Subgroups — C₅×#3340 #10616935

Order 640.

352 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#555	C₂	1	1	yes	maximal, characteristic
3	C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#485	C₂	1	2	yes	maximal, complemented
5	C₅×#482	C₂	1	2	yes	maximal, complemented
6	C₅×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#474	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#465	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#427	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×SD₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₈	C₂²	1	2	yes	—
14	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₅×#126	—	2	2	no	—
17	C₅×#124	—	2	1	no	—
18	C₅×#124	—	2	2	no	—
19	C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₅×#121	—	2	1	no	—
21	C₅×#121	C₂²	1	2	yes	—
22	C₄×C₅×Q₈	—	2	1	no	—
23	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
24	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
25	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
26	C₅×#117	—	2	1	no	—
27	C₅×#117	C₂²	1	2	yes	—
28	C₅×#117	C₂²	1	2	yes	—
29	C₅×#107	C₂²	1	2	yes	—
30	C₅×#107	C₂²	1	2	yes	—
31	C₅×#106	C₂²	1	1	yes	characteristic
32	C₅×#106	C₂²	1	2	yes	—
33	C₅×#106	C₂²	1	2	yes	—
34	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₅×#103	C₂²	1	1	yes	characteristic
36	C₅×#103	C₂²	1	2	yes	—
37	C₅×#102	C₂²	1	1	yes	characteristic
38	C₅×#102	C₂²	1	2	yes	—
39	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
41	#3340	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₅×Q₈	—	2	1	no	—
43	C₂×C₅×Q₈	—	2	1	no	—
44	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
45	C₂×C₅×D₄	—	2	2	no	—
46	C₂×C₅×D₄	D₄	1	1	yes	characteristic
47	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	1	no	—
49	C₂²×C₄×C₅	C₂³	1	2	yes	—
50	C₅×SD₁₆	—	2	2	no	—
51	C₂×C₅×C₈	—	2	4	no	—
52	C₂×C₅×C₈	C₂³	1	2	yes	—
53	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₅×C₄⋊C₄	—	2	1	no	—
56	C₅×C₄⋊C₄	—	2	2	no	—
57	C₅×C₄⋊C₄	—	2	1	no	—
58	C₅×C₄⋊C₄	—	2	1	no	—
59	C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
60	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
61	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
62	C₅×C₄⋊C₄	C₂³	1	2	yes	—
63	C₅×C₄⋊C₄	C₂³	1	2	yes	—
64	C₅×C₂²⋊C₄	—	2	2	no	—
65	C₅×C₂²⋊C₄	—	2	2	no	—
66	C₅×C₂²⋊C₄	—	2	2	no	—
67	C₅×C₂²⋊C₄	—	2	2	no	—
68	C₅×C₂²⋊C₄	—	2	1	no	—
69	C₅×C₂²⋊C₄	D₄	1	2	yes	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	—	2	1	no	—
73	C₄²×C₅	C₂³	1	1	yes	characteristic
74	#556	C₂×C₅	1	1	yes	characteristic, complemented
75	#555	C₂×C₅	1	1	yes	characteristic
76	#498	C₂×C₅	1	1	yes	characteristic, complemented
77	#485	C₂×C₅	1	2	yes	complemented
78	#482	C₂×C₅	1	2	yes	complemented
79	#477	C₂×C₅	1	1	yes	characteristic, complemented
80	#474	C₂×C₅	1	1	yes	characteristic, complemented
81	#465	C₂×C₅	1	2	yes	complemented
82	C₄×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
83	C₈×D₄	C₂×C₅	1	1	yes	characteristic, complemented
84	#427	C₂×C₅	1	2	yes	complemented
85	C₂³×C₅	—	2	1	no	—
86	C₂³×C₅	C₂×D₄	1	2	yes	—
87	C₅×Q₈	—	2	1	no	—
88	C₅×Q₈	—	2	1	no	—
89	C₅×Q₈	—	4	2	no	—
90	C₅×Q₈	—	8	1	no	—
91	C₅×D₄	—	2	2	no	—
92	C₅×D₄	—	2	1	no	—
93	C₅×D₄	—	2	1	no	—
94	C₅×D₄	—	8	2	no	—
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
107	C₅×C₈	—	2	2	no	—
108	C₅×C₈	—	2	1	no	—
109	C₅×C₈	#41	1	2	yes	—
110	C₂×SD₁₆	C₂²×C₅	1	1	yes	characteristic
111	C₂²×C₈	C₂²×C₅	1	2	yes	—
112	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
113	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
114	#126	—	2	2	no	—
115	#124	—	2	1	no	—
116	#124	—	2	2	no	—
117	C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
118	#121	—	2	1	no	—
119	#121	C₂²×C₅	1	2	yes	—
120	C₄×Q₈	—	2	1	no	—
121	C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
122	C₄×D₄	C₂²×C₅	1	1	yes	characteristic
123	C₄×D₄	C₂²×C₅	1	1	yes	characteristic
124	#117	—	2	1	no	—
125	#117	C₂²×C₅	1	2	yes	—
126	#117	C₂²×C₅	1	2	yes	—
127	#107	C₂²×C₅	1	2	yes	—
128	#107	C₂²×C₅	1	2	yes	—
129	#106	C₂²×C₅	1	1	yes	characteristic
130	#106	C₂²×C₅	1	2	yes	—
131	#106	C₂²×C₅	1	2	yes	—
132	C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
133	#103	C₂²×C₅	1	1	yes	characteristic
134	#103	C₂²×C₅	1	2	yes	—
135	#102	C₂²×C₅	1	1	yes	characteristic
136	#102	C₂²×C₅	1	2	yes	—
137	C₂²⋊C₈	C₂²×C₅	1	2	yes	—
138	C₄×C₈	C₂²×C₅	1	1	yes	characteristic
139	C₂²×C₅	—	2	2	no	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	—	2	2	no	—
142	C₂²×C₅	—	4	1	no	—
143	C₂²×C₅	—	4	1	no	—
144	C₂²×C₅	—	4	1	no	—
145	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	4	1	no	—
149	C₄×C₅	—	4	1	no	—
150	C₄×C₅	—	4	2	no	—
151	C₄×C₅	—	4	1	no	—
152	C₄×C₅	—	4	2	no	—
153	C₄×C₅	#135	1	2	yes	—
154	C₄×C₅	C₂×#41	1	1	yes	characteristic
155	C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	C₅×D₄	1	1	yes	characteristic
161	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	C₂³×C₅	1	2	yes	—
164	SD₁₆	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
167	C₂×C₈	C₂³×C₅	1	2	yes	—
168	C₂×C₈	C₂³×C₅	1	1	yes	characteristic
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
174	C₄⋊C₄	C₂³×C₅	1	2	yes	—
175	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
176	C₄⋊C₄	C₂³×C₅	1	2	yes	—
177	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	C₅×D₄	1	2	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	1	no	—
187	C₄²	C₂³×C₅	1	1	yes	characteristic
188	C₂×C₅	—	4	2	no	—
189	C₂×C₅	—	8	1	no	—
190	C₂×C₅	#547	1	1	yes	characteristic
191	C₂×C₅	C₂×#135	1	1	yes	characteristic
192	C₂×C₅	#577	1	1	yes	characteristic
193	C₂³	—	2	1	no	—
194	C₂³	C₂×C₅×D₄	1	2	yes	—
195	Q₈	—	2	1	no	—
196	Q₈	—	2	1	no	—
197	Q₈	—	4	2	no	—
198	Q₈	—	8	1	no	—
199	D₄	—	2	2	no	—
200	D₄	—	2	1	no	—
201	D₄	—	2	1	no	—
202	D₄	—	8	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
212	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₈	C₅×#41	1	2	yes	—
218	C₅	#3340	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	1	no	—
233	C₄	C₅×#135	1	2	yes	—
234	C₄	C₂×C₅×#41	1	1	yes	characteristic
235	C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
236	C₂	—	4	2	no	—
237	C₂	—	8	1	no	—
238	C₂	C₅×#547	1	1	yes	characteristic
239	C₂	C₂×C₅×#135	1	1	yes	characteristic
240	C₂	C₅×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₅×#555	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#556	C₂	semidirect
3. C₅×#498	C₂	semidirect
4. C₅×#485	C₂	semidirect
5. C₅×#482	C₂	semidirect
6. C₅×#477	C₂	semidirect
7. C₅×#474	C₂	semidirect
8. C₅×#465	C₂	semidirect
9. C₄×C₅×SD₁₆	C₂	semidirect
10. C₅×C₈×D₄	C₂	semidirect
11. C₅×#427	C₂	semidirect
41. #3340	C₅	direct (N × H)
74. #556	C₂×C₅	semidirect
76. #498	C₂×C₅	semidirect
77. #485	C₂×C₅	semidirect
78. #482	C₂×C₅	semidirect
79. #477	C₂×C₅	semidirect
80. #474	C₂×C₅	semidirect
81. #465	C₂×C₅	semidirect
82. C₄×SD₁₆	C₂×C₅	semidirect
83. C₈×D₄	C₂×C₅	semidirect
84. #427	C₂×C₅	semidirect
218. C₅	#3340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#556	2
C₅×#555	2
C₅×#498	2
C₅×#485	2
C₅×#482	2
C₅×#477	2
C₅×#474	2
C₅×#465	2
C₄×C₅×SD₁₆	2
C₅×C₈×D₄	2
C₅×#427	2
#3340	5

Subgroups — C5×#3340 #10616935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3340 #10616935