Subgroups — C₅×#69766 #14299105

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#818	C₂	1	2	yes	maximal
3	C₂×C₅×#817	C₂	1	2	yes	maximal
4	C₅×#4936	C₂	1	2	yes	maximal
5	C₅×#1846	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊Dic₃	C₂²	1	4	yes	—
10	C₃×C₄×C₅×C₈	C₄	1	4	yes	complemented
11	C₅×C₃⋊#95	—	2	4	no	—
12	C₅×#818	—	2	4	no	—
13	C₅×#817	—	2	4	no	—
14	#69766	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×#107	—	3	2	no	—
16	C₂×C₅×#106	—	3	2	no	—
17	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
18	C₅×#382	—	3	2	no	—
19	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
20	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×C₅×Dic₃	—	2	4	no	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
25	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
26	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
27	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
28	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₃×C₄²×C₅	Q₈	1	2	yes	—
30	C₅×C₄⋊Dic₃	—	2	4	no	—
31	C₅×C₄⋊Dic₃	—	2	4	no	—
32	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
33	C₂×#818	C₂×C₅	1	2	yes	—
34	C₂×#817	C₂×C₅	1	2	yes	—
35	#4936	C₂×C₅	1	2	yes	—
36	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
38	C₂×C₅×C₄⋊C₄	—	3	4	no	—
39	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
40	C₅×#107	—	6	4	no	—
41	C₅×#106	—	6	4	no	—
42	C₄×C₅×C₈	Dic₃	1	4	yes	complemented
43	C₅×#95	—	6	4	no	—
44	#1846	—	3	1	no	Sylow
45	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	—	2	4	no	—
47	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
48	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₅×Dic₃	—	4	4	no	—
55	C₂×C₅×Dic₃	—	4	8	no	—
56	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	—
57	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	—
58	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
60	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
61	C₂×C₄⋊Dic₃	C₂²×C₅	1	4	yes	—
62	C₃×C₄×C₈	C₄×C₅	1	4	yes	complemented
63	C₃⋊#95	—	2	4	no	—
64	#818	—	2	4	no	—
65	#817	—	2	4	no	—
66	C₂²×C₄×C₅	—	6	4	no	—
67	C₂²×C₄×C₅	D₁₂	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
69	C₂×C₅×C₈	Dic₆	1	2	yes	—
70	C₂×C₅×C₈	Dic₆	1	2	yes	—
71	C₂×C₅×C₈	D₁₂	1	2	yes	—
72	C₂×C₅×C₈	D₁₂	1	2	yes	—
73	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
74	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
75	C₅×C₄⋊C₄	—	6	4	no	—
76	C₅×C₄⋊C₄	—	6	4	no	—
77	C₄²×C₅	Dic₆	1	2	yes	—
78	C₄²×C₅	C₂×Dic₃	1	2	yes	—
79	C₂×#107	—	3	2	no	—
80	C₂×#106	—	3	2	no	—
81	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
82	#382	—	3	2	no	—
83	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₃×C₅	#127	1	1	yes	characteristic
85	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₃×C₅	#135	1	4	yes	—
87	C₃×C₄×C₅	—	2	4	no	—
88	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
91	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
92	C₅×Dic₃	—	8	4	no	—
93	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
94	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
95	C₂²×Dic₃	—	2	4	no	—
96	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
97	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
98	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
99	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
100	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
101	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
102	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
103	C₃×C₄²	C₅×Q₈	1	2	yes	—
104	C₄⋊Dic₃	—	2	4	no	—
105	C₄⋊Dic₃	—	2	4	no	—
106	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
107	C₂×C₄×C₅	—	2	4	no	—
108	C₂×C₄×C₅	—	12	4	no	—
109	C₂×C₄×C₅	—	12	8	no	—
110	C₂×C₄×C₅	C₄⋊Dic₃	1	4	yes	—
111	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
117	C₅×C₈	C₄⋊Dic₃	1	4	yes	—
118	C₅×C₈	C₄⋊Dic₃	1	4	yes	—
119	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
120	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	3	4	no	—
122	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
123	#107	—	6	4	no	—
124	#106	—	6	4	no	—
125	C₄×C₈	C₅×Dic₃	1	4	yes	complemented
126	#95	—	6	4	no	—
127	C₂×C₃×C₅	#384	1	1	yes	characteristic
128	C₂×C₃×C₅	#427	1	2	yes	—
129	C₂×C₃×C₅	#495	1	2	yes	—
130	C₂×C₃×C₅	#499	1	2	yes	—
131	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	C₅×C₄⋊C₄	1	4	yes	—
134	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
140	C₂×Dic₃	—	4	4	no	—
141	C₂×Dic₃	—	4	8	no	—
142	C₃×C₈	C₅×C₄⋊C₄	1	4	yes	—
143	C₃×C₈	C₅×C₄⋊C₄	1	4	yes	—
144	C₂²×C₅	#869	1	1	yes	characteristic
145	C₂²×C₅	#874	1	1	yes	characteristic
146	C₂²×C₅	#904	1	4	yes	—
147	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
148	C₄×C₅	—	2	4	no	—
149	C₄×C₅	—	24	4	no	—
150	C₄×C₅	#869	1	1	yes	characteristic
151	C₄×C₅	#869	1	1	yes	characteristic
152	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
153	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
154	C₂²×C₄	—	6	4	no	—
155	C₂²×C₄	C₅×D₁₂	1	2	yes	—
156	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
157	C₂×C₈	C₅×Dic₆	1	2	yes	—
158	C₂×C₈	C₅×Dic₆	1	2	yes	—
159	C₂×C₈	C₅×D₁₂	1	2	yes	—
160	C₂×C₈	C₅×D₁₂	1	2	yes	—
161	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
162	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
163	C₄⋊C₄	—	6	4	no	—
164	C₄⋊C₄	—	6	4	no	—
165	C₄²	C₅×Dic₆	1	2	yes	—
166	C₄²	C₂×C₅×Dic₃	1	2	yes	—
167	C₃×C₅	#1846	1	1	yes	characteristic, complemented, Hall
168	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₃	C₅×#127	1	1	yes	characteristic
170	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
171	C₂²×C₃	C₅×#135	1	4	yes	—
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
174	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
175	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
176	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
177	Dic₃	—	8	4	no	—
178	C₂×C₅	#4683	1	2	yes	—
179	C₂×C₅	#4696	1	2	yes	—
180	C₂×C₅	#4935	1	1	yes	characteristic
181	C₂×C₅	#5106	1	2	yes	—
182	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	12	4	no	—
185	C₂×C₄	—	12	8	no	—
186	C₂×C₄	C₅×C₄⋊Dic₃	1	4	yes	—
187	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
188	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
189	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
190	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
191	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
192	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
193	C₈	C₅×C₄⋊Dic₃	1	4	yes	—
194	C₈	C₅×C₄⋊Dic₃	1	4	yes	—
195	C₂×C₃	C₅×#384	1	1	yes	characteristic
196	C₂×C₃	C₅×#427	1	2	yes	—
197	C₂×C₃	C₅×#495	1	2	yes	—
198	C₂×C₃	C₅×#499	1	2	yes	—
199	C₅	#69766	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	C₅×#869	1	1	yes	characteristic
201	C₂²	C₅×#874	1	1	yes	characteristic
202	C₂²	C₅×#904	1	4	yes	—
203	C₂²	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	24	4	no	—
206	C₄	C₅×#869	1	1	yes	characteristic
207	C₄	C₅×#869	1	1	yes	characteristic
208	C₄	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
209	C₄	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
210	C₃	C₅×#1846	1	1	yes	characteristic, complemented, Sylow
211	C₂	C₅×#4683	1	2	yes	—
212	C₂	C₅×#4696	1	2	yes	—
213	C₂	C₅×#4935	1	1	yes	characteristic
214	C₂	C₅×#5106	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×#1846
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
10. C₃×C₄×C₅×C₈	C₄	semidirect
14. #69766	C₅	direct (N × H)
42. C₄×C₅×C₈	Dic₃	semidirect
62. C₃×C₄×C₈	C₄×C₅	semidirect
125. C₄×C₈	C₅×Dic₃	semidirect
167. C₃×C₅	#1846	semidirect
199. C₅	#69766	direct (N × H)
210. C₃	C₅×#1846	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#818	2
C₂×C₅×#817	2
C₅×#4936	2
C₅×#1846	3
#69766	5

Subgroups — C5×#69766 #14299105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69766 #14299105