Subgroups — C₅×#69771 #14305772

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#816	C₂	1	4	yes	maximal
4	C₅×#4935	C₂	1	1	yes	maximal, characteristic
5	C₅×#1956	—	3	1	no	maximal, Hall
6	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
10	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
11	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₅×#869	—	2	2	no	—
13	C₅×#869	C₄	1	4	yes	complemented
14	C₃×C₄×C₅×C₈	—	2	2	no	—
15	C₅×#816	—	2	8	no	—
16	#69771	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
18	C₂×C₅×#103	—	3	4	no	—
19	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
20	C₅×#384	—	3	1	no	—
21	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅×Dic₃	—	2	2	no	—
24	C₂²×C₅×Dic₃	—	2	2	no	—
25	C₂×C₅×Dic₆	—	2	4	no	—
26	C₂×C₅×Dic₆	—	2	4	no	—
27	C₂×C₅×Dic₆	—	2	2	no	—
28	C₂×C₅×Dic₆	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₃×C₄²×C₅	D₄	1	2	yes	—
34	C₅×C₄⋊Dic₃	—	2	8	no	—
35	C₅×C₄⋊Dic₃	—	2	4	no	—
36	C₅×C₄⋊Dic₃	—	4	4	no	—
37	C₂×#869	C₂×C₅	1	1	yes	characteristic
38	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
39	C₂×#816	C₂×C₅	1	4	yes	—
40	#4935	C₂×C₅	1	1	yes	characteristic
41	C₂²×C₅×Q₈	—	3	2	no	—
42	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
43	C₅×C₄⋊Q₈	—	3	4	no	—
44	C₅×C₄⋊Q₈	—	6	2	no	—
45	C₂×C₅×C₄⋊C₄	—	3	2	no	—
46	C₂×C₅×C₄⋊C₄	—	6	2	no	—
47	C₂×C₅×C₄⋊C₄	—	6	2	no	—
48	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
49	C₅×#103	—	6	8	no	—
50	C₄×C₅×C₈	—	2	2	no	—
51	#1956	—	3	1	no	Sylow
52	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
54	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
55	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	#41	1	2	yes	—
60	C₂×C₅×Dic₃	—	2	4	no	—
61	C₂×C₅×Dic₃	—	4	2	no	—
62	C₂×C₅×Dic₃	—	4	2	no	—
63	C₂×C₅×Dic₃	—	4	2	no	—
64	C₂×C₅×Dic₃	—	4	4	no	—
65	C₅×Dic₆	—	2	8	no	—
66	C₅×Dic₆	—	4	4	no	—
67	C₃×C₅×C₈	—	2	4	no	—
68	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
69	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
70	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₄⋊Dic₃	—	2	2	no	—
72	C₂×C₄⋊Dic₃	—	2	2	no	—
73	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
74	#869	—	2	2	no	—
75	#869	C₄×C₅	1	4	yes	complemented
76	C₃×C₄×C₈	—	2	2	no	—
77	#816	—	2	8	no	—
78	C₂×C₅×Q₈	—	3	4	no	—
79	C₂×C₅×Q₈	—	6	2	no	—
80	C₂×C₅×Q₈	—	6	4	no	—
81	C₂×C₅×Q₈	—	6	4	no	—
82	C₂²×C₄×C₅	—	6	2	no	—
83	C₂²×C₄×C₅	—	6	2	no	—
84	C₂²×C₄×C₅	D₁₂	1	2	yes	—
85	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	D₁₂	1	4	yes	—
89	C₅×C₄⋊C₄	—	6	8	no	—
90	C₅×C₄⋊C₄	—	6	4	no	—
91	C₅×C₄⋊C₄	—	12	4	no	—
92	C₄²×C₅	C₄×D₃	1	2	yes	—
93	C₄²×C₅	#67	1	2	yes	—
94	C₂×C₄⋊Q₈	—	3	1	no	—
95	C₂×#103	—	3	4	no	—
96	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
97	#384	—	3	1	no	—
98	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₃×C₅	#124	1	1	yes	characteristic
100	C₂²×C₃×C₅	#127	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
102	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
103	C₃×C₄×C₅	#103	1	8	yes	complemented
104	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
105	C₃×C₄×C₅	#121	1	2	yes	—
106	C₅×Dic₃	—	4	4	no	—
107	C₅×Dic₃	—	8	2	no	—
108	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
109	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
110	C₂²×Dic₃	—	2	2	no	—
111	C₂²×Dic₃	—	2	2	no	—
112	C₂×Dic₆	—	2	4	no	—
113	C₂×Dic₆	—	2	4	no	—
114	C₂×Dic₆	—	2	2	no	—
115	C₂×Dic₆	C₂×C₄×C₅	1	4	yes	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
119	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
120	C₃×C₄²	C₅×D₄	1	2	yes	—
121	C₄⋊Dic₃	—	2	8	no	—
122	C₄⋊Dic₃	—	2	4	no	—
123	C₄⋊Dic₃	—	4	4	no	—
124	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
125	C₅×Q₈	—	6	8	no	—
126	C₅×Q₈	—	12	4	no	—
127	C₂×C₄×C₅	—	6	4	no	—
128	C₂×C₄×C₅	—	12	2	no	—
129	C₂×C₄×C₅	—	12	2	no	—
130	C₂×C₄×C₅	—	12	2	no	—
131	C₂×C₄×C₅	—	12	4	no	—
132	C₂×C₄×C₅	#204	1	4	yes	—
133	C₂×C₄×C₅	Dic₁₂	1	4	yes	—
134	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
135	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
137	C₂×C₄×C₅	#235	1	2	yes	—
138	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
139	C₅×C₈	—	2	4	no	—
140	C₂²×Q₈	—	3	2	no	—
141	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
142	C₄⋊Q₈	—	3	4	no	—
143	C₄⋊Q₈	—	6	2	no	—
144	C₂×C₄⋊C₄	—	3	2	no	—
145	C₂×C₄⋊C₄	—	6	2	no	—
146	C₂×C₄⋊C₄	—	6	2	no	—
147	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
148	#103	—	6	8	no	—
149	C₄×C₈	—	2	2	no	—
150	C₂×C₃×C₅	#390	1	1	yes	characteristic
151	C₂×C₃×C₅	C₂×#103	1	2	yes	—
152	C₂×C₃×C₅	#487	1	2	yes	—
153	C₂×C₃×C₅	#492	1	1	yes	characteristic
154	C₂×C₃×C₅	#494	1	1	yes	characteristic
155	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
157	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
158	C₂×C₃×C₄	C₅×Q₁₆	1	4	yes	—
159	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
161	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₅×#41	1	2	yes	—
163	C₂×Dic₃	—	2	4	no	—
164	C₂×Dic₃	—	4	2	no	—
165	C₂×Dic₃	—	4	2	no	—
166	C₂×Dic₃	—	4	2	no	—
167	C₂×Dic₃	—	4	4	no	—
168	Dic₆	—	2	8	no	—
169	Dic₆	—	4	4	no	—
170	C₃×C₈	—	2	4	no	—
171	C₂²×C₅	#874	1	1	yes	characteristic
172	C₂²×C₅	#875	1	1	yes	characteristic
173	C₂²×C₅	C₂×#204	1	2	yes	—
174	C₂²×C₅	C₂×Dic₁₂	1	2	yes	—
175	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
176	C₄×C₅	—	12	4	no	—
177	C₄×C₅	—	24	2	no	—
178	C₄×C₅	#816	1	8	yes	complemented
179	C₄×C₅	C₄×D₁₂	1	2	yes	—
180	C₄×C₅	#930	1	2	yes	—
181	C₂×Q₈	—	3	4	no	—
182	C₂×Q₈	—	6	2	no	—
183	C₂×Q₈	—	6	4	no	—
184	C₂×Q₈	—	6	4	no	—
185	C₂²×C₄	—	6	2	no	—
186	C₂²×C₄	—	6	2	no	—
187	C₂²×C₄	C₅×D₁₂	1	2	yes	—
188	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₅×D₁₂	1	4	yes	—
192	C₄⋊C₄	—	6	8	no	—
193	C₄⋊C₄	—	6	4	no	—
194	C₄⋊C₄	—	12	4	no	—
195	C₄²	C₄×C₅×D₃	1	2	yes	—
196	C₄²	C₅×#67	1	2	yes	—
197	C₃×C₅	#1956	1	1	yes	characteristic, complemented, Hall
198	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₃	C₅×#124	1	1	yes	characteristic
200	C₂²×C₃	C₅×#127	1	1	yes	characteristic
201	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
202	C₂²×C₃	C₂×C₅×Q₁₆	1	2	yes	—
203	C₃×C₄	C₅×#103	1	8	yes	complemented
204	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
205	C₃×C₄	C₅×#121	1	2	yes	—
206	Dic₃	—	4	4	no	—
207	Dic₃	—	8	2	no	—
208	C₂×C₅	#4681	1	2	yes	—
209	C₂×C₅	#4693	1	1	yes	characteristic
210	C₂×C₅	#4699	1	1	yes	characteristic
211	C₂×C₅	#4939	1	1	yes	characteristic
212	C₂×C₅	C₂×#816	1	2	yes	—
213	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
214	Q₈	—	6	8	no	—
215	Q₈	—	12	4	no	—
216	C₂×C₄	—	6	4	no	—
217	C₂×C₄	—	12	2	no	—
218	C₂×C₄	—	12	2	no	—
219	C₂×C₄	—	12	2	no	—
220	C₂×C₄	—	12	4	no	—
221	C₂×C₄	C₅×#204	1	4	yes	—
222	C₂×C₄	C₅×Dic₁₂	1	4	yes	—
223	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
224	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
225	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
226	C₂×C₄	C₅×#235	1	2	yes	—
227	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
228	C₈	—	2	4	no	—
229	C₂×C₃	C₅×#390	1	1	yes	characteristic
230	C₂×C₃	C₂×C₅×#103	1	2	yes	—
231	C₂×C₃	C₅×#487	1	2	yes	—
232	C₂×C₃	C₅×#492	1	1	yes	characteristic
233	C₂×C₃	C₅×#494	1	1	yes	characteristic
234	C₅	#69771	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	C₅×#874	1	1	yes	characteristic
236	C₂²	C₅×#875	1	1	yes	characteristic
237	C₂²	C₂×C₅×#204	1	2	yes	—
238	C₂²	C₂×C₅×Dic₁₂	1	2	yes	—
239	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
240	C₄	—	12	4	no	—
241	C₄	—	24	2	no	—
242	C₄	C₅×#816	1	8	yes	complemented
243	C₄	C₄×C₅×D₁₂	1	2	yes	—
244	C₄	C₅×#930	1	2	yes	—
245	C₃	C₅×#1956	1	1	yes	characteristic, complemented, Sylow
246	C₂	C₅×#4681	1	2	yes	—
247	C₂	C₅×#4693	1	1	yes	characteristic
248	C₂	C₅×#4699	1	1	yes	characteristic
249	C₂	C₅×#4939	1	1	yes	characteristic
250	C₂	C₂×C₅×#816	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×#1956
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
13. C₅×#869	C₄	semidirect
16. #69771	C₅	direct (N × H)
75. #869	C₄×C₅	semidirect
103. C₃×C₄×C₅	#103	semidirect
178. C₄×C₅	#816	semidirect
197. C₃×C₅	#1956	semidirect
203. C₃×C₄	C₅×#103	semidirect
234. C₅	#69771	direct (N × H)
242. C₄	C₅×#816	semidirect
245. C₃	C₅×#1956	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#869	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#816	2
C₅×#4935	2
C₅×#1956	3
#69771	5

Subgroups — C5×#69771 #14305772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69771 #14305772