Subgroups — C₂×C₃×#64726 #14314668

Order 1920.

288 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	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3951	C₂	1	2	yes	maximal
3	C₃×#64726	C₂	1	4	yes	maximal, direct product factor
4	C₂×#64726	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₃×#681	—	2	2	no	—
7	C₂×C₃×C₅×C₁₆	C₄	1	2	yes	—
8	C₂×C₃×C₅×C₁₆	C₄	1	2	yes	—
9	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
10	C₃×#3951	—	2	2	no	—
11	C₃×#3951	C₂²	1	4	yes	—
12	C₂×C₃×#368	—	5	1	no	maximal, Hall
13	C₂²×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic
14	C₂×#3951	C₂×C₃	1	2	yes	—
15	#64726	C₂×C₃	1	4	yes	direct product factor
16	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
17	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
20	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₃×#681	—	2	4	no	—
24	C₃×#681	—	4	2	no	—
25	C₂×C₃×#168	—	4	2	no	—
26	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₃×C₁₆	D₅	1	1	yes	characteristic
28	C₂×C₃×#108	—	5	2	no	—
29	C₃×#368	—	5	4	no	—
30	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
31	C₂×#681	—	2	2	no	—
32	C₂×C₅×C₁₆	C₃×C₄	1	2	yes	—
33	C₂×C₅×C₁₆	C₃×C₄	1	2	yes	—
34	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
35	#3951	—	2	2	no	—
36	#3951	C₂²×C₃	1	4	yes	—
37	C₂×#368	—	5	1	no	Sylow
38	C₂³×C₃×C₅	Q₁₆	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	Q₁₆	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₃×#168	—	4	4	no	—
50	C₂×C₃×M₁₆	—	10	2	no	—
51	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
52	C₂×C₃×C₁₆	Dic₅	1	2	yes	—
53	C₂×C₃×C₁₆	Dic₅	1	2	yes	—
54	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	—
55	C₃×#108	—	5	4	no	—
56	C₃×#108	—	10	2	no	—
57	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₃×Q₈	1	1	yes	characteristic
62	C₂×C₅×C₈	C₃×Q₈	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
64	#681	—	2	4	no	—
65	#681	—	4	2	no	—
66	C₂×#168	—	4	2	no	—
67	C₅×C₁₆	C₂×C₃×C₄	1	4	yes	—
68	C₂²×C₁₆	C₃×D₅	1	1	yes	characteristic
69	C₂×#108	—	5	2	no	—
70	#368	—	5	4	no	—
71	C₂²×C₃×C₅	—	2	2	no	—
72	C₂²×C₃×C₅	#107	1	1	yes	characteristic
73	C₂²×C₃×C₅	#107	1	1	yes	characteristic
74	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
75	C₃×C₄×C₅	#107	1	1	yes	characteristic
76	C₃×C₄×C₅	#107	1	1	yes	characteristic
77	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
79	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
80	C₃×M₁₆	—	10	4	no	—
81	C₃×M₁₆	—	20	2	no	—
82	C₂×C₃×C₈	—	20	2	no	—
83	C₂×C₃×C₈	Dic₁₀	1	1	yes	characteristic
84	C₂×C₃×C₈	Dic₁₀	1	1	yes	characteristic
85	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
86	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
87	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
89	C₃×C₁₆	C₂×Dic₅	1	4	yes	—
90	C₂³×C₅	C₃×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₃×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
97	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
98	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
99	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
100	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
101	#168	—	4	4	no	—
102	C₂×M₁₆	—	10	2	no	—
103	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
104	C₂×C₁₆	C₃×Dic₅	1	2	yes	—
105	C₂×C₁₆	C₃×Dic₅	1	2	yes	—
106	C₂×C₁₆	C₂×C₃×D₅	1	2	yes	—
107	#108	—	5	4	no	—
108	#108	—	10	2	no	—
109	C₂×C₃×C₅	—	2	1	no	—
110	C₂×C₃×C₅	—	2	1	no	—
111	C₂×C₃×C₅	#368	1	2	yes	complemented
112	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
113	C₂³×C₃	Dic₂₀	1	1	yes	characteristic
114	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
115	C₂×C₃×C₄	D₄₀	1	1	yes	characteristic
116	C₂×C₃×C₄	Dic₂₀	1	1	yes	characteristic
117	C₂×C₃×C₄	C₄⋊Dic₅	1	1	yes	characteristic
118	C₂×C₃×C₄	C₄⋊Dic₅	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
120	C₃×C₈	—	20	4	no	—
121	C₃×C₈	C₄⋊Dic₅	1	1	yes	characteristic
122	C₃×C₈	C₄⋊Dic₅	1	1	yes	characteristic
123	C₃×C₈	C₂×Dic₁₀	1	1	yes	characteristic
124	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
125	C₂²×C₅	—	2	2	no	—
126	C₂²×C₅	C₃×#107	1	1	yes	characteristic
127	C₂²×C₅	C₃×#107	1	1	yes	characteristic
128	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
129	C₄×C₅	C₃×#107	1	1	yes	characteristic
130	C₄×C₅	C₃×#107	1	1	yes	characteristic
131	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
133	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
134	M₁₆	—	10	4	no	—
135	M₁₆	—	20	2	no	—
136	C₂×C₈	—	20	2	no	—
137	C₂×C₈	C₃×Dic₁₀	1	1	yes	characteristic
138	C₂×C₈	C₃×Dic₁₀	1	1	yes	characteristic
139	C₂×C₈	C₃×D₂₀	1	1	yes	characteristic
140	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
141	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
142	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
143	C₁₆	C₂×C₃×Dic₅	1	4	yes	—
144	C₃×C₅	C₂×#368	1	1	yes	characteristic, complemented, Hall
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	#3950	1	1	yes	characteristic
147	C₂²×C₃	#3950	1	1	yes	characteristic
148	C₂²×C₃	C₂×Dic₂₀	1	1	yes	characteristic
149	C₃×C₄	#3950	1	1	yes	characteristic
150	C₃×C₄	#3950	1	1	yes	characteristic
151	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
152	C₃×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
153	C₂×C₅	—	2	1	no	—
154	C₂×C₅	—	2	1	no	—
155	C₂×C₅	C₃×#368	1	2	yes	complemented
156	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
157	C₂³	C₃×Dic₂₀	1	1	yes	characteristic
158	C₂×C₄	C₃×D₄₀	1	1	yes	characteristic
159	C₂×C₄	C₃×D₄₀	1	1	yes	characteristic
160	C₂×C₄	C₃×Dic₂₀	1	1	yes	characteristic
161	C₂×C₄	C₃×C₄⋊Dic₅	1	1	yes	characteristic
162	C₂×C₄	C₃×C₄⋊Dic₅	1	1	yes	characteristic
163	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
164	C₈	—	20	4	no	—
165	C₈	C₃×C₄⋊Dic₅	1	1	yes	characteristic
166	C₈	C₃×C₄⋊Dic₅	1	1	yes	characteristic
167	C₈	C₂×C₃×Dic₁₀	1	1	yes	characteristic
168	C₈	C₂²×C₃×Dic₅	1	1	yes	characteristic
169	C₂×C₃	—	2	1	no	—
170	C₂×C₃	—	2	1	no	—
171	C₂×C₃	#64726	1	2	yes	direct product factor
172	C₂×C₃	C₂×#3950	1	1	yes	characteristic
173	C₅	C₂×C₃×#368	1	1	yes	characteristic, complemented, Sylow
174	C₂²	—	2	2	no	—
175	C₂²	C₃×#3950	1	1	yes	characteristic
176	C₂²	C₃×#3950	1	1	yes	characteristic
177	C₂²	C₂×C₃×Dic₂₀	1	1	yes	characteristic
178	C₄	C₃×#3950	1	1	yes	characteristic
179	C₄	C₃×#3950	1	1	yes	characteristic
180	C₄	C₂×C₃×D₄₀	1	1	yes	characteristic
181	C₄	C₂×C₃×C₄⋊Dic₅	1	1	yes	characteristic
182	C₃	C₂×#64726	1	1	yes	characteristic, direct product factor, Sylow
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	C₃×#64726	1	2	yes	direct product factor
186	C₂	C₂×C₃×#3950	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₄₀
commutator	C₅×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₂²×C₃×C₅×C₁₆	C₂
socle	C₂²×C₃×C₅	#107
prime radical	C₂³×C₃×C₅	Q₁₆
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₂²×C₃×C₁₆	D₅
hypocenter	C₅	C₂×C₃×#368
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂²×C₃×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×#64726	C₂	direct (N × H)
4. C₂×#64726	C₃	direct (N × H)
15. #64726	C₂×C₃	direct (N × H)
111. C₂×C₃×C₅	#368	semidirect
144. C₃×C₅	C₂×#368	semidirect
155. C₂×C₅	C₃×#368	semidirect
171. C₂×C₃	#64726	direct (N × H)
173. C₅	C₂×C₃×#368	semidirect
182. C₃	C₂×#64726	direct (N × H)
185. C₂	C₃×#64726	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×#3951	2
C₃×#64726	2
C₂×#64726	3
C₂×C₃×#368	5

Subgroups — C2×C3×#64726 #14314668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64726 #14314668