Subgroups — C₅×#13240 #11935960

Order 1280.

430 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#358	C₂	1	2	yes	maximal
4	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
5	C₅×#1969	C₂	1	2	yes	maximal
6	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₅×Q₃₂	—	2	2	no	—
8	C₂×C₅×Q₃₂	—	2	2	no	—
9	C₂×C₅×Q₃₂	C₄	1	4	yes	complemented
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#103	—	2	2	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#391	—	2	2	no	—
16	C₅×#367	—	2	2	no	—
17	C₅×#358	—	2	4	no	—
18	C₅×#346	—	2	2	no	—
19	#13240	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×Q₈	—	2	2	no	—
21	C₂×C₅×Q₁₆	—	2	2	no	—
22	C₂×C₅×Q₁₆	—	2	2	no	—
23	C₂×C₅×Q₁₆	—	2	2	no	—
24	C₂×C₅×Q₁₆	—	2	2	no	—
25	C₂×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₅×Q₃₂	—	2	8	no	—
32	C₂×C₅×C₁₆	—	2	1	no	—
33	C₂×C₅×C₁₆	—	2	1	no	—
34	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₅×C₁₆	D₄	1	2	yes	—
36	C₂×C₅×C₁₆	D₄	1	2	yes	—
37	C₂×C₅×C₁₆	D₄	1	2	yes	—
38	C₅×#107	—	2	2	no	—
39	C₅×#107	—	2	2	no	—
40	C₅×#103	—	4	4	no	—
41	C₄×C₅×C₈	—	2	1	no	—
42	C₄×C₅×C₈	—	2	1	no	—
43	C₅×#95	—	4	4	no	—
44	C₂²×Q₃₂	C₂×C₅	1	1	yes	characteristic
45	C₂×#367	C₂×C₅	1	1	yes	characteristic
46	C₂×#358	C₂×C₅	1	2	yes	—
47	C₂×#346	C₂×C₅	1	1	yes	characteristic
48	#1969	C₂×C₅	1	2	yes	—
49	C₂×C₅×Q₈	—	2	4	no	—
50	C₂×C₅×Q₈	—	4	2	no	—
51	C₂×C₅×Q₈	—	4	2	no	—
52	C₂×C₅×Q₈	—	4	2	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂²×C₄×C₅	—	4	2	no	—
56	C₂²×C₄×C₅	—	4	2	no	—
57	C₂²×C₄×C₅	D₈	1	2	yes	—
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₅×Q₁₆	—	2	4	no	—
60	C₅×Q₁₆	—	2	4	no	—
61	C₅×Q₁₆	—	4	2	no	—
62	C₅×Q₁₆	—	4	2	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	D₈	1	2	yes	—
66	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₅×C₈	#41	1	1	yes	characteristic
71	C₂×C₅×C₈	#41	1	1	yes	characteristic
72	C₅×C₄⋊C₄	—	4	2	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₄²×C₅	—	2	1	no	—
75	C₄²×C₅	—	2	1	no	—
76	C₅×C₁₆	—	2	2	no	—
77	C₅×C₁₆	C₂²⋊C₄	1	4	yes	—
78	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
79	C₂×Q₃₂	—	2	2	no	—
80	C₂×Q₃₂	—	2	2	no	—
81	C₂×Q₃₂	C₄×C₅	1	4	yes	complemented
82	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
83	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
84	C₂×#107	C₂²×C₅	1	2	yes	—
85	C₂×#103	—	2	2	no	—
86	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
87	#391	—	2	2	no	—
88	#367	—	2	2	no	—
89	#358	—	2	4	no	—
90	#346	—	2	2	no	—
91	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
92	C₅×Q₈	—	4	4	no	—
93	C₅×Q₈	—	4	4	no	—
94	C₅×Q₈	—	8	2	no	—
95	C₅×Q₈	—	8	2	no	—
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	4	4	no	—
99	C₂×C₄×C₅	—	8	2	no	—
100	C₂×C₄×C₅	—	8	2	no	—
101	C₂×C₄×C₅	—	8	2	no	—
102	C₂×C₄×C₅	—	8	4	no	—
103	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅	#121	1	1	yes	characteristic
105	C₂×C₄×C₅	#127	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₅	#135	1	1	yes	characteristic
108	C₂×C₄×C₅	#135	1	1	yes	characteristic
109	C₅×C₈	—	2	2	no	—
110	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₅×C₈	C₄×D₄	1	1	yes	characteristic
112	C₅×C₈	#121	1	1	yes	characteristic
113	C₅×C₈	#124	1	1	yes	characteristic
114	C₂²×Q₈	—	2	2	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
121	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
125	Q₃₂	—	2	8	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
129	C₂×C₁₆	C₅×D₄	1	2	yes	—
130	C₂×C₁₆	C₅×D₄	1	2	yes	—
131	C₂×C₁₆	C₅×D₄	1	2	yes	—
132	#107	—	2	2	no	—
133	#107	—	2	2	no	—
134	#103	—	4	4	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	#95	—	4	4	no	—
138	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
139	C₂²×C₅	#466	1	1	yes	characteristic
140	C₂²×C₅	#493	1	1	yes	characteristic
141	C₂²×C₅	#510	1	4	yes	—
142	C₄×C₅	—	4	2	no	—
143	C₄×C₅	—	8	4	no	—
144	C₄×C₅	—	16	2	no	—
145	C₄×C₅	#390	1	1	yes	characteristic
146	C₄×C₅	C₄×D₈	1	1	yes	characteristic
147	C₄×C₅	#466	1	1	yes	characteristic
148	C₄×C₅	#495	1	1	yes	characteristic
149	C₂×Q₈	—	2	4	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	C₅×D₈	1	2	yes	—
158	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
159	Q₁₆	—	2	4	no	—
160	Q₁₆	—	2	4	no	—
161	Q₁₆	—	4	2	no	—
162	Q₁₆	—	4	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	C₅×D₈	1	2	yes	—
166	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
167	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
169	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
170	C₂×C₈	C₅×#41	1	1	yes	characteristic
171	C₂×C₈	C₅×#41	1	1	yes	characteristic
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₅×C₂²⋊C₄	1	4	yes	—
178	C₂×C₅	#1970	1	1	yes	characteristic
179	C₂×C₅	#2176	1	2	yes	—
180	C₂×C₅	#2219	1	2	yes	—
181	C₂×C₅	#2251	1	2	yes	—
182	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
183	Q₈	—	4	4	no	—
184	Q₈	—	4	4	no	—
185	Q₈	—	8	2	no	—
186	Q₈	—	8	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	4	no	—
194	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
195	C₂×C₄	C₅×#121	1	1	yes	characteristic
196	C₂×C₄	C₅×#127	1	1	yes	characteristic
197	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
198	C₂×C₄	C₅×#135	1	1	yes	characteristic
199	C₂×C₄	C₅×#135	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₈	C₄×C₅×D₄	1	1	yes	characteristic
203	C₈	C₅×#121	1	1	yes	characteristic
204	C₈	C₅×#124	1	1	yes	characteristic
205	C₅	#13240	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
207	C₂²	C₅×#466	1	1	yes	characteristic
208	C₂²	C₅×#493	1	1	yes	characteristic
209	C₂²	C₅×#510	1	4	yes	—
210	C₄	—	4	2	no	—
211	C₄	—	8	4	no	—
212	C₄	—	16	2	no	—
213	C₄	C₅×#390	1	1	yes	characteristic
214	C₄	C₄×C₅×D₈	1	1	yes	characteristic
215	C₄	C₅×#466	1	1	yes	characteristic
216	C₄	C₅×#495	1	1	yes	characteristic
217	C₂	C₅×#1970	1	1	yes	characteristic
218	C₂	C₅×#2176	1	2	yes	—
219	C₂	C₅×#2219	1	2	yes	—
220	C₂	C₅×#2251	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₅×Q₃₂	C₄	semidirect
19. #13240	C₅	direct (N × H)
81. C₂×Q₃₂	C₄×C₅	semidirect
205. C₅	#13240	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×Q₃₂	2
C₂×C₅×#367	2
C₂×C₅×#358	2
C₂×C₅×#346	2
C₅×#1969	2
#13240	5

Subgroups — C5×#13240 #11935960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13240 #11935960