Subgroups — C₂×#11746896 #14437762

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#870	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4021	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90970	C₂	1	2	yes	maximal, complemented
5	C₂×#90964	C₂	1	2	yes	maximal
6	#11746896	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65357	—	3	1	no	maximal, Hall
8	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊#168	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₅×Dic₃⋊C₄	—	2	2	no	—
14	C₅×#870	—	2	2	no	—
15	C₅×#870	C₂²	1	4	yes	complemented
16	C₂×C₃×#685	—	2	2	no	—
17	C₃×#4021	—	2	2	no	—
18	C₃×#4021	C₂²	1	4	yes	—
19	C₄×C₃⋊#168	C₂²	1	4	yes	complemented
20	#90970	C₂²	1	8	yes	complemented
21	#90964	C₂²	1	8	yes	—
22	C₂×#5056	—	5	1	no	maximal, Hall
23	C₂×C₅×#125	—	3	1	no	—
24	C₂×#4021	D₃	1	1	yes	characteristic
25	#65357	—	3	8	no	—
26	C₂×#3942	—	3	2	no	—
27	C₂×#3941	—	3	2	no	—
28	C₂×C₄×#168	—	3	1	no	—
29	C₂³×C₃×D₅	—	2	1	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×Dic₃	—	2	2	no	—
33	C₂²×C₃×Dic₅	—	2	1	no	—
34	C₂×C₃×D₂₀	—	2	4	no	—
35	C₂×C₃×D₂₀	—	2	1	no	—
36	C₂×C₃×D₂₀	C₂³	1	2	yes	—
37	C₂×C₃×Dic₁₀	—	2	4	no	—
38	C₂×C₃×Dic₁₀	—	2	1	no	—
39	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₃×C₄²×C₅	C₂³	1	2	yes	—
42	C₂×C₃⋊#168	—	2	4	no	—
43	C₂×C₃⋊#168	C₂³	1	4	yes	—
44	C₅×C₄⋊Dic₃	—	2	2	no	—
45	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
46	C₅×Dic₃⋊C₄	—	2	8	no	—
47	C₃×#685	—	2	8	no	—
48	C₂×C₃×#124	—	5	1	no	—
49	C₂×#870	D₅	1	1	yes	characteristic, complemented
50	C₂×#835	—	5	2	no	—
51	C₂×#832	—	5	2	no	—
52	#5056	—	5	8	no	—
53	C₂×C₄×C₃⋊C₈	—	5	1	no	—
54	C₂²×D₂₀	C₂×D₃	1	1	yes	characteristic
55	C₂²×Dic₁₀	C₂×D₃	1	1	yes	characteristic
56	C₅×#125	—	3	4	no	—
57	C₅×#125	—	6	2	no	—
58	C₂×C₅×C₄⋊C₄	—	3	2	no	—
59	C₂×C₅×C₄⋊C₄	—	6	2	no	—
60	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
61	C₂×#685	—	2	2	no	—
62	C₂²×#168	—	3	2	no	—
63	#4021	—	2	2	no	—
64	#4021	C₂×D₃	1	4	yes	—
65	#3942	—	3	8	no	—
66	#3941	—	3	8	no	—
67	C₄×#168	—	3	4	no	—
68	C₂×#488	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×D₅	—	2	2	no	—
71	C₂²×C₃×D₅	—	4	2	no	—
72	C₂²×C₃×D₅	—	4	4	no	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	#41	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×C₅×Dic₃	—	2	4	no	—
79	C₂×C₅×Dic₃	—	4	4	no	—
80	C₂×C₃×Dic₅	—	2	2	no	—
81	C₂×C₃×Dic₅	—	4	2	no	—
82	C₃×D₂₀	—	2	4	no	—
83	C₃×D₂₀	—	4	2	no	—
84	C₃×Dic₁₀	—	2	4	no	—
85	C₃×Dic₁₀	—	4	2	no	—
86	C₃⋊#168	—	2	4	no	—
87	C₂²×C₃×Q₈	—	5	1	no	—
88	C₂²×C₃×D₄	—	5	1	no	—
89	C₃×#124	—	5	4	no	—
90	C₃×#124	—	10	2	no	—
91	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
92	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
93	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
94	C₂×Dic₃⋊C₄	—	2	2	no	—
95	C₂²×C₃⋊C₈	—	5	2	no	—
96	#870	—	2	2	no	—
97	#870	C₂×D₅	1	4	yes	complemented
98	#835	—	5	8	no	—
99	#832	—	5	8	no	—
100	C₄×C₃⋊C₈	—	5	4	no	—
101	C₂³×D₅	—	2	1	no	—
102	C₂²×C₄×C₅	—	6	2	no	—
103	C₂²×C₄×C₅	#67	1	2	yes	—
104	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
105	C₂²×Dic₅	—	2	1	no	—
106	C₂×D₂₀	—	2	4	no	—
107	C₂×D₂₀	—	2	1	no	—
108	C₂×D₂₀	C₂²×D₃	1	2	yes	—
109	C₂×Dic₁₀	—	2	4	no	—
110	C₂×Dic₁₀	—	2	1	no	—
111	C₂×Dic₁₀	C₂²×D₃	1	2	yes	—
112	C₅×C₄⋊C₄	—	3	4	no	—
113	C₅×C₄⋊C₄	—	6	2	no	—
114	C₅×C₄⋊C₄	—	6	8	no	—
115	C₄²×C₅	#67	1	2	yes	—
116	C₄²×C₅	C₂²×D₃	1	2	yes	—
117	#685	—	2	8	no	—
118	C₂×#168	—	3	4	no	—
119	C₂×#168	—	6	4	no	—
120	C₂×#125	—	3	1	no	—
121	C₂×#124	—	5	1	no	—
122	#488	—	15	8	no	—
123	C₂×#103	—	15	2	no	—
124	C₂×#102	—	15	2	no	—
125	C₂×C₄×C₈	—	15	1	no	—
126	C₂²×C₃×C₅	#124	1	2	yes	—
127	C₂²×C₃×C₅	#135	1	4	yes	—
128	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₃×D₅	—	4	2	no	—
130	C₂×C₃×D₅	—	4	4	no	—
131	C₂×C₃×D₅	—	8	4	no	—
132	C₃×C₄×C₅	—	2	4	no	—
133	C₃×C₄×C₅	#124	1	2	yes	—
134	C₃×C₄×C₅	C₂×#41	1	2	yes	—
135	C₃×Dic₅	—	4	2	no	—
136	C₅×Dic₃	—	4	4	no	—
137	C₂⁴×C₃	—	10	1	no	—
138	C₂×C₃×Q₈	—	5	2	no	—
139	C₂×C₃×Q₈	—	10	1	no	—
140	C₂×C₃×Q₈	—	10	4	no	—
141	C₂×C₃×D₄	—	5	2	no	—
142	C₂×C₃×D₄	—	10	4	no	—
143	C₂×C₃×D₄	—	10	1	no	—
144	C₂²×C₃×C₄	—	10	1	no	—
145	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
146	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₂²×Dic₃	—	2	2	no	—
148	C₃×C₂²⋊C₄	—	10	8	no	—
149	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
150	C₃×C₄²	C₂²×D₅	1	2	yes	—
151	C₄⋊Dic₃	—	2	2	no	—
152	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
153	Dic₃⋊C₄	—	2	8	no	—
154	C₂×C₃⋊C₈	—	5	4	no	—
155	C₂×C₃⋊C₈	—	10	4	no	—
156	C₂³×C₅	C₂×#67	1	1	yes	characteristic
157	C₂²×D₅	—	2	2	no	—
158	C₂²×D₅	—	4	2	no	—
159	C₂²×D₅	—	4	4	no	—
160	C₂×C₄×C₅	—	2	4	no	—
161	C₂×C₄×C₅	—	6	4	no	—
162	C₂×C₄×C₅	—	12	4	no	—
163	C₂×C₄×C₅	#237	1	4	yes	—
164	C₂×C₄×C₅	C₂×#67	1	4	yes	—
165	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
167	C₂×Dic₅	—	2	2	no	—
168	C₂×Dic₅	—	4	2	no	—
169	D₂₀	—	2	4	no	—
170	D₂₀	—	4	2	no	—
171	Dic₁₀	—	2	4	no	—
172	Dic₁₀	—	4	2	no	—
173	#168	—	6	4	no	—
174	C₂²×Q₈	—	5	1	no	—
175	C₂²×D₄	—	5	1	no	—
176	C₂²×C₈	—	15	2	no	—
177	#125	—	3	4	no	—
178	#125	—	6	2	no	—
179	#124	—	5	4	no	—
180	#124	—	10	2	no	—
181	C₂×C₄⋊C₄	—	3	2	no	—
182	C₂×C₄⋊C₄	—	6	2	no	—
183	C₂×C₂²⋊C₄	—	10	2	no	—
184	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
185	#103	—	15	8	no	—
186	#102	—	15	8	no	—
187	C₄×C₈	—	15	4	no	—
188	C₂×C₃×C₅	#488	1	4	yes	complemented
189	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×#135	1	2	yes	—
191	C₃×D₅	—	8	2	no	—
192	C₂³×C₃	—	10	2	no	—
193	C₂³×C₃	—	20	4	no	—
194	C₂³×C₃	—	20	2	no	—
195	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
196	C₃×Q₈	—	10	4	no	—
197	C₃×Q₈	—	20	2	no	—
198	C₃×D₄	—	10	4	no	—
199	C₃×D₄	—	20	2	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	10	2	no	—
202	C₂×C₃×C₄	—	20	2	no	—
203	C₂×C₃×C₄	#713	1	4	yes	—
204	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
205	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
207	C₂×Dic₃	—	2	4	no	—
208	C₂×Dic₃	—	4	4	no	—
209	C₃⋊C₈	—	10	4	no	—
210	C₂²×C₅	#936	1	2	yes	—
211	C₂²×C₅	#950	1	4	yes	—
212	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
213	C₂×D₅	—	4	2	no	—
214	C₂×D₅	—	4	4	no	—
215	C₂×D₅	—	8	4	no	—
216	C₄×C₅	—	2	4	no	—
217	C₄×C₅	—	12	4	no	—
218	C₄×C₅	#936	1	2	yes	—
219	C₄×C₅	C₂×#237	1	2	yes	—
220	Dic₅	—	4	2	no	—
221	C₂⁴	—	10	1	no	—
222	C₂×Q₈	—	5	2	no	—
223	C₂×Q₈	—	10	1	no	—
224	C₂×Q₈	—	10	4	no	—
225	C₂×D₄	—	5	2	no	—
226	C₂×D₄	—	10	4	no	—
227	C₂×D₄	—	10	1	no	—
228	C₂²×C₄	—	6	2	no	—
229	C₂²×C₄	—	10	1	no	—
230	C₂²×C₄	#1201	1	2	yes	—
231	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
232	C₂×C₈	—	15	4	no	—
233	C₂×C₈	—	30	4	no	—
234	C₄⋊C₄	—	3	4	no	—
235	C₄⋊C₄	—	6	2	no	—
236	C₄⋊C₄	—	6	8	no	—
237	C₂²⋊C₄	—	10	8	no	—
238	C₄²	#1201	1	2	yes	—
239	C₄²	C₂×D₃×D₅	1	2	yes	—
240	C₃×C₅	C₂×#488	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	20	2	no	—
242	C₂²×C₃	—	20	4	no	—
243	C₂²×C₃	—	40	4	no	—
244	C₂²×C₃	#4093	1	2	yes	—
245	C₂²×C₃	#4096	1	4	yes	—
246	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
247	C₃×C₄	—	2	4	no	—
248	C₃×C₄	—	20	2	no	—
249	C₃×C₄	#4093	1	2	yes	—
250	C₃×C₄	C₂×#713	1	2	yes	—
251	Dic₃	—	4	4	no	—
252	C₂×C₅	#5056	1	4	yes	complemented
253	C₂×C₅	C₂×#936	1	1	yes	characteristic
254	C₂×C₅	C₂×#950	1	2	yes	—
255	D₅	—	8	2	no	—
256	C₂³	—	10	2	no	—
257	C₂³	—	20	4	no	—
258	C₂³	—	20	2	no	—
259	C₂³	C₂×#1201	1	1	yes	characteristic
260	Q₈	—	10	4	no	—
261	Q₈	—	20	2	no	—
262	D₄	—	10	4	no	—
263	D₄	—	20	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	6	4	no	—
266	C₂×C₄	—	10	2	no	—
267	C₂×C₄	—	12	4	no	—
268	C₂×C₄	—	20	2	no	—
269	C₂×C₄	#6766	1	4	yes	—
270	C₂×C₄	C₂×#1201	1	4	yes	—
271	C₂×C₄	C₂×#1201	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
273	C₈	—	30	4	no	—
274	C₂×C₃	—	40	2	no	—
275	C₂×C₃	#65357	1	4	yes	complemented
276	C₂×C₃	C₂×#4093	1	1	yes	characteristic
277	C₂×C₃	C₂×#4096	1	2	yes	—
278	C₅	C₂×#5056	1	1	yes	characteristic, complemented, Sylow
279	C₂²	—	20	2	no	—
280	C₂²	—	20	4	no	—
281	C₂²	—	40	4	no	—
282	C₂²	#91297	1	4	yes	—
283	C₂²	#91368	1	2	yes	—
284	C₂²	C₂²×#1201	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	12	4	no	—
287	C₄	—	20	2	no	—
288	C₄	#91368	1	2	yes	—
289	C₄	C₂×#6766	1	2	yes	—
290	C₃	C₂×#65357	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	40	2	no	—
292	C₂	#11746896	1	4	yes	direct product factor
293	C₂	C₂×#91297	1	2	yes	—
294	C₂	C₂×#91368	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	C₂²×C₃×D₂₀	C₂²
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₂×#488
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
1. C₂×C₅×#870	C₂	semidirect
3. C₂×C₄×C₃⋊#168	C₂	semidirect
4. C₂×#90970	C₂	semidirect
6. #11746896	C₂	direct (N × H)
15. C₅×#870	C₂²	semidirect
19. C₄×C₃⋊#168	C₂²	semidirect
20. #90970	C₂²	semidirect
49. C₂×#870	D₅	semidirect
97. #870	C₂×D₅	semidirect
188. C₂×C₃×C₅	#488	semidirect
240. C₃×C₅	C₂×#488	semidirect
252. C₂×C₅	#5056	semidirect
275. C₂×C₃	#65357	semidirect
278. C₅	C₂×#5056	semidirect
290. C₃	C₂×#65357	semidirect
292. C₂	#11746896	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×#870	2
C₂×C₃×#4021	2
C₂×C₄×C₃⋊#168	2
C₂×#90970	2
C₂×#90964	2
#11746896	2
C₂×#65357	3
C₂×#5056	5

Subgroups — C2×#11746896 #14437762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746896 #14437762