Subgroups — C₂²×C₅×#340 #11943064

Order 1280.

3006 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	C₂⁴×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₅×#340	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₅	—	2	2	no	—
5	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂³×C₅×C₈	—	2	8	no	—
7	C₂³×C₅×C₈	C₄	1	8	yes	complemented
8	C₂³×C₅×C₈	C₂²	1	6	yes	—
9	C₂²×C₅×C₄⋊C₄	—	2	6	no	—
10	C₂²×C₅×C₄⋊C₄	—	2	8	no	—
11	C₂²×C₅×C₄⋊C₄	C₄	1	16	yes	complemented
12	C₂²×C₅×C₄⋊C₄	C₂²	1	12	yes	—
13	C₅×#340	C₂²	1	16	yes	direct product factor
14	C₂²×#340	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂⁵×C₅	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₅	—	2	16	no	—
17	C₂³×C₄×C₅	—	2	12	no	—
18	C₂³×C₄×C₅	—	4	8	no	—
19	C₂³×C₄×C₅	—	4	8	no	—
20	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₅	D₄	1	3	yes	—
23	C₂³×C₄×C₅	D₄	1	8	yes	—
24	C₂³×C₄×C₅	Q₈	1	4	yes	—
25	C₂³×C₄×C₅	C₂³	1	3	yes	—
26	C₂²×C₅×C₈	—	2	32	no	—
27	C₂²×C₅×C₈	—	2	24	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	24	yes	complemented
29	C₂²×C₅×C₈	C₂³	1	4	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	48	no	—
31	C₂×C₅×C₄⋊C₄	—	2	12	no	—
32	C₂×C₅×C₄⋊C₄	—	2	24	no	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	48	yes	complemented
34	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
35	C₂⁴×C₈	C₂×C₅	1	1	yes	characteristic
36	C₂³×C₄⋊C₄	C₂×C₅	1	2	yes	—
37	C₂×#340	C₂×C₅	1	12	yes	direct product factor
38	C₂⁴×C₅	C₂²⋊C₄	1	8	yes	—
39	C₂⁴×C₅	C₄⋊C₄	1	4	yes	—
40	C₂⁴×C₅	D₈	1	4	yes	—
41	C₂⁴×C₅	SD₁₆	1	8	yes	—
42	C₂⁴×C₅	Q₁₆	1	4	yes	—
43	C₂⁴×C₅	C₂×D₄	1	3	yes	—
44	C₂²×C₄×C₅	—	2	48	no	—
45	C₂²×C₄×C₅	—	2	8	no	—
46	C₂²×C₄×C₅	—	4	64	no	—
47	C₂²×C₄×C₅	—	4	24	no	—
48	C₂²×C₄×C₅	—	4	24	no	—
49	C₂²×C₄×C₅	C₄²	1	16	yes	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	24	yes	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	16	yes	—
52	C₂²×C₄×C₅	C₄⋊C₄	1	12	yes	—
53	C₂²×C₄×C₅	C₄⋊C₄	1	32	yes	—
54	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
58	C₂²×C₄×C₅	C₂×Q₈	1	6	yes	—
59	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
60	C₂×C₅×C₈	—	2	48	no	—
61	C₂×C₅×C₈	—	2	8	no	—
62	C₂×C₅×C₈	C₂²×C₄	1	8	yes	complemented
63	C₅×C₄⋊C₄	—	2	48	no	—
64	C₅×C₄⋊C₄	—	2	8	no	—
65	C₅×C₄⋊C₄	C₂²×C₄	1	16	yes	complemented
66	C₂⁴×C₄	—	2	2	no	—
67	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
68	C₂³×C₈	—	2	8	no	—
69	C₂³×C₈	C₄×C₅	1	8	yes	complemented
70	C₂³×C₈	C₂²×C₅	1	6	yes	—
71	C₂²×C₄⋊C₄	—	2	6	no	—
72	C₂²×C₄⋊C₄	—	2	8	no	—
73	C₂²×C₄⋊C₄	C₄×C₅	1	16	yes	complemented
74	C₂²×C₄⋊C₄	C₂²×C₅	1	12	yes	—
75	#340	C₂²×C₅	1	16	yes	direct product factor
76	C₂³×C₅	#95	1	16	yes	—
77	C₂³×C₅	#102	1	32	yes	—
78	C₂³×C₅	#103	1	32	yes	—
79	C₂³×C₅	#106	1	16	yes	—
80	C₂³×C₅	#107	1	16	yes	—
81	C₂³×C₅	C₂×C₂²⋊C₄	1	12	yes	—
82	C₂³×C₅	C₂×C₄⋊C₄	1	6	yes	—
83	C₂³×C₅	C₂×D₈	1	6	yes	—
84	C₂³×C₅	C₂×SD₁₆	1	12	yes	—
85	C₂³×C₅	C₂×Q₁₆	1	6	yes	—
86	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅	—	2	16	no	—
88	C₂×C₄×C₅	—	4	8	no	—
89	C₂×C₄×C₅	—	4	8	no	—
90	C₂×C₄×C₅	—	4	96	no	—
91	C₂×C₄×C₅	#95	1	48	yes	—
92	C₂×C₄×C₅	C₂×C₄²	1	12	yes	—
93	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
94	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	24	yes	—
96	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
97	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
99	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
101	C₅×C₈	—	2	8	no	—
102	C₂⁵	C₅×D₄	1	1	yes	characteristic
103	C₂³×C₄	—	2	16	no	—
104	C₂³×C₄	—	2	12	no	—
105	C₂³×C₄	—	4	8	no	—
106	C₂³×C₄	—	4	8	no	—
107	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
108	C₂³×C₄	C₂×C₄×C₅	1	8	yes	—
109	C₂³×C₄	C₅×D₄	1	3	yes	—
110	C₂³×C₄	C₅×D₄	1	8	yes	—
111	C₂³×C₄	C₅×Q₈	1	4	yes	—
112	C₂³×C₄	C₂³×C₅	1	3	yes	—
113	C₂²×C₈	—	2	32	no	—
114	C₂²×C₈	—	2	24	no	—
115	C₂²×C₈	C₂×C₄×C₅	1	24	yes	complemented
116	C₂²×C₈	C₂³×C₅	1	4	yes	—
117	C₂×C₄⋊C₄	—	2	48	no	—
118	C₂×C₄⋊C₄	—	2	12	no	—
119	C₂×C₄⋊C₄	—	2	24	no	—
120	C₂×C₄⋊C₄	C₂×C₄×C₅	1	48	yes	complemented
121	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
122	C₂²×C₅	#340	1	64	yes	direct product factor
123	C₂²×C₅	C₂×#95	1	12	yes	—
124	C₂²×C₅	C₂×#102	1	24	yes	—
125	C₂²×C₅	C₂×#103	1	24	yes	—
126	C₂²×C₅	C₂×#106	1	12	yes	—
127	C₂²×C₅	C₂×#107	1	12	yes	—
128	C₂²×C₅	C₂²×C₂²⋊C₄	1	2	yes	—
129	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
131	C₂²×C₅	C₂²×SD₁₆	1	2	yes	—
132	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
133	C₄×C₅	—	4	16	no	—
134	C₄×C₅	C₂×#95	1	12	yes	—
135	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
136	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₅	C₂²×C₄⋊C₄	1	2	yes	—
138	C₂⁴	C₅×C₂²⋊C₄	1	8	yes	—
139	C₂⁴	C₅×C₄⋊C₄	1	4	yes	—
140	C₂⁴	C₅×D₈	1	4	yes	—
141	C₂⁴	C₅×SD₁₆	1	8	yes	—
142	C₂⁴	C₅×Q₁₆	1	4	yes	—
143	C₂⁴	C₂×C₅×D₄	1	3	yes	—
144	C₂²×C₄	—	2	48	no	—
145	C₂²×C₄	—	2	8	no	—
146	C₂²×C₄	—	4	24	no	—
147	C₂²×C₄	—	4	24	no	—
148	C₂²×C₄	—	4	64	no	—
149	C₂²×C₄	C₄²×C₅	1	16	yes	—
150	C₂²×C₄	C₅×C₂²⋊C₄	1	16	yes	—
151	C₂²×C₄	C₅×C₂²⋊C₄	1	24	yes	—
152	C₂²×C₄	C₅×C₄⋊C₄	1	32	yes	—
153	C₂²×C₄	C₅×C₄⋊C₄	1	12	yes	—
154	C₂²×C₄	C₂²×C₄×C₅	1	6	yes	—
155	C₂²×C₄	C₂²×C₄×C₅	1	12	yes	—
156	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
157	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
158	C₂²×C₄	C₂×C₅×Q₈	1	6	yes	—
159	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
160	C₂×C₈	—	2	48	no	—
161	C₂×C₈	—	2	8	no	—
162	C₂×C₈	C₂²×C₄×C₅	1	8	yes	complemented
163	C₄⋊C₄	—	2	48	no	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	C₂²×C₄×C₅	1	16	yes	complemented
166	C₂×C₅	C₂×#340	1	24	yes	direct product factor
167	C₂×C₅	C₂²×#95	1	1	yes	characteristic
168	C₂×C₅	C₂²×#102	1	2	yes	—
169	C₂×C₅	C₂²×#103	1	2	yes	—
170	C₂×C₅	C₂²×#106	1	1	yes	characteristic
171	C₂×C₅	C₂²×#107	1	1	yes	characteristic
172	C₂³	C₅×#95	1	16	yes	—
173	C₂³	C₅×#102	1	32	yes	—
174	C₂³	C₅×#103	1	32	yes	—
175	C₂³	C₅×#106	1	16	yes	—
176	C₂³	C₅×#107	1	16	yes	—
177	C₂³	C₂×C₅×C₂²⋊C₄	1	12	yes	—
178	C₂³	C₂×C₅×C₄⋊C₄	1	6	yes	—
179	C₂³	C₂×C₅×D₈	1	6	yes	—
180	C₂³	C₂×C₅×SD₁₆	1	12	yes	—
181	C₂³	C₂×C₅×Q₁₆	1	6	yes	—
182	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	4	96	no	—
185	C₂×C₄	—	4	8	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	C₅×#95	1	48	yes	—
188	C₂×C₄	C₂×C₄²×C₅	1	12	yes	—
189	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	12	yes	—
190	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	12	yes	—
191	C₂×C₄	C₂×C₅×C₄⋊C₄	1	6	yes	—
192	C₂×C₄	C₂×C₅×C₄⋊C₄	1	24	yes	—
193	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
194	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
195	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
196	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
197	C₈	—	2	8	no	—
198	C₅	C₂²×#340	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	C₅×#340	1	64	yes	direct product factor
200	C₂²	C₂×C₅×#95	1	12	yes	—
201	C₂²	C₂×C₅×#102	1	24	yes	—
202	C₂²	C₂×C₅×#103	1	24	yes	—
203	C₂²	C₂×C₅×#106	1	12	yes	—
204	C₂²	C₂×C₅×#107	1	12	yes	—
205	C₂²	C₂²×C₅×C₂²⋊C₄	1	2	yes	—
206	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
207	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
208	C₂²	C₂²×C₅×SD₁₆	1	2	yes	—
209	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
210	C₄	—	4	16	no	—
211	C₄	C₂×C₅×#95	1	12	yes	—
212	C₄	C₂²×C₄²×C₅	1	1	yes	characteristic
213	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₄	C₂²×C₅×C₄⋊C₄	1	2	yes	—
215	C₂	C₂×C₅×#340	1	24	yes	direct product factor
216	C₂	C₂²×C₅×#95	1	1	yes	characteristic
217	C₂	C₂²×C₅×#102	1	2	yes	—
218	C₂	C₂²×C₅×#103	1	2	yes	—
219	C₂	C₂²×C₅×#106	1	1	yes	characteristic
220	C₂	C₂²×C₅×#107	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₅	D₄
commutator	C₄	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₅ ◁ C₂⁴×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#340	C₂	direct (N × H)
7. C₂³×C₅×C₈	C₄	semidirect
11. C₂²×C₅×C₄⋊C₄	C₄	semidirect
13. C₅×#340	C₂²	direct (N × H)
14. C₂²×#340	C₅	direct (N × H)
28. C₂²×C₅×C₈	C₂×C₄	semidirect
33. C₂×C₅×C₄⋊C₄	C₂×C₄	semidirect
37. C₂×#340	C₂×C₅	direct (N × H)
62. C₂×C₅×C₈	C₂²×C₄	semidirect
65. C₅×C₄⋊C₄	C₂²×C₄	semidirect
69. C₂³×C₈	C₄×C₅	semidirect
73. C₂²×C₄⋊C₄	C₄×C₅	semidirect
75. #340	C₂²×C₅	direct (N × H)
115. C₂²×C₈	C₂×C₄×C₅	semidirect
120. C₂×C₄⋊C₄	C₂×C₄×C₅	semidirect
122. C₂²×C₅	#340	direct (N × H)
162. C₂×C₈	C₂²×C₄×C₅	semidirect
165. C₄⋊C₄	C₂²×C₄×C₅	semidirect
166. C₂×C₅	C₂×#340	direct (N × H)
198. C₅	C₂²×#340	direct (N × H)
199. C₂²	C₅×#340	direct (N × H)
215. C₂	C₂×C₅×#340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×C₈	2
C₂³×C₅×C₄⋊C₄	2
C₂×C₅×#340	2
C₂²×#340	5

Subgroups — C22×C5×#340 #11943064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#340 #11943064