Subgroups — C₂×C₅×#1849 #11943525

Order 1280.

1006 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×#108	C₂	1	4	yes	maximal
2	C₂×C₅×#423	C₂	1	2	yes	maximal
3	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
4	C₅×#1849	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
6	C₂³×C₅×C₈	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	—	2	16	no	—
9	C₂×C₅×#108	C₂²	1	16	yes	—
10	C₅×#423	—	2	4	no	—
11	C₅×#423	C₂²	1	8	yes	—
12	C₂×C₅×C₄⋊C₈	—	2	4	no	—
13	C₂×C₅×#97	C₄	1	4	yes	—
14	C₂×C₅×#97	C₄	1	4	yes	—
15	C₂×C₅×#97	C₂²	1	4	yes	—
16	C₂×#1849	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₅	Q₈	1	2	yes	—
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂×C₅×M₁₆	—	2	4	no	—
20	C₂×C₅×M₁₆	—	2	16	no	—
21	C₂×C₅×M₁₆	C₂³	1	8	yes	—
22	C₂²×C₅×C₈	—	2	4	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₅×C₈	D₄	1	8	yes	—
26	C₂²×C₅×C₈	Q₈	1	8	yes	—
27	C₂²×C₅×C₈	C₂³	1	4	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	2	yes	—
31	C₂×C₄²×C₅	D₄	1	2	yes	—
32	C₂×C₄²×C₅	Q₈	1	2	yes	—
33	C₂×C₄²×C₅	C₂³	1	2	yes	—
34	C₅×#108	—	2	32	no	—
35	C₅×C₄⋊C₈	—	2	16	no	—
36	C₅×#97	C₂×C₄	1	16	yes	—
37	C₂²×#108	C₂×C₅	1	4	yes	—
38	C₂×#423	C₂×C₅	1	2	yes	—
39	C₂²×#97	C₂×C₅	1	1	yes	characteristic
40	#1849	C₂×C₅	1	8	yes	direct product factor
41	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅	—	4	4	no	—
43	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
54	C₅×M₁₆	—	2	16	no	—
55	C₅×M₁₆	—	4	8	no	—
56	C₂×C₅×C₈	—	2	8	no	—
57	C₂×C₅×C₈	—	4	8	no	—
58	C₂×C₅×C₈	C₄⋊C₄	1	16	yes	—
59	C₂×C₅×C₈	C₄⋊C₄	1	16	yes	—
60	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
61	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
62	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
63	C₄²×C₅	C₄⋊C₄	1	4	yes	—
64	C₄²×C₅	C₄⋊C₄	1	4	yes	—
65	C₄²×C₅	C₂²×C₄	1	4	yes	—
66	C₄²×C₅	C₂×D₄	1	4	yes	—
67	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
68	C₂³×C₈	C₂²×C₅	1	2	yes	—
69	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
70	C₂×#108	—	2	16	no	—
71	C₂×#108	C₂²×C₅	1	16	yes	—
72	#423	—	2	4	no	—
73	#423	C₂²×C₅	1	8	yes	—
74	C₂×C₄⋊C₈	—	2	4	no	—
75	C₂×#97	C₄×C₅	1	4	yes	—
76	C₂×#97	C₄×C₅	1	4	yes	—
77	C₂×#97	C₂²×C₅	1	4	yes	—
78	C₂³×C₅	—	2	4	no	—
79	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
82	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅	—	2	8	no	—
84	C₂×C₄×C₅	—	4	4	no	—
85	C₂×C₄×C₅	—	4	4	no	—
86	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
87	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	#127	1	2	yes	—
92	C₂×C₄×C₅	#127	1	2	yes	—
93	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
94	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
96	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
98	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
100	C₅×C₈	—	4	8	no	—
101	C₅×C₈	C₂×C₄⋊C₄	1	16	yes	—
102	C₂³×C₄	C₅×Q₈	1	2	yes	—
103	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
104	C₂×M₁₆	—	2	4	no	—
105	C₂×M₁₆	—	2	16	no	—
106	C₂×M₁₆	C₂³×C₅	1	8	yes	—
107	C₂²×C₈	—	2	4	no	—
108	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
109	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
110	C₂²×C₈	C₅×D₄	1	8	yes	—
111	C₂²×C₈	C₅×Q₈	1	8	yes	—
112	C₂²×C₈	C₂³×C₅	1	4	yes	—
113	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
115	C₂×C₄²	C₅×D₄	1	2	yes	—
116	C₂×C₄²	C₅×D₄	1	2	yes	—
117	C₂×C₄²	C₅×Q₈	1	2	yes	—
118	C₂×C₄²	C₂³×C₅	1	2	yes	—
119	#108	—	2	32	no	—
120	C₄⋊C₈	—	2	16	no	—
121	#97	C₂×C₄×C₅	1	16	yes	—
122	C₂²×C₅	—	2	8	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	—	2	2	no	—
125	C₂²×C₅	#384	1	2	yes	—
126	C₂²×C₅	#384	1	2	yes	—
127	C₂²×C₅	#430	1	4	yes	—
128	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₅	C₂×C₄⋊Q₈	1	2	yes	—
130	C₄×C₅	—	2	8	no	—
131	C₄×C₅	#384	1	2	yes	—
132	C₄×C₅	#384	1	2	yes	—
133	C₄×C₅	C₂²×C₄⋊C₄	1	2	yes	—
134	C₄×C₅	C₂×#127	1	1	yes	characteristic
135	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
139	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
140	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
141	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
143	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
146	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
148	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
149	M₁₆	—	2	16	no	—
150	M₁₆	—	4	8	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	4	8	no	—
153	C₂×C₈	C₅×C₄⋊C₄	1	16	yes	—
154	C₂×C₈	C₅×C₄⋊C₄	1	16	yes	—
155	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
156	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
157	C₂×C₈	C₂×C₅×Q₈	1	8	yes	—
158	C₄²	C₅×C₄⋊C₄	1	4	yes	—
159	C₄²	C₅×C₄⋊C₄	1	4	yes	—
160	C₄²	C₂²×C₄×C₅	1	4	yes	—
161	C₄²	C₂×C₅×D₄	1	4	yes	—
162	C₂×C₅	—	2	2	no	—
163	C₂×C₅	—	2	2	no	—
164	C₂×C₅	#1849	1	4	yes	direct product factor
165	C₂×C₅	C₂×#384	1	1	yes	characteristic
166	C₂×C₅	C₂×#430	1	2	yes	—
167	C₂³	—	2	4	no	—
168	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
170	C₂³	C₅×C₄⋊Q₈	1	4	yes	—
171	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
176	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
179	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
180	C₂×C₄	C₅×#127	1	2	yes	—
181	C₂×C₄	C₅×#127	1	2	yes	—
182	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
183	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
184	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
185	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
186	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
187	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
189	C₈	—	4	8	no	—
190	C₈	C₂×C₅×C₄⋊C₄	1	16	yes	—
191	C₅	C₂×#1849	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	—	2	8	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	C₅×#384	1	2	yes	—
196	C₂²	C₅×#384	1	2	yes	—
197	C₂²	C₅×#430	1	4	yes	—
198	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
199	C₂²	C₂×C₅×C₄⋊Q₈	1	2	yes	—
200	C₄	—	2	8	no	—
201	C₄	C₅×#384	1	2	yes	—
202	C₄	C₅×#384	1	2	yes	—
203	C₄	C₂²×C₅×C₄⋊C₄	1	2	yes	—
204	C₄	C₂×C₅×#127	1	1	yes	characteristic
205	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	C₅×#1849	1	4	yes	direct product factor
209	C₂	C₂×C₅×#384	1	1	yes	characteristic
210	C₂	C₂×C₅×#430	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1849	C₂	direct (N × H)
16. C₂×#1849	C₅	direct (N × H)
40. #1849	C₂×C₅	direct (N × H)
164. C₂×C₅	#1849	direct (N × H)
191. C₅	C₂×#1849	direct (N × H)
208. C₂	C₅×#1849	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×#108	2
C₂×C₅×#423	2
C₂²×C₅×#97	2
C₅×#1849	2
C₂×#1849	5

Subgroups — C2×C5×#1849 #11943525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1849 #11943525