Subgroups — C₂×#11751172 #14437913

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×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×#91759	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91116	C₂	1	4	yes	maximal, complemented
4	#11751172	C₂	1	8	yes	maximal, direct product factor
5	C₂×#91087	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#65360	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×D₄	—	2	2	no	—
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₃×C₅×#127	—	2	2	no	—
10	C₃×C₅×#127	C₂²	1	4	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
13	C₂×#6712	—	2	2	no	—
14	C₂²×C₃⋊#168	C₂²	1	2	yes	—
15	#91759	—	2	2	no	—
16	#91759	C₂²	1	4	yes	complemented
17	#91116	C₂²	1	16	yes	complemented
18	#91087	C₂²	1	4	yes	complemented
19	C₂×#5071	—	5	1	no	maximal, Hall
20	C₂×C₅×#127	D₃	1	1	yes	characteristic
21	C₂×#4016	—	3	1	no	—
22	C₂×#3964	—	3	4	no	—
23	#65360	—	3	8	no	—
24	C₂×#3935	—	3	1	no	—
25	C₂⁴×C₃×C₅	—	2	2	no	—
26	C₂×C₃×C₅×D₄	—	2	8	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	8	no	—
29	C₂×C₃×C₅×D₄	—	4	8	no	—
30	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
31	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₅	—	2	2	no	—
34	C₃×C₄²×C₅	D₄	1	2	yes	—
35	C₃×C₄²×C₅	C₂³	1	2	yes	—
36	C₄⋊Dic₁₅	—	2	2	no	—
37	C₄⋊Dic₁₅	C₂³	1	4	yes	—
38	#6712	—	2	8	no	—
39	C₂×C₃⋊#168	—	2	4	no	—
40	C₂×C₃⋊#168	C₂³	1	4	yes	—
41	C₂×C₃×#127	D₅	1	1	yes	characteristic
42	C₂×#870	—	5	1	no	—
43	C₂×#832	—	5	4	no	—
44	#5071	—	5	8	no	—
45	C₂×#803	—	5	1	no	—
46	C₂²×C₅×D₄	—	2	2	no	—
47	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
48	C₅×#127	—	2	2	no	—
49	C₅×#127	C₂×D₃	1	4	yes	—
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄⋊Dic₅	—	3	2	no	—
52	C₂×#683	—	6	2	no	—
53	C₂²×#168	—	3	2	no	—
54	#4016	—	3	4	no	—
55	#4016	—	6	2	no	—
56	#3964	—	3	16	no	—
57	#3935	—	3	4	no	—
58	C₂×#490	—	15	1	no	Sylow
59	C₂³×C₃×C₅	—	2	4	no	—
60	C₂³×C₃×C₅	—	4	4	no	—
61	C₂³×C₃×C₅	—	4	8	no	—
62	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
63	C₃×C₅×D₄	—	2	8	no	—
64	C₃×C₅×D₄	—	4	16	no	—
65	C₃×C₅×D₄	—	4	4	no	—
66	C₂×C₃×C₄×C₅	—	2	4	no	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	#41	1	4	yes	—
70	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
71	C₂×Dic₁₅	—	2	4	no	—
72	C₂×Dic₁₅	—	4	4	no	—
73	C₃⋊#168	—	2	4	no	—
74	C₂²×C₃×D₄	—	2	2	no	—
75	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
76	C₃×#127	—	2	2	no	—
77	C₃×#127	C₂×D₅	1	4	yes	—
78	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
79	C₂×C₄⋊Dic₃	—	5	2	no	—
80	C₂×Dic₃⋊C₄	—	10	2	no	—
81	C₂²×C₃⋊C₈	—	5	2	no	—
82	#870	—	5	4	no	—
83	#870	—	10	2	no	—
84	#832	—	5	16	no	—
85	#803	—	5	4	no	—
86	C₂⁴×C₅	—	2	2	no	—
87	C₂×C₅×D₄	—	2	8	no	—
88	C₂×C₅×D₄	—	2	2	no	—
89	C₂×C₅×D₄	—	2	8	no	—
90	C₂×C₅×D₄	—	4	8	no	—
91	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
92	C₂²×C₄×C₅	#67	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₂²×Dic₅	—	6	2	no	—
95	C₄²×C₅	#67	1	2	yes	—
96	C₄²×C₅	C₂²×D₃	1	2	yes	—
97	C₄⋊Dic₅	—	3	4	no	—
98	C₄⋊Dic₅	—	6	2	no	—
99	#683	—	6	8	no	—
100	C₂×#168	—	3	4	no	—
101	C₂×#168	—	6	4	no	—
102	C₂×#127	D₁₅	1	1	yes	characteristic
103	C₂×#125	—	15	1	no	—
104	#490	—	15	8	no	—
105	C₂×#102	—	15	4	no	—
106	C₂×#97	—	15	1	no	—
107	C₂²×C₃×C₅	—	4	4	no	—
108	C₂²×C₃×C₅	—	4	8	no	—
109	C₂²×C₃×C₅	—	8	8	no	—
110	C₂²×C₃×C₅	#124	1	2	yes	—
111	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
112	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
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	Dic₁₅	—	4	4	no	—
117	C₂⁴×C₃	—	2	2	no	—
118	C₂×C₃×D₄	—	2	8	no	—
119	C₂×C₃×D₄	—	2	2	no	—
120	C₂×C₃×D₄	—	2	8	no	—
121	C₂×C₃×D₄	—	4	8	no	—
122	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
123	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
124	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
125	C₂²×Dic₃	—	10	2	no	—
126	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
127	C₃×C₄²	C₂²×D₅	1	2	yes	—
128	C₄⋊Dic₃	—	5	4	no	—
129	C₄⋊Dic₃	—	10	2	no	—
130	Dic₃⋊C₄	—	10	8	no	—
131	C₂×C₃⋊C₈	—	5	4	no	—
132	C₂×C₃⋊C₈	—	10	4	no	—
133	C₂³×C₅	—	2	4	no	—
134	C₂³×C₅	—	4	8	no	—
135	C₂³×C₅	—	4	4	no	—
136	C₂³×C₅	C₂×#67	1	1	yes	characteristic
137	C₅×D₄	—	2	8	no	—
138	C₅×D₄	—	4	16	no	—
139	C₅×D₄	—	4	4	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	#237	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	—	6	4	no	—
146	C₂×Dic₅	—	12	4	no	—
147	#168	—	6	4	no	—
148	C₂²×D₄	—	2	2	no	—
149	C₂²×D₄	C₂×D₁₅	1	2	yes	—
150	C₂²×C₈	—	15	2	no	—
151	#127	—	2	2	no	—
152	#127	C₂×D₁₅	1	4	yes	—
153	#125	—	15	4	no	—
154	#125	—	30	2	no	—
155	C₂×C₄⋊C₄	—	15	2	no	—
156	C₂×C₄⋊C₄	—	30	2	no	—
157	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
158	#102	—	15	16	no	—
159	#97	—	15	4	no	—
160	C₂×C₃×C₅	—	8	4	no	—
161	C₂×C₃×C₅	#490	1	4	yes	complemented
162	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
163	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
164	C₂³×C₃	—	2	4	no	—
165	C₂³×C₃	—	4	8	no	—
166	C₂³×C₃	—	4	4	no	—
167	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
168	C₃×D₄	—	2	8	no	—
169	C₃×D₄	—	4	16	no	—
170	C₃×D₄	—	4	4	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	#711	1	4	yes	—
173	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
174	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
176	C₂×Dic₃	—	10	4	no	—
177	C₂×Dic₃	—	20	4	no	—
178	C₃⋊C₈	—	10	4	no	—
179	C₂²×C₅	—	4	8	no	—
180	C₂²×C₅	—	4	4	no	—
181	C₂²×C₅	—	8	8	no	—
182	C₂²×C₅	#932	1	4	yes	—
183	C₂²×C₅	#936	1	2	yes	—
184	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	#936	1	2	yes	—
187	C₄×C₅	C₂×#237	1	2	yes	—
188	Dic₅	—	12	4	no	—
189	C₂⁴	—	2	2	no	—
190	C₂×D₄	—	2	8	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	8	no	—
193	C₂×D₄	—	4	8	no	—
194	C₂×D₄	C₂²×D₁₅	1	4	yes	—
195	C₂²×C₄	—	30	2	no	—
196	C₂²×C₄	#1220	1	2	yes	—
197	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
198	C₂×C₈	—	15	4	no	—
199	C₂×C₈	—	30	4	no	—
200	C₄⋊C₄	—	15	4	no	—
201	C₄⋊C₄	—	30	2	no	—
202	C₄⋊C₄	—	30	8	no	—
203	C₄²	#1220	1	2	yes	—
204	C₄²	C₂²×D₁₅	1	2	yes	—
205	C₃×C₅	C₂×#490	1	1	yes	characteristic, complemented, Hall
206	C₂²×C₃	—	4	8	no	—
207	C₂²×C₃	—	4	4	no	—
208	C₂²×C₃	—	8	8	no	—
209	C₂²×C₃	#4078	1	4	yes	—
210	C₂²×C₃	#4082	1	2	yes	—
211	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	#4082	1	2	yes	—
214	C₃×C₄	C₂×#711	1	2	yes	—
215	Dic₃	—	20	4	no	—
216	C₂×C₅	—	8	4	no	—
217	C₂×C₅	#5071	1	4	yes	complemented
218	C₂×C₅	C₂×#932	1	2	yes	—
219	C₂×C₅	C₂×#936	1	1	yes	characteristic
220	C₂³	—	2	4	no	—
221	C₂³	—	4	8	no	—
222	C₂³	—	4	4	no	—
223	C₂³	C₂×#1220	1	1	yes	characteristic
224	D₄	—	2	8	no	—
225	D₄	—	4	16	no	—
226	D₄	—	4	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	30	4	no	—
229	C₂×C₄	—	60	4	no	—
230	C₂×C₄	#6819	1	4	yes	—
231	C₂×C₄	C₂×#1220	1	4	yes	—
232	C₂×C₄	C₂×#1220	1	1	yes	characteristic
233	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
234	C₈	—	30	4	no	—
235	C₂×C₃	—	8	4	no	—
236	C₂×C₃	#65360	1	4	yes	complemented
237	C₂×C₃	C₂×#4078	1	2	yes	—
238	C₂×C₃	C₂×#4082	1	1	yes	characteristic
239	C₅	C₂×#5071	1	1	yes	characteristic, complemented, Sylow
240	C₂²	—	4	8	no	—
241	C₂²	—	4	4	no	—
242	C₂²	—	8	8	no	—
243	C₂²	#91821	1	4	yes	—
244	C₂²	#91825	1	2	yes	—
245	C₂²	C₂²×#1220	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	60	4	no	—
248	C₄	#91825	1	2	yes	—
249	C₄	C₂×#6819	1	2	yes	—
250	C₃	C₂×#65360	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	8	4	no	—
252	C₂	#11751172	1	4	yes	direct product factor
253	C₂	C₂×#91821	1	2	yes	—
254	C₂	C₂×#91825	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1220
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting, prime radical	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×#127	D₁₅
hypocenter	C₃×C₅	C₂×#490
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
2. C₂×#91759	C₂	semidirect
3. C₂×#91116	C₂	semidirect
4. #11751172	C₂	direct (N × H)
5. C₂×#91087	C₂	semidirect
16. #91759	C₂²	semidirect
17. #91116	C₂²	semidirect
18. #91087	C₂²	semidirect
161. C₂×C₃×C₅	#490	semidirect
205. C₃×C₅	C₂×#490	semidirect
217. C₂×C₅	#5071	semidirect
236. C₂×C₃	#65360	semidirect
239. C₅	C₂×#5071	semidirect
250. C₃	C₂×#65360	semidirect
252. C₂	#11751172	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×C₃×C₅×#127	2
C₂×#91759	2
C₂×#91116	2
#11751172	2
C₂×#91087	2
C₂×#65360	3
C₂×#5071	5

Subgroups — C2×#11751172 #14437913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751172 #14437913