²	semidirect
22. C₂×C₅×SD₁₆	C₂²	semidirect
26. C₅×C₂²⋊Q₈	C₂²	semidirect
30. C₅×#103	C₂²	semidirect
31. C₅×#102	C₂²	semidirect
32. C₅×C₂²⋊C₈	C₂²	semidirect
33. C₂×#452	C₅	direct (N × H)
75. C₂²×Q₁₆	C₂×C₅	semidirect
76. C₂²×SD₁₆	C₂×C₅	semidirect
77. C₂×C₂²⋊Q₈	C₂×C₅	semidirect
78. #452	C₂×C₅	direct (N × H)
79. C₂×#103	C₂×C₅	semidirect
80. C₂×#102	C₂×C₅	semidirect
81. C₂×C₂²⋊C₈	C₂×C₅	semidirect
130. C₂×Q₁₆	C₂²×C₅	semidirect
132. C₂×SD₁₆	C₂²×C₅	semidirect
136. C₂²⋊Q₈	C₂²×C₅	semidirect
140. #103	C₂²×C₅	semidirect
141. #102	C₂²×C₅	semidirect
142. C₂²⋊C₈	C₂²×C₅	semidirect
204. C₂×C₅	#452	direct (N × H)
245. C₅	C₂×#452	direct (N × H)
267. C₂	C₅×#452	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#41	2
C₂²×C₅×Q₁₆	2
C₂²×C₅×SD₁₆	2
C₂×C₅×C₂²⋊Q₈	2
C₅×#452	2
C₂×C₅×#103	2
C₂×C₅×#102	2
C₂×C₅×C₂²⋊C₈	2
C₂×#452	5

Subgroups — C2×C5×#452 #10616596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#452 #10616596