Subgroups — C₅×#69388 #14294867

Order 1920.

592 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₃⋊C₈	C₂	1	2	yes	maximal
3	C₅×#4524	C₂	1	4	yes	maximal
4	C₅×#1744	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
6	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₃⋊C₈	C₂²	1	4	yes	—
10	C₄×C₅×C₃⋊C₈	C₄	1	8	yes	complemented
11	#69388	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₄×C₅×C₈	—	3	2	no	—
13	C₄³×C₅	D₃	1	1	yes	characteristic
14	C₅×#336	—	3	4	no	—
15	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
16	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₃×C₄²×C₅	—	2	4	no	—
20	C₃×C₄²×C₅	—	2	4	no	—
21	C₃×C₄²×C₅	C₈	1	8	yes	complemented
22	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
23	C₂×C₅×C₃⋊C₈	—	2	8	no	—
24	C₂×C₅×C₃⋊C₈	C₂×C₄	1	8	yes	—
25	C₃×C₄³	C₂×C₅	1	1	yes	characteristic
26	C₂×C₄×C₃⋊C₈	C₂×C₅	1	2	yes	—
27	#4524	C₂×C₅	1	4	yes	—
28	C₂²×C₅×C₈	—	3	4	no	—
29	C₂×C₄²×C₅	Dic₃	1	2	yes	—
30	C₂×C₄²×C₅	Dic₃	1	2	yes	—
31	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
32	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
33	C₄×C₅×C₈	—	3	8	no	—
34	#1744	—	3	1	no	Sylow
35	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₅	—	2	4	no	—
37	C₂×C₃×C₄×C₅	—	2	4	no	—
38	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
39	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂×C₈	1	8	yes	—
41	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
42	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
43	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
44	C₂×C₃×C₄×C₅	#41	1	4	yes	—
45	C₅×C₃⋊C₈	—	2	8	no	—
46	C₂×C₃×C₄²	C₄×C₅	1	2	yes	—
47	C₂×C₃×C₄²	C₄×C₅	1	2	yes	—
48	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
49	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
50	C₂²×C₃⋊C₈	C₂²×C₅	1	4	yes	—
51	C₄×C₃⋊C₈	C₄×C₅	1	8	yes	complemented
52	C₂²×C₄×C₅	C₂×Dic₃	1	4	yes	—
53	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
56	C₂×C₅×C₈	—	3	8	no	—
57	C₂×C₅×C₈	—	6	8	no	—
58	C₄²×C₅	—	2	4	no	—
59	C₄²×C₅	—	2	4	no	—
60	C₄²×C₅	C₃⋊C₈	1	8	yes	complemented
61	C₄²×C₅	C₄×D₃	1	4	yes	—
62	C₂×C₄×C₈	—	3	2	no	—
63	C₄³	C₅×D₃	1	1	yes	characteristic
64	#336	—	3	4	no	—
65	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₃×C₅	#117	1	2	yes	—
67	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
68	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
69	C₃×C₄×C₅	—	2	8	no	—
70	C₃×C₄×C₅	C₄×C₈	1	4	yes	—
71	C₃×C₄×C₅	#97	1	4	yes	—
72	C₃×C₄×C₅	#117	1	4	yes	—
73	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
74	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
77	C₃×C₄²	—	2	4	no	—
78	C₃×C₄²	—	2	4	no	—
79	C₃×C₄²	C₅×C₈	1	8	yes	complemented
80	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
81	C₂×C₃⋊C₈	—	2	8	no	—
82	C₂×C₃⋊C₈	C₂×C₄×C₅	1	8	yes	—
83	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	4	no	—
86	C₂×C₄×C₅	C₂×C₃⋊C₈	1	8	yes	—
87	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
88	C₂×C₄×C₅	#208	1	4	yes	—
89	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
90	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
91	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
92	C₂×C₄×C₅	#235	1	4	yes	—
93	C₅×C₈	—	6	8	no	—
94	C₂²×C₈	—	3	4	no	—
95	C₂×C₄²	C₅×Dic₃	1	2	yes	—
96	C₂×C₄²	C₅×Dic₃	1	2	yes	—
97	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
98	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
99	C₄×C₈	—	3	8	no	—
100	C₂×C₃×C₅	#376	1	1	yes	characteristic
101	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
102	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
103	C₂×C₃×C₅	#431	1	4	yes	—
104	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
108	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
109	C₂×C₃×C₄	C₂×C₅×C₈	1	8	yes	—
110	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
111	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
112	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
113	C₂×C₃×C₄	C₅×#41	1	4	yes	—
114	C₃⋊C₈	—	2	8	no	—
115	C₂²×C₅	C₂²×C₃⋊C₈	1	2	yes	—
116	C₂²×C₅	C₂×#208	1	2	yes	—
117	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
118	C₂²×C₅	#926	1	2	yes	—
119	C₄×C₅	—	2	8	no	—
120	C₄×C₅	C₄×C₃⋊C₈	1	4	yes	—
121	C₄×C₅	#803	1	4	yes	—
122	C₄×C₅	#872	1	4	yes	—
123	C₂²×C₄	C₂×C₅×Dic₃	1	4	yes	—
124	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
127	C₂×C₈	—	3	8	no	—
128	C₂×C₈	—	6	8	no	—
129	C₄²	—	2	4	no	—
130	C₄²	—	2	4	no	—
131	C₄²	C₅×C₃⋊C₈	1	8	yes	complemented
132	C₄²	C₄×C₅×D₃	1	4	yes	—
133	C₃×C₅	#1744	1	1	yes	characteristic, complemented, Hall
134	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
135	C₂²×C₃	C₅×#117	1	2	yes	—
136	C₂²×C₃	C₂²×C₅×C₈	1	2	yes	—
137	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
138	C₃×C₄	—	2	8	no	—
139	C₃×C₄	C₄×C₅×C₈	1	4	yes	—
140	C₃×C₄	C₅×#97	1	4	yes	—
141	C₃×C₄	C₅×#117	1	4	yes	—
142	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
143	C₂×C₅	C₂×#803	1	1	yes	characteristic
144	C₂×C₅	#4925	1	4	yes	—
145	C₂×C₅	#4932	1	1	yes	characteristic
146	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	C₂×C₅×C₃⋊C₈	1	8	yes	—
150	C₂×C₄	C₂×C₅×C₃⋊C₈	1	4	yes	—
151	C₂×C₄	C₅×#208	1	4	yes	—
152	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
153	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
154	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
155	C₂×C₄	C₅×#235	1	4	yes	—
156	C₈	—	6	8	no	—
157	C₂×C₃	C₅×#376	1	1	yes	characteristic
158	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
159	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
160	C₂×C₃	C₅×#431	1	4	yes	—
161	C₅	#69388	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²	C₂²×C₅×C₃⋊C₈	1	2	yes	—
163	C₂²	C₂×C₅×#208	1	2	yes	—
164	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
165	C₂²	C₅×#926	1	2	yes	—
166	C₄	—	2	8	no	—
167	C₄	C₄×C₅×C₃⋊C₈	1	4	yes	—
168	C₄	C₅×#803	1	4	yes	—
169	C₄	C₅×#872	1	4	yes	—
170	C₃	C₅×#1744	1	1	yes	characteristic, complemented, Sylow
171	C₂	C₂×C₄×C₅×C₃⋊C₈	1	1	yes	characteristic
172	C₂	C₂×C₅×#803	1	1	yes	characteristic
173	C₂	C₅×#4925	1	4	yes	—
174	C₂	C₅×#4932	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₅	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅×C₈
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₄³×C₅	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₄³×C₅	D₃
hypocenter	C₃	C₅×#1744
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄²×C₅ ◁ C₄³×C₅

Complements

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

N	H ≅ G/N	type
10. C₄×C₅×C₃⋊C₈	C₄	semidirect
11. #69388	C₅	direct (N × H)
21. C₃×C₄²×C₅	C₈	semidirect
51. C₄×C₃⋊C₈	C₄×C₅	semidirect
60. C₄²×C₅	C₃⋊C₈	semidirect
79. C₃×C₄²	C₅×C₈	semidirect
131. C₄²	C₅×C₃⋊C₈	semidirect
133. C₃×C₅	#1744	semidirect
161. C₅	#69388	direct (N × H)
170. C₃	C₅×#1744	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄³×C₅	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#4524	2
C₅×#1744	3
#69388	5

Subgroups — C5×#69388 #14294867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69388 #14294867