Subgroups — C₅×#69303 #14314427

Order 1920.

304 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₃×M₃₂	C₂	1	2	yes	maximal
2	C₅×#4689	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4511	C₂	1	2	yes	maximal, complemented
5	C₅×#4463	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2153	—	3	1	no	maximal, Hall
7	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
8	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
9	C₅×#872	C₄	1	1	yes	characteristic
10	C₃×C₅×M₃₂	C₂²	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#815	C₄	1	1	yes	characteristic
13	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
14	C₅×#812	C₂²	1	2	yes	—
15	C₅×#803	C₄	1	1	yes	characteristic
16	C₅×#798	—	2	2	no	—
17	C₅×D₃×C₁₆	—	2	2	no	—
18	#69303	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×M₃₂	—	3	2	no	—
20	C₅×#405	—	3	1	no	—
21	C₅×#364	—	3	2	no	—
22	C₅×#364	—	3	1	no	—
23	C₅×#364	D₃	1	1	yes	characteristic, complemented
24	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
29	C₅×Dic₃⋊C₄	—	2	1	no	—
30	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₅×ΓL(1,𝔽₂₅)	C₈	1	4	yes	—
34	C₅×D₃×C₈	D₄	1	2	yes	—
35	C₅×D₃×C₈	Q₈	1	2	yes	—
36	C₃×C₅×C₁₆	—	2	2	no	—
37	C₅×C₃⋊C₁₆	—	2	2	no	—
38	D₃×M₃₂	C₂×C₅	1	2	yes	—
39	#4689	C₂×C₅	1	1	yes	characteristic
40	C₃×#364	C₂×C₅	1	1	yes	characteristic, complemented
41	#4511	C₂×C₅	1	2	yes	complemented
42	#4463	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂×C₅×M₁₆	—	3	1	no	—
44	C₂²×C₅×C₈	—	3	1	no	—
45	C₅×#117	—	3	1	no	—
46	C₅×M₃₂	—	3	2	no	—
47	C₅×M₃₂	—	6	2	no	—
48	C₅×M₃₂	C₂×D₃	1	2	yes	—
49	C₂×C₅×C₁₆	—	6	2	no	—
50	C₅×#97	—	3	1	no	—
51	C₅×#97	—	3	1	no	—
52	C₄×C₅×C₈	—	3	1	no	—
53	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
54	#2153	—	3	1	no	Sylow
55	C₂²×C₅×D₃	M₁₆	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₅×Dic₃	—	2	1	no	—
59	C₂×C₅×Dic₃	M₁₆	1	1	yes	characteristic
60	C₄×C₅×D₃	C₄⋊C₄	1	2	yes	—
61	C₄×C₅×D₃	C₂×C₈	1	2	yes	—
62	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
63	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₅×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
66	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	—
67	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	1	yes	characteristic
68	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
69	#872	C₄×C₅	1	1	yes	characteristic
70	C₃×M₃₂	C₂²×C₅	1	2	yes	—
71	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	#815	C₄×C₅	1	1	yes	characteristic
73	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
74	#812	C₂²×C₅	1	2	yes	—
75	#803	C₄×C₅	1	1	yes	characteristic
76	#798	—	2	2	no	—
77	D₃×C₁₆	—	2	2	no	—
78	C₂²×C₄×C₅	—	3	1	no	—
79	C₅×M₁₆	—	3	4	no	—
80	C₂×C₅×C₈	—	3	1	no	—
81	C₂×C₅×C₈	—	3	1	no	—
82	C₂×C₅×C₈	—	3	2	no	—
83	C₂×C₅×C₈	—	3	2	no	—
84	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	6	1	no	—
87	C₅×C₂²⋊C₄	—	6	1	no	—
88	C₄²×C₅	—	3	1	no	—
89	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
90	C₅×C₁₆	—	2	2	no	—
91	C₅×C₁₆	—	6	2	no	—
92	C₂×M₃₂	—	3	2	no	—
93	#405	—	3	1	no	—
94	#364	—	3	2	no	—
95	#364	—	3	1	no	—
96	#364	C₅×D₃	1	1	yes	characteristic, complemented
97	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₅×D₃	—	4	1	no	—
99	C₂×C₅×D₃	C₄⋊C₈	1	2	yes	—
100	C₃×C₄×C₅	—	4	1	no	—
101	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₅×Dic₃	—	4	1	no	—
104	C₅×Dic₃	C₄⋊C₈	1	2	yes	—
105	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
108	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
109	(C₂×D₃)⋊C₄	—	2	1	no	—
110	Dic₃⋊C₄	—	2	1	no	—
111	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
113	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
114	ΓL(1,𝔽₂₅)	C₅×C₈	1	4	yes	—
115	D₃×C₈	C₅×D₄	1	2	yes	—
116	D₃×C₈	C₅×Q₈	1	2	yes	—
117	C₃×C₁₆	—	2	2	no	—
118	C₃⋊C₁₆	—	2	2	no	—
119	C₂³×C₅	—	3	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	3	1	no	—
122	C₂×C₄×C₅	—	3	2	no	—
123	C₂×C₄×C₅	—	3	2	no	—
124	C₂×C₄×C₅	—	6	1	no	—
125	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
126	C₅×C₈	—	3	2	no	—
127	C₅×C₈	—	3	2	no	—
128	C₅×C₈	D₃×C₈	1	2	yes	—
129	C₅×C₈	D₃×D₄	1	1	yes	characteristic
130	C₅×C₈	D₃×Q₈	1	1	yes	characteristic
131	C₂×M₁₆	—	3	1	no	—
132	C₂²×C₈	—	3	1	no	—
133	#117	—	3	1	no	—
134	M₃₂	—	3	2	no	—
135	M₃₂	—	6	2	no	—
136	M₃₂	C₂×C₅×D₃	1	2	yes	—
137	C₂×C₁₆	—	6	2	no	—
138	#97	—	3	1	no	—
139	#97	—	3	1	no	—
140	C₄×C₈	—	3	1	no	—
141	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
142	C₂×C₃×C₅	—	2	1	no	—
143	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₅×D₃	—	2	2	no	—
145	C₂²×D₃	C₅×M₁₆	1	1	yes	characteristic
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
148	C₂×Dic₃	—	2	1	no	—
149	C₂×Dic₃	C₅×M₁₆	1	1	yes	characteristic
150	C₄×D₃	C₅×C₄⋊C₄	1	2	yes	—
151	C₄×D₃	C₂×C₅×C₈	1	2	yes	—
152	C₃×C₈	C₂×C₅×C₈	1	2	yes	—
153	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
154	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
155	C₃⋊C₈	C₅×C₄⋊C₄	1	2	yes	—
156	C₃⋊C₈	C₂×C₅×C₈	1	2	yes	—
157	C₂²×C₅	—	3	2	no	—
158	C₂²×C₅	—	12	1	no	—
159	C₂²×C₅	D₃×M₁₆	1	1	yes	characteristic
160	C₄×C₅	—	3	2	no	—
161	C₄×C₅	—	4	1	no	—
162	C₄×C₅	—	12	1	no	—
163	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
165	C₂²×C₄	—	3	1	no	—
166	M₁₆	—	3	4	no	—
167	C₂×C₈	—	3	1	no	—
168	C₂×C₈	—	3	1	no	—
169	C₂×C₈	—	3	2	no	—
170	C₂×C₈	—	3	2	no	—
171	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
173	C₄⋊C₄	—	6	1	no	—
174	C₂²⋊C₄	—	6	1	no	—
175	C₄²	—	3	1	no	—
176	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
177	C₁₆	—	2	2	no	—
178	C₁₆	—	6	2	no	—
179	C₃×C₅	#2153	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
181	C₂×D₃	—	4	1	no	—
182	C₂×D₃	C₅×C₄⋊C₈	1	2	yes	—
183	C₃×C₄	—	4	1	no	—
184	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
185	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
186	Dic₃	—	4	1	no	—
187	Dic₃	C₅×C₄⋊C₈	1	2	yes	—
188	C₂×C₅	—	2	1	no	—
189	C₂×C₅	—	6	2	no	—
190	C₂×C₅	D₃×C₄⋊C₈	1	1	yes	characteristic
191	C₂³	—	3	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	3	1	no	—
194	C₂×C₄	—	3	2	no	—
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	—	6	1	no	—
197	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
198	C₈	—	3	2	no	—
199	C₈	—	3	2	no	—
200	C₈	C₅×D₃×C₈	1	2	yes	—
201	C₈	C₅×D₃×D₄	1	1	yes	characteristic
202	C₈	C₅×D₃×Q₈	1	1	yes	characteristic
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
205	D₃	—	2	2	no	—
206	C₅	#69303	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	3	2	no	—
208	C₂²	—	12	1	no	—
209	C₂²	C₅×D₃×M₁₆	1	1	yes	characteristic
210	C₄	—	3	2	no	—
211	C₄	—	4	1	no	—
212	C₄	—	12	1	no	—
213	C₄	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
214	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
215	C₃	C₅×#2153	1	1	yes	characteristic, complemented, Sylow
216	C₂	—	2	1	no	—
217	C₂	—	6	2	no	—
218	C₂	C₅×D₃×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	D₃×D₄
commutator	C₃×C₄	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#364	C₂
socle	C₂×C₃×C₅	C₂×C₄⋊C₈
prime radical	C₂²×C₅×D₃	M₁₆
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#364	D₃
hypocenter	C₃	C₅×#2153
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×C₈ ◁ C₅×#364

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#364	C₂	semidirect
4. C₅×#4511	C₂	semidirect
5. C₅×#4463	C₂	semidirect
18. #69303	C₅	direct (N × H)
23. C₅×#364	D₃	semidirect
40. C₃×#364	C₂×C₅	semidirect
41. #4511	C₂×C₅	semidirect
42. #4463	C₂×C₅	semidirect
96. #364	C₅×D₃	semidirect
179. C₃×C₅	#2153	semidirect
206. C₅	#69303	direct (N × H)
215. C₃	C₅×#2153	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₃×M₃₂	2
C₅×#4689	2
C₃×C₅×#364	2
C₅×#4511	2
C₅×#4463	2
C₅×#2153	3
#69303	5

Subgroups — C5×#69303 #14314427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69303 #14314427