Subgroups — C₂×C₅×#2150 #11956115

Order 1280.

798 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#2150	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×C₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₂×C₅×M₃₂	—	2	8	no	—
9	C₂×C₅×M₃₂	C₂²	1	8	yes	—
10	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
11	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
12	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
15	C₅×C₄⋊C₁₆	C₂²	1	16	yes	complemented
16	C₂×#2150	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₅	C₈	1	2	yes	—
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	—	2	4	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	D₄	1	4	yes	—
25	C₂²×C₅×C₈	Q₈	1	4	yes	—
26	C₂²×C₅×C₈	C₂³	1	2	yes	—
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₈	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₈	1	4	yes	—
30	C₂×C₄²×C₅	C₈	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₅×M₃₂	—	2	16	no	—
34	C₂×C₅×C₁₆	—	2	8	no	—
35	C₂×C₅×C₁₆	C₂³	1	8	yes	—
36	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
37	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
38	C₄×C₅×C₈	C₂³	1	4	yes	—
39	C₂²×M₃₂	C₂×C₅	1	2	yes	—
40	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
41	#2150	C₂×C₅	1	8	yes	direct product factor
42	C₂×C₄⋊C₁₆	C₂×C₅	1	4	yes	complemented
43	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	—	2	8	no	—
45	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₅	C₂×C₈	1	8	yes	—
48	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₅	M₁₆	1	4	yes	—
52	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₅×C₈	—	2	8	no	—
55	C₂×C₅×C₈	—	2	4	no	—
56	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
57	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
58	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
59	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
62	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
64	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
65	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
66	C₄²×C₅	C₂×C₈	1	4	yes	—
67	C₄²×C₅	C₂×C₈	1	4	yes	—
68	C₄²×C₅	C₂×C₈	1	4	yes	—
69	C₄²×C₅	C₂²×C₄	1	4	yes	—
70	C₅×C₁₆	—	2	8	no	—
71	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
72	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
74	C₂×M₃₂	—	2	8	no	—
75	C₂×M₃₂	C₂²×C₅	1	8	yes	—
76	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
77	C₂×C₄×C₈	C₄×C₅	1	4	yes	—
78	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
79	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
80	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
81	C₄⋊C₁₆	C₂²×C₅	1	16	yes	complemented
82	C₂³×C₅	—	2	4	no	—
83	C₂³×C₅	C₄⋊C₈	1	4	yes	—
84	C₂³×C₅	C₂²×C₈	1	2	yes	—
85	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	—	2	16	no	—
88	C₂×C₄×C₅	C₄⋊C₈	1	8	yes	—
89	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
90	C₂×C₄×C₅	M₃₂	1	8	yes	complemented
91	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
97	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
98	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
100	C₅×C₈	—	2	4	no	—
101	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
102	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
103	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
104	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
105	C₂³×C₄	C₅×C₈	1	2	yes	—
106	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂²×C₈	—	2	4	no	—
108	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
109	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
110	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
111	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
112	C₂²×C₈	C₅×D₄	1	4	yes	—
113	C₂²×C₈	C₅×Q₈	1	4	yes	—
114	C₂²×C₈	C₂³×C₅	1	2	yes	—
115	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
116	C₂×C₄²	C₅×C₈	1	1	yes	characteristic
117	C₂×C₄²	C₅×C₈	1	4	yes	—
118	C₂×C₄²	C₅×C₈	1	1	yes	characteristic
119	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
120	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
121	M₃₂	—	2	16	no	—
122	C₂×C₁₆	—	2	8	no	—
123	C₂×C₁₆	C₂³×C₅	1	8	yes	—
124	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
125	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
126	C₄×C₈	C₂³×C₅	1	4	yes	—
127	C₂²×C₅	—	2	4	no	—
128	C₂²×C₅	—	2	8	no	—
129	C₂²×C₅	C₂×C₄⋊C₈	1	4	yes	—
130	C₂²×C₅	C₂×C₄⋊C₈	1	2	yes	—
131	C₂²×C₅	C₂×M₃₂	1	4	yes	—
132	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₅	—	2	4	no	—
134	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
135	C₄×C₅	C₂×C₄⋊C₈	1	4	yes	—
136	C₄×C₅	C₂×M₃₂	1	8	yes	complemented
137	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
139	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
142	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
143	C₂²×C₄	C₂×C₅×C₈	1	8	yes	—
144	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
145	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
147	C₂²×C₄	C₅×M₁₆	1	4	yes	—
148	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
149	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
153	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
154	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
155	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
156	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
157	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
158	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
159	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
160	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
161	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
162	C₄²	C₂×C₅×C₈	1	4	yes	—
163	C₄²	C₂×C₅×C₈	1	4	yes	—
164	C₄²	C₂×C₅×C₈	1	4	yes	—
165	C₄²	C₂²×C₄×C₅	1	4	yes	—
166	C₁₆	—	2	8	no	—
167	C₂×C₅	—	2	4	no	—
168	C₂×C₅	#2150	1	4	yes	direct product factor
169	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
170	C₂×C₅	C₂²×M₃₂	1	2	yes	—
171	C₂³	—	2	4	no	—
172	C₂³	C₅×C₄⋊C₈	1	4	yes	—
173	C₂³	C₂²×C₅×C₈	1	2	yes	—
174	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	16	no	—
177	C₂×C₄	C₅×C₄⋊C₈	1	8	yes	—
178	C₂×C₄	C₅×C₄⋊C₈	1	4	yes	—
179	C₂×C₄	C₅×M₃₂	1	8	yes	complemented
180	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
181	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
182	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
183	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
184	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
186	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
187	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
188	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
189	C₈	—	2	4	no	—
190	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
191	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
192	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
193	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
194	C₅	C₂×#2150	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	—	2	8	no	—
196	C₂²	—	2	4	no	—
197	C₂²	C₂×C₅×C₄⋊C₈	1	4	yes	—
198	C₂²	C₂×C₅×C₄⋊C₈	1	2	yes	—
199	C₂²	C₂×C₅×M₃₂	1	4	yes	—
200	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	C₂×C₅×C₄⋊C₈	1	4	yes	—
203	C₄	C₂×C₅×C₄⋊C₈	1	2	yes	—
204	C₄	C₂×C₅×M₃₂	1	8	yes	complemented
205	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
206	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	C₅×#2150	1	4	yes	direct product factor
209	C₂	C₂²×C₅×C₄⋊C₈	1	1	yes	characteristic
210	C₂	C₂²×C₅×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2150	C₂	direct (N × H)
4. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
15. C₅×C₄⋊C₁₆	C₂²	semidirect
16. C₂×#2150	C₅	direct (N × H)
41. #2150	C₂×C₅	direct (N × H)
42. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
81. C₄⋊C₁₆	C₂²×C₅	semidirect
90. C₂×C₄×C₅	M₃₂	semidirect
136. C₄×C₅	C₂×M₃₂	semidirect
168. C₂×C₅	#2150	direct (N × H)
179. C₂×C₄	C₅×M₃₂	semidirect
194. C₅	C₂×#2150	direct (N × H)
204. C₄	C₂×C₅×M₃₂	semidirect
208. C₂	C₅×#2150	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×M₃₂	2
C₂²×C₄×C₅×C₈	2
C₅×#2150	2
C₂×C₅×C₄⋊C₁₆	2
C₂×#2150	5

Subgroups — C2×C5×#2150 #11956115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2150 #11956115