Subgroups — C₃×C₅×#2177 #14275211

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×D₈	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#357	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2177	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
8	C₃×C₄×C₅×D₄	—	2	2	no	—
9	C₃×C₅×D₁₆	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₃×C₅×#107	C₂²	1	2	yes	—
13	C₃×C₅×#102	—	2	2	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×#2177	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×D₁₆	C₂×C₃	1	1	yes	characteristic
17	C₄×C₅×D₈	C₂×C₃	1	2	yes	complemented
18	C₅×#367	C₂×C₃	1	1	yes	characteristic, complemented
19	C₅×#357	C₂×C₃	1	2	yes	complemented
20	C₅×#346	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₃×C₅×D₄	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	4	2	no	—
23	C₃×C₅×D₈	—	2	2	no	—
24	C₃×C₅×D₈	C₂×C₄	1	4	yes	complemented
25	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₃×C₅×C₄⋊C₄	—	2	2	no	—
28	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
29	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₃×C₅×C₁₆	—	2	1	no	—
31	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
32	C₂×C₃×D₁₆	C₂×C₅	1	1	yes	characteristic
33	C₃×C₄×D₈	C₂×C₅	1	2	yes	complemented
34	C₃×#367	C₂×C₅	1	1	yes	characteristic, complemented
35	C₃×#357	C₂×C₅	1	2	yes	complemented
36	C₃×#346	C₂×C₅	1	1	yes	characteristic, complemented
37	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
38	C₄×C₅×D₄	—	2	2	no	—
39	C₅×D₁₆	C₃×C₄	1	4	yes	complemented
40	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
42	C₅×#107	C₂²×C₃	1	2	yes	—
43	C₅×#102	—	2	2	no	—
44	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	#2177	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₅	—	4	2	no	—
47	C₃×C₅×D₄	—	2	4	no	—
48	C₃×C₅×D₄	—	4	2	no	—
49	C₂×C₃×C₄×C₅	—	4	2	no	—
50	C₂×C₃×C₄×C₅	—	8	4	no	—
51	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₃×C₅×C₈	—	2	1	no	—
54	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₃×C₅×C₈	#41	1	1	yes	characteristic
56	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
57	C₃×C₄×D₄	—	2	2	no	—
58	C₃×D₁₆	C₄×C₅	1	4	yes	complemented
59	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
60	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
61	C₃×#107	C₂²×C₅	1	2	yes	—
62	C₃×#102	—	2	2	no	—
63	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₅×D₄	—	2	2	no	—
65	C₂²×C₄×C₅	—	4	2	no	—
66	C₅×D₈	—	2	2	no	—
67	C₅×D₈	C₂×C₃×C₄	1	4	yes	complemented
68	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₂²⋊C₄	—	4	2	no	—
72	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
73	C₅×C₁₆	—	2	1	no	—
74	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	complemented
75	C₂×D₁₆	C₂×C₃×C₅	1	1	yes	characteristic
76	C₄×D₈	C₂×C₃×C₅	1	2	yes	complemented
77	#367	C₂×C₃×C₅	1	1	yes	characteristic, complemented
78	#357	C₂×C₃×C₅	1	2	yes	complemented
79	#346	C₂×C₃×C₅	1	1	yes	characteristic, complemented
80	C₂²×C₃×C₅	—	4	4	no	—
81	C₂²×C₃×C₅	—	8	2	no	—
82	C₂²×C₃×C₅	—	8	2	no	—
83	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
84	C₃×C₄×C₅	—	2	2	no	—
85	C₃×C₄×C₅	—	8	2	no	—
86	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
87	C₃×C₄×C₅	#135	1	1	yes	characteristic
88	C₂×C₃×D₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	4	2	no	—
90	C₃×D₈	—	2	2	no	—
91	C₃×D₈	C₂×C₄×C₅	1	4	yes	complemented
92	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
94	C₃×C₄⋊C₄	—	2	2	no	—
95	C₃×C₂²⋊C₄	—	4	2	no	—
96	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
97	C₃×C₁₆	—	2	1	no	—
98	C₃×C₁₆	C₂×C₄×C₅	1	2	yes	complemented
99	C₂³×C₅	—	4	2	no	—
100	C₅×D₄	—	2	4	no	—
101	C₅×D₄	—	4	2	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	—	8	4	no	—
104	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
105	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
106	C₅×C₈	—	2	1	no	—
107	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
108	C₅×C₈	C₃×#41	1	1	yes	characteristic
109	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
110	C₄×D₄	—	2	2	no	—
111	D₁₆	C₃×C₄×C₅	1	4	yes	complemented
112	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
113	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
114	#107	C₂²×C₃×C₅	1	2	yes	—
115	#102	—	2	2	no	—
116	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
117	C₂×C₃×C₅	—	8	4	no	—
118	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
119	C₂×C₃×C₅	#509	1	2	yes	—
120	C₂³×C₃	—	4	2	no	—
121	C₃×D₄	—	2	4	no	—
122	C₃×D₄	—	4	2	no	—
123	C₂×C₃×C₄	—	4	2	no	—
124	C₂×C₃×C₄	—	8	4	no	—
125	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
126	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
127	C₃×C₈	—	2	1	no	—
128	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
129	C₃×C₈	C₅×#41	1	1	yes	characteristic
130	C₂²×C₅	—	4	4	no	—
131	C₂²×C₅	—	8	2	no	—
132	C₂²×C₅	—	8	2	no	—
133	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
134	C₄×C₅	—	2	2	no	—
135	C₄×C₅	—	8	2	no	—
136	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
137	C₄×C₅	C₃×#135	1	1	yes	characteristic
138	C₂×D₄	—	2	2	no	—
139	C₂²×C₄	—	4	2	no	—
140	D₈	—	2	2	no	—
141	D₈	C₂×C₃×C₄×C₅	1	4	yes	complemented
142	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
143	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
144	C₄⋊C₄	—	2	2	no	—
145	C₂²⋊C₄	—	4	2	no	—
146	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
147	C₁₆	—	2	1	no	—
148	C₁₆	C₂×C₃×C₄×C₅	1	2	yes	complemented
149	C₃×C₅	#2177	1	1	yes	characteristic, direct product factor, Hall
150	C₂²×C₃	—	4	4	no	—
151	C₂²×C₃	—	8	2	no	—
152	C₂²×C₃	—	8	2	no	—
153	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
154	C₃×C₄	—	2	2	no	—
155	C₃×C₄	—	8	2	no	—
156	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
157	C₃×C₄	C₅×#135	1	1	yes	characteristic
158	C₂×C₅	—	8	4	no	—
159	C₂×C₅	C₃×C₄×D₈	1	1	yes	characteristic
160	C₂×C₅	C₃×#509	1	2	yes	—
161	C₂³	—	4	2	no	—
162	D₄	—	2	4	no	—
163	D₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	8	4	no	—
166	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
167	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
170	C₈	C₃×C₅×#41	1	1	yes	characteristic
171	C₂×C₃	—	8	4	no	—
172	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
173	C₂×C₃	C₅×#509	1	2	yes	—
174	C₅	C₃×#2177	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	—	4	4	no	—
176	C₂²	—	8	2	no	—
177	C₂²	—	8	2	no	—
178	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	8	2	no	—
181	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
182	C₄	C₃×C₅×#135	1	1	yes	characteristic
183	C₃	C₅×#2177	1	1	yes	characteristic, direct product factor, Sylow
184	C₂	—	8	4	no	—
185	C₂	C₃×C₄×C₅×D₈	1	1	yes	characteristic
186	C₂	C₃×C₅×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×D₁₆	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×D₈	C₂	semidirect
3. C₃×C₅×#367	C₂	semidirect
4. C₃×C₅×#357	C₂	semidirect
5. C₃×C₅×#346	C₂	semidirect
6. C₅×#2177	C₃	direct (N × H)
9. C₃×C₅×D₁₆	C₄	semidirect
15. C₃×#2177	C₅	direct (N × H)
17. C₄×C₅×D₈	C₂×C₃	semidirect
18. C₅×#367	C₂×C₃	semidirect
19. C₅×#357	C₂×C₃	semidirect
20. C₅×#346	C₂×C₃	semidirect
24. C₃×C₅×D₈	C₂×C₄	semidirect
31. C₃×C₅×C₁₆	C₂×C₄	semidirect
33. C₃×C₄×D₈	C₂×C₅	semidirect
34. C₃×#367	C₂×C₅	semidirect
35. C₃×#357	C₂×C₅	semidirect
36. C₃×#346	C₂×C₅	semidirect
39. C₅×D₁₆	C₃×C₄	semidirect
45. #2177	C₃×C₅	direct (N × H)
58. C₃×D₁₆	C₄×C₅	semidirect
67. C₅×D₈	C₂×C₃×C₄	semidirect
74. C₅×C₁₆	C₂×C₃×C₄	semidirect
76. C₄×D₈	C₂×C₃×C₅	semidirect
77. #367	C₂×C₃×C₅	semidirect
78. #357	C₂×C₃×C₅	semidirect
79. #346	C₂×C₃×C₅	semidirect
91. C₃×D₈	C₂×C₄×C₅	semidirect
98. C₃×C₁₆	C₂×C₄×C₅	semidirect
111. D₁₆	C₃×C₄×C₅	semidirect
141. D₈	C₂×C₃×C₄×C₅	semidirect
148. C₁₆	C₂×C₃×C₄×C₅	semidirect
149. C₃×C₅	#2177	direct (N × H)
174. C₅	C₃×#2177	direct (N × H)
183. C₃	C₅×#2177	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×D₁₆	2
C₃×C₄×C₅×D₈	2
C₃×C₅×#367	2
C₃×C₅×#357	2
C₃×C₅×#346	2
C₅×#2177	3
C₃×#2177	5

Subgroups — C3×C5×#2177 #14275211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2177 #14275211