Subgroups — C₅×#68697 #14294963

Order 1920.

400 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×#5122	C₂	1	1	yes	maximal, characteristic
2	C₅×#4997	C₂	1	2	yes	maximal
3	C₃×C₅×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
4	C₅×C₃⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₅×#4531	C₂	1	2	yes	maximal
6	C₅×#1747	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	C₂²	1	2	yes	—
9	C₅×#926	C₂²	1	1	yes	characteristic
10	C₂×C₅×#208	C₂²	1	2	yes	—
11	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₅×#815	C₄	1	2	yes	complemented
13	C₅×C₈×Dic₃	C₄	1	2	yes	complemented
14	C₅×#803	C₄	1	4	yes	complemented
15	C₄×C₅×C₃⋊C₈	C₄	1	4	yes	complemented
16	#68697	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#405	—	3	1	no	—
18	C₅×#405	—	3	2	no	—
19	C₅×#338	—	3	1	no	—
20	C₅×H(3,ℤ/4)	—	3	2	no	—
21	C₅×H(3,ℤ/4)	D₃	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₃×C₅×M₁₆	—	2	2	no	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₃×C₅×C₄⋊C₄	—	2	2	no	—
26	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
27	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
28	C₅×#217	—	2	1	no	—
29	C₅×C₄⋊Dic₃	—	2	1	no	—
30	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
31	C₅×#208	—	2	2	no	—
32	C₅×#208	—	2	2	no	—
33	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
34	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	#5122	C₂×C₅	1	1	yes	characteristic
39	#4997	C₂×C₅	1	2	yes	—
40	C₃×H(3,ℤ/4)	C₂×C₅	1	1	yes	characteristic
41	C₃⋊#338	C₂×C₅	1	1	yes	characteristic
42	#4531	C₂×C₅	1	2	yes	—
43	C₂×C₅×M₁₆	—	3	2	no	—
44	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₅×C₈	—	3	1	no	—
46	C₅×#117	—	3	1	no	—
47	C₅×#117	C₂×D₃	1	2	yes	—
48	C₅×#97	—	3	2	no	—
49	C₅×#97	—	3	4	no	—
50	C₄×C₅×C₈	—	3	2	no	—
51	C₄×C₅×C₈	—	3	4	no	—
52	#1747	—	3	1	no	Sylow
53	C₂³×C₃×C₅	#41	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	#41	1	2	yes	—
58	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
59	C₂×C₅×Dic₃	—	2	2	no	—
60	C₃×C₅×C₈	—	2	2	no	—
61	C₅×C₃⋊C₈	—	2	4	no	—
62	C₅×C₃⋊C₈	C₄²	1	1	yes	characteristic
63	C₅×C₃⋊C₈	C₄²	1	2	yes	—
64	C₅×C₃⋊C₈	C₄²	1	1	yes	characteristic
65	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
66	C₃×#117	C₂²×C₅	1	2	yes	—
67	#926	C₂²×C₅	1	1	yes	characteristic
68	C₂×#208	C₂²×C₅	1	2	yes	—
69	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
70	#815	C₄×C₅	1	2	yes	complemented
71	C₈×Dic₃	C₄×C₅	1	2	yes	complemented
72	#803	C₄×C₅	1	4	yes	complemented
73	C₄×C₃⋊C₈	C₄×C₅	1	4	yes	complemented
74	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
75	C₅×M₁₆	—	2	2	no	—
76	C₅×M₁₆	—	6	2	no	—
77	C₅×M₁₆	—	6	2	no	—
78	C₂×C₅×C₈	—	3	1	no	—
79	C₂×C₅×C₈	—	3	2	no	—
80	C₂×C₅×C₈	—	3	1	no	—
81	C₂×C₅×C₈	—	3	2	no	—
82	C₂×C₅×C₈	—	3	4	no	—
83	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
84	C₅×C₄⋊C₄	—	2	2	no	—
85	C₅×C₄⋊C₄	—	6	1	no	—
86	C₅×C₂²⋊C₄	—	2	2	no	—
87	C₅×C₂²⋊C₄	—	6	1	no	—
88	C₄²×C₅	—	3	2	no	—
89	C₄²×C₅	C₄×D₃	1	4	yes	—
90	#405	—	3	1	no	—
91	#405	—	3	2	no	—
92	#338	—	3	1	no	—
93	H(3,ℤ/4)	—	3	2	no	—
94	H(3,ℤ/4)	C₅×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₅	—	4	1	no	—
96	C₂²×C₃×C₅	#117	1	2	yes	—
97	C₂²×C₃×C₅	#117	1	1	yes	characteristic
98	C₃×C₄×C₅	—	4	4	no	—
99	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
100	C₃×C₄×C₅	#117	1	1	yes	characteristic
101	C₃×C₄×C₅	#117	1	2	yes	—
102	C₅×Dic₃	—	4	2	no	—
103	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
104	C₃×M₁₆	—	2	2	no	—
105	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₂²⋊C₄	—	2	2	no	—
108	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
109	#217	—	2	1	no	—
110	C₄⋊Dic₃	—	2	1	no	—
111	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
112	#208	—	2	2	no	—
113	#208	—	2	2	no	—
114	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
115	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
116	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
117	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
118	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
119	C₂³×C₅	#237	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	6	2	no	—
122	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
124	C₂×C₄×C₅	#237	1	2	yes	—
125	C₂×C₄×C₅	#239	1	1	yes	characteristic
126	C₅×C₈	—	2	2	no	—
127	C₅×C₈	—	3	1	no	—
128	C₅×C₈	—	3	2	no	—
129	C₅×C₈	—	3	1	no	—
130	C₅×C₈	—	6	4	no	—
131	C₂×M₁₆	—	3	2	no	—
132	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
133	C₂²×C₈	—	3	1	no	—
134	#117	—	3	1	no	—
135	#117	C₂×C₅×D₃	1	2	yes	—
136	#97	—	3	2	no	—
137	#97	—	3	4	no	—
138	C₄×C₈	—	3	2	no	—
139	C₄×C₈	—	3	4	no	—
140	C₂×C₃×C₅	—	2	1	no	—
141	C₂×C₃×C₅	—	2	2	no	—
142	C₂×C₃×C₅	#376	1	1	yes	characteristic
143	C₂³×C₃	C₅×#41	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
146	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
147	C₂×C₃×C₄	C₅×#41	1	2	yes	—
148	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
149	C₂×Dic₃	—	2	2	no	—
150	C₃×C₈	—	2	2	no	—
151	C₃⋊C₈	—	2	4	no	—
152	C₃⋊C₈	C₄²×C₅	1	1	yes	characteristic
153	C₃⋊C₈	C₄²×C₅	1	2	yes	—
154	C₃⋊C₈	C₄²×C₅	1	1	yes	characteristic
155	C₂²×C₅	—	4	1	no	—
156	C₂²×C₅	#877	1	2	yes	—
157	C₂²×C₅	#892	1	1	yes	characteristic
158	C₄×C₅	—	4	4	no	—
159	C₄×C₅	—	12	2	no	—
160	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
161	C₄×C₅	#877	1	1	yes	characteristic
162	C₄×C₅	#892	1	2	yes	—
163	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
164	M₁₆	—	2	2	no	—
165	M₁₆	—	6	2	no	—
166	M₁₆	—	6	2	no	—
167	C₂×C₈	—	3	1	no	—
168	C₂×C₈	—	3	2	no	—
169	C₂×C₈	—	3	1	no	—
170	C₂×C₈	—	3	2	no	—
171	C₂×C₈	—	3	4	no	—
172	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	6	1	no	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	6	1	no	—
177	C₄²	—	3	2	no	—
178	C₄²	C₄×C₅×D₃	1	4	yes	—
179	C₃×C₅	#1747	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	—	4	1	no	—
181	C₂²×C₃	C₅×#117	1	2	yes	—
182	C₂²×C₃	C₅×#117	1	1	yes	characteristic
183	C₃×C₄	—	4	4	no	—
184	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
185	C₃×C₄	C₅×#117	1	1	yes	characteristic
186	C₃×C₄	C₅×#117	1	2	yes	—
187	Dic₃	—	4	2	no	—
188	C₂×C₅	—	2	2	no	—
189	C₂×C₅	—	2	1	no	—
190	C₂×C₅	#4648	1	1	yes	characteristic
191	C₂³	C₅×#237	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	6	2	no	—
194	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
196	C₂×C₄	C₅×#237	1	2	yes	—
197	C₂×C₄	C₅×#239	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	3	1	no	—
200	C₈	—	3	1	no	—
201	C₈	—	3	2	no	—
202	C₈	—	6	4	no	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	2	no	—
205	C₂×C₃	C₅×#376	1	1	yes	characteristic
206	C₅	#68697	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	4	1	no	—
208	C₂²	C₅×#877	1	2	yes	—
209	C₂²	C₅×#892	1	1	yes	characteristic
210	C₄	—	4	4	no	—
211	C₄	—	12	2	no	—
212	C₄	C₄²×C₅×D₃	1	1	yes	characteristic
213	C₄	C₅×#877	1	1	yes	characteristic
214	C₄	C₅×#892	1	2	yes	—
215	C₃	C₅×#1747	1	1	yes	characteristic, complemented, Sylow
216	C₂	—	2	2	no	—
217	C₂	—	2	1	no	—
218	C₂	C₅×#4648	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₄²×D₃
commutator	C₃×C₄	C₂×C₄²×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×H(3,ℤ/4)	C₂
socle	C₂×C₃×C₅	#376
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×H(3,ℤ/4)	D₃
hypocenter	C₃	C₅×#1747
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₅×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
12. C₅×#815	C₄	semidirect
13. C₅×C₈×Dic₃	C₄	semidirect
14. C₅×#803	C₄	semidirect
15. C₄×C₅×C₃⋊C₈	C₄	semidirect
16. #68697	C₅	direct (N × H)
70. #815	C₄×C₅	semidirect
71. C₈×Dic₃	C₄×C₅	semidirect
72. #803	C₄×C₅	semidirect
73. C₄×C₃⋊C₈	C₄×C₅	semidirect
179. C₃×C₅	#1747	semidirect
206. C₅	#68697	direct (N × H)
215. C₃	C₅×#1747	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5122	2
C₅×#4997	2
C₃×C₅×H(3,ℤ/4)	2
C₅×C₃⋊#338	2
C₅×#4531	2
C₅×#1747	3
#68697	5

Subgroups — C5×#68697 #14294963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68697 #14294963