Subgroups — C₅×#2742 #10616337

Order 640.

416 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#400	C₂	1	2	yes	maximal, complemented
2	C₅×#399	C₂	1	1	yes	maximal, characteristic
3	C₅×#395	C₂	1	2	yes	maximal, complemented
4	C₅×#393	C₂	1	1	yes	maximal, characteristic
5	C₅×#389	C₂	1	2	yes	maximal, complemented
6	C₅×#388	C₂	1	2	yes	maximal
7	C₅×#387	C₂	1	2	yes	maximal, complemented
8	C₅×#385	C₂	1	1	yes	maximal, characteristic
9	C₅×#380	C₂	1	2	yes	maximal
10	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂²	1	2	yes	—
23	C₅×#95	C₂²	1	2	yes	—
24	C₅×#95	C₂²	1	2	yes	—
25	C₅×#95	C₂²	1	2	yes	—
26	C₅×#95	C₂²	1	1	yes	characteristic
27	C₅×#95	C₂²	1	2	yes	—
28	C₅×#95	C₂²	1	2	yes	—
29	C₅×#95	C₂²	1	2	yes	—
30	C₅×#95	C₂²	1	1	yes	characteristic
31	#2742	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×Q₈	—	4	2	no	—
34	C₂×C₅×Q₈	—	4	1	no	—
35	C₂²×C₄×C₅	—	2	4	no	—
36	C₂²×C₄×C₅	—	4	1	no	—
37	C₂²×C₄×C₅	C₂³	1	2	yes	—
38	C₂²×C₄×C₅	C₂³	1	2	yes	—
39	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂³	1	2	yes	—
41	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂³	1	2	yes	—
43	C₂²×C₄×C₅	C₂³	1	2	yes	—
44	C₂²×C₄×C₅	C₂³	1	2	yes	—
45	C₅×C₄⋊C₄	—	2	4	no	—
46	C₅×C₄⋊C₄	—	2	4	no	—
47	C₅×C₄⋊C₄	—	4	2	no	—
48	C₅×C₄⋊C₄	—	4	2	no	—
49	C₅×C₄⋊C₄	—	4	2	no	—
50	C₅×C₄⋊C₄	—	4	1	no	—
51	C₅×C₂²⋊C₄	—	4	2	no	—
52	C₅×C₂²⋊C₄	—	4	2	no	—
53	C₅×C₂²⋊C₄	—	4	1	no	—
54	C₅×C₂²⋊C₄	Q₈	1	4	yes	complemented
55	C₄²×C₅	—	4	2	no	—
56	#400	C₂×C₅	1	2	yes	complemented
57	#399	C₂×C₅	1	1	yes	characteristic
58	#395	C₂×C₅	1	2	yes	complemented
59	#393	C₂×C₅	1	1	yes	characteristic
60	#389	C₂×C₅	1	2	yes	complemented
61	#388	C₂×C₅	1	2	yes	—
62	#387	C₂×C₅	1	2	yes	complemented
63	#385	C₂×C₅	1	1	yes	characteristic
64	#380	C₂×C₅	1	2	yes	—
65	C₂³×C₅	—	2	2	no	—
66	C₂³×C₅	—	2	2	no	—
67	C₂³×C₅	—	2	1	no	—
68	C₂³×C₅	—	2	1	no	—
69	C₂³×C₅	C₂×Q₈	1	2	yes	—
70	C₂³×C₅	C₂⁴	1	1	yes	characteristic
71	C₅×Q₈	—	4	4	no	—
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	1	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	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	—	4	4	no	—
93	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
94	C₂×C₄×C₅	#41	1	4	yes	—
95	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
96	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
99	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
101	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
102	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
104	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
105	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
106	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
107	C₂×C₄²	C₂²×C₅	1	2	yes	—
108	#95	C₂²×C₅	1	2	yes	—
109	#95	C₂²×C₅	1	2	yes	—
110	#95	C₂²×C₅	1	2	yes	—
111	#95	C₂²×C₅	1	1	yes	characteristic
112	#95	C₂²×C₅	1	2	yes	—
113	#95	C₂²×C₅	1	2	yes	—
114	#95	C₂²×C₅	1	2	yes	—
115	#95	C₂²×C₅	1	1	yes	characteristic
116	C₂²×C₅	—	2	2	no	—
117	C₂²×C₅	—	2	4	no	—
118	C₂²×C₅	—	4	2	no	—
119	C₂²×C₅	—	4	1	no	—
120	C₂²×C₅	—	4	1	no	—
121	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
122	C₂²×C₅	C₂×#41	1	2	yes	—
123	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
124	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
126	C₄×C₅	—	2	4	no	—
127	C₄×C₅	—	2	4	no	—
128	C₄×C₅	—	4	2	no	—
129	C₄×C₅	—	4	2	no	—
130	C₄×C₅	—	4	1	no	—
131	C₄×C₅	—	4	2	no	—
132	C₄×C₅	—	4	2	no	—
133	C₄×C₅	—	4	1	no	—
134	C₂⁴	C₂³×C₅	1	1	yes	characteristic
135	C₂×Q₈	—	4	2	no	—
136	C₂×Q₈	—	4	1	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	4	1	no	—
139	C₂²×C₄	C₂³×C₅	1	2	yes	—
140	C₂²×C₄	C₂³×C₅	1	2	yes	—
141	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₅	1	2	yes	—
143	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₅	1	2	yes	—
145	C₂²×C₄	C₂³×C₅	1	2	yes	—
146	C₂²×C₄	C₂³×C₅	1	2	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₂²⋊C₄	—	4	2	no	—
154	C₂²⋊C₄	—	4	2	no	—
155	C₂²⋊C₄	—	4	1	no	—
156	C₂²⋊C₄	C₅×Q₈	1	4	yes	complemented
157	C₄²	—	4	2	no	—
158	C₂×C₅	—	4	2	no	—
159	C₂×C₅	#542	1	2	yes	—
160	C₂×C₅	#544	1	1	yes	characteristic
161	C₂×C₅	#551	1	1	yes	characteristic
162	C₂×C₅	#554	1	2	yes	—
163	C₂×C₅	#560	1	1	yes	characteristic
164	C₂³	—	2	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	1	no	—
167	C₂³	—	2	1	no	—
168	C₂³	C₂×C₅×Q₈	1	2	yes	—
169	C₂³	C₂⁴×C₅	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₂×C₅×Q₈	1	4	yes	—
193	C₂×C₄	C₅×#41	1	4	yes	—
194	C₅	#2742	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	—	2	2	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	1	no	—
199	C₂²	—	4	1	no	—
200	C₂²	C₂²×C₅×Q₈	1	1	yes	characteristic
201	C₂²	C₂×C₅×#41	1	2	yes	—
202	C₂²	C₅×H(5,𝔽₂)	1	2	yes	—
203	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂²	C₅×2₋¹⁺⁴	1	1	yes	characteristic
205	C₄	—	2	4	no	—
206	C₄	—	2	4	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	1	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	1	no	—
212	C₄	—	4	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₅×#542	1	2	yes	—
215	C₂	C₅×#544	1	1	yes	characteristic
216	C₂	C₅×#551	1	1	yes	characteristic
217	C₂	C₅×#554	1	2	yes	—
218	C₂	C₅×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#400	C₂	semidirect
3. C₅×#395	C₂	semidirect
5. C₅×#389	C₂	semidirect
7. C₅×#387	C₂	semidirect
31. #2742	C₅	direct (N × H)
54. C₅×C₂²⋊C₄	Q₈	semidirect
56. #400	C₂×C₅	semidirect
58. #395	C₂×C₅	semidirect
60. #389	C₂×C₅	semidirect
62. #387	C₂×C₅	semidirect
156. C₂²⋊C₄	C₅×Q₈	semidirect
194. C₅	#2742	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×#400	2
C₅×#399	2
C₅×#395	2
C₅×#393	2
C₅×#389	2
C₅×#388	2
C₅×#387	2
C₅×#385	2
C₅×#380	2
#2742	5

Subgroups — C5×#2742 #10616337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2742 #10616337