Subgroups — C₅×#69279 #14315127

Order 1920.

324 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×C₅×#798	C₂	1	2	yes	maximal, complemented
2	C₅×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×(C₂×D₃)⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₄×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2169	—	3	1	no	maximal, Hall
7	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
8	C₄×C₅×D₁₂	C₄	1	1	yes	characteristic
9	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
12	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
15	C₅×#798	—	2	4	no	—
16	#69279	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×M₃₂	—	3	2	no	—
18	C₅×C₈×D₄	—	3	1	no	—
19	C₅×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
20	C₅×C₂²⋊C₁₆	—	3	2	no	—
21	C₄×C₅×C₁₆	—	3	1	no	—
22	C₂×C₅×D₁₂	C₈	1	1	yes	characteristic
23	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₅×(C₂×D₃)⋊C₄	C₈	1	2	yes	—
28	C₅×C₄⋊Dic₃	C₈	1	1	yes	characteristic
29	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₅×D₃×C₈	—	2	2	no	—
31	C₅×D₃×C₈	—	2	2	no	—
32	C₃×C₅×C₁₆	—	2	2	no	—
33	C₅×C₃⋊C₁₆	—	2	1	no	—
34	C₅×C₃⋊C₁₆	D₄	1	2	yes	complemented
35	C₂×#798	C₂×C₅	1	2	yes	complemented
36	C₈×D₁₂	C₂×C₅	1	1	yes	characteristic
37	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
38	(C₂×D₃)⋊C₁₆	C₂×C₅	1	2	yes	complemented
39	C₄×C₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂²×C₅×C₈	—	3	2	no	—
41	C₄×C₅×D₄	—	3	1	no	—
42	C₅×M₃₂	—	6	4	no	—
43	C₂×C₅×C₁₆	—	3	1	no	—
44	C₂×C₅×C₁₆	—	3	1	no	—
45	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
46	C₅×C₄⋊C₈	—	3	1	no	—
47	C₅×C₂²⋊C₈	—	3	2	no	—
48	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	#2169	—	3	1	no	Sylow
50	C₂²×C₅×D₃	C₂×C₈	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
55	C₅×D₁₂	—	2	2	no	—
56	C₄×C₅×D₃	—	2	2	no	—
57	C₄×C₅×D₃	—	2	2	no	—
58	C₃×C₅×C₈	—	2	1	no	—
59	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
60	C₃×C₅×C₈	#41	1	1	yes	characteristic
61	C₅×C₃⋊C₈	—	2	2	no	—
62	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
63	C₄×D₁₂	C₄×C₅	1	1	yes	characteristic
64	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
65	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
66	(C₂×D₃)⋊C₈	C₄×C₅	1	2	yes	—
67	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
68	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
69	C₄⋊(C₃⋊C₈)	C₄×C₅	1	1	yes	characteristic
70	#798	—	2	4	no	—
71	C₂×C₅×D₄	—	3	1	no	—
72	C₂²×C₄×C₅	—	3	2	no	—
73	C₂×C₅×C₈	—	3	2	no	—
74	C₂×C₅×C₈	—	6	2	no	—
75	C₂×C₅×C₈	—	6	2	no	—
76	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
78	C₅×C₄⋊C₄	—	3	1	no	—
79	C₅×C₂²⋊C₄	—	3	2	no	—
80	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
81	C₅×C₁₆	—	2	2	no	—
82	C₅×C₁₆	—	3	2	no	—
83	C₅×C₁₆	—	6	1	no	—
84	C₂×M₃₂	—	3	2	no	—
85	C₈×D₄	—	3	1	no	—
86	C₄⋊C₁₆	C₅×D₃	1	1	yes	characteristic, complemented
87	C₂²⋊C₁₆	—	3	2	no	—
88	C₄×C₁₆	—	3	1	no	—
89	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₅×D₃	—	2	2	no	—
91	C₂×C₅×D₃	—	2	2	no	—
92	C₂×C₅×D₃	—	2	2	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	M₃₂	1	2	yes	complemented
95	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
96	C₃×C₄×C₅	#131	1	1	yes	characteristic
97	C₅×Dic₃	—	2	2	no	—
98	C₂×D₁₂	C₅×C₈	1	1	yes	characteristic
99	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	—
100	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
102	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
103	(C₂×D₃)⋊C₄	C₅×C₈	1	2	yes	—
104	C₄⋊Dic₃	C₅×C₈	1	1	yes	characteristic
105	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
106	D₃×C₈	—	2	2	no	—
107	D₃×C₈	—	2	2	no	—
108	C₃×C₁₆	—	2	2	no	—
109	C₃⋊C₁₆	—	2	1	no	—
110	C₃⋊C₁₆	C₅×D₄	1	2	yes	complemented
111	C₂³×C₅	—	3	2	no	—
112	C₅×D₄	—	6	2	no	—
113	C₂×C₄×C₅	—	3	2	no	—
114	C₂×C₄×C₅	—	6	2	no	—
115	C₂×C₄×C₅	—	6	2	no	—
116	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	6	2	no	—
121	C₅×C₈	D₃×D₄	1	1	yes	characteristic
122	C₅×C₈	#239	1	1	yes	characteristic
123	C₂²×C₈	—	3	2	no	—
124	C₄×D₄	—	3	1	no	—
125	M₃₂	—	6	4	no	—
126	C₂×C₁₆	—	3	1	no	—
127	C₂×C₁₆	—	3	1	no	—
128	C₂×C₁₆	C₂×C₅×D₃	1	2	yes	—
129	C₄⋊C₈	—	3	1	no	—
130	C₂²⋊C₈	—	3	2	no	—
131	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
132	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
133	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
134	C₂×C₃×C₅	#504	1	1	yes	characteristic
135	C₅×D₃	—	4	2	no	—
136	C₂²×D₃	C₂×C₅×C₈	1	2	yes	—
137	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
140	C₂×Dic₃	C₂×C₅×C₈	1	2	yes	—
141	D₁₂	—	2	2	no	—
142	C₄×D₃	—	2	2	no	—
143	C₄×D₃	—	2	2	no	—
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
146	C₃×C₈	C₅×#41	1	1	yes	characteristic
147	C₃⋊C₈	—	2	2	no	—
148	C₂²×C₅	—	6	2	no	—
149	C₂²×C₅	—	6	2	no	—
150	C₂²×C₅	—	6	2	no	—
151	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	6	2	no	—
154	C₄×C₅	#798	1	2	yes	complemented
155	C₄×C₅	#893	1	1	yes	characteristic
156	C₄×C₅	#907	1	1	yes	characteristic
157	C₂×D₄	—	3	1	no	—
158	C₂²×C₄	—	3	2	no	—
159	C₂×C₈	—	3	2	no	—
160	C₂×C₈	—	6	2	no	—
161	C₂×C₈	—	6	2	no	—
162	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
163	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
164	C₄⋊C₄	—	3	1	no	—
165	C₂²⋊C₄	—	3	2	no	—
166	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
167	C₁₆	—	2	2	no	—
168	C₁₆	—	3	2	no	—
169	C₁₆	—	6	1	no	—
170	C₃×C₅	#2169	1	1	yes	characteristic, complemented, Hall
171	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
172	C₂×D₃	—	2	2	no	—
173	C₂×D₃	—	2	2	no	—
174	C₂×D₃	—	2	2	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	C₅×M₃₂	1	2	yes	complemented
177	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
178	C₃×C₄	C₅×#131	1	1	yes	characteristic
179	Dic₃	—	2	2	no	—
180	C₂×C₅	—	12	2	no	—
181	C₂×C₅	#4834	1	1	yes	characteristic
182	C₂×C₅	C₂×#798	1	1	yes	characteristic
183	C₂×C₅	#4907	1	1	yes	characteristic
184	C₂³	—	3	2	no	—
185	D₄	—	6	2	no	—
186	C₂×C₄	—	3	2	no	—
187	C₂×C₄	—	6	2	no	—
188	C₂×C₄	—	6	2	no	—
189	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
190	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	6	2	no	—
194	C₈	C₅×D₃×D₄	1	1	yes	characteristic
195	C₈	C₅×#239	1	1	yes	characteristic
196	C₂×C₃	C₅×C₈×D₄	1	1	yes	characteristic
197	C₂×C₃	C₂×C₅×M₃₂	1	1	yes	characteristic
198	C₂×C₃	C₅×#504	1	1	yes	characteristic
199	D₃	—	4	2	no	—
200	C₅	#69279	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	—	6	2	no	—
202	C₂²	—	6	2	no	—
203	C₂²	—	6	2	no	—
204	C₂²	C₂×C₅×D₃×C₈	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	6	2	no	—
207	C₄	C₅×#798	1	2	yes	complemented
208	C₄	C₅×#893	1	1	yes	characteristic
209	C₄	C₅×#907	1	1	yes	characteristic
210	C₃	C₅×#2169	1	1	yes	characteristic, complemented, Sylow
211	C₂	—	12	2	no	—
212	C₂	C₅×#4834	1	1	yes	characteristic
213	C₂	C₂×C₅×#798	1	1	yes	characteristic
214	C₂	C₅×#4907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×C₄⋊C₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₅×D₁₂	C₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×C₄⋊C₁₆	D₃
hypocenter	C₃	C₅×#2169
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₅×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#798	C₂	semidirect
3. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
4. C₅×(C₂×D₃)⋊C₁₆	C₂	semidirect
5. C₄×C₅×C₃⋊C₁₆	C₂	semidirect
16. #69279	C₅	direct (N × H)
19. C₅×C₄⋊C₁₆	D₃	semidirect
34. C₅×C₃⋊C₁₆	D₄	semidirect
35. C₂×#798	C₂×C₅	semidirect
37. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
38. (C₂×D₃)⋊C₁₆	C₂×C₅	semidirect
39. C₄×C₃⋊C₁₆	C₂×C₅	semidirect
86. C₄⋊C₁₆	C₅×D₃	semidirect
94. C₃×C₄×C₅	M₃₂	semidirect
110. C₃⋊C₁₆	C₅×D₄	semidirect
154. C₄×C₅	#798	semidirect
170. C₃×C₅	#2169	semidirect
176. C₃×C₄	C₅×M₃₂	semidirect
200. C₅	#69279	direct (N × H)
207. C₄	C₅×#798	semidirect
210. C₃	C₅×#2169	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#798	2
C₅×C₈×D₁₂	2
C₃×C₅×C₄⋊C₁₆	2
C₅×(C₂×D₃)⋊C₁₆	2
C₄×C₅×C₃⋊C₁₆	2
C₅×#2169	3
#69279	5

Subgroups — C5×#69279 #14315127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69279 #14315127