Subgroups — C₂×C₃×C₅×#366 #14275190

Order 1920.

336 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#107	C₂	1	2	yes	maximal
3	C₃×C₅×#366	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×#366	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
7	C₂×C₃×C₅×C₁₆	C₄	1	4	yes	complemented
8	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
9	C₃×C₅×#107	—	2	2	no	—
10	C₃×C₅×#107	C₂²	1	4	yes	—
11	C₂×C₃×#366	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂²×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic
13	C₂×C₅×#107	C₂×C₃	1	2	yes	—
14	C₅×#366	C₂×C₃	1	4	yes	direct product factor
15	C₂²×C₃×C₄×C₅	—	4	2	no	—
16	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
17	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
19	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
20	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	—	2	4	no	—
22	C₃×C₅×C₄⋊C₄	—	4	2	no	—
23	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
24	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic
25	C₂×C₃×#107	C₂×C₅	1	2	yes	—
26	C₃×#366	C₂×C₅	1	4	yes	direct product factor
27	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₁₆	C₃×C₄	1	4	yes	complemented
30	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
31	C₅×#107	—	2	2	no	—
32	C₅×#107	C₂²×C₃	1	4	yes	—
33	C₂×#366	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
34	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
35	C₂×C₃×C₄×C₅	—	4	4	no	—
36	C₂×C₃×C₄×C₅	—	8	4	no	—
37	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
38	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
41	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
42	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
45	C₂×C₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₃×C₁₆	C₄×C₅	1	4	yes	complemented
47	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
48	C₃×#107	—	2	2	no	—
49	C₃×#107	C₂²×C₅	1	4	yes	—
50	C₂²×C₄×C₅	—	4	2	no	—
51	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
53	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
55	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
56	C₅×C₄⋊C₄	—	2	4	no	—
57	C₅×C₄⋊C₄	—	4	2	no	—
58	C₅×C₁₆	C₂×C₃×C₄	1	4	yes	complemented
59	C₂²×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
60	C₂×#107	C₂×C₃×C₅	1	2	yes	—
61	#366	C₂×C₃×C₅	1	4	yes	direct product factor
62	C₂²×C₃×C₅	#107	1	2	yes	—
63	C₂²×C₃×C₅	D₁₆	1	2	yes	—
64	C₂²×C₃×C₅	Q₃₂	1	2	yes	—
65	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
66	C₃×C₄×C₅	—	8	4	no	—
67	C₃×C₄×C₅	#107	1	2	yes	—
68	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₃×C₄	—	4	2	no	—
71	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
73	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
75	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
76	C₃×C₄⋊C₄	—	2	4	no	—
77	C₃×C₄⋊C₄	—	4	2	no	—
78	C₃×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
79	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
80	C₂×C₄×C₅	—	4	4	no	—
81	C₂×C₄×C₅	—	8	4	no	—
82	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
85	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
86	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
87	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
88	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₁₆	C₃×C₄×C₅	1	4	yes	complemented
92	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
93	#107	—	2	2	no	—
94	#107	C₂²×C₃×C₅	1	4	yes	—
95	C₂×C₃×C₅	#366	1	4	yes	direct product factor
96	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
97	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
98	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
99	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
100	C₂×C₃×C₄	—	4	4	no	—
101	C₂×C₃×C₄	—	8	4	no	—
102	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
103	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
104	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
105	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
106	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
107	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
108	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
109	C₂²×C₅	C₃×#107	1	2	yes	—
110	C₂²×C₅	C₃×D₁₆	1	2	yes	—
111	C₂²×C₅	C₃×Q₃₂	1	2	yes	—
112	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
113	C₄×C₅	—	8	4	no	—
114	C₄×C₅	C₃×#107	1	2	yes	—
115	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
119	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
120	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
121	C₂×C₈	C₃×C₅×Q₈	1	2	yes	—
122	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₁₆	C₂×C₃×C₄×C₅	1	4	yes	complemented
126	C₃×C₅	C₂×#366	1	1	yes	characteristic, direct product factor, Hall
127	C₂²×C₃	C₅×#107	1	2	yes	—
128	C₂²×C₃	C₅×D₁₆	1	2	yes	—
129	C₂²×C₃	C₅×Q₃₂	1	2	yes	—
130	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
131	C₃×C₄	—	8	4	no	—
132	C₃×C₄	C₅×#107	1	2	yes	—
133	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
134	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
135	C₂×C₅	C₃×#366	1	4	yes	direct product factor
136	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
137	C₂×C₅	C₂×C₃×D₁₆	1	1	yes	characteristic
138	C₂×C₅	C₂×C₃×Q₃₂	1	1	yes	characteristic
139	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
143	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
144	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
145	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
146	C₈	C₃×C₅×C₄⋊C₄	1	2	yes	—
147	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
148	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
149	C₂×C₃	C₅×#366	1	4	yes	direct product factor
150	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
151	C₂×C₃	C₂×C₅×D₁₆	1	1	yes	characteristic
152	C₂×C₃	C₂×C₅×Q₃₂	1	1	yes	characteristic
153	C₅	C₂×C₃×#366	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²	C₃×C₅×#107	1	2	yes	—
155	C₂²	C₃×C₅×D₁₆	1	2	yes	—
156	C₂²	C₃×C₅×Q₃₂	1	2	yes	—
157	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
158	C₄	—	8	4	no	—
159	C₄	C₃×C₅×#107	1	2	yes	—
160	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
161	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
162	C₃	C₂×C₅×#366	1	1	yes	characteristic, direct product factor, Sylow
163	C₂	C₃×C₅×#366	1	4	yes	direct product factor
164	C₂	C₂×C₃×C₅×#107	1	1	yes	characteristic
165	C₂	C₂×C₃×C₅×D₁₆	1	1	yes	characteristic
166	C₂	C₂×C₃×C₅×Q₃₂	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#366	C₂	direct (N × H)
4. C₂×C₅×#366	C₃	direct (N × H)
7. C₂×C₃×C₅×C₁₆	C₄	semidirect
11. C₂×C₃×#366	C₅	direct (N × H)
14. C₅×#366	C₂×C₃	direct (N × H)
23. C₃×C₅×C₁₆	C₂×C₄	semidirect
26. C₃×#366	C₂×C₅	direct (N × H)
29. C₂×C₅×C₁₆	C₃×C₄	semidirect
33. C₂×#366	C₃×C₅	direct (N × H)
46. C₂×C₃×C₁₆	C₄×C₅	semidirect
58. C₅×C₁₆	C₂×C₃×C₄	semidirect
61. #366	C₂×C₃×C₅	direct (N × H)
78. C₃×C₁₆	C₂×C₄×C₅	semidirect
91. C₂×C₁₆	C₃×C₄×C₅	semidirect
95. C₂×C₃×C₅	#366	direct (N × H)
125. C₁₆	C₂×C₃×C₄×C₅	semidirect
126. C₃×C₅	C₂×#366	direct (N × H)
135. C₂×C₅	C₃×#366	direct (N × H)
149. C₂×C₃	C₅×#366	direct (N × H)
153. C₅	C₂×C₃×#366	direct (N × H)
162. C₃	C₂×C₅×#366	direct (N × H)
163. C₂	C₃×C₅×#366	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×C₁₆	2
C₂×C₃×C₅×#107	2
C₃×C₅×#366	2
C₂×C₅×#366	3
C₂×C₃×#366	5

Subgroups — C2×C3×C5×#366 #14275190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#366 #14275190