Subgroups — C₅×#68928 #14298887

Order 1920.

376 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5118	C₂	1	2	yes	maximal
2	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#341	C₂	1	1	yes	maximal, characteristic
4	C₅×#4558	C₂	1	2	yes	maximal
5	C₅×#4552	C₂	1	1	yes	maximal, characteristic
6	C₅×#1837	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×Dic₃	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×Dic₃⋊C₈	—	2	4	no	—
13	C₅×C₈×Dic₃	C₄	1	2	yes	—
14	C₅×C₈×Dic₃	C₄	1	2	yes	—
15	#68928	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₅×#423	—	3	2	no	—
17	C₂×C₄×C₅×C₈	—	3	1	no	—
18	C₅×#341	—	3	2	no	—
19	C₅×#341	—	3	1	no	—
20	C₅×#341	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×C₅×Dic₃	Q₈	1	2	yes	—
23	C₃×C₅×M₁₆	—	2	2	no	—
24	C₃×C₅×M₁₆	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	2	no	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₅×Dic₃	D₄	1	2	yes	—
31	C₅×#208	—	2	2	no	—
32	C₅×#208	—	2	2	no	—
33	C₂×C₅×C₃⋊C₈	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	#5118	C₂×C₅	1	2	yes	—
37	C₂×C₈×Dic₃	C₂×C₅	1	1	yes	characteristic
38	C₃×#341	C₂×C₅	1	1	yes	characteristic
39	#4558	C₂×C₅	1	2	yes	—
40	#4552	C₂×C₅	1	1	yes	characteristic
41	C₂×C₅×M₁₆	—	3	2	no	—
42	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
43	C₂²×C₅×C₈	—	3	1	no	—
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄²×C₅	—	3	1	no	—
46	C₅×C₄⋊C₈	—	6	4	no	—
47	C₄×C₅×C₈	—	3	2	no	—
48	C₄×C₅×C₈	—	3	2	no	—
49	#1837	—	3	1	no	Sylow
50	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	#41	1	2	yes	—
54	C₂×C₃×C₄×C₅	#41	1	2	yes	—
55	C₂×C₅×Dic₃	—	2	4	no	—
56	C₂×C₅×Dic₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₅×Dic₃	C₄⋊C₄	1	2	yes	—
58	C₃×C₅×C₈	—	2	2	no	—
59	C₃×C₅×C₈	—	4	2	no	—
60	C₅×C₃⋊C₈	—	2	2	no	—
61	C₅×C₃⋊C₈	—	4	2	no	—
62	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
63	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
65	C₂×#208	C₂²×C₅	1	2	yes	—
66	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
67	Dic₃⋊C₈	—	2	4	no	—
68	C₈×Dic₃	C₄×C₅	1	2	yes	—
69	C₈×Dic₃	C₄×C₅	1	2	yes	—
70	C₂²×C₄×C₅	—	3	2	no	—
71	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
72	C₅×M₁₆	—	2	2	no	—
73	C₅×M₁₆	—	2	2	no	—
74	C₅×M₁₆	—	6	2	no	—
75	C₅×M₁₆	—	6	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	3	2	no	—
79	C₂×C₅×C₈	—	6	2	no	—
80	C₂×C₅×C₈	—	6	2	no	—
81	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
82	C₄²×C₅	—	3	1	no	—
83	C₄²×C₅	—	3	1	no	—
84	C₄²×C₅	—	3	2	no	—
85	#423	—	3	2	no	—
86	C₂×C₄×C₈	—	3	1	no	—
87	#341	—	3	2	no	—
88	#341	—	3	1	no	—
89	#341	C₅×D₃	1	1	yes	characteristic
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₃×C₅	#125	1	2	yes	—
93	C₃×C₄×C₅	#117	1	1	yes	characteristic
94	C₃×C₄×C₅	#117	1	1	yes	characteristic
95	C₃×C₄×C₅	#124	1	2	yes	—
96	C₅×Dic₃	—	2	2	no	—
97	C₅×Dic₃	#108	1	4	yes	—
98	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
99	C₂²×Dic₃	C₅×Q₈	1	2	yes	—
100	C₃×M₁₆	—	2	2	no	—
101	C₃×M₁₆	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
105	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
106	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
107	C₄×Dic₃	C₅×D₄	1	2	yes	—
108	#208	—	2	2	no	—
109	#208	—	2	2	no	—
110	C₂×C₃⋊C₈	—	2	2	no	—
111	C₂×C₃⋊C₈	—	2	2	no	—
112	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
113	C₂³×C₅	D₃×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅	—	3	2	no	—
115	C₂×C₄×C₅	—	3	2	no	—
116	C₂×C₄×C₅	—	6	4	no	—
117	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
118	C₂×C₄×C₅	#235	1	2	yes	—
119	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅	#237	1	2	yes	—
121	C₅×C₈	—	2	2	no	—
122	C₅×C₈	—	4	2	no	—
123	C₅×C₈	—	6	2	no	—
124	C₅×C₈	—	12	2	no	—
125	C₂×M₁₆	—	3	2	no	—
126	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
127	C₂²×C₈	—	3	1	no	—
128	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
129	C₂×C₄²	—	3	1	no	—
130	C₄⋊C₈	—	6	4	no	—
131	C₄×C₈	—	3	2	no	—
132	C₄×C₈	—	3	2	no	—
133	C₂×C₃×C₅	—	2	2	no	—
134	C₂×C₃×C₅	#382	1	1	yes	characteristic
135	C₂×C₃×C₅	C₂×#108	1	2	yes	—
136	C₂³×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₅×#41	1	2	yes	—
140	C₂×C₃×C₄	C₅×#41	1	2	yes	—
141	C₂×Dic₃	—	2	4	no	—
142	C₂×Dic₃	C₅×C₄⋊C₄	1	2	yes	—
143	C₂×Dic₃	C₅×C₄⋊C₄	1	2	yes	—
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	—	4	2	no	—
146	C₃⋊C₈	—	2	2	no	—
147	C₃⋊C₈	—	4	2	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	#889	1	2	yes	—
150	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₅	—	3	4	no	—
152	C₄×C₅	—	6	2	no	—
153	C₄×C₅	#872	1	1	yes	characteristic
154	C₄×C₅	#877	1	1	yes	characteristic
155	C₄×C₅	#885	1	2	yes	—
156	C₂²×C₄	—	3	2	no	—
157	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
158	M₁₆	—	2	2	no	—
159	M₁₆	—	2	2	no	—
160	M₁₆	—	6	2	no	—
161	M₁₆	—	6	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	3	2	no	—
165	C₂×C₈	—	6	2	no	—
166	C₂×C₈	—	6	2	no	—
167	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
168	C₄²	—	3	1	no	—
169	C₄²	—	3	1	no	—
170	C₄²	—	3	2	no	—
171	C₃×C₅	#1837	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₃	C₅×#125	1	2	yes	—
175	C₃×C₄	C₅×#117	1	1	yes	characteristic
176	C₃×C₄	C₅×#117	1	1	yes	characteristic
177	C₃×C₄	C₅×#124	1	2	yes	—
178	Dic₃	—	2	2	no	—
179	Dic₃	C₅×#108	1	4	yes	—
180	C₂×C₅	—	2	2	no	—
181	C₂×C₅	#4650	1	1	yes	characteristic
182	C₂×C₅	D₃×#108	1	2	yes	—
183	C₂³	C₅×D₃×Q₈	1	1	yes	characteristic
184	C₂×C₄	—	3	2	no	—
185	C₂×C₄	—	3	2	no	—
186	C₂×C₄	—	6	4	no	—
187	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
188	C₂×C₄	C₅×#235	1	2	yes	—
189	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
190	C₂×C₄	C₅×#237	1	2	yes	—
191	C₈	—	2	2	no	—
192	C₈	—	4	2	no	—
193	C₈	—	6	2	no	—
194	C₈	—	12	2	no	—
195	C₂×C₃	—	2	2	no	—
196	C₂×C₃	C₅×#382	1	1	yes	characteristic
197	C₂×C₃	C₂×C₅×#108	1	2	yes	—
198	C₅	#68928	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	—	2	2	no	—
200	C₂²	C₅×#889	1	2	yes	—
201	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
202	C₄	—	3	4	no	—
203	C₄	—	6	2	no	—
204	C₄	C₅×#872	1	1	yes	characteristic
205	C₄	C₅×#877	1	1	yes	characteristic
206	C₄	C₅×#885	1	2	yes	—
207	C₃	C₅×#1837	1	1	yes	characteristic, complemented, Sylow
208	C₂	—	2	2	no	—
209	C₂	C₅×#4650	1	1	yes	characteristic
210	C₂	C₅×D₃×#108	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#341	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#341	D₃
hypocenter	C₃	C₅×#1837
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#341

Complements

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

N	H ≅ G/N	type
15. #68928	C₅	direct (N × H)
171. C₃×C₅	#1837	semidirect
198. C₅	#68928	direct (N × H)
207. C₃	C₅×#1837	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5118	2
C₂×C₅×C₈×Dic₃	2
C₃×C₅×#341	2
C₅×#4558	2
C₅×#4552	2
C₅×#1837	3
#68928	5

Subgroups — C5×#68928 #14298887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68928 #14298887