Subgroups — C₃×C₄×C₅×Q₃₂ #14275208

Order 1920.

300 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×Q₁₆	C₂	1	2	yes	maximal
3	C₃×C₅×#366	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#358	C₂	1	2	yes	maximal
5	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×C₅×Q₃₂	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
8	C₃×C₄×C₅×Q₈	—	2	2	no	—
9	C₃×C₅×Q₃₂	C₄	1	4	yes	direct product factor
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×#103	—	2	2	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×Q₃₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×Q₃₂	C₂×C₃	1	1	yes	characteristic
17	C₄×C₅×Q₁₆	C₂×C₃	1	2	yes	—
18	C₅×#366	C₂×C₃	1	1	yes	characteristic
19	C₅×#358	C₂×C₃	1	2	yes	—
20	C₄×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₅×Q₈	—	2	2	no	—
22	C₃×C₅×Q₁₆	—	2	2	no	—
23	C₃×C₅×Q₁₆	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₅×C₄⋊C₄	—	2	2	no	—
27	C₃×C₅×C₄⋊C₄	—	4	2	no	—
28	C₃×C₄²×C₅	—	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	—
32	C₂×C₃×Q₃₂	C₂×C₅	1	1	yes	characteristic
33	C₃×C₄×Q₁₆	C₂×C₅	1	2	yes	—
34	C₃×#366	C₂×C₅	1	1	yes	characteristic
35	C₃×#358	C₂×C₅	1	2	yes	—
36	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
37	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
38	C₄×C₅×Q₈	—	2	2	no	—
39	C₅×Q₃₂	C₃×C₄	1	4	yes	direct product factor
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×#103	—	2	2	no	—
44	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₄×Q₃₂	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₃×C₅×Q₈	—	2	4	no	—
47	C₃×C₅×Q₈	—	4	2	no	—
48	C₂×C₃×C₄×C₅	—	4	2	no	—
49	C₂×C₃×C₄×C₅	—	4	2	no	—
50	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
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×Q₁₆	C₂²×C₅	1	2	yes	—
57	C₃×C₄×Q₈	—	2	2	no	—
58	C₃×Q₃₂	C₄×C₅	1	4	yes	direct product factor
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×#103	—	2	2	no	—
63	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₅×Q₈	—	2	2	no	—
65	C₅×Q₁₆	—	2	2	no	—
66	C₅×Q₁₆	C₂×C₃×C₄	1	4	yes	—
67	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
69	C₅×C₄⋊C₄	—	2	2	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₄²×C₅	—	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	—
75	C₂×Q₃₂	C₂×C₃×C₅	1	1	yes	characteristic
76	C₄×Q₁₆	C₂×C₃×C₅	1	2	yes	—
77	#366	C₂×C₃×C₅	1	1	yes	characteristic
78	#358	C₂×C₃×C₅	1	2	yes	—
79	C₄×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
80	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
81	C₃×C₄×C₅	—	2	1	no	—
82	C₃×C₄×C₅	—	4	4	no	—
83	C₃×C₄×C₅	—	8	2	no	—
84	C₃×C₄×C₅	Q₃₂	1	2	yes	direct product factor
85	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
86	C₃×C₄×C₅	#135	1	1	yes	characteristic
87	C₂×C₃×Q₈	—	2	2	no	—
88	C₃×Q₁₆	—	2	2	no	—
89	C₃×Q₁₆	C₂×C₄×C₅	1	4	yes	—
90	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
92	C₃×C₄⋊C₄	—	2	2	no	—
93	C₃×C₄⋊C₄	—	4	2	no	—
94	C₃×C₄²	—	4	2	no	—
95	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
96	C₃×C₁₆	—	2	1	no	—
97	C₃×C₁₆	C₂×C₄×C₅	1	2	yes	—
98	C₅×Q₈	—	2	4	no	—
99	C₅×Q₈	—	4	2	no	—
100	C₂×C₄×C₅	—	4	2	no	—
101	C₂×C₄×C₅	—	4	2	no	—
102	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
105	C₅×C₈	—	2	1	no	—
106	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
107	C₅×C₈	C₃×#41	1	1	yes	characteristic
108	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
109	C₄×Q₈	—	2	2	no	—
110	Q₃₂	C₃×C₄×C₅	1	4	yes	direct product factor
111	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
112	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
113	#107	C₂²×C₃×C₅	1	2	yes	—
114	#103	—	2	2	no	—
115	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
116	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
117	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
118	C₂×C₃×C₅	#508	1	1	yes	characteristic
119	C₃×Q₈	—	2	4	no	—
120	C₃×Q₈	—	4	2	no	—
121	C₂×C₃×C₄	—	4	2	no	—
122	C₂×C₃×C₄	—	4	2	no	—
123	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
124	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
126	C₃×C₈	—	2	1	no	—
127	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
128	C₃×C₈	C₅×#41	1	1	yes	characteristic
129	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
130	C₄×C₅	—	2	1	no	—
131	C₄×C₅	—	4	4	no	—
132	C₄×C₅	—	8	2	no	—
133	C₄×C₅	C₃×Q₃₂	1	2	yes	direct product factor
134	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
135	C₄×C₅	C₃×#135	1	1	yes	characteristic
136	C₂×Q₈	—	2	2	no	—
137	Q₁₆	—	2	2	no	—
138	Q₁₆	C₂×C₃×C₄×C₅	1	4	yes	—
139	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
140	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄²	—	4	2	no	—
144	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
145	C₁₆	—	2	1	no	—
146	C₁₆	C₂×C₃×C₄×C₅	1	2	yes	—
147	C₃×C₅	C₄×Q₃₂	1	1	yes	characteristic, direct product factor, Hall
148	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
149	C₃×C₄	—	2	1	no	—
150	C₃×C₄	—	4	4	no	—
151	C₃×C₄	—	8	2	no	—
152	C₃×C₄	C₅×Q₃₂	1	2	yes	direct product factor
153	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
154	C₃×C₄	C₅×#135	1	1	yes	characteristic
155	C₂×C₅	C₃×C₄×D₈	1	1	yes	characteristic
156	C₂×C₅	C₂×C₃×Q₃₂	1	1	yes	characteristic
157	C₂×C₅	C₃×#508	1	1	yes	characteristic
158	Q₈	—	2	4	no	—
159	Q₈	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
163	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
164	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
167	C₈	C₃×C₅×#41	1	1	yes	characteristic
168	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
169	C₂×C₃	C₂×C₅×Q₃₂	1	1	yes	characteristic
170	C₂×C₃	C₅×#508	1	1	yes	characteristic
171	C₅	C₃×C₄×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	4	4	no	—
175	C₄	—	8	2	no	—
176	C₄	C₃×C₅×Q₃₂	1	2	yes	direct product factor
177	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
178	C₄	C₃×C₅×#135	1	1	yes	characteristic
179	C₃	C₄×C₅×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
180	C₂	C₃×C₄×C₅×D₈	1	1	yes	characteristic
181	C₂	C₂×C₃×C₅×Q₃₂	1	1	yes	characteristic
182	C₂	C₃×C₅×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle, prime radical	C₂²×C₃×C₅	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×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
6. C₄×C₅×Q₃₂	C₃	direct (N × H)
9. C₃×C₅×Q₃₂	C₄	direct (N × H)
15. C₃×C₄×Q₃₂	C₅	direct (N × H)
39. C₅×Q₃₂	C₃×C₄	direct (N × H)
45. C₄×Q₃₂	C₃×C₅	direct (N × H)
58. C₃×Q₃₂	C₄×C₅	direct (N × H)
84. C₃×C₄×C₅	Q₃₂	direct (N × H)
110. Q₃₂	C₃×C₄×C₅	direct (N × H)
133. C₄×C₅	C₃×Q₃₂	direct (N × H)
147. C₃×C₅	C₄×Q₃₂	direct (N × H)
152. C₃×C₄	C₅×Q₃₂	direct (N × H)
171. C₅	C₃×C₄×Q₃₂	direct (N × H)
176. C₄	C₃×C₅×Q₃₂	direct (N × H)
179. C₃	C₄×C₅×Q₃₂	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×Q₃₂	2
C₃×C₄×C₅×Q₁₆	2
C₃×C₅×#366	2
C₃×C₅×#358	2
C₃×C₄×C₅×C₁₆	2
C₄×C₅×Q₃₂	3
C₃×C₄×Q₃₂	5

Subgroups — C3×C4×C5×Q32 #14275208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₅×Q₃₂ #14275208