Subgroups — C₂×C₅×#2641 #11957340

Order 1280.

3326 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×#127	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2641	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#394	C₂	1	8	yes	maximal, complemented
5	C₂×C₅×#390	C₂	1	4	yes	maximal, complemented
6	C₂×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×D₄	C₂²	1	2	yes	—
8	C₂³×C₅×D₄	C₂²	1	4	yes	—
9	C₂×C₅×#127	—	2	8	no	—
10	C₂×C₅×#127	C₂²	1	8	yes	—
11	C₂×C₅×#125	—	2	8	no	—
12	C₂×C₅×#125	C₂²	1	8	yes	complemented
13	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂²×C₅×C₂²⋊C₄	C₂²	1	16	yes	—
16	C₂²×C₄²×C₅	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₅×#394	C₂²	1	64	yes	complemented
19	C₅×#390	C₂²	1	32	yes	complemented
20	C₅×#376	C₂²	1	8	yes	complemented
21	C₂×C₅×#95	C₂²	1	4	yes	—
22	C₂×#2641	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂⁵×C₅	C₂³	1	4	yes	—
24	C₂²×C₅×D₄	—	2	16	no	—
25	C₂²×C₅×D₄	—	2	8	no	—
26	C₂²×C₅×D₄	—	2	16	no	—
27	C₂²×C₅×D₄	—	4	16	no	—
28	C₂²×C₅×D₄	C₂³	1	8	yes	—
29	C₂²×C₅×D₄	C₂³	1	16	yes	—
30	C₂³×C₄×C₅	C₂³	1	4	yes	—
31	C₂³×C₄×C₅	C₂³	1	4	yes	—
32	C₂³×C₄×C₅	C₂³	1	2	yes	—
33	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₅×#127	—	2	32	no	—
35	C₅×#125	—	2	32	no	—
36	C₂×C₅×C₄⋊C₄	—	2	16	no	—
37	C₂×C₅×C₄⋊C₄	—	2	8	no	—
38	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
39	C₂×C₅×C₄⋊C₄	C₂³	1	16	yes	—
40	C₂×C₅×C₂²⋊C₄	—	4	32	no	—
41	C₂×C₅×C₂²⋊C₄	C₂³	1	64	yes	—
42	C₂×C₄²×C₅	—	2	8	no	—
43	C₂×C₄²×C₅	D₄	1	8	yes	—
44	C₂×C₄²×C₅	C₂³	1	8	yes	—
45	C₂×C₄²×C₅	C₂³	1	4	yes	—
46	C₅×#95	C₂³	1	16	yes	—
47	C₂²×#127	C₂×C₅	1	1	yes	characteristic
48	C₂²×#125	C₂×C₅	1	1	yes	characteristic, complemented
49	#2641	C₂×C₅	1	16	yes	direct product factor
50	C₂×#394	C₂×C₅	1	8	yes	complemented
51	C₂×#390	C₂×C₅	1	4	yes	complemented
52	C₂×#376	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂⁴×C₅	—	2	8	no	—
54	C₂⁴×C₅	—	2	16	no	—
55	C₂⁴×C₅	—	2	32	no	—
56	C₂⁴×C₅	C₂⁴	1	8	yes	—
57	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
58	C₂×C₅×D₄	—	2	32	no	—
59	C₂×C₅×D₄	—	2	16	no	—
60	C₂×C₅×D₄	—	2	32	no	—
61	C₂×C₅×D₄	—	4	32	no	—
62	C₂×C₅×D₄	—	4	64	no	—
63	C₂×C₅×D₄	—	4	32	no	—
64	C₂²×C₄×C₅	—	2	8	no	—
65	C₂²×C₄×C₅	—	2	8	no	—
66	C₂²×C₄×C₅	—	2	8	no	—
67	C₂²×C₄×C₅	—	2	16	no	—
68	C₂²×C₄×C₅	—	2	16	no	—
69	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₅	#41	1	8	yes	—
72	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
73	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
74	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
75	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
76	C₅×C₄⋊C₄	—	2	32	no	—
77	C₅×C₄⋊C₄	—	2	16	no	—
78	C₅×C₂²⋊C₄	—	4	64	no	—
79	C₄²×C₅	—	2	16	no	—
80	C₄²×C₅	C₂×D₄	1	16	yes	—
81	C₂³×D₄	C₂²×C₅	1	2	yes	—
82	C₂³×D₄	C₂²×C₅	1	4	yes	—
83	C₂×#127	—	2	8	no	—
84	C₂×#127	C₂²×C₅	1	8	yes	—
85	C₂×#125	—	2	8	no	—
86	C₂×#125	C₂²×C₅	1	8	yes	complemented
87	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
88	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
89	C₂²×C₂²⋊C₄	C₂²×C₅	1	16	yes	—
90	C₂²×C₄²	C₂²×C₅	1	2	yes	—
91	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
92	#394	C₂²×C₅	1	64	yes	complemented
93	#390	C₂²×C₅	1	32	yes	complemented
94	#376	C₂²×C₅	1	8	yes	complemented
95	C₂×#95	C₂²×C₅	1	4	yes	—
96	C₂³×C₅	—	2	8	no	—
97	C₂³×C₅	—	2	16	no	—
98	C₂³×C₅	—	2	32	no	—
99	C₂³×C₅	—	4	16	no	—
100	C₂³×C₅	—	4	32	no	—
101	C₂³×C₅	—	4	64	no	—
102	C₂³×C₅	C₂²×D₄	1	2	yes	—
103	C₂³×C₅	C₂×#41	1	4	yes	—
104	C₂³×C₅	H(5,𝔽₂)	1	8	yes	—
105	C₂³×C₅	C₂⁵	1	1	yes	characteristic
106	C₅×D₄	—	4	64	no	—
107	C₅×D₄	—	4	32	no	—
108	C₂×C₄×C₅	—	2	16	no	—
109	C₂×C₄×C₅	—	2	8	no	—
110	C₂×C₄×C₅	—	2	8	no	—
111	C₂×C₄×C₅	—	2	8	no	—
112	C₂×C₄×C₅	—	2	16	no	—
113	C₂×C₄×C₅	—	2	16	no	—
114	C₂×C₄×C₅	—	4	16	no	—
115	C₂×C₄×C₅	—	4	16	no	—
116	C₂×C₄×C₅	#124	1	16	yes	—
117	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
118	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
119	C₂×C₄×C₅	C₂×#41	1	8	yes	—
120	C₂⁵	C₂³×C₅	1	4	yes	—
121	C₂²×D₄	—	2	16	no	—
122	C₂²×D₄	—	2	8	no	—
123	C₂²×D₄	—	2	16	no	—
124	C₂²×D₄	—	4	16	no	—
125	C₂²×D₄	C₂³×C₅	1	8	yes	—
126	C₂²×D₄	C₂³×C₅	1	16	yes	—
127	C₂³×C₄	C₂³×C₅	1	4	yes	—
128	C₂³×C₄	C₂³×C₅	1	4	yes	—
129	C₂³×C₄	C₂³×C₅	1	2	yes	—
130	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
131	#127	—	2	32	no	—
132	#125	—	2	32	no	—
133	C₂×C₄⋊C₄	—	2	16	no	—
134	C₂×C₄⋊C₄	—	2	8	no	—
135	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
136	C₂×C₄⋊C₄	C₂³×C₅	1	16	yes	—
137	C₂×C₂²⋊C₄	—	4	32	no	—
138	C₂×C₂²⋊C₄	C₂³×C₅	1	64	yes	—
139	C₂×C₄²	—	2	8	no	—
140	C₂×C₄²	C₅×D₄	1	8	yes	—
141	C₂×C₄²	C₂³×C₅	1	8	yes	—
142	C₂×C₄²	C₂³×C₅	1	4	yes	—
143	#95	C₂³×C₅	1	16	yes	—
144	C₂²×C₅	—	4	8	no	—
145	C₂²×C₅	—	4	16	no	—
146	C₂²×C₅	—	4	32	no	—
147	C₂²×C₅	—	8	32	no	—
148	C₂²×C₅	C₂×#124	1	4	yes	—
149	C₂²×C₅	#535	1	8	yes	—
150	C₂²×C₅	#540	1	16	yes	—
151	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
152	C₂²×C₅	C₂²×#41	1	2	yes	—
153	C₂²×C₅	C₂×H(5,𝔽₂)	1	4	yes	—
154	C₄×C₅	—	2	8	no	—
155	C₄×C₅	—	4	8	no	—
156	C₄×C₅	—	4	8	no	—
157	C₄×C₅	C₂×#124	1	8	yes	—
158	C₂⁴	—	2	8	no	—
159	C₂⁴	—	2	16	no	—
160	C₂⁴	—	2	32	no	—
161	C₂⁴	C₂⁴×C₅	1	8	yes	—
162	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
163	C₂×D₄	—	2	32	no	—
164	C₂×D₄	—	2	16	no	—
165	C₂×D₄	—	2	32	no	—
166	C₂×D₄	—	4	32	no	—
167	C₂×D₄	—	4	64	no	—
168	C₂×D₄	—	4	32	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	8	no	—
171	C₂²×C₄	—	2	8	no	—
172	C₂²×C₄	—	2	16	no	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
175	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
176	C₂²×C₄	C₅×#41	1	8	yes	—
177	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
178	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
179	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
180	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
181	C₄⋊C₄	—	2	32	no	—
182	C₄⋊C₄	—	2	16	no	—
183	C₂²⋊C₄	—	4	64	no	—
184	C₄²	—	2	16	no	—
185	C₄²	C₂×C₅×D₄	1	16	yes	—
186	C₂×C₅	—	8	8	no	—
187	C₂×C₅	#2641	1	8	yes	direct product factor
188	C₂×C₅	C₂²×#124	1	1	yes	characteristic
189	C₂×C₅	C₂×#535	1	2	yes	—
190	C₂×C₅	C₂×#540	1	4	yes	—
191	C₂³	—	2	8	no	—
192	C₂³	—	2	16	no	—
193	C₂³	—	2	32	no	—
194	C₂³	—	4	16	no	—
195	C₂³	—	4	32	no	—
196	C₂³	—	4	64	no	—
197	C₂³	C₂²×C₅×D₄	1	2	yes	—
198	C₂³	C₂×C₅×#41	1	4	yes	—
199	C₂³	C₅×H(5,𝔽₂)	1	8	yes	—
200	C₂³	C₂⁵×C₅	1	1	yes	characteristic
201	D₄	—	4	64	no	—
202	D₄	—	4	32	no	—
203	C₂×C₄	—	2	16	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	16	no	—
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	4	16	no	—
210	C₂×C₄	—	4	16	no	—
211	C₂×C₄	C₅×#124	1	16	yes	—
212	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
213	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
214	C₂×C₄	C₂×C₅×#41	1	8	yes	—
215	C₅	C₂×#2641	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	4	8	no	—
217	C₂²	—	4	16	no	—
218	C₂²	—	4	32	no	—
219	C₂²	—	8	32	no	—
220	C₂²	C₂×C₅×#124	1	4	yes	—
221	C₂²	C₅×#535	1	8	yes	—
222	C₂²	C₅×#540	1	16	yes	—
223	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
224	C₂²	C₂²×C₅×#41	1	2	yes	—
225	C₂²	C₂×C₅×H(5,𝔽₂)	1	4	yes	—
226	C₄	—	2	8	no	—
227	C₄	—	4	8	no	—
228	C₄	—	4	8	no	—
229	C₄	C₂×C₅×#124	1	8	yes	—
230	C₂	—	8	8	no	—
231	C₂	C₅×#2641	1	8	yes	direct product factor
232	C₂	C₂²×C₅×#124	1	1	yes	characteristic
233	C₂	C₂×C₅×#535	1	2	yes	—
234	C₂	C₂×C₅×#540	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂²×C₅×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×#125	C₂	semidirect
3. C₅×#2641	C₂	direct (N × H)
4. C₂×C₅×#394	C₂	semidirect
5. C₂×C₅×#390	C₂	semidirect
6. C₂×C₅×#376	C₂	semidirect
12. C₂×C₅×#125	C₂²	semidirect
18. C₅×#394	C₂²	semidirect
19. C₅×#390	C₂²	semidirect
20. C₅×#376	C₂²	semidirect
22. C₂×#2641	C₅	direct (N × H)
48. C₂²×#125	C₂×C₅	semidirect
49. #2641	C₂×C₅	direct (N × H)
50. C₂×#394	C₂×C₅	semidirect
51. C₂×#390	C₂×C₅	semidirect
52. C₂×#376	C₂×C₅	semidirect
86. C₂×#125	C₂²×C₅	semidirect
92. #394	C₂²×C₅	semidirect
93. #390	C₂²×C₅	semidirect
94. #376	C₂²×C₅	semidirect
187. C₂×C₅	#2641	direct (N × H)
215. C₅	C₂×#2641	direct (N × H)
231. C₂	C₅×#2641	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×#127	2
C₂²×C₅×#125	2
C₅×#2641	2
C₂×C₅×#394	2
C₂×C₅×#390	2
C₂×C₅×#376	2
C₂×#2641	5

Subgroups — C2×C5×#2641 #11957340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2641 #11957340