Subgroups — C₅×#12843 #11935563

Order 1280.

366 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1816	C₂	1	2	yes	maximal
5	C₅×#1387	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₂²	1	1	yes	characteristic
7	C₅×#504	—	2	2	no	—
8	C₅×#504	C₄	1	4	yes	complemented
9	C₂×C₅×M₃₂	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₁₆	—	2	1	no	—
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₅×#108	C₂²	1	2	yes	—
14	C₂×C₅×#107	C₂²	1	2	yes	—
15	C₅×#417	—	2	2	no	—
16	C₅×#368	—	2	2	no	—
17	C₅×#367	—	2	1	no	—
18	C₅×#367	—	2	1	no	—
19	C₅×#343	—	2	2	no	—
20	#12843	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×#41	D₄	1	1	yes	characteristic
22	C₅×#131	—	2	1	no	—
23	C₅×#131	—	2	1	no	—
24	C₅×#131	C₂×C₄	1	2	yes	—
25	C₅×#131	Q₈	1	2	yes	—
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₅×M₃₂	—	2	1	no	—
33	C₅×M₃₂	—	2	4	no	—
34	C₅×M₃₂	C₂×C₄	1	2	yes	—
35	C₂×C₅×C₁₆	—	2	1	no	—
36	C₂×C₅×C₁₆	—	2	4	no	—
37	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
39	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₅×C₁₆	D₄	1	2	yes	—
41	C₅×#108	—	2	2	no	—
42	C₅×#108	—	2	2	no	—
43	C₅×#107	—	2	2	no	—
44	C₅×#107	—	2	2	no	—
45	C₅×#103	—	4	2	no	—
46	C₅×#102	—	4	2	no	—
47	C₂×#504	C₂×C₅	1	1	yes	characteristic
48	C₂×#368	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂×#367	C₂×C₅	1	1	yes	characteristic, complemented
50	#1816	C₂×C₅	1	2	yes	—
51	#1387	C₂×C₅	1	2	yes	complemented
52	C₅×#41	—	2	1	no	—
53	C₅×#41	—	2	1	no	—
54	C₅×#41	C₄⋊C₄	1	2	yes	—
55	C₅×#41	Q₁₆	1	2	yes	—
56	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
57	C₂×C₅×D₄	—	2	1	no	—
58	C₂×C₅×D₄	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	4	2	no	—
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₅×M₁₆	—	2	1	no	—
63	C₅×M₁₆	—	2	2	no	—
64	C₅×M₁₆	—	2	2	no	—
65	C₅×M₁₆	—	4	2	no	—
66	C₅×M₁₆	—	4	2	no	—
67	C₅×M₁₆	C₄⋊C₄	1	2	yes	—
68	C₂×C₅×C₈	—	2	1	no	—
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₅×C₈	#41	1	1	yes	characteristic
77	C₂×C₅×C₈	#41	1	1	yes	characteristic
78	C₅×C₄⋊C₄	—	4	2	no	—
79	C₅×C₄⋊C₄	—	4	2	no	—
80	C₅×C₁₆	—	2	2	no	—
81	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
82	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
83	C₂×#131	C₂²×C₅	1	1	yes	characteristic
84	#504	—	2	2	no	—
85	#504	C₄×C₅	1	4	yes	complemented
86	C₂×M₃₂	—	2	1	no	—
87	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
88	C₂²×C₁₆	—	2	1	no	—
89	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
90	C₂×#108	C₂²×C₅	1	2	yes	—
91	C₂×#107	C₂²×C₅	1	2	yes	—
92	#417	—	2	2	no	—
93	#368	—	2	2	no	—
94	#367	—	2	1	no	—
95	#367	—	2	1	no	—
96	#343	—	2	2	no	—
97	C₂³×C₅	—	2	1	no	—
98	C₂³×C₅	#135	1	1	yes	characteristic
99	C₅×Q₈	—	2	1	no	—
100	C₅×Q₈	#107	1	2	yes	—
101	C₅×D₄	—	2	2	no	—
102	C₅×D₄	—	2	2	no	—
103	C₅×D₄	—	2	1	no	—
104	C₅×D₄	#107	1	2	yes	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	8	2	no	—
109	C₂×C₄×C₅	—	16	2	no	—
110	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	#121	1	1	yes	characteristic
112	C₂×C₄×C₅	#121	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅	#135	1	1	yes	characteristic
116	C₅×C₈	—	2	2	no	—
117	C₅×C₈	—	8	2	no	—
118	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₅×C₈	#117	1	1	yes	characteristic
120	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
121	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×#41	C₅×D₄	1	1	yes	characteristic
123	#131	—	2	1	no	—
124	#131	—	2	1	no	—
125	#131	C₂×C₄×C₅	1	2	yes	—
126	#131	C₅×Q₈	1	2	yes	—
127	C₂×M₁₆	—	2	1	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
130	C₂²×C₈	—	2	1	no	—
131	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
132	C₂×C₄⋊C₄	—	2	2	no	—
133	M₃₂	—	2	1	no	—
134	M₃₂	—	2	4	no	—
135	M₃₂	C₂×C₄×C₅	1	2	yes	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	2	4	no	—
138	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
139	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
140	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
141	C₂×C₁₆	C₅×D₄	1	2	yes	—
142	#108	—	2	2	no	—
143	#108	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	—	2	2	no	—
146	#103	—	4	2	no	—
147	#102	—	4	2	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	—	4	2	no	—
151	C₂²×C₅	#427	1	1	yes	characteristic
152	C₂²×C₅	#466	1	1	yes	characteristic
153	C₂²×C₅	#467	1	1	yes	characteristic
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	16	2	no	—
156	C₄×C₅	#380	1	1	yes	characteristic
157	C₄×C₅	C₂×#107	1	1	yes	characteristic
158	C₄×C₅	#466	1	1	yes	characteristic
159	C₄×C₅	#467	1	1	yes	characteristic
160	#41	—	2	1	no	—
161	#41	—	2	1	no	—
162	#41	C₅×C₄⋊C₄	1	2	yes	—
163	#41	C₅×Q₁₆	1	2	yes	—
164	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	C₅×D₈	1	1	yes	characteristic
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
170	M₁₆	—	2	1	no	—
171	M₁₆	—	2	2	no	—
172	M₁₆	—	2	2	no	—
173	M₁₆	—	4	2	no	—
174	M₁₆	—	4	2	no	—
175	M₁₆	C₅×C₄⋊C₄	1	2	yes	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
181	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
182	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
183	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
184	C₂×C₈	C₅×#41	1	1	yes	characteristic
185	C₂×C₈	C₅×#41	1	1	yes	characteristic
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₁₆	—	2	2	no	—
189	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
190	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
191	C₂×C₅	—	2	1	no	—
192	C₂×C₅	—	2	1	no	—
193	C₂×C₅	—	4	2	no	—
194	C₂×C₅	#1808	1	1	yes	characteristic
195	C₂×C₅	#2223	1	2	yes	—
196	C₂³	—	2	1	no	—
197	C₂³	C₅×#135	1	1	yes	characteristic
198	Q₈	—	2	1	no	—
199	Q₈	C₅×#107	1	2	yes	—
200	D₄	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	1	no	—
203	D₄	C₅×#107	1	2	yes	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	—	16	2	no	—
209	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₄	C₅×#121	1	1	yes	characteristic
211	C₂×C₄	C₅×#121	1	1	yes	characteristic
212	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
213	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
214	C₂×C₄	C₅×#135	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	8	2	no	—
217	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
218	C₈	C₅×#117	1	1	yes	characteristic
219	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
220	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
221	C₅	#12843	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	C₅×#427	1	1	yes	characteristic
226	C₂²	C₅×#466	1	1	yes	characteristic
227	C₂²	C₅×#467	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	16	2	no	—
230	C₄	C₅×#380	1	1	yes	characteristic
231	C₄	C₂×C₅×#107	1	1	yes	characteristic
232	C₄	C₅×#466	1	1	yes	characteristic
233	C₄	C₅×#467	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	4	2	no	—
237	C₂	C₅×#1808	1	1	yes	characteristic
238	C₂	C₅×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#368	C₂	semidirect
3. C₂×C₅×#367	C₂	semidirect
5. C₅×#1387	C₂	semidirect
8. C₅×#504	C₄	semidirect
20. #12843	C₅	direct (N × H)
48. C₂×#368	C₂×C₅	semidirect
49. C₂×#367	C₂×C₅	semidirect
51. #1387	C₂×C₅	semidirect
85. #504	C₄×C₅	semidirect
221. C₅	#12843	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×#504	2
C₂×C₅×#368	2
C₂×C₅×#367	2
C₅×#1816	2
C₅×#1387	2
#12843	5

Subgroups — C5×#12843 #11935563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12843 #11935563