Subgroups — C₅×#70704 #14305220

Order 1920.

416 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₅×#5119	C₂	1	2	yes	maximal
3	C₅×#5106	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₅×#4555	C₂	1	2	yes	maximal
6	C₅×#1945	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#926	C₂²	1	2	yes	—
10	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#108	—	2	2	no	—
13	C₅×#818	C₄	1	2	yes	complemented
14	C₅×#817	C₄	1	2	yes	complemented
15	C₅×Dic₃⋊C₈	—	2	4	no	—
16	C₅×C₈×Dic₃	—	2	2	no	—
17	#70704	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#108	D₃	1	1	yes	characteristic
19	C₅×#427	—	3	1	no	—
20	C₅×#424	—	3	2	no	—
21	C₂×C₄×C₅×C₈	—	3	1	no	—
22	C₅×#339	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×Dic₃	—	2	1	no	—
25	C₃×C₅×M₁₆	—	2	2	no	—
26	C₃×C₅×M₁₆	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
32	C₅×#217	—	2	2	no	—
33	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
34	C₄×C₅×Dic₃	—	2	1	no	—
35	C₄×C₅×Dic₃	—	2	2	no	—
36	C₄×C₅×Dic₃	Q₈	1	2	yes	—
37	C₂×C₅×C₃⋊C₈	—	2	1	no	—
38	C₂×C₅×C₃⋊C₈	—	2	1	no	—
39	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
40	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
41	#5119	C₂×C₅	1	2	yes	—
42	#5106	C₂×C₅	1	1	yes	characteristic
43	C₂×C₈×Dic₃	C₂×C₅	1	1	yes	characteristic
44	#4555	C₂×C₅	1	2	yes	—
45	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
46	C₂²×C₅×C₈	—	3	1	no	—
47	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
48	C₅×#117	—	3	2	no	—
49	C₂×C₄²×C₅	—	3	1	no	—
50	C₅×#108	—	2	2	no	—
51	C₅×#107	—	3	2	no	—
52	C₅×#106	—	3	2	no	—
53	C₅×C₄⋊C₈	—	6	4	no	—
54	C₄×C₅×C₈	—	6	2	no	—
55	#1945	—	3	1	no	Sylow
56	C₂³×C₃×C₅	#41	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
63	C₂×C₅×Dic₃	—	2	4	no	—
64	C₂×C₅×Dic₃	—	2	2	no	—
65	C₂×C₅×Dic₃	—	2	4	no	—
66	C₃×C₅×C₈	—	4	2	no	—
67	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
68	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
69	C₅×C₃⋊C₈	—	2	2	no	—
70	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
71	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
72	#926	C₂²×C₅	1	2	yes	—
73	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
74	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
75	C₃×#108	—	2	2	no	—
76	#818	C₄×C₅	1	2	yes	complemented
77	#817	C₄×C₅	1	2	yes	complemented
78	Dic₃⋊C₈	—	2	4	no	—
79	C₈×Dic₃	—	2	2	no	—
80	C₂²×C₄×C₅	—	6	1	no	—
81	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
82	C₅×M₁₆	—	2	2	no	—
83	C₅×M₁₆	—	2	2	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	3	2	no	—
86	C₂×C₅×C₈	—	6	1	no	—
87	C₂×C₅×C₈	—	6	1	no	—
88	C₂×C₅×C₈	Dic₆	1	2	yes	—
89	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
90	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
91	C₂×C₅×C₈	#67	1	2	yes	—
92	C₅×C₄⋊C₄	—	3	4	no	—
93	C₅×C₂²⋊C₄	—	6	2	no	—
94	C₄²×C₅	—	3	2	no	—
95	C₄²×C₅	—	6	2	no	—
96	C₄²×C₅	—	6	1	no	—
97	C₂×#108	C₅×D₃	1	1	yes	characteristic
98	#427	—	3	1	no	—
99	#424	—	3	2	no	—
100	C₂×C₄×C₈	—	3	1	no	—
101	#339	—	3	2	no	—
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂²×C₃×C₅	#125	1	1	yes	characteristic
106	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
108	C₃×C₄×C₅	#121	1	1	yes	characteristic
109	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
110	C₅×Dic₃	—	2	4	no	—
111	C₅×Dic₃	—	4	4	no	—
112	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
113	C₂²×Dic₃	—	2	1	no	—
114	C₃×M₁₆	—	2	2	no	—
115	C₃×M₁₆	—	2	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
119	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
120	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
121	#217	—	2	2	no	—
122	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
123	C₄×Dic₃	—	2	1	no	—
124	C₄×Dic₃	—	2	2	no	—
125	C₄×Dic₃	C₅×Q₈	1	2	yes	—
126	C₂×C₃⋊C₈	—	2	1	no	—
127	C₂×C₃⋊C₈	—	2	1	no	—
128	C₂×C₃⋊C₈	C₅×D₄	1	2	yes	—
129	C₂³×C₅	#239	1	1	yes	characteristic
130	C₂×C₄×C₅	—	6	4	no	—
131	C₂×C₄×C₅	—	6	2	no	—
132	C₂×C₄×C₅	—	6	4	no	—
133	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	#237	1	1	yes	characteristic
137	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
139	C₅×C₈	—	4	2	no	—
140	C₅×C₈	—	6	2	no	—
141	C₅×C₈	Dic₃⋊C₄	1	2	yes	—
142	C₅×C₈	Dic₃⋊C₄	1	2	yes	—
143	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
144	C₂²×C₈	—	3	1	no	—
145	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
146	#117	—	3	2	no	—
147	C₂×C₄²	—	3	1	no	—
148	#108	—	2	2	no	—
149	#107	—	3	2	no	—
150	#106	—	3	2	no	—
151	C₄⋊C₈	—	6	4	no	—
152	C₄×C₈	—	6	2	no	—
153	C₂×C₃×C₅	—	2	1	no	—
154	C₂×C₃×C₅	—	2	1	no	—
155	C₂×C₃×C₅	#389	1	1	yes	characteristic
156	C₂×C₃×C₅	#443	1	2	yes	—
157	C₂³×C₃	C₅×#41	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
163	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
164	C₂×Dic₃	—	2	4	no	—
165	C₂×Dic₃	—	2	2	no	—
166	C₂×Dic₃	—	2	4	no	—
167	C₃×C₈	—	4	2	no	—
168	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
169	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
170	C₃⋊C₈	—	2	2	no	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	#890	1	1	yes	characteristic
173	C₂²×C₅	#893	1	1	yes	characteristic
174	C₂²×C₅	#946	1	1	yes	characteristic
175	C₄×C₅	—	6	4	no	—
176	C₄×C₅	—	12	4	no	—
177	C₄×C₅	#887	1	1	yes	characteristic
178	C₄×C₅	#888	1	1	yes	characteristic
179	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
180	C₄×C₅	#938	1	1	yes	characteristic
181	C₂²×C₄	—	6	1	no	—
182	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
183	M₁₆	—	2	2	no	—
184	M₁₆	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	3	2	no	—
187	C₂×C₈	—	6	1	no	—
188	C₂×C₈	—	6	1	no	—
189	C₂×C₈	C₅×Dic₆	1	2	yes	—
190	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
191	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
192	C₂×C₈	C₅×#67	1	2	yes	—
193	C₄⋊C₄	—	3	4	no	—
194	C₂²⋊C₄	—	6	2	no	—
195	C₄²	—	3	2	no	—
196	C₄²	—	6	2	no	—
197	C₄²	—	6	1	no	—
198	C₃×C₅	#1945	1	1	yes	characteristic, complemented, Hall
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
201	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
202	C₂²×C₃	C₅×#125	1	1	yes	characteristic
203	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
204	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
205	C₃×C₄	C₅×#121	1	1	yes	characteristic
206	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
207	Dic₃	—	2	4	no	—
208	Dic₃	—	4	4	no	—
209	C₂×C₅	—	2	1	no	—
210	C₂×C₅	—	2	1	no	—
211	C₂×C₅	#4895	1	2	yes	—
212	C₂×C₅	#4972	1	1	yes	characteristic
213	C₂³	C₅×#239	1	1	yes	characteristic
214	C₂×C₄	—	6	4	no	—
215	C₂×C₄	—	6	2	no	—
216	C₂×C₄	—	6	4	no	—
217	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
219	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
220	C₂×C₄	C₅×#237	1	1	yes	characteristic
221	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
222	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
223	C₈	—	4	2	no	—
224	C₈	—	6	2	no	—
225	C₈	C₅×Dic₃⋊C₄	1	2	yes	—
226	C₈	C₅×Dic₃⋊C₄	1	2	yes	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	C₅×#389	1	1	yes	characteristic
230	C₂×C₃	C₅×#443	1	2	yes	—
231	C₅	#70704	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	2	2	no	—
233	C₂²	C₅×#890	1	1	yes	characteristic
234	C₂²	C₅×#893	1	1	yes	characteristic
235	C₂²	C₅×#946	1	1	yes	characteristic
236	C₄	—	6	4	no	—
237	C₄	—	12	4	no	—
238	C₄	C₅×#887	1	1	yes	characteristic
239	C₄	C₅×#888	1	1	yes	characteristic
240	C₄	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
241	C₄	C₅×#938	1	1	yes	characteristic
242	C₃	C₅×#1945	1	1	yes	characteristic, complemented, Sylow
243	C₂	—	2	1	no	—
244	C₂	—	2	1	no	—
245	C₂	C₅×#4895	1	2	yes	—
246	C₂	C₅×#4972	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#108	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#108	D₃
hypocenter	C₃	C₅×#1945
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#108

Complements

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

N	H ≅ G/N	type
13. C₅×#818	C₄	semidirect
14. C₅×#817	C₄	semidirect
17. #70704	C₅	direct (N × H)
76. #818	C₄×C₅	semidirect
77. #817	C₄×C₅	semidirect
198. C₃×C₅	#1945	semidirect
231. C₅	#70704	direct (N × H)
242. C₃	C₅×#1945	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#108	2
C₅×#5119	2
C₅×#5106	2
C₂×C₅×C₈×Dic₃	2
C₅×#4555	2
C₅×#1945	3
#70704	5

Subgroups — C5×#70704 #14305220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70704 #14305220