Subgroups — C₂×C₅×#1848 #11943524

Order 1280.

1046 conjugacy classes of subgroups.

The subgroup lattice has 230 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×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#1848	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×C₄×C₅×C₈	C₄	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
15	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
17	C₂×#1848	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	Q₈	1	2	yes	—
19	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₅×M₁₆	—	2	4	no	—
21	C₂×C₅×M₁₆	—	2	16	no	—
22	C₂×C₅×M₁₆	C₂³	1	8	yes	—
23	C₂²×C₅×C₈	—	2	4	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	D₄	1	8	yes	—
27	C₂²×C₅×C₈	Q₈	1	8	yes	—
28	C₂²×C₅×C₈	C₂³	1	4	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	D₄	1	2	yes	—
33	C₂×C₄²×C₅	D₄	1	2	yes	—
34	C₂×C₄²×C₅	Q₈	1	2	yes	—
35	C₂×C₄²×C₅	C₂³	1	2	yes	—
36	C₅×#108	—	2	32	no	—
37	C₅×C₄⋊C₈	—	2	16	no	—
38	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
39	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
40	C₂²×#108	C₂×C₅	1	4	yes	—
41	C₂×#423	C₂×C₅	1	2	yes	—
42	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
43	#1848	C₂×C₅	1	8	yes	direct product factor
44	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	—	2	8	no	—
46	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
58	C₅×M₁₆	—	2	16	no	—
59	C₅×M₁₆	—	4	8	no	—
60	C₂×C₅×C₈	—	2	8	no	—
61	C₂×C₅×C₈	—	4	8	no	—
62	C₂×C₅×C₈	C₄⋊C₄	1	16	yes	—
63	C₂×C₅×C₈	C₄⋊C₄	1	16	yes	—
64	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
65	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
66	C₂×C₅×C₈	C₂×Q₈	1	8	yes	—
67	C₄²×C₅	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅	C₄⋊C₄	1	4	yes	—
69	C₄²×C₅	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅	C₂²×C₄	1	2	yes	—
71	C₄²×C₅	C₂²×C₄	1	2	yes	—
72	C₄²×C₅	C₂×D₄	1	4	yes	—
73	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
74	C₂³×C₈	C₂²×C₅	1	2	yes	—
75	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
76	C₂×#108	—	2	16	no	—
77	C₂×#108	C₂²×C₅	1	16	yes	—
78	#423	—	2	4	no	—
79	#423	C₂²×C₅	1	8	yes	—
80	C₂×C₄⋊C₈	—	2	4	no	—
81	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
82	C₂×C₄×C₈	C₄×C₅	1	4	yes	—
83	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
84	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
85	C₂³×C₅	—	2	4	no	—
86	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂³×C₅	C₄⋊Q₈	1	4	yes	—
89	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅	—	2	4	no	—
91	C₂×C₄×C₅	—	2	8	no	—
92	C₂×C₄×C₅	—	2	8	no	—
93	C₂×C₄×C₅	#108	1	8	yes	complemented
94	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₅	#127	1	2	yes	—
101	C₂×C₄×C₅	#127	1	2	yes	—
102	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
105	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
107	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
109	C₅×C₈	—	4	8	no	—
110	C₅×C₈	C₂×C₄⋊C₄	1	16	yes	—
111	C₂³×C₄	C₅×Q₈	1	2	yes	—
112	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
113	C₂×M₁₆	—	2	4	no	—
114	C₂×M₁₆	—	2	16	no	—
115	C₂×M₁₆	C₂³×C₅	1	8	yes	—
116	C₂²×C₈	—	2	4	no	—
117	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
118	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
119	C₂²×C₈	C₅×D₄	1	8	yes	—
120	C₂²×C₈	C₅×Q₈	1	8	yes	—
121	C₂²×C₈	C₂³×C₅	1	4	yes	—
122	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
123	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
124	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂×C₄²	C₅×D₄	1	2	yes	—
126	C₂×C₄²	C₅×D₄	1	2	yes	—
127	C₂×C₄²	C₅×Q₈	1	2	yes	—
128	C₂×C₄²	C₂³×C₅	1	2	yes	—
129	#108	—	2	32	no	—
130	C₄⋊C₈	—	2	16	no	—
131	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
132	C₄×C₈	C₂×C₄×C₅	1	8	yes	—
133	C₂²×C₅	—	2	8	no	—
134	C₂²×C₅	—	2	2	no	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	#384	1	2	yes	—
137	C₂²×C₅	#384	1	2	yes	—
138	C₂²×C₅	C₂×#108	1	4	yes	—
139	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₅	C₂×C₄⋊Q₈	1	2	yes	—
141	C₄×C₅	—	2	4	no	—
142	C₄×C₅	#384	1	2	yes	—
143	C₄×C₅	#384	1	2	yes	—
144	C₄×C₅	C₂×#108	1	8	yes	complemented
145	C₄×C₅	C₂²×C₄⋊C₄	1	2	yes	—
146	C₄×C₅	C₂×#127	1	1	yes	characteristic
147	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
151	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
152	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
153	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
154	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
156	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
159	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
161	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
162	M₁₆	—	2	16	no	—
163	M₁₆	—	4	8	no	—
164	C₂×C₈	—	2	8	no	—
165	C₂×C₈	—	4	8	no	—
166	C₂×C₈	C₅×C₄⋊C₄	1	16	yes	—
167	C₂×C₈	C₅×C₄⋊C₄	1	16	yes	—
168	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
169	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
170	C₂×C₈	C₂×C₅×Q₈	1	8	yes	—
171	C₄²	C₅×C₄⋊C₄	1	2	yes	—
172	C₄²	C₅×C₄⋊C₄	1	4	yes	—
173	C₄²	C₅×C₄⋊C₄	1	2	yes	—
174	C₄²	C₂²×C₄×C₅	1	2	yes	—
175	C₄²	C₂²×C₄×C₅	1	2	yes	—
176	C₄²	C₂×C₅×D₄	1	4	yes	—
177	C₂×C₅	—	2	2	no	—
178	C₂×C₅	—	2	2	no	—
179	C₂×C₅	#1848	1	4	yes	direct product factor
180	C₂×C₅	C₂×#384	1	1	yes	characteristic
181	C₂×C₅	C₂²×#108	1	2	yes	—
182	C₂³	—	2	4	no	—
183	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
185	C₂³	C₅×C₄⋊Q₈	1	4	yes	—
186	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	C₅×#108	1	8	yes	complemented
191	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
192	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
195	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
196	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
197	C₂×C₄	C₅×#127	1	2	yes	—
198	C₂×C₄	C₅×#127	1	2	yes	—
199	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
200	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
201	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
202	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
203	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
204	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
206	C₈	—	4	8	no	—
207	C₈	C₂×C₅×C₄⋊C₄	1	16	yes	—
208	C₅	C₂×#1848	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²	—	2	8	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	C₅×#384	1	2	yes	—
213	C₂²	C₅×#384	1	2	yes	—
214	C₂²	C₂×C₅×#108	1	4	yes	—
215	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
216	C₂²	C₂×C₅×C₄⋊Q₈	1	2	yes	—
217	C₄	—	2	4	no	—
218	C₄	C₅×#384	1	2	yes	—
219	C₄	C₅×#384	1	2	yes	—
220	C₄	C₂×C₅×#108	1	8	yes	complemented
221	C₄	C₂²×C₅×C₄⋊C₄	1	2	yes	—
222	C₄	C₂×C₅×#127	1	1	yes	characteristic
223	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	2	2	no	—
226	C₂	C₅×#1848	1	4	yes	direct product factor
227	C₂	C₂×C₅×#384	1	1	yes	characteristic
228	C₂	C₂²×C₅×#108	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×#1848	C₂	direct (N × H)
17. C₂×#1848	C₅	direct (N × H)
43. #1848	C₂×C₅	direct (N × H)
93. C₂×C₄×C₅	#108	semidirect
144. C₄×C₅	C₂×#108	semidirect
179. C₂×C₅	#1848	direct (N × H)
190. C₂×C₄	C₅×#108	semidirect
208. C₅	C₂×#1848	direct (N × H)
220. C₄	C₂×C₅×#108	semidirect
226. C₂	C₅×#1848	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×C₄×C₅×C₈	2
C₅×#1848	2
C₂×#1848	5

Subgroups — C2×C5×#1848 #11943524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1848 #11943524