Subgroups — C₅×#3094 #10616689

Order 640.

462 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×C₅×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#482	C₂	1	4	yes	maximal, complemented
4	C₅×#481	C₂	1	2	yes	maximal, complemented
5	C₅×#480	C₂	1	2	yes	maximal, complemented
6	C₅×#428	C₂	1	2	yes	maximal, complemented
7	C₅×#418	C₂	1	2	yes	maximal, complemented
8	C₅×#407	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×D₄	—	2	1	no	—
10	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×M₁₆	C₂²	1	2	yes	—
13	C₅×#121	—	2	1	no	—
14	C₅×#121	—	2	1	no	—
15	C₅×#121	C₂²	1	2	yes	—
16	C₅×#121	C₂²	1	2	yes	—
17	C₅×#117	—	2	1	no	—
18	C₅×#117	—	2	1	no	—
19	C₅×#117	C₂²	1	2	yes	—
20	C₅×#117	C₂²	1	2	yes	complemented
21	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
25	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
26	C₂×C₄²×C₅	—	2	1	no	—
27	C₅×#107	C₂²	1	4	yes	complemented
28	C₅×#106	C₂²	1	4	yes	complemented
29	C₅×#102	C₂²	1	4	yes	complemented
30	C₅×#102	C₂²	1	4	yes	complemented
31	C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
32	C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	#3094	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	C₂×C₅×D₄	—	2	2	no	—
37	C₂×C₅×D₄	—	2	2	no	—
38	C₂×C₅×D₄	—	2	1	no	—
39	C₂×C₅×D₄	—	2	4	no	—
40	C₂×C₅×D₄	—	4	4	no	—
41	C₂×C₅×D₄	C₂³	1	2	yes	—
42	C₂²×C₄×C₅	—	2	4	no	—
43	C₂²×C₄×C₅	—	2	1	no	—
44	C₂²×C₄×C₅	—	2	1	no	—
45	C₂²×C₄×C₅	—	2	1	no	—
46	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₅×M₁₆	—	2	4	no	—
53	C₂×C₅×C₈	C₂³	1	4	yes	—
54	C₅×C₄⋊C₄	—	2	1	no	—
55	C₅×C₄⋊C₄	—	2	1	no	—
56	C₅×C₄⋊C₄	—	2	1	no	—
57	C₅×C₄⋊C₄	C₂³	1	2	yes	—
58	C₅×C₄⋊C₄	C₂³	1	2	yes	—
59	C₅×C₄⋊C₄	C₂³	1	2	yes	—
60	C₅×C₂²⋊C₄	—	2	2	no	—
61	C₅×C₂²⋊C₄	—	2	2	no	—
62	C₅×C₂²⋊C₄	—	2	2	no	—
63	C₅×C₂²⋊C₄	—	2	2	no	—
64	C₄²×C₅	—	2	2	no	—
65	C₄²×C₅	—	2	2	no	—
66	C₂×#121	C₂×C₅	1	1	yes	characteristic
67	C₂×#117	C₂×C₅	1	1	yes	characteristic, complemented
68	#482	C₂×C₅	1	4	yes	complemented
69	#481	C₂×C₅	1	2	yes	complemented
70	#480	C₂×C₅	1	2	yes	complemented
71	#428	C₂×C₅	1	2	yes	complemented
72	#418	C₂×C₅	1	2	yes	complemented
73	#407	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂³×C₅	—	2	2	no	—
75	C₂³×C₅	—	2	2	no	—
76	C₂³×C₅	—	2	2	no	—
77	C₂³×C₅	—	4	1	no	—
78	C₂³×C₅	—	4	2	no	—
79	C₂³×C₅	—	4	2	no	—
80	C₂³×C₅	—	4	2	no	—
81	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
84	C₅×D₄	—	2	4	no	—
85	C₅×D₄	—	4	4	no	—
86	C₅×D₄	—	4	4	no	—
87	C₅×D₄	—	4	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	4	2	no	—
97	C₂×C₄×C₅	—	4	2	no	—
98	C₂×C₄×C₅	—	4	2	no	—
99	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅	#41	1	4	yes	—
103	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
104	C₅×C₈	—	2	4	no	—
105	C₂²×D₄	—	2	1	no	—
106	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
107	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
108	C₂×M₁₆	C₂²×C₅	1	2	yes	—
109	#121	—	2	1	no	—
110	#121	—	2	1	no	—
111	#121	C₂²×C₅	1	2	yes	—
112	#121	C₂²×C₅	1	2	yes	—
113	#117	—	2	1	no	—
114	#117	—	2	1	no	—
115	#117	C₂²×C₅	1	2	yes	—
116	#117	C₂²×C₅	1	2	yes	complemented
117	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
119	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
120	C₂×C₂²⋊C₄	—	2	1	no	—
121	C₂×C₂²⋊C₄	—	2	1	no	—
122	C₂×C₄²	—	2	1	no	—
123	#107	C₂²×C₅	1	4	yes	complemented
124	#106	C₂²×C₅	1	4	yes	complemented
125	#102	C₂²×C₅	1	4	yes	complemented
126	#102	C₂²×C₅	1	4	yes	complemented
127	C₂²⋊C₈	C₂²×C₅	1	2	yes	complemented
128	C₂²⋊C₈	C₂²×C₅	1	2	yes	complemented
129	C₂²×C₅	—	2	1	no	—
130	C₂²×C₅	—	2	1	no	—
131	C₂²×C₅	—	2	1	no	—
132	C₂²×C₅	—	2	1	no	—
133	C₂²×C₅	—	2	1	no	—
134	C₂²×C₅	—	2	1	no	—
135	C₂²×C₅	—	2	1	no	—
136	C₂²×C₅	—	4	4	no	—
137	C₂²×C₅	—	4	2	no	—
138	C₂²×C₅	—	4	2	no	—
139	C₂²×C₅	—	4	2	no	—
140	C₂²×C₅	—	8	4	no	—
141	C₂²×C₅	#123	1	2	yes	—
142	C₂²×C₅	AGL(1,ℤ/8)	1	2	yes	—
143	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	4	2	no	—
147	C₄×C₅	—	4	2	no	—
148	C₄×C₅	—	4	2	no	—
149	C₄×C₅	#123	1	2	yes	—
150	C₄×C₅	C₂×#41	1	2	yes	—
151	C₂⁴	—	2	1	no	—
152	C₂⁴	C₅×D₄	1	1	yes	characteristic
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	—	4	4	no	—
158	C₂×D₄	C₂³×C₅	1	2	yes	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	C₅×D₄	1	1	yes	characteristic
164	C₂²×C₄	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₄	C₅×D₄	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
169	M₁₆	—	2	4	no	—
170	C₂×C₈	C₂³×C₅	1	4	yes	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	C₂³×C₅	1	2	yes	—
175	C₄⋊C₄	C₂³×C₅	1	2	yes	—
176	C₄⋊C₄	C₂³×C₅	1	2	yes	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₂×C₅	—	2	2	no	—
184	C₂×C₅	—	2	2	no	—
185	C₂×C₅	—	4	1	no	—
186	C₂×C₅	—	8	2	no	—
187	C₂×C₅	C₂×#123	1	1	yes	characteristic
188	C₂×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂×C₅	#575	1	1	yes	characteristic
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	4	1	no	—
194	C₂³	—	4	2	no	—
195	C₂³	—	4	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
200	D₄	—	2	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂×C₄	C₅×#41	1	4	yes	—
219	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
220	C₈	—	2	4	no	—
221	C₅	#3094	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	4	4	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	8	4	no	—
234	C₂²	C₅×#123	1	2	yes	—
235	C₂²	C₅×AGL(1,ℤ/8)	1	2	yes	—
236	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₄	C₅×#123	1	2	yes	—
243	C₄	C₂×C₅×#41	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	2	2	no	—
246	C₂	—	4	1	no	—
247	C₂	—	8	2	no	—
248	C₂	C₂×C₅×#123	1	1	yes	characteristic
249	C₂	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₂	C₅×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#117	C₂	semidirect
3. C₅×#482	C₂	semidirect
4. C₅×#481	C₂	semidirect
5. C₅×#480	C₂	semidirect
6. C₅×#428	C₂	semidirect
7. C₅×#418	C₂	semidirect
8. C₅×#407	C₂	semidirect
20. C₅×#117	C₂²	semidirect
22. C₂×C₅×C₄⋊C₄	C₂²	semidirect
23. C₂×C₅×C₄⋊C₄	C₂²	semidirect
27. C₅×#107	C₂²	semidirect
28. C₅×#106	C₂²	semidirect
29. C₅×#102	C₂²	semidirect
30. C₅×#102	C₂²	semidirect
31. C₅×C₂²⋊C₈	C₂²	semidirect
32. C₅×C₂²⋊C₈	C₂²	semidirect
33. #3094	C₅	direct (N × H)
67. C₂×#117	C₂×C₅	semidirect
68. #482	C₂×C₅	semidirect
69. #481	C₂×C₅	semidirect
70. #480	C₂×C₅	semidirect
71. #428	C₂×C₅	semidirect
72. #418	C₂×C₅	semidirect
73. #407	C₂×C₅	semidirect
116. #117	C₂²×C₅	semidirect
118. C₂×C₄⋊C₄	C₂²×C₅	semidirect
119. C₂×C₄⋊C₄	C₂²×C₅	semidirect
123. #107	C₂²×C₅	semidirect
124. #106	C₂²×C₅	semidirect
125. #102	C₂²×C₅	semidirect
126. #102	C₂²×C₅	semidirect
127. C₂²⋊C₈	C₂²×C₅	semidirect
128. C₂²⋊C₈	C₂²×C₅	semidirect
221. C₅	#3094	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#121	2
C₂×C₅×#117	2
C₅×#482	2
C₅×#481	2
C₅×#480	2
C₅×#428	2
C₅×#418	2
C₅×#407	2
#3094	5

Subgroups — C5×#3094 #10616689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3094 #10616689