Subgroups — C₄×C₅×#827 #14315346

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#827	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4562	C₂	1	1	yes	maximal, characteristic
5	C₅×#4492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×SD₃₂	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅×D₄	—	2	1	no	—
11	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic
12	C₄×C₅×Dic₆	—	2	1	no	—
13	C₃×C₅×#107	C₂²	1	1	yes	characteristic
14	C₃×C₅×#102	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#827	C₄	1	4	yes	direct product factor
17	C₅×#818	C₂²	1	1	yes	characteristic
18	C₅×#816	—	2	1	no	—
19	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₄×#827	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×SD₃₂	—	3	1	no	—
23	C₄×C₅×Q₁₆	—	3	1	no	—
24	C₄×C₅×D₈	D₃	1	1	yes	characteristic
25	C₅×#367	—	3	1	no	—
26	C₅×#358	—	3	1	no	—
27	C₅×#357	—	3	1	no	—
28	C₄×C₅×C₁₆	—	3	1	no	—
29	C₂×C₃×C₅×D₄	—	2	1	no	—
30	C₂²×C₃×C₄×C₅	—	4	1	no	—
31	C₂×C₅×Dic₆	—	2	1	no	—
32	C₃×C₅×D₈	—	2	1	no	—
33	C₃×C₅×D₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₃×C₅×C₄⋊C₄	—	2	1	no	—
37	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₅×C₄⋊Dic₃	—	2	1	no	—
40	C₅×Dic₃⋊C₄	—	4	1	no	—
41	C₄×C₅×Dic₃	—	4	1	no	—
42	C₅×Dic₁₂	—	2	1	no	—
43	C₅×Dic₁₂	C₂×C₄	1	2	yes	complemented
44	C₅×C₃⋊C₁₆	—	2	1	no	—
45	C₅×C₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
46	C₃×C₄×D₈	C₂×C₅	1	1	yes	characteristic
47	C₂×#827	C₂×C₅	1	1	yes	characteristic
48	C₄×Dic₁₂	C₂×C₅	1	1	yes	characteristic, complemented
49	#4562	C₂×C₅	1	1	yes	characteristic
50	#4492	C₂×C₅	1	1	yes	characteristic, complemented
51	#4490	C₂×C₅	1	1	yes	characteristic, complemented
52	C₄×C₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂×C₅×Q₁₆	—	3	1	no	—
54	C₂×C₅×D₈	C₂×D₃	1	1	yes	characteristic
55	C₄×C₅×Q₈	—	6	1	no	—
56	C₄×C₅×D₄	—	2	1	no	—
57	C₅×SD₃₂	—	3	4	no	—
58	C₂×C₅×C₁₆	—	3	1	no	—
59	C₂×C₅×C₁₆	—	3	1	no	—
60	C₅×#107	—	3	1	no	—
61	C₅×#107	C₂×D₃	1	1	yes	characteristic
62	C₅×#103	—	6	1	no	—
63	C₅×#102	—	2	1	no	—
64	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
65	C₄×SD₃₂	—	3	1	no	Sylow
66	C₂³×C₃×C₅	—	4	1	no	—
67	C₃×C₅×D₄	—	2	2	no	—
68	C₃×C₅×D₄	—	4	1	no	—
69	C₂×C₃×C₄×C₅	—	4	1	no	—
70	C₂×C₃×C₄×C₅	—	8	2	no	—
71	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×Dic₃	—	4	1	no	—
75	C₂×C₅×Dic₃	—	4	1	no	—
76	C₅×Dic₆	—	2	2	no	—
77	C₅×Dic₆	—	4	1	no	—
78	C₃×C₅×C₈	—	2	1	no	—
79	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₅×C₈	#41	1	1	yes	characteristic
81	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
82	C₃×C₄×D₄	—	2	1	no	—
83	C₂×Dic₁₂	C₂²×C₅	1	1	yes	characteristic
84	C₄×Dic₆	—	2	1	no	—
85	C₃×#107	C₂²×C₅	1	1	yes	characteristic
86	C₃×#102	—	2	1	no	—
87	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
88	#827	C₄×C₅	1	4	yes	direct product factor
89	#818	C₂²×C₅	1	1	yes	characteristic
90	#816	—	2	1	no	—
91	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₅×Q₈	—	6	1	no	—
94	C₂×C₅×D₄	—	2	1	no	—
95	C₂²×C₄×C₅	—	4	1	no	—
96	C₅×Q₁₆	—	3	2	no	—
97	C₅×Q₁₆	—	6	1	no	—
98	C₅×D₈	—	2	1	no	—
99	C₅×D₈	C₄×D₃	1	2	yes	—
100	C₂×C₅×C₈	#67	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₄⋊C₄	—	6	1	no	—
104	C₅×C₄⋊C₄	—	12	1	no	—
105	C₅×C₂²⋊C₄	—	4	1	no	—
106	C₄²×C₅	—	12	1	no	—
107	C₄²×C₅	#67	1	1	yes	characteristic
108	C₅×C₁₆	—	3	2	no	—
109	C₅×C₁₆	—	6	1	no	—
110	C₂×SD₃₂	—	3	1	no	—
111	C₄×Q₁₆	—	3	1	no	—
112	C₄×D₈	C₅×D₃	1	1	yes	characteristic
113	#367	—	3	1	no	—
114	#358	—	3	1	no	—
115	#357	—	3	1	no	—
116	C₄×C₁₆	—	3	1	no	—
117	C₂²×C₃×C₅	—	4	2	no	—
118	C₂²×C₃×C₅	—	8	1	no	—
119	C₂²×C₃×C₅	—	8	1	no	—
120	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
121	C₃×C₄×C₅	—	2	1	no	—
122	C₃×C₄×C₅	—	8	1	no	—
123	C₃×C₄×C₅	SD₃₂	1	2	yes	complemented
124	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
125	C₃×C₄×C₅	#135	1	1	yes	characteristic
126	C₅×Dic₃	—	4	2	no	—
127	C₅×Dic₃	—	8	1	no	—
128	C₂×C₃×D₄	—	2	1	no	—
129	C₂²×C₃×C₄	—	4	1	no	—
130	C₂×Dic₆	—	2	1	no	—
131	C₃×D₈	—	2	1	no	—
132	C₃×D₈	C₂×C₄×C₅	1	2	yes	—
133	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
135	C₃×C₄⋊C₄	—	2	1	no	—
136	C₃×C₂²⋊C₄	—	4	1	no	—
137	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
138	C₄⋊Dic₃	—	2	1	no	—
139	Dic₃⋊C₄	—	4	1	no	—
140	C₄×Dic₃	—	4	1	no	—
141	Dic₁₂	—	2	1	no	—
142	Dic₁₂	C₂×C₄×C₅	1	2	yes	complemented
143	C₃⋊C₁₆	—	2	1	no	—
144	C₃⋊C₁₆	C₂×C₄×C₅	1	2	yes	complemented
145	C₂³×C₅	—	4	1	no	—
146	C₅×Q₈	—	6	2	no	—
147	C₅×Q₈	—	12	1	no	—
148	C₅×D₄	—	2	2	no	—
149	C₅×D₄	—	4	1	no	—
150	C₂×C₄×C₅	—	4	1	no	—
151	C₂×C₄×C₅	—	8	2	no	—
152	C₂×C₄×C₅	—	12	1	no	—
153	C₂×C₄×C₅	—	12	1	no	—
154	C₂×C₄×C₅	#213	1	1	yes	characteristic
155	C₂×C₄×C₅	#213	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
157	C₅×C₈	—	2	1	no	—
158	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
159	C₅×C₈	#235	1	1	yes	characteristic
160	C₂×Q₁₆	—	3	1	no	—
161	C₂×D₈	C₂×C₅×D₃	1	1	yes	characteristic
162	C₄×Q₈	—	6	1	no	—
163	C₄×D₄	—	2	1	no	—
164	SD₃₂	—	3	4	no	—
165	C₂×C₁₆	—	3	1	no	—
166	C₂×C₁₆	—	3	1	no	—
167	#107	—	3	1	no	—
168	#107	C₂×C₅×D₃	1	1	yes	characteristic
169	#103	—	6	1	no	—
170	#102	—	2	1	no	—
171	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
172	C₂×C₃×C₅	—	8	2	no	—
173	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
174	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
175	C₂×C₃×C₅	#508	1	1	yes	characteristic
176	C₂³×C₃	—	4	1	no	—
177	C₃×D₄	—	2	2	no	—
178	C₃×D₄	—	4	1	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	—	8	2	no	—
181	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
182	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
184	C₂×Dic₃	—	4	1	no	—
185	C₂×Dic₃	—	4	1	no	—
186	Dic₆	—	2	2	no	—
187	Dic₆	—	4	1	no	—
188	C₃×C₈	—	2	1	no	—
189	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
190	C₃×C₈	C₅×#41	1	1	yes	characteristic
191	C₂²×C₅	—	4	2	no	—
192	C₂²×C₅	—	8	1	no	—
193	C₂²×C₅	—	8	1	no	—
194	C₂²×C₅	C₂×#213	1	1	yes	characteristic
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	8	1	no	—
197	C₄×C₅	—	12	2	no	—
198	C₄×C₅	—	24	1	no	—
199	C₄×C₅	#827	1	2	yes	direct product factor
200	C₄×C₅	C₄×#67	1	1	yes	characteristic
201	C₄×C₅	#950	1	1	yes	characteristic
202	C₂×Q₈	—	6	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	Q₁₆	—	3	2	no	—
206	Q₁₆	—	6	1	no	—
207	D₈	—	2	1	no	—
208	D₈	C₄×C₅×D₃	1	2	yes	—
209	C₂×C₈	C₅×#67	1	1	yes	characteristic
210	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	6	1	no	—
213	C₄⋊C₄	—	12	1	no	—
214	C₂²⋊C₄	—	4	1	no	—
215	C₄²	—	12	1	no	—
216	C₄²	C₅×#67	1	1	yes	characteristic
217	C₁₆	—	3	2	no	—
218	C₁₆	—	6	1	no	—
219	C₃×C₅	C₄×SD₃₂	1	1	yes	characteristic, complemented, Hall
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	—	8	1	no	—
222	C₂²×C₃	—	8	1	no	—
223	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	8	1	no	—
226	C₃×C₄	C₅×SD₃₂	1	2	yes	complemented
227	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
228	C₃×C₄	C₅×#135	1	1	yes	characteristic
229	Dic₃	—	4	2	no	—
230	Dic₃	—	8	1	no	—
231	C₂×C₅	—	8	2	no	—
232	C₂×C₅	C₄×#213	1	1	yes	characteristic
233	C₂×C₅	C₂×#827	1	1	yes	characteristic
234	C₂×C₅	#5194	1	1	yes	characteristic
235	C₂³	—	4	1	no	—
236	Q₈	—	6	2	no	—
237	Q₈	—	12	1	no	—
238	D₄	—	2	2	no	—
239	D₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	—	12	1	no	—
243	C₂×C₄	—	12	1	no	—
244	C₂×C₄	C₅×#213	1	1	yes	characteristic
245	C₂×C₄	C₅×#213	1	1	yes	characteristic
246	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
247	C₈	—	2	1	no	—
248	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
249	C₈	C₅×#235	1	1	yes	characteristic
250	C₂×C₃	—	8	2	no	—
251	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
252	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
253	C₂×C₃	C₅×#508	1	1	yes	characteristic
254	C₅	C₄×#827	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²	—	4	2	no	—
256	C₂²	—	8	1	no	—
257	C₂²	—	8	1	no	—
258	C₂²	C₂×C₅×#213	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	8	1	no	—
261	C₄	—	12	2	no	—
262	C₄	—	24	1	no	—
263	C₄	C₅×#827	1	2	yes	direct product factor
264	C₄	C₄×C₅×#67	1	1	yes	characteristic
265	C₄	C₅×#950	1	1	yes	characteristic
266	C₃	C₄×C₅×SD₃₂	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	8	2	no	—
268	C₂	C₄×C₅×#213	1	1	yes	characteristic
269	C₂	C₂×C₅×#827	1	1	yes	characteristic
270	C₂	C₅×#5194	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#213
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×D₈	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₃×C₅×D₈	C₂²
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₄×C₅×D₈	D₃
hypocenter	C₃	C₄×C₅×SD₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×D₈

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×Dic₁₂	C₂	semidirect
5. C₅×#4492	C₂	semidirect
6. C₅×#4490	C₂	semidirect
7. C₄×C₅×C₃⋊C₁₆	C₂	semidirect
16. C₅×#827	C₄	direct (N × H)
21. C₄×#827	C₅	direct (N × H)
43. C₅×Dic₁₂	C₂×C₄	semidirect
45. C₅×C₃⋊C₁₆	C₂×C₄	semidirect
48. C₄×Dic₁₂	C₂×C₅	semidirect
50. #4492	C₂×C₅	semidirect
51. #4490	C₂×C₅	semidirect
52. C₄×C₃⋊C₁₆	C₂×C₅	semidirect
88. #827	C₄×C₅	direct (N × H)
123. C₃×C₄×C₅	SD₃₂	semidirect
142. Dic₁₂	C₂×C₄×C₅	semidirect
144. C₃⋊C₁₆	C₂×C₄×C₅	semidirect
199. C₄×C₅	#827	direct (N × H)
219. C₃×C₅	C₄×SD₃₂	semidirect
226. C₃×C₄	C₅×SD₃₂	semidirect
254. C₅	C₄×#827	direct (N × H)
263. C₄	C₅×#827	direct (N × H)
266. C₃	C₄×C₅×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₈	2
C₂×C₅×#827	2
C₄×C₅×Dic₁₂	2
C₅×#4562	2
C₅×#4492	2
C₅×#4490	2
C₄×C₅×C₃⋊C₁₆	2
C₄×C₅×SD₃₂	3
C₄×#827	5

Subgroups — C4×C5×#827 #14315346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#827 #14315346