Subgroups — C₂×#11746905 #14437936

Order 1920.

976 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#874	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4016	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91087	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90966	C₂	1	4	yes	maximal, complemented
5	#11746905	C₂	1	8	yes	maximal, direct product factor
6	C₂×#65360	—	3	1	no	maximal, Hall
7	C₂²×C₅×D₁₂	—	2	2	no	—
8	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊#168	C₂²	1	2	yes	—
11	C₅×#874	—	2	2	no	—
12	C₅×#874	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₂×C₃×#683	—	2	2	no	—
15	C₃×#4016	—	2	2	no	—
16	C₃×#4016	C₂²	1	4	yes	complemented
17	#91087	C₂²	1	4	yes	complemented
18	#90966	C₂²	1	16	yes	complemented
19	C₂×#5067	—	5	1	no	maximal, Hall
20	C₂×C₅×#127	—	3	1	no	—
21	C₂×#4016	D₃	1	1	yes	characteristic, complemented
22	C₂×#3964	—	3	4	no	—
23	#65360	—	3	8	no	—
24	C₂×#3935	—	3	1	no	—
25	C₂³×C₅×D₃	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₅×D₁₂	—	2	8	no	—
29	C₂×C₅×D₁₂	—	2	2	no	—
30	C₂×C₅×D₁₂	—	2	8	no	—
31	C₂×C₅×D₁₂	—	4	8	no	—
32	C₂×C₅×D₁₂	C₂³	1	4	yes	—
33	C₂²×C₃×Dic₅	—	2	2	no	—
34	C₃×C₄²×C₅	D₄	1	2	yes	—
35	C₃×C₄²×C₅	C₂³	1	2	yes	—
36	C₂×C₃⋊#168	—	2	4	no	—
37	C₂×C₃⋊#168	C₂³	1	4	yes	—
38	C₃×C₄⋊Dic₅	—	2	2	no	—
39	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
40	C₃×#683	—	2	8	no	—
41	C₂×C₃×#125	—	5	1	no	—
42	C₂×#874	D₅	1	1	yes	characteristic
43	#5067	—	5	8	no	—
44	C₂×#809	—	5	4	no	—
45	C₂×#803	—	5	1	no	—
46	C₂²×C₅×D₄	—	3	2	no	—
47	C₂²×C₅×D₄	—	6	2	no	—
48	C₅×#127	—	3	4	no	—
49	C₅×#127	—	6	2	no	—
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
52	C₂×#683	—	2	2	no	—
53	C₂²×#168	—	3	2	no	—
54	#4016	—	2	2	no	—
55	#4016	C₂×D₃	1	4	yes	complemented
56	#3964	—	3	16	no	—
57	#3935	—	3	4	no	—
58	C₂×#490	—	15	1	no	Sylow
59	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₅×D₃	—	2	4	no	—
61	C₂²×C₅×D₃	—	4	4	no	—
62	C₂²×C₅×D₃	—	4	8	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	#41	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
68	C₅×D₁₂	—	2	8	no	—
69	C₅×D₁₂	—	4	16	no	—
70	C₅×D₁₂	—	4	4	no	—
71	C₂×C₃×Dic₅	—	2	4	no	—
72	C₂×C₃×Dic₅	—	4	4	no	—
73	C₃⋊#168	—	2	4	no	—
74	C₂²×D₁₂	—	2	2	no	—
75	C₂²×D₁₂	C₂×D₅	1	2	yes	—
76	C₃×#125	—	5	4	no	—
77	C₃×#125	—	10	2	no	—
78	C₂×C₃×C₄⋊C₄	—	5	2	no	—
79	C₂×C₃×C₄⋊C₄	—	10	2	no	—
80	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
81	C₂²×C₃⋊C₈	—	5	2	no	—
82	#874	—	2	2	no	—
83	#874	C₂×D₅	1	4	yes	—
84	#809	—	5	16	no	—
85	#803	—	5	4	no	—
86	C₂⁴×C₅	—	6	2	no	—
87	C₂×C₅×D₄	—	3	4	no	—
88	C₂×C₅×D₄	—	6	8	no	—
89	C₂×C₅×D₄	—	6	8	no	—
90	C₂×C₅×D₄	—	6	2	no	—
91	C₂×C₅×D₄	—	12	8	no	—
92	C₂²×C₄×C₅	#67	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₂²×Dic₅	—	2	2	no	—
95	C₄²×C₅	#67	1	2	yes	—
96	C₄²×C₅	C₂²×D₃	1	2	yes	—
97	C₄⋊Dic₅	—	2	2	no	—
98	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
99	#683	—	2	8	no	—
100	C₂×#168	—	3	4	no	—
101	C₂×#168	—	6	4	no	—
102	C₂×#127	—	3	1	no	—
103	C₂×#125	—	5	1	no	—
104	#490	—	15	8	no	—
105	C₂×#102	—	15	4	no	—
106	C₂×#97	—	15	1	no	—
107	C₂²×C₃×C₅	#124	1	2	yes	—
108	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
109	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₅×D₃	—	4	4	no	—
111	C₂×C₅×D₃	—	4	8	no	—
112	C₂×C₅×D₃	—	8	8	no	—
113	C₃×C₄×C₅	—	2	4	no	—
114	C₃×C₄×C₅	#124	1	2	yes	—
115	C₃×C₄×C₅	C₂×#41	1	2	yes	—
116	C₃×Dic₅	—	4	4	no	—
117	C₂³×D₃	—	2	2	no	—
118	C₂²×C₃×C₄	—	10	2	no	—
119	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
120	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
121	C₂×D₁₂	—	2	8	no	—
122	C₂×D₁₂	—	2	2	no	—
123	C₂×D₁₂	—	2	8	no	—
124	C₂×D₁₂	—	4	8	no	—
125	C₂×D₁₂	C₂²×D₅	1	4	yes	—
126	C₃×C₄⋊C₄	—	5	4	no	—
127	C₃×C₄⋊C₄	—	10	8	no	—
128	C₃×C₄⋊C₄	—	10	2	no	—
129	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
130	C₃×C₄²	C₂²×D₅	1	2	yes	—
131	C₂×C₃⋊C₈	—	5	4	no	—
132	C₂×C₃⋊C₈	—	10	4	no	—
133	C₂³×C₅	—	6	4	no	—
134	C₂³×C₅	—	12	4	no	—
135	C₂³×C₅	—	12	8	no	—
136	C₂³×C₅	C₂×#67	1	1	yes	characteristic
137	C₅×D₄	—	6	8	no	—
138	C₅×D₄	—	12	16	no	—
139	C₅×D₄	—	12	4	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	#239	1	4	yes	—
142	C₂×C₄×C₅	C₂×#67	1	4	yes	—
143	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
145	C₂×Dic₅	—	2	4	no	—
146	C₂×Dic₅	—	4	4	no	—
147	#168	—	6	4	no	—
148	C₂²×D₄	—	3	2	no	—
149	C₂²×D₄	—	6	2	no	—
150	C₂²×C₈	—	15	2	no	—
151	#127	—	3	4	no	—
152	#127	—	6	2	no	—
153	#125	—	5	4	no	—
154	#125	—	10	2	no	—
155	C₂×C₄⋊C₄	—	5	2	no	—
156	C₂×C₄⋊C₄	—	10	2	no	—
157	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
158	#102	—	15	16	no	—
159	#97	—	15	4	no	—
160	C₂×C₃×C₅	#490	1	4	yes	complemented
161	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
162	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
163	C₅×D₃	—	8	4	no	—
164	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
165	C₂²×D₃	—	2	4	no	—
166	C₂²×D₃	—	4	8	no	—
167	C₂²×D₃	—	4	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	10	4	no	—
170	C₂×C₃×C₄	—	20	4	no	—
171	C₂×C₃×C₄	#711	1	4	yes	—
172	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
173	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
175	D₁₂	—	2	8	no	—
176	D₁₂	—	4	16	no	—
177	D₁₂	—	4	4	no	—
178	C₃⋊C₈	—	10	4	no	—
179	C₂²×C₅	—	12	4	no	—
180	C₂²×C₅	—	12	8	no	—
181	C₂²×C₅	—	24	8	no	—
182	C₂²×C₅	#947	1	2	yes	—
183	C₂²×C₅	#949	1	4	yes	—
184	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	#947	1	2	yes	—
187	C₄×C₅	C₂×#239	1	2	yes	—
188	Dic₅	—	4	4	no	—
189	C₂⁴	—	6	2	no	—
190	C₂×D₄	—	3	4	no	—
191	C₂×D₄	—	6	8	no	—
192	C₂×D₄	—	6	8	no	—
193	C₂×D₄	—	6	2	no	—
194	C₂×D₄	—	12	8	no	—
195	C₂²×C₄	—	10	2	no	—
196	C₂²×C₄	#1201	1	2	yes	—
197	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
198	C₂×C₈	—	15	4	no	—
199	C₂×C₈	—	30	4	no	—
200	C₄⋊C₄	—	5	4	no	—
201	C₄⋊C₄	—	10	2	no	—
202	C₄⋊C₄	—	10	8	no	—
203	C₄²	#1201	1	2	yes	—
204	C₄²	C₂×D₃×D₅	1	2	yes	—
205	C₃×C₅	C₂×#490	1	1	yes	characteristic, complemented, Hall
206	C₂²×C₃	#4078	1	4	yes	—
207	C₂²×C₃	#4082	1	2	yes	—
208	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
209	C₂×D₃	—	4	8	no	—
210	C₂×D₃	—	4	4	no	—
211	C₂×D₃	—	8	8	no	—
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	20	4	no	—
214	C₃×C₄	#4082	1	2	yes	—
215	C₃×C₄	C₂×#711	1	2	yes	—
216	C₂×C₅	—	24	4	no	—
217	C₂×C₅	#5067	1	4	yes	complemented
218	C₂×C₅	C₂×#947	1	1	yes	characteristic
219	C₂×C₅	C₂×#949	1	2	yes	—
220	C₂³	—	6	4	no	—
221	C₂³	—	12	8	no	—
222	C₂³	—	12	4	no	—
223	C₂³	C₂×#1201	1	1	yes	characteristic
224	D₄	—	6	8	no	—
225	D₄	—	12	16	no	—
226	D₄	—	12	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	10	4	no	—
229	C₂×C₄	—	20	4	no	—
230	C₂×C₄	#6768	1	4	yes	—
231	C₂×C₄	C₂×#1201	1	4	yes	—
232	C₂×C₄	C₂×#1201	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
234	C₈	—	30	4	no	—
235	C₂×C₃	#65360	1	4	yes	complemented
236	C₂×C₃	C₂×#4078	1	2	yes	—
237	C₂×C₃	C₂×#4082	1	1	yes	characteristic
238	D₃	—	8	4	no	—
239	C₅	C₂×#5067	1	1	yes	characteristic, complemented, Sylow
240	C₂²	—	12	4	no	—
241	C₂²	—	12	8	no	—
242	C₂²	—	24	8	no	—
243	C₂²	#91299	1	4	yes	—
244	C₂²	#91370	1	2	yes	—
245	C₂²	C₂²×#1201	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	20	4	no	—
248	C₄	#91370	1	2	yes	—
249	C₄	C₂×#6768	1	2	yes	—
250	C₃	C₂×#65360	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	24	4	no	—
252	C₂	#11746905	1	4	yes	direct product factor
253	C₂	C₂×#91299	1	2	yes	—
254	C₂	C₂×#91370	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1201
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×#874	C₂
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₂×#490
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#4016	C₂	semidirect
3. C₂×#91087	C₂	semidirect
4. C₂×#90966	C₂	semidirect
5. #11746905	C₂	direct (N × H)
16. C₃×#4016	C₂²	semidirect
17. #91087	C₂²	semidirect
18. #90966	C₂²	semidirect
21. C₂×#4016	D₃	semidirect
55. #4016	C₂×D₃	semidirect
160. C₂×C₃×C₅	#490	semidirect
205. C₃×C₅	C₂×#490	semidirect
217. C₂×C₅	#5067	semidirect
235. C₂×C₃	#65360	semidirect
239. C₅	C₂×#5067	semidirect
250. C₃	C₂×#65360	semidirect
252. C₂	#11746905	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#874	2
C₂×C₃×#4016	2
C₂×#91087	2
C₂×#90966	2
#11746905	2
C₂×#65360	3
C₂×#5067	5

Subgroups — C2×#11746905 #14437936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746905 #14437936