Subgroups — C₂×C₃×C₅×#333 #14274575

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	2	yes	maximal
3	C₃×C₅×#333	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×#333	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×Q₈	C₄	1	2	yes	—
6	C₂²×C₃×C₅×C₈	—	2	2	no	—
7	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
8	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	—	2	2	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₃×#333	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂×C₃	1	2	yes	—
17	C₅×#333	C₂×C₃	1	4	yes	direct product factor
18	C₂×C₃×C₅×Q₈	—	2	2	no	—
19	C₂×C₃×C₅×Q₈	—	4	4	no	—
20	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	—	2	4	no	—
25	C₂×C₃×C₅×C₈	—	4	4	no	—
26	C₃×C₅×C₄⋊C₄	—	2	4	no	—
27	C₃×C₅×C₄⋊C₄	—	2	4	no	—
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	D₄	1	2	yes	—
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
32	C₂×C₃×C₄⋊C₈	C₂×C₅	1	2	yes	—
33	C₃×#333	C₂×C₅	1	4	yes	direct product factor
34	C₂²×C₅×Q₈	C₃×C₄	1	2	yes	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₅×C₄⋊Q₈	—	2	2	no	—
37	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	—
38	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
43	C₂×#333	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
45	C₃×C₅×Q₈	—	4	4	no	—
46	C₃×C₅×Q₈	—	4	4	no	—
47	C₂×C₃×C₄×C₅	—	2	2	no	—
48	C₂×C₃×C₄×C₅	—	2	4	no	—
49	C₂×C₃×C₄×C₅	—	4	4	no	—
50	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
53	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
56	C₃×C₅×C₈	—	4	4	no	—
57	C₂²×C₃×Q₈	C₄×C₅	1	2	yes	—
58	C₂²×C₃×C₈	—	2	2	no	—
59	C₃×C₄⋊Q₈	—	2	2	no	—
60	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	—
61	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
62	C₂×C₃×C₄⋊C₄	—	2	2	no	—
63	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
64	C₃×C₄⋊C₈	—	2	2	no	—
65	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
66	C₂×C₅×Q₈	—	2	2	no	—
67	C₂×C₅×Q₈	—	4	4	no	—
68	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
72	C₂×C₅×C₈	—	2	4	no	—
73	C₂×C₅×C₈	—	4	4	no	—
74	C₅×C₄⋊C₄	—	2	4	no	—
75	C₅×C₄⋊C₄	—	2	4	no	—
76	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
77	C₄²×C₅	C₃×D₄	1	2	yes	—
78	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
79	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
80	C₂×C₄⋊C₈	C₂×C₃×C₅	1	2	yes	—
81	#333	C₂×C₃×C₅	1	4	yes	direct product factor
82	C₂²×C₃×C₅	#100	1	2	yes	—
83	C₂²×C₃×C₅	#103	1	4	yes	—
84	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₃×C₄×C₅	—	2	1	no	—
86	C₃×C₄×C₅	—	2	1	no	—
87	C₃×C₄×C₅	—	4	4	no	—
88	C₃×C₄×C₅	#103	1	4	yes	—
89	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
90	C₃×C₄×C₅	C₂×Q₁₆	1	2	yes	—
91	C₂×C₃×Q₈	—	2	2	no	—
92	C₂×C₃×Q₈	—	4	4	no	—
93	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
94	C₂²×C₃×C₄	—	2	2	no	—
95	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	—	4	4	no	—
99	C₃×C₄⋊C₄	—	2	4	no	—
100	C₃×C₄⋊C₄	—	2	4	no	—
101	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
102	C₃×C₄²	C₅×D₄	1	2	yes	—
103	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
104	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
105	C₅×Q₈	—	4	4	no	—
106	C₅×Q₈	—	4	4	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	4	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
113	C₂×C₄×C₅	C₃×Q₁₆	1	4	yes	—
114	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
116	C₅×C₈	—	4	4	no	—
117	C₂²×Q₈	C₃×C₄×C₅	1	2	yes	—
118	C₂²×C₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	—
121	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
126	C₂×C₃×C₅	#333	1	4	yes	direct product factor
127	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂×#103	1	2	yes	—
129	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
130	C₃×Q₈	—	4	4	no	—
131	C₃×Q₈	—	4	4	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	4	4	no	—
135	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
137	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
138	C₂×C₃×C₄	C₅×Q₁₆	1	4	yes	—
139	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
141	C₃×C₈	—	4	4	no	—
142	C₂²×C₅	C₃×#100	1	2	yes	—
143	C₂²×C₅	C₃×#103	1	4	yes	—
144	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	—	4	4	no	—
148	C₄×C₅	C₃×#103	1	4	yes	—
149	C₄×C₅	C₂×C₃×SD₁₆	1	2	yes	—
150	C₄×C₅	C₂×C₃×Q₁₆	1	2	yes	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	C₂×C₃×C₄×C₅	1	4	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	4	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
162	C₄²	C₃×C₅×D₄	1	2	yes	—
163	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
164	C₃×C₅	C₂×#333	1	1	yes	characteristic, direct product factor, Hall
165	C₂²×C₃	C₅×#100	1	2	yes	—
166	C₂²×C₃	C₅×#103	1	4	yes	—
167	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	4	4	no	—
171	C₃×C₄	C₅×#103	1	4	yes	—
172	C₃×C₄	C₂×C₅×SD₁₆	1	2	yes	—
173	C₃×C₄	C₂×C₅×Q₁₆	1	2	yes	—
174	C₂×C₅	C₃×#333	1	4	yes	direct product factor
175	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
176	C₂×C₅	C₂×C₃×#103	1	2	yes	—
177	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
178	Q₈	—	4	4	no	—
179	Q₈	—	4	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
185	C₂×C₄	C₃×C₅×SD₁₆	1	4	yes	—
186	C₂×C₄	C₃×C₅×Q₁₆	1	4	yes	—
187	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
188	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
189	C₈	—	4	4	no	—
190	C₂×C₃	C₅×#333	1	4	yes	direct product factor
191	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
192	C₂×C₃	C₂×C₅×#103	1	2	yes	—
193	C₅	C₂×C₃×#333	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²	C₃×C₅×#100	1	2	yes	—
195	C₂²	C₃×C₅×#103	1	4	yes	—
196	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	4	4	no	—
200	C₄	C₃×C₅×#103	1	4	yes	—
201	C₄	C₂×C₃×C₅×SD₁₆	1	2	yes	—
202	C₄	C₂×C₃×C₅×Q₁₆	1	2	yes	—
203	C₃	C₂×C₅×#333	1	1	yes	characteristic, direct product factor, Sylow
204	C₂	C₃×C₅×#333	1	4	yes	direct product factor
205	C₂	C₂×C₃×C₅×#100	1	1	yes	characteristic
206	C₂	C₂×C₃×C₅×#103	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#333	C₂	direct (N × H)
4. C₂×C₅×#333	C₃	direct (N × H)
14. C₂×C₃×#333	C₅	direct (N × H)
17. C₅×#333	C₂×C₃	direct (N × H)
33. C₃×#333	C₂×C₅	direct (N × H)
43. C₂×#333	C₃×C₅	direct (N × H)
81. #333	C₂×C₃×C₅	direct (N × H)
126. C₂×C₃×C₅	#333	direct (N × H)
164. C₃×C₅	C₂×#333	direct (N × H)
174. C₂×C₅	C₃×#333	direct (N × H)
190. C₂×C₃	C₅×#333	direct (N × H)
193. C₅	C₂×C₃×#333	direct (N × H)
203. C₃	C₂×C₅×#333	direct (N × H)
204. C₂	C₃×C₅×#333	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×C₄⋊Q₈	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×C₅×#333	2
C₂×C₅×#333	3
C₂×C₃×#333	5

Subgroups — C2×C3×C5×#333 #14274575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#333 #14274575