Subgroups — C₅×#69827 #14307342

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#870	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
3	C₅×#4935	C₂	1	1	yes	maximal, characteristic
4	C₅×#4551	C₂	1	4	yes	maximal
5	C₅×#1992	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
9	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₅×Dic₃⋊C₄	—	2	2	no	—
12	C₅×#870	—	2	2	no	—
13	C₅×#870	C₄	1	4	yes	complemented
14	C₃×C₅×#97	—	2	2	no	—
15	#69827	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×#125	—	3	1	no	—
17	C₂×C₅×#97	D₃	1	1	yes	characteristic
18	C₅×#384	—	3	1	no	—
19	C₅×#340	—	3	4	no	—
20	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×C₅×Dic₃	—	2	2	no	—
23	C₂²×C₅×Dic₃	—	2	2	no	—
24	C₂×C₃×C₅×C₈	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	2	no	—
26	C₂×C₃×C₅×C₈	Q₈	1	4	yes	—
27	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₃×C₄²×C₅	D₄	1	2	yes	—
29	C₅×C₄⋊Dic₃	—	2	2	no	—
30	C₅×C₄⋊Dic₃	—	2	4	no	—
31	C₅×C₄⋊Dic₃	—	4	4	no	—
32	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
33	C₅×Dic₃⋊C₄	—	2	8	no	—
34	C₂×#870	C₂×C₅	1	1	yes	characteristic
35	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
36	#4935	C₂×C₅	1	1	yes	characteristic
37	#4551	C₂×C₅	1	4	yes	—
38	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
39	C₅×#125	—	3	4	no	—
40	C₅×#125	—	6	2	no	—
41	C₂×C₅×C₄⋊C₄	—	3	2	no	—
42	C₂×C₅×C₄⋊C₄	—	3	2	no	—
43	C₂×C₅×C₄⋊C₄	—	6	2	no	—
44	C₂×C₅×C₄⋊C₄	—	6	2	no	—
45	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
46	C₅×#97	—	2	2	no	—
47	#1992	—	3	1	no	Sylow
48	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
52	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
55	C₂×C₃×C₄×C₅	#41	1	2	yes	—
56	C₂×C₅×Dic₃	—	2	4	no	—
57	C₂×C₅×Dic₃	—	4	2	no	—
58	C₂×C₅×Dic₃	—	4	2	no	—
59	C₂×C₅×Dic₃	—	4	2	no	—
60	C₂×C₅×Dic₃	—	4	4	no	—
61	C₃×C₅×C₈	—	2	4	no	—
62	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
63	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₄⋊Dic₃	—	2	2	no	—
65	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
66	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
67	C₂×Dic₃⋊C₄	—	2	2	no	—
68	#870	—	2	2	no	—
69	#870	C₄×C₅	1	4	yes	complemented
70	C₃×#97	—	2	2	no	—
71	C₂²×C₄×C₅	—	6	2	no	—
72	C₂²×C₄×C₅	—	6	2	no	—
73	C₂²×C₄×C₅	D₁₂	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	Dic₆	1	4	yes	—
78	C₅×C₄⋊C₄	—	3	4	no	—
79	C₅×C₄⋊C₄	—	6	8	no	—
80	C₅×C₄⋊C₄	—	6	2	no	—
81	C₅×C₄⋊C₄	—	6	4	no	—
82	C₅×C₄⋊C₄	—	12	4	no	—
83	C₄²×C₅	C₄×D₃	1	2	yes	—
84	C₄²×C₅	#67	1	2	yes	—
85	C₂×#125	—	3	1	no	—
86	C₂×#97	C₅×D₃	1	1	yes	characteristic
87	#384	—	3	1	no	—
88	#340	—	3	4	no	—
89	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₃×C₅	#124	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
92	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
93	C₂²×C₃×C₅	#137	1	2	yes	—
94	C₃×C₄×C₅	—	2	4	no	—
95	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
96	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
97	C₅×Dic₃	—	4	4	no	—
98	C₅×Dic₃	—	8	2	no	—
99	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
100	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
101	C₂²×Dic₃	—	2	2	no	—
102	C₂²×Dic₃	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	C₅×Q₈	1	4	yes	—
106	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
107	C₃×C₄²	C₅×D₄	1	2	yes	—
108	C₄⋊Dic₃	—	2	2	no	—
109	C₄⋊Dic₃	—	2	4	no	—
110	C₄⋊Dic₃	—	4	4	no	—
111	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
112	Dic₃⋊C₄	—	2	8	no	—
113	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	6	4	no	—
117	C₂×C₄×C₅	—	12	2	no	—
118	C₂×C₄×C₅	—	12	2	no	—
119	C₂×C₄×C₅	—	12	2	no	—
120	C₂×C₄×C₅	—	12	4	no	—
121	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
122	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
123	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
124	C₂×C₄×C₅	#235	1	2	yes	—
125	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
126	C₅×C₈	—	2	4	no	—
127	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
128	#125	—	3	4	no	—
129	#125	—	6	2	no	—
130	C₂×C₄⋊C₄	—	3	2	no	—
131	C₂×C₄⋊C₄	—	3	2	no	—
132	C₂×C₄⋊C₄	—	6	2	no	—
133	C₂×C₄⋊C₄	—	6	2	no	—
134	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
135	#97	—	2	2	no	—
136	C₂×C₃×C₅	#391	1	1	yes	characteristic
137	C₂×C₃×C₅	#417	1	2	yes	—
138	C₂×C₃×C₅	#490	1	1	yes	characteristic
139	C₂×C₃×C₅	#491	1	1	yes	characteristic
140	C₂×C₃×C₅	#501	1	2	yes	—
141	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
145	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
148	C₂×C₃×C₄	C₅×#41	1	2	yes	—
149	C₂×Dic₃	—	2	4	no	—
150	C₂×Dic₃	—	4	2	no	—
151	C₂×Dic₃	—	4	2	no	—
152	C₂×Dic₃	—	4	2	no	—
153	C₂×Dic₃	—	4	4	no	—
154	C₃×C₈	—	2	4	no	—
155	C₂²×C₅	#869	1	1	yes	characteristic
156	C₂²×C₅	#875	1	1	yes	characteristic
157	C₂²×C₅	#908	1	2	yes	—
158	C₂²×C₅	#909	1	2	yes	—
159	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
160	C₄×C₅	—	2	4	no	—
161	C₄×C₅	—	12	4	no	—
162	C₄×C₅	—	24	2	no	—
163	C₄×C₅	C₄×Dic₆	1	2	yes	—
164	C₄×C₅	#924	1	2	yes	—
165	C₂²×C₄	—	6	2	no	—
166	C₂²×C₄	—	6	2	no	—
167	C₂²×C₄	C₅×D₁₂	1	2	yes	—
168	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₅×Dic₆	1	4	yes	—
172	C₄⋊C₄	—	3	4	no	—
173	C₄⋊C₄	—	6	8	no	—
174	C₄⋊C₄	—	6	2	no	—
175	C₄⋊C₄	—	6	4	no	—
176	C₄⋊C₄	—	12	4	no	—
177	C₄²	C₄×C₅×D₃	1	2	yes	—
178	C₄²	C₅×#67	1	2	yes	—
179	C₃×C₅	#1992	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₃	C₅×#124	1	1	yes	characteristic
182	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
183	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
184	C₂²×C₃	C₅×#137	1	2	yes	—
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
187	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
188	Dic₃	—	4	4	no	—
189	Dic₃	—	8	2	no	—
190	C₂×C₅	#4702	1	2	yes	—
191	C₂×C₅	#4703	1	1	yes	characteristic
192	C₂×C₅	#4713	1	1	yes	characteristic
193	C₂×C₅	#4929	1	1	yes	characteristic
194	C₂×C₅	#5133	1	2	yes	—
195	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	6	4	no	—
199	C₂×C₄	—	12	2	no	—
200	C₂×C₄	—	12	2	no	—
201	C₂×C₄	—	12	2	no	—
202	C₂×C₄	—	12	4	no	—
203	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
204	C₂×C₄	C₂×C₅×Dic₆	1	2	yes	—
205	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
206	C₂×C₄	C₅×#235	1	2	yes	—
207	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
208	C₈	—	2	4	no	—
209	C₂×C₃	C₅×#391	1	1	yes	characteristic
210	C₂×C₃	C₅×#417	1	2	yes	—
211	C₂×C₃	C₅×#490	1	1	yes	characteristic
212	C₂×C₃	C₅×#491	1	1	yes	characteristic
213	C₂×C₃	C₅×#501	1	2	yes	—
214	C₅	#69827	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	C₅×#869	1	1	yes	characteristic
216	C₂²	C₅×#875	1	1	yes	characteristic
217	C₂²	C₅×#908	1	2	yes	—
218	C₂²	C₅×#909	1	2	yes	—
219	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	12	4	no	—
222	C₄	—	24	2	no	—
223	C₄	C₄×C₅×Dic₆	1	2	yes	—
224	C₄	C₅×#924	1	2	yes	—
225	C₃	C₅×#1992	1	1	yes	characteristic, complemented, Sylow
226	C₂	C₅×#4702	1	2	yes	—
227	C₂	C₅×#4703	1	1	yes	characteristic
228	C₂	C₅×#4713	1	1	yes	characteristic
229	C₂	C₅×#4929	1	1	yes	characteristic
230	C₂	C₅×#5133	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×C₅×#97	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#97	D₃
hypocenter	C₃	C₅×#1992
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#97

Complements

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

N	H ≅ G/N	type
13. C₅×#870	C₄	semidirect
15. #69827	C₅	direct (N × H)
69. #870	C₄×C₅	semidirect
179. C₃×C₅	#1992	semidirect
214. C₅	#69827	direct (N × H)
225. C₃	C₅×#1992	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#870	2
C₂×C₃×C₅×#97	2
C₅×#4935	2
C₅×#4551	2
C₅×#1992	3
#69827	5

Subgroups — C5×#69827 #14307342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69827 #14307342