Subgroups — C₅×#3037 #10616632

Order 640.

484 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	C₂×C₅×#137	C₂	1	2	yes	maximal, complemented
2	C₅×#532	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#464	C₂	1	4	yes	maximal, complemented
5	C₅×#462	C₂	1	2	yes	maximal, complemented
6	C₅×#460	C₂	1	2	yes	maximal, complemented
7	C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#417	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#41	C₂²	1	2	yes	—
10	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₅×#137	—	2	8	no	—
12	C₂×C₅×Q₁₆	C₂²	1	4	yes	—
13	C₂×C₅×SD₁₆	C₂²	1	4	yes	—
14	C₂×C₅×M₁₆	C₂²	1	2	yes	—
15	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₅×#127	C₂²	1	1	yes	characteristic
17	C₅×#124	C₂²	1	2	yes	—
18	C₅×#121	—	2	1	no	—
19	C₅×#121	—	2	1	no	—
20	C₄×C₅×Q₈	—	2	1	no	—
21	C₄×C₅×Q₈	—	2	1	no	—
22	C₄×C₅×Q₈	C₂²	1	2	yes	—
23	C₄×C₅×Q₈	C₂²	1	2	yes	—
24	C₄×C₅×D₄	—	2	1	no	—
25	C₄×C₅×D₄	—	2	1	no	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₅	—	2	1	no	—
29	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
30	C₅×C₄⋊C₈	C₂²	1	2	yes	—
31	C₅×C₄⋊C₈	C₂²	1	2	yes	—
32	C₅×#103	C₂²	1	4	yes	—
33	C₅×#102	C₂²	1	4	yes	—
34	#3037	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₅×#41	—	2	4	no	—
36	C₅×#41	—	2	4	no	—
37	C₂×C₅×Q₈	—	2	1	no	—
38	C₂×C₅×Q₈	—	2	2	no	—
39	C₂×C₅×Q₈	D₄	1	4	yes	complemented
40	C₂×C₅×Q₈	C₂³	1	2	yes	—
41	C₂×C₅×Q₈	C₂³	1	2	yes	—
42	C₂×C₅×D₄	—	2	2	no	—
43	C₂×C₅×D₄	—	2	1	no	—
44	C₂×C₅×D₄	—	2	1	no	—
45	C₂×C₅×D₄	C₂³	1	2	yes	—
46	C₂²×C₄×C₅	—	2	1	no	—
47	C₂²×C₄×C₅	—	2	1	no	—
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₅×Q₁₆	—	2	8	no	—
53	C₅×SD₁₆	—	2	8	no	—
54	C₅×M₁₆	—	2	4	no	—
55	C₂×C₅×C₈	C₂³	1	4	yes	—
56	C₅×C₄⋊C₄	—	2	2	no	—
57	C₅×C₄⋊C₄	—	2	2	no	—
58	C₅×C₄⋊C₄	—	2	1	no	—
59	C₅×C₄⋊C₄	—	2	1	no	—
60	C₅×C₄⋊C₄	—	2	1	no	—
61	C₅×C₄⋊C₄	C₂³	1	2	yes	—
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	C₄²×C₅	D₄	1	1	yes	characteristic
67	C₄²×C₅	D₄	1	1	yes	characteristic
68	C₄²×C₅	C₂³	1	1	yes	characteristic
69	C₄²×C₅	C₂³	1	1	yes	characteristic
70	C₂×#137	C₂×C₅	1	2	yes	complemented
71	#532	C₂×C₅	1	1	yes	characteristic
72	C₂×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
73	#464	C₂×C₅	1	4	yes	complemented
74	#462	C₂×C₅	1	2	yes	complemented
75	#460	C₂×C₅	1	2	yes	complemented
76	#423	C₂×C₅	1	1	yes	characteristic, complemented
77	#417	C₂×C₅	1	2	yes	complemented
78	C₂³×C₅	—	2	2	no	—
79	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
80	C₅×Q₈	—	2	4	no	—
81	C₅×Q₈	—	2	2	no	—
82	C₅×Q₈	—	2	2	no	—
83	C₅×Q₈	—	2	2	no	—
84	C₅×Q₈	C₂×D₄	1	4	yes	—
85	C₅×D₄	—	2	2	no	—
86	C₅×D₄	—	2	2	no	—
87	C₅×D₄	—	4	2	no	—
88	C₅×D₄	—	4	4	no	—
89	C₅×D₄	—	4	2	no	—
90	C₂×C₄×C₅	—	2	4	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	—	4	1	no	—
96	C₂×C₄×C₅	—	4	2	no	—
97	C₂×C₄×C₅	—	4	4	no	—
98	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅	#41	1	2	yes	—
105	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
106	C₅×C₈	—	2	4	no	—
107	C₂×#41	C₂²×C₅	1	2	yes	—
108	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
109	#137	—	2	8	no	—
110	C₂×Q₁₆	C₂²×C₅	1	4	yes	—
111	C₂×SD₁₆	C₂²×C₅	1	4	yes	—
112	C₂×M₁₆	C₂²×C₅	1	2	yes	—
113	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
114	#127	C₂²×C₅	1	1	yes	characteristic
115	#124	C₂²×C₅	1	2	yes	—
116	#121	—	2	1	no	—
117	#121	—	2	1	no	—
118	C₄×Q₈	—	2	1	no	—
119	C₄×Q₈	—	2	1	no	—
120	C₄×Q₈	C₂²×C₅	1	2	yes	—
121	C₄×Q₈	C₂²×C₅	1	2	yes	—
122	C₄×D₄	—	2	1	no	—
123	C₄×D₄	—	2	1	no	—
124	C₂×C₄⋊C₄	—	2	1	no	—
125	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
128	C₄⋊C₈	C₂²×C₅	1	2	yes	—
129	C₄⋊C₈	C₂²×C₅	1	2	yes	—
130	#103	C₂²×C₅	1	4	yes	—
131	#102	C₂²×C₅	1	4	yes	—
132	C₂²×C₅	—	2	1	no	—
133	C₂²×C₅	—	2	1	no	—
134	C₂²×C₅	—	4	2	no	—
135	C₂²×C₅	—	4	2	no	—
136	C₂²×C₅	—	4	2	no	—
137	C₂²×C₅	#121	1	2	yes	—
138	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₅	—	2	4	no	—
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	—	2	1	no	—
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	4	2	no	—
144	C₄×C₅	—	4	2	no	—
145	C₄×C₅	#121	1	2	yes	—
146	C₄×C₅	#137	1	2	yes	complemented
147	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₅	C₂×#41	1	1	yes	characteristic
149	#41	—	2	4	no	—
150	#41	—	2	4	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	C₅×D₄	1	4	yes	complemented
154	C₂×Q₈	C₂³×C₅	1	2	yes	—
155	C₂×Q₈	C₂³×C₅	1	2	yes	—
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	C₂³×C₅	1	2	yes	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₅×D₄	1	1	yes	characteristic
164	C₂²×C₄	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
166	Q₁₆	—	2	8	no	—
167	SD₁₆	—	2	8	no	—
168	M₁₆	—	2	4	no	—
169	C₂×C₈	C₂³×C₅	1	4	yes	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	C₂³×C₅	1	2	yes	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	C₅×D₄	1	1	yes	characteristic
181	C₄²	C₅×D₄	1	1	yes	characteristic
182	C₄²	C₂³×C₅	1	1	yes	characteristic
183	C₄²	C₂³×C₅	1	1	yes	characteristic
184	C₂×C₅	—	2	2	no	—
185	C₂×C₅	—	8	2	no	—
186	C₂×C₅	C₂×#121	1	1	yes	characteristic
187	C₂×C₅	C₂×#137	1	1	yes	characteristic
188	C₂×C₅	#575	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
191	Q₈	—	2	4	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	2	2	no	—
195	Q₈	C₂×C₅×D₄	1	4	yes	—
196	D₄	—	2	2	no	—
197	D₄	—	2	2	no	—
198	D₄	—	4	2	no	—
199	D₄	—	4	4	no	—
200	D₄	—	4	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
214	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₄	C₅×#41	1	2	yes	—
216	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
217	C₈	—	2	4	no	—
218	C₅	#3037	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₅×#121	1	2	yes	—
225	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	C₅×#121	1	2	yes	—
233	C₄	C₅×#137	1	2	yes	complemented
234	C₄	C₂²×C₅×D₄	1	1	yes	characteristic
235	C₄	C₂×C₅×#41	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	—	8	2	no	—
238	C₂	C₂×C₅×#121	1	1	yes	characteristic
239	C₂	C₂×C₅×#137	1	1	yes	characteristic
240	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	C₅×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#137	C₂	semidirect
3. C₂×C₄×C₅×Q₈	C₂	semidirect
4. C₅×#464	C₂	semidirect
5. C₅×#462	C₂	semidirect
6. C₅×#460	C₂	semidirect
7. C₅×#423	C₂	semidirect
8. C₅×#417	C₂	semidirect
34. #3037	C₅	direct (N × H)
39. C₂×C₅×Q₈	D₄	semidirect
70. C₂×#137	C₂×C₅	semidirect
72. C₂×C₄×Q₈	C₂×C₅	semidirect
73. #464	C₂×C₅	semidirect
74. #462	C₂×C₅	semidirect
75. #460	C₂×C₅	semidirect
76. #423	C₂×C₅	semidirect
77. #417	C₂×C₅	semidirect
146. C₄×C₅	#137	semidirect
153. C₂×Q₈	C₅×D₄	semidirect
218. C₅	#3037	direct (N × H)
233. C₄	C₅×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#137	2
C₅×#532	2
C₂×C₄×C₅×Q₈	2
C₅×#464	2
C₅×#462	2
C₅×#460	2
C₅×#423	2
C₅×#417	2
#3037	5

Subgroups — C5×#3037 #10616632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3037 #10616632