Subgroups — C₅×#13245 #11935965

Order 1280.

442 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#361	C₂	1	2	yes	maximal, complemented
4	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1978	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×D₈	C₂²	1	2	yes	—
7	C₂×C₅×D₁₆	—	2	1	no	—
8	C₂×C₅×D₁₆	—	2	1	no	—
9	C₂×C₅×D₁₆	—	2	2	no	—
10	C₂×C₅×D₁₆	C₄	1	4	yes	complemented
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₅×#108	C₂²	1	2	yes	—
14	C₅×#418	—	2	2	no	—
15	C₂×C₅×#100	—	2	2	no	—
16	C₅×#405	C₂²	1	1	yes	characteristic
17	C₅×#368	—	2	2	no	—
18	C₅×#361	—	2	4	no	—
19	C₅×#346	—	2	1	no	—
20	C₄×C₅×C₁₆	—	2	1	no	—
21	#13245	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₅×D₄	—	2	2	no	—
23	C₂×C₅×D₈	—	2	2	no	—
24	C₂×C₅×D₈	—	2	2	no	—
25	C₂×C₅×D₈	—	2	2	no	—
26	C₂×C₅×D₈	—	2	2	no	—
27	C₂×C₅×D₈	—	2	2	no	—
28	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₅×#117	D₄	1	1	yes	characteristic
33	C₅×D₁₆	—	2	4	no	—
34	C₅×D₁₆	—	2	4	no	—
35	C₅×M₃₂	—	2	1	no	—
36	C₅×M₃₂	D₄	1	2	yes	—
37	C₂×C₅×C₁₆	—	2	1	no	—
38	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₅×C₁₆	D₄	1	2	yes	—
42	C₅×#108	—	2	2	no	—
43	C₅×#108	—	2	2	no	—
44	C₅×#102	—	4	4	no	—
45	C₅×#100	—	4	4	no	—
46	C₅×#97	—	2	1	no	—
47	C₄×C₅×C₈	—	2	1	no	—
48	C₂²×D₁₆	C₂×C₅	1	1	yes	characteristic
49	C₂×#368	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×#361	C₂×C₅	1	2	yes	complemented
51	#2108	C₂×C₅	1	1	yes	characteristic, complemented
52	#1978	C₂×C₅	1	2	yes	complemented
53	C₂⁴×C₅	—	4	2	no	—
54	C₂×C₅×D₄	—	2	4	no	—
55	C₂×C₅×D₄	—	4	2	no	—
56	C₂×C₅×D₄	—	4	2	no	—
57	C₂×C₅×D₄	—	4	2	no	—
58	C₂×C₅×D₄	—	4	2	no	—
59	C₂×C₅×D₄	—	4	2	no	—
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₅×D₈	—	2	4	no	—
62	C₅×D₈	—	2	4	no	—
63	C₅×D₈	—	4	2	no	—
64	C₅×D₈	—	4	2	no	—
65	C₅×M₁₆	—	2	1	no	—
66	C₅×M₁₆	—	4	2	no	—
67	C₅×M₁₆	—	4	2	no	—
68	C₅×M₁₆	D₈	1	2	yes	—
69	C₂×C₅×C₈	—	2	1	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂×D₄	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	#41	1	1	yes	characteristic
76	C₂×C₅×C₈	#41	1	1	yes	characteristic
77	C₅×C₄⋊C₄	D₈	1	2	yes	—
78	C₅×C₂²⋊C₄	—	2	1	no	—
79	C₄²×C₅	—	2	1	no	—
80	C₅×C₁₆	—	2	2	no	—
81	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
82	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
83	C₂²×D₈	C₂²×C₅	1	2	yes	—
84	C₂×D₁₆	—	2	1	no	—
85	C₂×D₁₆	—	2	1	no	—
86	C₂×D₁₆	—	2	2	no	—
87	C₂×D₁₆	C₄×C₅	1	4	yes	complemented
88	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
90	C₂×#108	C₂²×C₅	1	2	yes	—
91	#418	—	2	2	no	—
92	C₂×#100	—	2	2	no	—
93	#405	C₂²×C₅	1	1	yes	characteristic
94	#368	—	2	2	no	—
95	#361	—	2	4	no	—
96	#346	—	2	1	no	—
97	C₄×C₁₆	—	2	1	no	—
98	C₂³×C₅	—	4	4	no	—
99	C₂³×C₅	—	8	2	no	—
100	C₂³×C₅	—	8	2	no	—
101	C₂³×C₅	—	16	4	no	—
102	C₂³×C₅	#135	1	1	yes	characteristic
103	C₅×D₄	—	4	4	no	—
104	C₅×D₄	—	4	4	no	—
105	C₅×D₄	—	8	2	no	—
106	C₅×D₄	—	8	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅	#121	1	1	yes	characteristic
110	C₂×C₄×C₅	#127	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₅	#135	1	1	yes	characteristic
114	C₅×C₈	—	2	2	no	—
115	C₅×C₈	—	8	2	no	—
116	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₅×C₈	C₄×D₄	1	1	yes	characteristic
118	C₅×C₈	#121	1	1	yes	characteristic
119	C₅×C₈	#124	1	1	yes	characteristic
120	C₂²×D₄	—	2	2	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×D₈	—	2	2	no	—
124	C₂×D₈	—	2	2	no	—
125	C₂×D₈	—	2	2	no	—
126	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
130	#117	C₅×D₄	1	1	yes	characteristic
131	D₁₆	—	2	4	no	—
132	D₁₆	—	2	4	no	—
133	M₃₂	—	2	1	no	—
134	M₃₂	C₅×D₄	1	2	yes	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
137	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
138	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
139	C₂×C₁₆	C₅×D₄	1	2	yes	—
140	#108	—	2	2	no	—
141	#108	—	2	2	no	—
142	#102	—	4	4	no	—
143	#100	—	4	4	no	—
144	#97	—	2	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	—	8	4	no	—
148	C₂²×C₅	—	16	4	no	—
149	C₂²×C₅	—	16	4	no	—
150	C₂²×C₅	—	16	4	no	—
151	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
152	C₂²×C₅	#466	1	1	yes	characteristic
153	C₂²×C₅	#495	1	1	yes	characteristic
154	C₄×C₅	—	4	2	no	—
155	C₄×C₅	#390	1	1	yes	characteristic
156	C₄×C₅	C₄×D₈	1	1	yes	characteristic
157	C₄×C₅	#466	1	1	yes	characteristic
158	C₄×C₅	#493	1	1	yes	characteristic
159	C₂⁴	—	4	2	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
167	D₈	—	2	4	no	—
168	D₈	—	2	4	no	—
169	D₈	—	4	2	no	—
170	D₈	—	4	2	no	—
171	M₁₆	—	2	1	no	—
172	M₁₆	—	4	2	no	—
173	M₁₆	—	4	2	no	—
174	M₁₆	C₅×D₈	1	2	yes	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
178	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
179	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
180	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
181	C₂×C₈	C₅×#41	1	1	yes	characteristic
182	C₂×C₈	C₅×#41	1	1	yes	characteristic
183	C₄⋊C₄	C₅×D₈	1	2	yes	—
184	C₂²⋊C₄	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₁₆	—	2	2	no	—
187	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
188	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
189	C₂×C₅	—	2	1	no	—
190	C₂×C₅	—	2	1	no	—
191	C₂×C₅	—	16	4	no	—
192	C₂×C₅	#1970	1	1	yes	characteristic
193	C₂×C₅	#2221	1	2	yes	—
194	C₂³	—	4	4	no	—
195	C₂³	—	8	2	no	—
196	C₂³	—	8	2	no	—
197	C₂³	—	16	4	no	—
198	C₂³	C₅×#135	1	1	yes	characteristic
199	D₄	—	4	4	no	—
200	D₄	—	4	4	no	—
201	D₄	—	8	2	no	—
202	D₄	—	8	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
205	C₂×C₄	C₅×#121	1	1	yes	characteristic
206	C₂×C₄	C₅×#127	1	1	yes	characteristic
207	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
208	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
209	C₂×C₄	C₅×#135	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	8	2	no	—
212	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₈	C₄×C₅×D₄	1	1	yes	characteristic
214	C₈	C₅×#121	1	1	yes	characteristic
215	C₈	C₅×#124	1	1	yes	characteristic
216	C₅	#13245	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	2	2	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	16	4	no	—
220	C₂²	—	16	4	no	—
221	C₂²	—	16	4	no	—
222	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
223	C₂²	C₅×#466	1	1	yes	characteristic
224	C₂²	C₅×#495	1	1	yes	characteristic
225	C₄	—	4	2	no	—
226	C₄	C₅×#390	1	1	yes	characteristic
227	C₄	C₄×C₅×D₈	1	1	yes	characteristic
228	C₄	C₅×#466	1	1	yes	characteristic
229	C₄	C₅×#493	1	1	yes	characteristic
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	—	16	4	no	—
233	C₂	C₅×#1970	1	1	yes	characteristic
234	C₂	C₅×#2221	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	C₂²×C₅×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ 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×#361	C₂	semidirect
4. C₅×#2108	C₂	semidirect
5. C₅×#1978	C₂	semidirect
10. C₂×C₅×D₁₆	C₄	semidirect
21. #13245	C₅	direct (N × H)
49. C₂×#368	C₂×C₅	semidirect
50. C₂×#361	C₂×C₅	semidirect
51. #2108	C₂×C₅	semidirect
52. #1978	C₂×C₅	semidirect
87. C₂×D₁₆	C₄×C₅	semidirect
216. C₅	#13245	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₁₆	2
C₂×C₅×#368	2
C₂×C₅×#361	2
C₅×#2108	2
C₅×#1978	2
#13245	5

Subgroups — C5×#13245 #11935965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13245 #11935965