Subgroups — C₄×#90990 #14296639

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#90990	C₂	1	2	yes	maximal
5	#11745980	C₂	1	2	yes	maximal
6	C₄×#3950	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₅	—	2	1	no	—
10	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
13	C₂×C₃×C₄×Dic₅	—	2	1	no	—
14	#91115	—	2	1	no	—
15	C₄×C₅×C₃⋊C₈	C₄	1	4	yes	complemented
16	C₃×#3963	—	2	1	no	—
17	#90990	C₄	1	8	yes	direct product factor
18	C₄×#807	—	5	1	no	maximal, Hall
19	C₂×C₄×C₅×C₈	—	3	1	no	—
20	C₂×#3950	—	3	2	no	—
21	C₄×C₄⋊Dic₅	—	3	1	no	—
22	C₄×C₄⋊Dic₅	D₃	1	1	yes	characteristic
23	#64771	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₅	—	2	2	no	—
28	C₂²×C₃×Dic₅	—	2	2	no	—
29	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₃×C₄²×C₅	Q₈	1	2	yes	—
31	C₄⋊Dic₁₅	—	2	2	no	—
32	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₅	—	4	2	no	—
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
38	C₃×C₄⋊Dic₅	—	2	2	no	—
39	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
40	C₃×C₄×Dic₅	—	4	2	no	—
41	C₃×C₄×C₄⋊C₄	—	5	1	no	—
42	C₂×#807	—	5	2	no	—
43	C₄×C₄⋊Dic₃	—	5	1	no	—
44	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic
45	#4529	—	5	2	no	—
46	C₂²×C₅×C₈	—	3	1	no	—
47	C₂²×C₅×C₈	—	3	1	no	—
48	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
49	C₂×C₄⋊Dic₅	—	3	2	no	—
50	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
51	C₂×C₄×Dic₅	—	2	1	no	—
52	C₂×C₄×Dic₅	—	6	1	no	—
53	C₄×C₅×C₈	—	3	4	no	—
54	#3963	—	2	1	no	—
55	#3963	—	6	1	no	—
56	#3950	—	3	8	no	—
57	C₄×#107	—	15	1	no	Sylow
58	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
63	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	2	yes	—
68	C₂×Dic₁₅	—	2	4	no	—
69	C₂×Dic₁₅	—	4	2	no	—
70	C₂×Dic₁₅	—	4	2	no	—
71	C₂×C₃×Dic₅	—	2	4	no	—
72	C₂×C₃×Dic₅	—	4	2	no	—
73	C₂×C₃×Dic₅	—	4	2	no	—
74	C₅×C₃⋊C₈	—	2	2	no	—
75	C₅×C₃⋊C₈	C₄²	1	4	yes	complemented
76	C₂×C₃×C₄⋊C₄	—	5	2	no	—
77	C₂×C₃×C₄²	—	10	1	no	—
78	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
79	C₂×C₄⋊Dic₃	—	5	2	no	—
80	C₂×C₄×Dic₃	—	10	1	no	—
81	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
82	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
83	C₃×#95	—	10	1	no	—
84	C₃⋊#95	—	10	1	no	—
85	#807	—	5	8	no	—
86	C₄×C₃⋊C₈	Dic₅	1	4	yes	complemented
87	C₂²×C₄×C₅	#67	1	1	yes	characteristic
88	C₂²×C₄×C₅	#67	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
90	C₂²×Dic₅	—	2	2	no	—
91	C₂²×Dic₅	—	6	2	no	—
92	C₂×C₅×C₈	—	3	4	no	—
93	C₂×C₅×C₈	—	3	2	no	—
94	C₂×C₅×C₈	—	3	2	no	—
95	C₂×C₅×C₈	—	6	2	no	—
96	C₄²×C₅	Dic₆	1	2	yes	—
97	C₄²×C₅	C₄×D₃	1	2	yes	—
98	C₄⋊Dic₅	—	2	2	no	—
99	C₄⋊Dic₅	—	3	4	no	—
100	C₄⋊Dic₅	—	6	2	no	—
101	C₄⋊Dic₅	C₄×D₃	1	4	yes	—
102	C₄×Dic₅	—	4	2	no	—
103	C₄×Dic₅	—	12	2	no	—
104	C₂×#107	—	15	2	no	—
105	C₂×C₄×C₈	—	3	1	no	—
106	C₄×C₄⋊C₄	—	5	1	no	—
107	C₄×C₄⋊C₄	—	15	1	no	—
108	#340	—	15	2	no	—
109	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
111	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
113	C₂²×C₃×C₅	#135	1	2	yes	—
114	C₃×C₄×C₅	—	2	2	no	—
115	C₃×C₄×C₅	#107	1	4	yes	complemented
116	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
117	C₃×C₄×C₅	#117	1	1	yes	characteristic
118	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
119	Dic₁₅	—	4	4	no	—
120	C₃×Dic₅	—	4	4	no	—
121	C₂²×C₃×C₄	—	10	2	no	—
122	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
123	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
125	C₂²×Dic₃	—	10	2	no	—
126	C₃×C₄⋊C₄	—	5	4	no	—
127	C₃×C₄⋊C₄	—	10	2	no	—
128	C₃×C₄²	—	20	2	no	—
129	C₃×C₄²	Dic₁₀	1	2	yes	—
130	C₃×C₄²	C₂×Dic₅	1	2	yes	—
131	C₄⋊Dic₃	—	5	4	no	—
132	C₄⋊Dic₃	—	10	2	no	—
133	C₄×Dic₃	—	20	2	no	—
134	C₂×C₃⋊C₈	—	2	2	no	—
135	C₂×C₃⋊C₈	C₄×D₅	1	4	yes	—
136	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
137	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
138	C₂³×C₅	C₂×#67	1	1	yes	characteristic
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
141	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
142	C₂×C₄×C₅	#213	1	2	yes	—
143	C₂×C₄×C₅	#216	1	2	yes	—
144	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂×C₄×C₅	#235	1	2	yes	—
148	C₂×Dic₅	—	2	4	no	—
149	C₂×Dic₅	—	4	2	no	—
150	C₂×Dic₅	—	4	2	no	—
151	C₂×Dic₅	—	6	4	no	—
152	C₂×Dic₅	—	12	2	no	—
153	C₂×Dic₅	—	12	2	no	—
154	C₅×C₈	—	3	4	no	—
155	C₅×C₈	—	6	2	no	—
156	C₂²×C₈	—	3	1	no	—
157	C₂²×C₈	—	3	1	no	—
158	C₂×C₄⋊C₄	—	5	2	no	—
159	C₂×C₄⋊C₄	—	15	2	no	—
160	C₂×C₄²	—	10	1	no	—
161	C₂×C₄²	—	30	1	no	—
162	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
163	#107	—	15	8	no	—
164	C₄×C₈	—	3	4	no	—
165	#95	—	10	1	no	—
166	#95	—	30	1	no	—
167	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
169	C₂×C₃×C₅	#427	1	1	yes	characteristic
170	C₂×C₃×C₅	C₄×D₈	1	2	yes	—
171	C₂×C₃×C₅	C₄×Q₁₆	1	2	yes	—
172	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	10	4	no	—
175	C₂×C₃×C₄	—	20	2	no	—
176	C₂×C₃×C₄	—	20	2	no	—
177	C₂×C₃×C₄	D₄₀	1	2	yes	—
178	C₂×C₃×C₄	Dic₂₀	1	2	yes	—
179	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
180	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
181	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
183	C₂×C₃×C₄	#709	1	2	yes	—
184	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
185	C₂×Dic₃	—	10	4	no	—
186	C₂×Dic₃	—	20	2	no	—
187	C₂×Dic₃	—	20	2	no	—
188	C₃⋊C₈	—	2	2	no	—
189	C₃⋊C₈	C₄×Dic₅	1	4	yes	complemented
190	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
191	C₂²×C₅	C₄×#67	1	2	yes	—
192	C₂²×C₅	C₂×#213	1	1	yes	characteristic
193	C₂²×C₅	C₂×#216	1	1	yes	characteristic
194	C₂²×C₅	#950	1	2	yes	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	#807	1	4	yes	complemented
197	C₄×C₅	C₄×Dic₆	1	2	yes	—
198	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
199	C₄×C₅	#872	1	1	yes	characteristic
200	Dic₅	—	4	4	no	—
201	Dic₅	—	12	4	no	—
202	C₂²×C₄	—	10	2	no	—
203	C₂²×C₄	—	30	2	no	—
204	C₂²×C₄	#1202	1	1	yes	characteristic
205	C₂²×C₄	#1202	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
207	C₂×C₈	—	3	4	no	—
208	C₂×C₈	—	3	2	no	—
209	C₂×C₈	—	3	2	no	—
210	C₂×C₈	—	6	2	no	—
211	C₄⋊C₄	—	5	4	no	—
212	C₄⋊C₄	—	10	2	no	—
213	C₄⋊C₄	—	15	4	no	—
214	C₄⋊C₄	—	30	2	no	—
215	C₄²	—	20	2	no	—
216	C₄²	—	60	2	no	—
217	C₄²	D₃×Dic₅	1	2	yes	—
218	C₄²	#1204	1	2	yes	—
219	C₃×C₅	C₄×#107	1	1	yes	characteristic, complemented, Hall
220	C₂²×C₃	C₄×D₂₀	1	2	yes	—
221	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
222	C₂²×C₃	#4050	1	2	yes	—
223	C₂²×C₃	C₂×Dic₂₀	1	1	yes	characteristic
224	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	20	4	no	—
227	C₃×C₄	#3950	1	4	yes	complemented
228	C₃×C₄	C₄×Dic₁₀	1	2	yes	—
229	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
230	C₃×C₄	#4072	1	1	yes	characteristic
231	Dic₃	—	20	4	no	—
232	C₂×C₅	C₄×Dic₃⋊C₄	1	1	yes	characteristic
233	C₂×C₅	C₂×#807	1	1	yes	characteristic
234	C₂×C₅	#4998	1	1	yes	characteristic
235	C₂×C₅	C₄×#213	1	2	yes	—
236	C₂×C₅	C₄×#216	1	2	yes	—
237	C₂³	C₂×#1202	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	10	4	no	—
240	C₂×C₄	—	20	2	no	—
241	C₂×C₄	—	20	2	no	—
242	C₂×C₄	—	30	4	no	—
243	C₂×C₄	—	60	2	no	—
244	C₂×C₄	—	60	2	no	—
245	C₂×C₄	#6653	1	2	yes	—
246	C₂×C₄	#6662	1	2	yes	—
247	C₂×C₄	Dic₃⋊Dic₅	1	2	yes	—
248	C₂×C₄	Dic₃⋊Dic₅	1	2	yes	—
249	C₂×C₄	#6771	1	2	yes	—
250	C₂×C₄	C₄×D₃×D₅	1	2	yes	—
251	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
252	C₂×C₄	C₂×#1204	1	1	yes	characteristic
253	C₈	—	3	4	no	—
254	C₈	—	6	2	no	—
255	C₂×C₃	C₄×D₄₀	1	2	yes	—
256	C₂×C₃	C₄×Dic₂₀	1	2	yes	—
257	C₂×C₃	C₄×C₄⋊Dic₅	1	1	yes	characteristic
258	C₂×C₃	C₂×#3950	1	1	yes	characteristic
259	C₂×C₃	#65395	1	1	yes	characteristic
260	C₅	C₄×#807	1	1	yes	characteristic, complemented, Sylow
261	C₂²	C₂×#6653	1	1	yes	characteristic
262	C₂²	#91311	1	2	yes	—
263	C₂²	C₂×#6662	1	1	yes	characteristic
264	C₂²	C₄×#1202	1	2	yes	—
265	C₂²	C₂×Dic₃⋊Dic₅	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	20	4	no	—
268	C₄	—	60	4	no	—
269	C₄	#90990	1	4	yes	direct product factor
270	C₄	#91371	1	1	yes	characteristic
271	C₄	C₄×D₃×Dic₅	1	1	yes	characteristic
272	C₄	C₄×#1204	1	2	yes	—
273	C₃	C₄×#3950	1	1	yes	characteristic, complemented, Sylow
274	C₂	C₄×#6653	1	2	yes	—
275	C₂	C₄×#6662	1	2	yes	—
276	C₂	#11748451	1	1	yes	characteristic
277	C₂	C₂×#90990	1	1	yes	characteristic
278	C₂	C₄×Dic₃⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#1202
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₄×#107
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
15. C₄×C₅×C₃⋊C₈	C₄	semidirect
17. #90990	C₄	direct (N × H)
75. C₅×C₃⋊C₈	C₄²	semidirect
86. C₄×C₃⋊C₈	Dic₅	semidirect
115. C₃×C₄×C₅	#107	semidirect
189. C₃⋊C₈	C₄×Dic₅	semidirect
196. C₄×C₅	#807	semidirect
219. C₃×C₅	C₄×#107	semidirect
227. C₃×C₄	#3950	semidirect
260. C₅	C₄×#807	semidirect
269. C₄	#90990	direct (N × H)
273. C₃	C₄×#3950	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Dic₁₅	2
C₂×C₄×C₅×C₃⋊C₈	2
C₃×C₄×C₄⋊Dic₅	2
C₂×#90990	2
#11745980	2
C₄×#3950	3
C₄×#807	5

Subgroups — C4×#90990 #14296639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#90990 #14296639