Subgroups — C₅×#12930 #11935650

Order 1280.

414 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#365	C₂	1	2	yes	maximal, complemented
3	C₅×#2112	C₂	1	1	yes	maximal, characteristic
4	C₅×#1810	C₂	1	1	yes	maximal, characteristic
5	C₅×#1383	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	—	2	1	no	—
8	C₂×C₅×M₃₂	C₂²	1	2	yes	—
9	C₂×C₅×#108	C₂²	1	1	yes	characteristic
10	C₂×C₅×#108	C₂²	1	1	yes	characteristic
11	C₅×#427	—	2	4	no	—
12	C₂×C₅×#107	—	2	1	no	—
13	C₂×C₅×#106	C₂²	1	2	yes	—
14	C₅×#407	—	2	1	no	—
15	C₅×#365	—	2	4	no	—
16	C₅×C₂²⋊C₁₆	C₄	1	4	yes	complemented
17	C₅×#341	—	2	1	no	—
18	#12930	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂×C₅×M₁₆	—	2	1	no	—
21	C₂×C₅×M₁₆	—	2	1	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	Q₈	1	2	yes	—
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₅×#117	—	2	4	no	—
29	C₅×#117	—	4	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₅	—	4	1	no	—
33	C₅×M₃₂	—	2	2	no	—
34	C₅×M₃₂	—	2	2	no	—
35	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
36	C₅×#108	—	2	2	no	—
37	C₅×#108	—	2	2	no	—
38	C₅×#107	—	2	4	no	—
39	C₅×#106	—	2	2	no	—
40	C₅×#106	D₄	1	4	yes	—
41	C₅×C₂²⋊C₈	—	4	2	no	—
42	C₂×#427	C₂×C₅	1	1	yes	characteristic
43	C₂×#365	C₂×C₅	1	2	yes	complemented
44	#2112	C₂×C₅	1	1	yes	characteristic
45	#1810	C₂×C₅	1	1	yes	characteristic
46	#1383	C₂×C₅	1	2	yes	complemented
47	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	1	no	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	—	2	1	no	—
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	—	4	2	no	—
53	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₅×M₁₆	—	4	1	no	—
58	C₅×M₁₆	—	4	1	no	—
59	C₅×M₁₆	—	4	1	no	—
60	C₅×M₁₆	—	4	1	no	—
61	C₂×C₅×C₈	—	2	1	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	—	2	4	no	—
65	C₂×C₅×C₈	—	4	1	no	—
66	C₂×C₅×C₈	—	4	1	no	—
67	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
70	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₅×C₈	#41	1	2	yes	—
72	C₅×C₄⋊C₄	—	2	4	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₂²⋊C₄	—	4	4	no	—
75	C₄²×C₅	—	4	4	no	—
76	C₅×C₁₆	—	2	4	no	—
77	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
78	C₂×#117	—	2	1	no	—
79	C₂×M₃₂	C₂²×C₅	1	2	yes	—
80	C₂×#108	C₂²×C₅	1	1	yes	characteristic
81	C₂×#108	C₂²×C₅	1	1	yes	characteristic
82	#427	—	2	4	no	—
83	C₂×#107	—	2	1	no	—
84	C₂×#106	C₂²×C₅	1	2	yes	—
85	#407	—	2	1	no	—
86	#365	—	2	4	no	—
87	C₂²⋊C₁₆	C₄×C₅	1	4	yes	complemented
88	#341	—	2	1	no	—
89	C₂³×C₅	—	2	1	no	—
90	C₂³×C₅	—	2	1	no	—
91	C₂³×C₅	—	4	2	no	—
92	C₂³×C₅	#106	1	2	yes	—
93	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	4	4	no	—
101	C₂×C₄×C₅	—	8	4	no	—
102	C₂×C₄×C₅	#106	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	#120	1	1	yes	characteristic
105	C₂×C₄×C₅	#123	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₅	#137	1	1	yes	characteristic
109	C₅×C₈	—	2	2	no	—
110	C₅×C₈	—	8	1	no	—
111	C₅×C₈	—	8	1	no	—
112	C₅×C₈	C₄×D₄	1	2	yes	—
113	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
114	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
115	C₂×M₁₆	—	2	1	no	—
116	C₂×M₁₆	—	2	1	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
120	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
121	C₂²×C₈	C₅×Q₈	1	2	yes	—
122	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
123	#117	—	2	4	no	—
124	#117	—	4	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₂²⋊C₄	—	4	1	no	—
127	C₂×C₄²	—	4	1	no	—
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
131	#108	—	2	2	no	—
132	#108	—	2	2	no	—
133	#107	—	2	4	no	—
134	#106	—	2	2	no	—
135	#106	C₅×D₄	1	4	yes	—
136	C₂²⋊C₈	—	4	2	no	—
137	C₂²×C₅	—	2	2	no	—
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	—	4	2	no	—
141	C₂²×C₅	—	4	2	no	—
142	C₂²×C₅	C₂×#106	1	1	yes	characteristic
143	C₂²×C₅	#450	1	1	yes	characteristic
144	C₂²×C₅	#483	1	1	yes	characteristic
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	8	4	no	—
147	C₄×C₅	#385	1	1	yes	characteristic
148	C₄×C₅	#428	1	1	yes	characteristic
149	C₄×C₅	#448	1	1	yes	characteristic
150	C₄×C₅	#481	1	1	yes	characteristic
151	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
158	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
159	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
161	M₁₆	—	4	1	no	—
162	M₁₆	—	4	1	no	—
163	M₁₆	—	4	1	no	—
164	M₁₆	—	4	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
172	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
173	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
174	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
175	C₂×C₈	C₅×#41	1	2	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₂²⋊C₄	—	4	4	no	—
179	C₄²	—	4	4	no	—
180	C₁₆	—	2	4	no	—
181	C₂×C₅	—	2	1	no	—
182	C₂×C₅	—	2	1	no	—
183	C₂×C₅	—	4	2	no	—
184	C₂×C₅	#1853	1	1	yes	characteristic
185	C₂×C₅	#2155	1	2	yes	—
186	C₂³	—	2	1	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	4	2	no	—
189	C₂³	C₅×#106	1	2	yes	—
190	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	8	4	no	—
199	C₂×C₄	C₅×#106	1	2	yes	—
200	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
201	C₂×C₄	C₅×#120	1	1	yes	characteristic
202	C₂×C₄	C₅×#123	1	1	yes	characteristic
203	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
204	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂×C₄	C₅×#137	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	8	1	no	—
208	C₈	—	8	1	no	—
209	C₈	C₄×C₅×D₄	1	2	yes	—
210	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
211	C₅	#12930	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₂×C₅×#106	1	1	yes	characteristic
218	C₂²	C₅×#450	1	1	yes	characteristic
219	C₂²	C₅×#483	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	8	4	no	—
222	C₄	C₅×#385	1	1	yes	characteristic
223	C₄	C₅×#428	1	1	yes	characteristic
224	C₄	C₅×#448	1	1	yes	characteristic
225	C₄	C₅×#481	1	1	yes	characteristic
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	4	2	no	—
229	C₂	C₅×#1853	1	1	yes	characteristic
230	C₂	C₅×#2155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#365	C₂	semidirect
5. C₅×#1383	C₂	semidirect
16. C₅×C₂²⋊C₁₆	C₄	semidirect
18. #12930	C₅	direct (N × H)
43. C₂×#365	C₂×C₅	semidirect
46. #1383	C₂×C₅	semidirect
87. C₂²⋊C₁₆	C₄×C₅	semidirect
211. C₅	#12930	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×#427	2
C₂×C₅×#365	2
C₅×#2112	2
C₅×#1810	2
C₅×#1383	2
#12930	5

Subgroups — C5×#12930 #11935650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12930 #11935650