Subgroups — C₅×#70853 #14291772

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#5008	C₂	1	2	yes	maximal
3	C₅×#4682	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4481	C₂	1	2	yes	maximal, complemented
5	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1675	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	—	2	1	no	—
8	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₅×#869	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#832	—	2	2	no	—
14	C₅×#818	—	2	1	no	—
15	C₅×#818	—	2	1	no	—
16	C₅×#808	—	2	2	no	—
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	#70853	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#500	—	3	1	no	—
20	C₅×#476	—	3	2	no	—
21	C₅×C₈×D₄	D₃	1	1	yes	characteristic
22	C₅×#335	—	3	1	no	—
23	C₅×#333	—	3	2	no	—
24	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	1	no	—
26	C₂×C₅×Dic₆	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	4	2	no	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
32	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
33	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊Dic₃	—	2	2	no	—
35	C₅×C₄⋊Dic₃	—	2	2	no	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₃×C₈×D₄	C₂×C₅	1	1	yes	characteristic
38	#5008	C₂×C₅	1	2	yes	—
39	#4682	C₂×C₅	1	1	yes	characteristic, complemented
40	#4481	C₂×C₅	1	2	yes	complemented
41	#4458	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂²×C₅×C₈	—	2	1	no	—
43	C₅×C₄⋊Q₈	—	3	2	no	—
44	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
45	C₅×#107	—	6	1	no	—
46	C₅×#107	—	6	1	no	—
47	C₅×#106	—	6	2	no	—
48	C₅×C₄⋊C₈	—	3	2	no	—
49	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
50	C₅×#102	—	6	2	no	—
51	C₅×C₂²⋊C₈	—	2	1	no	—
52	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
53	#1675	—	3	1	no	Sylow
54	C₂³×C₃×C₅	—	2	1	no	—
55	C₃×C₅×D₄	—	2	1	no	—
56	C₃×C₅×D₄	Q₁₆	1	2	yes	—
57	C₂×C₃×C₄×C₅	—	2	1	no	—
58	C₂×C₃×C₄×C₅	—	4	1	no	—
59	C₂×C₃×C₄×C₅	—	4	1	no	—
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
63	C₂×C₅×Dic₃	—	4	2	no	—
64	C₅×Dic₆	—	8	2	no	—
65	C₃×C₅×C₈	—	2	1	no	—
66	C₃×C₅×C₈	—	4	1	no	—
67	C₃×C₅×C₈	SD₁₆	1	2	yes	—
68	C₅×C₃⋊C₈	—	4	2	no	—
69	C₂²×C₃×C₈	—	2	1	no	—
70	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
71	#869	C₂²×C₅	1	2	yes	—
72	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
73	C₃×C₂²⋊C₈	—	2	1	no	—
74	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
75	#832	—	2	2	no	—
76	#818	—	2	1	no	—
77	#818	—	2	1	no	—
78	#808	—	2	2	no	—
79	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
80	C₂×C₅×Q₈	—	6	2	no	—
81	C₂×C₅×D₄	D₁₂	1	1	yes	characteristic
82	C₂²×C₄×C₅	—	2	1	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₂×C₅×C₈	—	6	2	no	—
86	C₂×C₅×C₈	#67	1	1	yes	characteristic
87	C₂×C₅×C₈	#67	1	1	yes	characteristic
88	C₅×C₄⋊C₄	—	6	2	no	—
89	C₅×C₄⋊C₄	—	6	2	no	—
90	C₅×C₄⋊C₄	D₁₂	1	1	yes	characteristic
91	C₅×C₂²⋊C₄	—	2	1	no	—
92	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
93	#500	—	3	1	no	—
94	#476	—	3	2	no	—
95	C₈×D₄	C₅×D₃	1	1	yes	characteristic
96	#335	—	3	1	no	—
97	#333	—	3	2	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	—	4	1	no	—
100	C₂²×C₃×C₅	—	4	1	no	—
101	C₂²×C₃×C₅	#121	1	1	yes	characteristic
102	C₃×C₄×C₅	—	2	1	no	—
103	C₃×C₄×C₅	—	4	1	no	—
104	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
105	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
106	C₃×C₄×C₅	#135	1	1	yes	characteristic
107	C₃×C₄×C₅	#137	1	1	yes	characteristic
108	C₅×Dic₃	—	8	2	no	—
109	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂×Dic₆	—	2	2	no	—
112	C₂×C₃×C₈	—	2	1	no	—
113	C₂×C₃×C₈	—	4	2	no	—
114	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
115	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
116	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
117	C₃×C₂²⋊C₄	—	2	1	no	—
118	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
119	C₄⋊Dic₃	—	2	2	no	—
120	C₄⋊Dic₃	—	2	2	no	—
121	C₂×C₃⋊C₈	—	2	2	no	—
122	C₂³×C₅	—	2	1	no	—
123	C₅×Q₈	—	24	2	no	—
124	C₅×D₄	—	2	1	no	—
125	C₅×D₄	Dic₁₂	1	2	yes	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	12	2	no	—
130	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
131	C₂×C₄×C₅	#235	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
133	C₅×C₈	—	2	1	no	—
134	C₅×C₈	—	4	1	no	—
135	C₅×C₈	—	12	2	no	—
136	C₅×C₈	#214	1	2	yes	—
137	C₂²×C₈	—	2	1	no	—
138	C₄⋊Q₈	—	3	2	no	—
139	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
140	#107	—	6	1	no	—
141	#107	—	6	1	no	—
142	#106	—	6	2	no	—
143	C₄⋊C₈	—	3	2	no	—
144	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
145	#102	—	6	2	no	—
146	C₂²⋊C₈	—	2	1	no	—
147	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
148	C₂×C₃×C₅	—	4	2	no	—
149	C₂×C₃×C₅	#461	1	1	yes	characteristic
150	C₂×C₃×C₅	#467	1	1	yes	characteristic
151	C₂×C₃×C₅	#472	1	1	yes	characteristic
152	C₂³×C₃	—	2	1	no	—
153	C₃×D₄	—	2	1	no	—
154	C₃×D₄	C₅×Q₁₆	1	2	yes	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	4	1	no	—
157	C₂×C₃×C₄	—	4	1	no	—
158	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
161	C₂×Dic₃	—	4	2	no	—
162	Dic₆	—	8	2	no	—
163	C₃×C₈	—	2	1	no	—
164	C₃×C₈	—	4	1	no	—
165	C₃×C₈	C₅×SD₁₆	1	2	yes	—
166	C₃⋊C₈	—	4	2	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	4	1	no	—
169	C₂²×C₅	—	4	1	no	—
170	C₂²×C₅	#930	1	1	yes	characteristic
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	—	4	1	no	—
173	C₄×C₅	—	24	2	no	—
174	C₄×C₅	#904	1	1	yes	characteristic
175	C₄×C₅	C₂×Dic₁₂	1	1	yes	characteristic
176	C₄×C₅	C₂×#214	1	1	yes	characteristic
177	C₄×C₅	#951	1	1	yes	characteristic
178	C₂×Q₈	—	6	2	no	—
179	C₂×D₄	C₅×D₁₂	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	6	2	no	—
184	C₂×C₈	C₅×#67	1	1	yes	characteristic
185	C₂×C₈	C₅×#67	1	1	yes	characteristic
186	C₄⋊C₄	—	6	2	no	—
187	C₄⋊C₄	—	6	2	no	—
188	C₄⋊C₄	C₅×D₁₂	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	1	no	—
190	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
191	C₃×C₅	#1675	1	1	yes	characteristic, complemented, Hall
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	4	1	no	—
194	C₂²×C₃	—	4	1	no	—
195	C₂²×C₃	C₅×#121	1	1	yes	characteristic
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	4	1	no	—
198	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
199	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
200	C₃×C₄	C₅×#135	1	1	yes	characteristic
201	C₃×C₄	C₅×#137	1	1	yes	characteristic
202	Dic₃	—	8	2	no	—
203	C₂×C₅	—	4	2	no	—
204	C₂×C₅	#5019	1	1	yes	characteristic
205	C₂×C₅	#5116	1	1	yes	characteristic
206	C₂×C₅	#5142	1	1	yes	characteristic
207	C₂³	—	2	1	no	—
208	Q₈	—	24	2	no	—
209	D₄	—	2	1	no	—
210	D₄	C₅×Dic₁₂	1	2	yes	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	12	2	no	—
215	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
216	C₂×C₄	C₅×#235	1	1	yes	characteristic
217	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	4	1	no	—
220	C₈	—	12	2	no	—
221	C₈	C₅×#214	1	2	yes	—
222	C₂×C₃	—	4	2	no	—
223	C₂×C₃	C₅×#461	1	1	yes	characteristic
224	C₂×C₃	C₅×#467	1	1	yes	characteristic
225	C₂×C₃	C₅×#472	1	1	yes	characteristic
226	C₅	#70853	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	C₅×#930	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	24	2	no	—
234	C₄	C₅×#904	1	1	yes	characteristic
235	C₄	C₂×C₅×Dic₁₂	1	1	yes	characteristic
236	C₄	C₂×C₅×#214	1	1	yes	characteristic
237	C₄	C₅×#951	1	1	yes	characteristic
238	C₃	C₅×#1675	1	1	yes	characteristic, complemented, Sylow
239	C₂	—	4	2	no	—
240	C₂	C₅×#5019	1	1	yes	characteristic
241	C₂	C₅×#5116	1	1	yes	characteristic
242	C₂	C₅×#5142	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#930
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈×D₄	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₂×C₃×C₅×D₄	D₄
hypercenter	C₅×C₈×D₄	D₃
hypocenter	C₃	C₅×#1675
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×D₄

Complements

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

N	H ≅ G/N	type
3. C₅×#4682	C₂	semidirect
4. C₅×#4481	C₂	semidirect
5. C₅×#4458	C₂	semidirect
18. #70853	C₅	direct (N × H)
39. #4682	C₂×C₅	semidirect
40. #4481	C₂×C₅	semidirect
41. #4458	C₂×C₅	semidirect
191. C₃×C₅	#1675	semidirect
226. C₅	#70853	direct (N × H)
238. C₃	C₅×#1675	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₈×D₄	2
C₅×#5008	2
C₅×#4682	2
C₅×#4481	2
C₅×#4458	2
C₅×#1675	3
#70853	5

Subgroups — C5×#70853 #14291772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70853 #14291772