Subgroups — C₅×#3042 #10616637

Order 640.

494 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#536	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#463	C₂	1	4	yes	maximal, complemented
6	C₅×#461	C₂	1	4	yes	maximal, complemented
7	C₅×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#419	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#416	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#41	C₂²	1	1	yes	characteristic
11	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₅×AGL(1,ℤ/8)	—	2	4	no	—
14	C₂×C₅×SD₁₆	—	2	4	no	—
15	C₂×C₅×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₅×SD₁₆	C₂²	1	4	yes	—
17	C₂×C₅×D₈	C₂²	1	2	yes	—
18	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
21	C₅×#124	C₂²	1	2	yes	complemented
22	C₅×#123	—	2	2	no	—
23	C₅×C₂²⋊Q₈	—	2	2	no	—
24	C₄×C₅×D₄	—	2	2	no	—
25	C₄×C₅×D₄	C₂²	1	4	yes	—
26	C₅×#117	C₂²	1	1	yes	characteristic
27	C₅×#117	C₂²	1	1	yes	characteristic
28	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
29	C₅×C₄⋊C₈	C₂²	1	4	yes	complemented
30	C₅×#103	C₂²	1	2	yes	complemented
31	C₅×#103	C₂²	1	4	yes	complemented
32	C₅×#102	C₂²	1	2	yes	complemented
33	#3042	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂⁴×C₅	—	2	1	no	—
35	C₅×#41	—	2	2	no	—
36	C₅×#41	—	2	2	no	—
37	C₂×C₅×Q₈	—	2	1	no	—
38	C₂×C₅×Q₈	—	2	2	no	—
39	C₂×C₅×Q₈	—	2	2	no	—
40	C₂×C₅×Q₈	C₂³	1	2	yes	—
41	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
42	C₂×C₅×D₄	—	2	1	no	—
43	C₂×C₅×D₄	—	2	1	no	—
44	C₂×C₅×D₄	—	2	2	no	—
45	C₂×C₅×D₄	D₄	1	4	yes	—
46	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
47	C₂×C₅×D₄	C₂³	1	2	yes	—
48	C₂²×C₄×C₅	—	2	1	no	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	—	2	4	no	—
51	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₅×SD₁₆	—	2	8	no	—
53	C₅×SD₁₆	—	2	4	no	—
54	C₅×D₈	—	2	4	no	—
55	C₅×M₁₆	—	2	2	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	C₂³	1	2	yes	—
58	C₂×C₅×C₈	C₂³	1	2	yes	—
59	C₅×C₄⋊C₄	—	2	4	no	—
60	C₅×C₄⋊C₄	D₄	1	2	yes	—
61	C₅×C₄⋊C₄	C₂³	1	2	yes	—
62	C₅×C₂²⋊C₄	—	2	1	no	—
63	C₅×C₂²⋊C₄	—	2	4	no	—
64	C₅×C₂²⋊C₄	—	2	4	no	—
65	C₅×C₂²⋊C₄	D₄	1	2	yes	—
66	C₄²×C₅	—	2	1	no	—
67	C₄²×C₅	C₂³	1	2	yes	—
68	C₂×AGL(1,ℤ/8)	C₂×C₅	1	1	yes	characteristic
69	C₂²×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
70	#536	C₂×C₅	1	1	yes	characteristic, complemented
71	#518	C₂×C₅	1	1	yes	characteristic, complemented
72	#463	C₂×C₅	1	4	yes	complemented
73	#461	C₂×C₅	1	4	yes	complemented
74	#424	C₂×C₅	1	1	yes	characteristic, complemented
75	#419	C₂×C₅	1	1	yes	characteristic, complemented
76	#416	C₂×C₅	1	1	yes	characteristic, complemented
77	C₂³×C₅	—	2	2	no	—
78	C₂³×C₅	—	2	4	no	—
79	C₂³×C₅	—	2	1	no	—
80	C₂³×C₅	—	8	2	no	—
81	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
82	C₅×Q₈	—	2	2	no	—
83	C₅×Q₈	—	2	1	no	—
84	C₅×Q₈	—	2	2	no	—
85	C₅×Q₈	—	2	1	no	—
86	C₅×Q₈	—	4	2	no	—
87	C₅×D₄	—	2	4	no	—
88	C₅×D₄	—	2	1	no	—
89	C₅×D₄	—	2	2	no	—
90	C₅×D₄	—	2	1	no	—
91	C₅×D₄	—	4	2	no	—
92	C₅×D₄	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	4	2	no	—
97	C₂×C₄×C₅	—	4	2	no	—
98	C₂×C₄×C₅	—	8	4	no	—
99	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
100	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₅	#41	1	2	yes	—
103	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
104	C₅×C₈	—	2	2	no	—
105	C₅×C₈	—	2	2	no	—
106	C₂×#41	C₂²×C₅	1	1	yes	characteristic
107	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
108	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
109	AGL(1,ℤ/8)	—	2	4	no	—
110	C₂×SD₁₆	—	2	4	no	—
111	C₂×SD₁₆	C₂²×C₅	1	4	yes	—
112	C₂×SD₁₆	C₂²×C₅	1	2	yes	—
113	C₂×D₈	C₂²×C₅	1	2	yes	—
114	C₂×M₁₆	C₂²×C₅	1	1	yes	characteristic
115	C₂²×C₈	C₂²×C₅	1	1	yes	characteristic
116	C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
117	#124	C₂²×C₅	1	2	yes	complemented
118	#123	—	2	2	no	—
119	C₂²⋊Q₈	—	2	2	no	—
120	C₄×D₄	—	2	2	no	—
121	C₄×D₄	C₂²×C₅	1	4	yes	—
122	#117	C₂²×C₅	1	1	yes	characteristic
123	#117	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₂²⋊C₄	—	2	2	no	—
125	C₄⋊C₈	C₂²×C₅	1	4	yes	complemented
126	#103	C₂²×C₅	1	2	yes	complemented
127	#103	C₂²×C₅	1	4	yes	complemented
128	#102	C₂²×C₅	1	2	yes	complemented
129	C₂²×C₅	—	2	2	no	—
130	C₂²×C₅	—	2	4	no	—
131	C₂²×C₅	—	4	4	no	—
132	C₂²×C₅	—	4	1	no	—
133	C₂²×C₅	—	4	2	no	—
134	C₂²×C₅	—	8	4	no	—
135	C₂²×C₅	#121	1	2	yes	—
136	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₅	—	2	4	no	—
138	C₄×C₅	—	4	1	no	—
139	C₄×C₅	—	4	2	no	—
140	C₄×C₅	—	4	2	no	—
141	C₄×C₅	#121	1	2	yes	—
142	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₅	C₂×#41	1	1	yes	characteristic
144	C₂⁴	—	2	1	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	C₂³×C₅	1	2	yes	—
151	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	C₅×D₄	1	4	yes	—
156	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
157	C₂×D₄	C₂³×C₅	1	2	yes	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
162	SD₁₆	—	2	8	no	—
163	SD₁₆	—	2	4	no	—
164	D₈	—	2	4	no	—
165	M₁₆	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₂³×C₅	1	2	yes	—
168	C₂×C₈	C₂³×C₅	1	2	yes	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	C₅×D₄	1	2	yes	—
171	C₄⋊C₄	C₂³×C₅	1	2	yes	—
172	C₂²⋊C₄	—	2	1	no	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	C₅×D₄	1	2	yes	—
176	C₄²	—	2	1	no	—
177	C₄²	C₂³×C₅	1	2	yes	—
178	C₂×C₅	—	2	2	no	—
179	C₂×C₅	—	4	4	no	—
180	C₂×C₅	—	8	1	no	—
181	C₂×C₅	C₂×#121	1	1	yes	characteristic
182	C₂×C₅	#577	1	2	yes	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	4	no	—
185	C₂³	—	2	1	no	—
186	C₂³	—	8	2	no	—
187	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
188	Q₈	—	2	2	no	—
189	Q₈	—	2	1	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	2	1	no	—
192	Q₈	—	4	2	no	—
193	D₄	—	2	4	no	—
194	D₄	—	2	1	no	—
195	D₄	—	2	2	no	—
196	D₄	—	2	1	no	—
197	D₄	—	4	2	no	—
198	D₄	C₂×C₅×D₄	1	4	yes	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	C₂×C₅×D₄	1	4	yes	—
206	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
207	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₄	C₅×#41	1	2	yes	—
209	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₅	#3042	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	4	no	—
216	C₂²	—	4	1	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	8	4	no	—
219	C₂²	C₅×#121	1	2	yes	—
220	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	C₅×#121	1	2	yes	—
226	C₄	C₂²×C₅×D₄	1	1	yes	characteristic
227	C₄	C₂×C₅×#41	1	1	yes	characteristic
228	C₂	—	2	2	no	—
229	C₂	—	4	4	no	—
230	C₂	—	8	1	no	—
231	C₂	C₂×C₅×#121	1	1	yes	characteristic
232	C₂	C₅×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂×C₅×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×SD₁₆	C₂	semidirect
3. C₅×#536	C₂	semidirect
4. C₅×#518	C₂	semidirect
5. C₅×#463	C₂	semidirect
6. C₅×#461	C₂	semidirect
7. C₅×#424	C₂	semidirect
8. C₅×#419	C₂	semidirect
9. C₅×#416	C₂	semidirect
20. C₅×C₄⋊Q₈	C₂²	semidirect
21. C₅×#124	C₂²	semidirect
29. C₅×C₄⋊C₈	C₂²	semidirect
30. C₅×#103	C₂²	semidirect
31. C₅×#103	C₂²	semidirect
32. C₅×#102	C₂²	semidirect
33. #3042	C₅	direct (N × H)
69. C₂²×SD₁₆	C₂×C₅	semidirect
70. #536	C₂×C₅	semidirect
71. #518	C₂×C₅	semidirect
72. #463	C₂×C₅	semidirect
73. #461	C₂×C₅	semidirect
74. #424	C₂×C₅	semidirect
75. #419	C₂×C₅	semidirect
76. #416	C₂×C₅	semidirect
116. C₄⋊Q₈	C₂²×C₅	semidirect
117. #124	C₂²×C₅	semidirect
125. C₄⋊C₈	C₂²×C₅	semidirect
126. #103	C₂²×C₅	semidirect
127. #103	C₂²×C₅	semidirect
128. #102	C₂²×C₅	semidirect
212. C₅	#3042	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×AGL(1,ℤ/8)	2
C₂²×C₅×SD₁₆	2
C₅×#536	2
C₅×#518	2
C₅×#463	2
C₅×#461	2
C₅×#424	2
C₅×#419	2
C₅×#416	2
#3042	5

Subgroups — C5×#3042 #10616637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3042 #10616637