Subgroups — C₃×#10602213 #14314326

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3944	C₂	1	1	yes	maximal, characteristic
4	C₃×#64721	C₂	1	4	yes	maximal, complemented
5	#10602213	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
8	C₂²×C₃×#168	C₄	1	1	yes	characteristic
9	C₃×C₅×M₃₂	—	2	2	no	—
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₃×C₈×Dic₅	C₄	1	2	yes	—
12	C₃×C₈×Dic₅	C₂²	1	2	yes	—
13	C₃×#3944	—	2	2	no	—
14	C₂×C₃×#672	C₂²	1	2	yes	—
15	C₃×#2150	—	5	1	no	maximal, Hall
16	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic
17	C₂×C₈×Dic₅	C₂×C₃	1	1	yes	characteristic
18	C₂×#3944	C₂×C₃	1	1	yes	characteristic
19	#64721	C₂×C₃	1	4	yes	complemented
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×Dic₅	C₈	1	2	yes	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
25	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₃×C₄×Dic₅	C₈	1	1	yes	characteristic
27	C₃×C₄×Dic₅	C₈	1	1	yes	characteristic
28	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
29	C₂×C₃×#168	—	2	2	no	—
30	C₂×C₃×#168	C₂×C₄	1	2	yes	—
31	C₃×C₅×C₁₆	—	2	2	no	—
32	C₃×#672	—	2	2	no	—
33	C₂×C₃×M₃₂	—	5	1	no	—
34	C₂×C₃×M₃₂	D₅	1	1	yes	characteristic
35	C₂×C₃×C₄×C₈	—	5	1	no	—
36	C₃×C₄⋊C₁₆	—	5	4	no	—
37	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
38	C₂×C₄×Dic₅	C₃×C₄	1	1	yes	characteristic
39	C₂²×#168	C₃×C₄	1	1	yes	characteristic
40	C₅×M₃₂	—	2	2	no	—
41	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
42	C₈×Dic₅	C₃×C₄	1	2	yes	—
43	C₈×Dic₅	C₂²×C₃	1	2	yes	—
44	#3944	—	2	2	no	—
45	C₂×#672	C₂²×C₃	1	2	yes	—
46	#2150	—	5	1	no	Sylow
47	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×Dic₅	—	2	4	no	—
53	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
54	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
55	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
56	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₃×#168	—	2	2	no	—
59	C₂²×C₃×C₈	—	5	1	no	—
60	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
61	C₂×C₃×C₄²	—	5	1	no	—
62	C₃×M₃₂	—	2	2	no	—
63	C₃×M₃₂	—	10	2	no	—
64	C₂×C₃×C₁₆	—	5	2	no	—
65	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	—
66	C₃×C₄×C₈	—	5	2	no	—
67	C₃×C₄×C₈	—	5	2	no	—
68	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂²×Dic₅	C₃×C₈	1	2	yes	—
70	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
72	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
73	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
74	C₄×Dic₅	C₃×C₈	1	1	yes	characteristic
75	C₄×Dic₅	C₃×C₈	1	1	yes	characteristic
76	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
77	C₂×#168	—	2	2	no	—
78	C₂×#168	C₂×C₃×C₄	1	2	yes	—
79	C₅×C₁₆	—	2	2	no	—
80	#672	—	2	2	no	—
81	C₂×M₃₂	—	5	1	no	—
82	C₂×M₃₂	C₃×D₅	1	1	yes	characteristic
83	C₂×C₄×C₈	—	5	1	no	—
84	C₄⋊C₁₆	—	5	4	no	—
85	C₂²×C₃×C₅	—	2	2	no	—
86	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
87	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
88	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
89	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
91	C₃×Dic₅	—	2	2	no	—
92	C₃×Dic₅	M₃₂	1	4	yes	complemented
93	C₂²×C₃×C₄	—	5	2	no	—
94	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
95	C₂×C₃×C₈	—	5	2	no	—
96	C₂×C₃×C₈	—	10	2	no	—
97	C₂×C₃×C₈	Dic₁₀	1	2	yes	—
98	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
100	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
101	C₃×C₄²	—	5	1	no	—
102	C₃×C₄²	—	5	1	no	—
103	C₃×C₄²	—	5	2	no	—
104	C₃×C₁₆	—	2	2	no	—
105	C₃×C₁₆	—	10	2	no	—
106	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
110	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
111	C₂×Dic₅	—	2	4	no	—
112	C₂×Dic₅	C₂×C₃×C₈	1	2	yes	—
113	C₂×Dic₅	C₂×C₃×C₈	1	2	yes	—
114	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
115	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
116	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
117	#168	—	2	2	no	—
118	C₂²×C₈	—	5	1	no	—
119	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
120	C₂×C₄²	—	5	1	no	—
121	M₃₂	—	2	2	no	—
122	M₃₂	—	10	2	no	—
123	C₂×C₁₆	—	5	2	no	—
124	C₂×C₁₆	C₂×C₃×D₅	1	2	yes	—
125	C₄×C₈	—	5	2	no	—
126	C₄×C₈	—	5	2	no	—
127	C₂×C₃×C₅	—	2	2	no	—
128	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
130	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
131	C₂×C₃×C₄	—	5	2	no	—
132	C₂×C₃×C₄	—	5	2	no	—
133	C₂×C₃×C₄	—	10	4	no	—
134	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
135	C₂×C₃×C₄	#676	1	2	yes	—
136	C₂×C₃×C₄	#683	1	2	yes	—
137	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
138	C₃×C₈	—	10	2	no	—
139	C₃×C₈	#683	1	2	yes	—
140	C₃×C₈	C₂×Dic₁₀	1	1	yes	characteristic
141	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
142	C₂²×C₅	—	2	2	no	—
143	C₂²×C₅	C₃×C₄⋊C₈	1	2	yes	—
144	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
145	C₄×C₅	C₃×C₄⋊C₈	1	2	yes	—
146	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
148	Dic₅	—	2	2	no	—
149	Dic₅	C₃×M₃₂	1	4	yes	complemented
150	C₂²×C₄	—	5	2	no	—
151	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
152	C₂×C₈	—	5	2	no	—
153	C₂×C₈	—	10	2	no	—
154	C₂×C₈	C₃×Dic₁₀	1	2	yes	—
155	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
156	C₂×C₈	C₃×C₂²⋊D₅	1	2	yes	—
157	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
158	C₄²	—	5	1	no	—
159	C₄²	—	5	1	no	—
160	C₄²	—	5	2	no	—
161	C₁₆	—	2	2	no	—
162	C₁₆	—	10	2	no	—
163	C₃×C₅	#2150	1	1	yes	characteristic, complemented, Hall
164	C₂²×C₃	—	2	2	no	—
165	C₂²×C₃	#3946	1	2	yes	—
166	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
167	C₃×C₄	—	5	4	no	—
168	C₃×C₄	—	10	2	no	—
169	C₃×C₄	#3946	1	2	yes	—
170	C₃×C₄	C₂×#676	1	1	yes	characteristic
171	C₃×C₄	C₂×#683	1	1	yes	characteristic
172	C₂×C₅	—	2	2	no	—
173	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
174	C₂×C₅	C₂×C₃×M₃₂	1	2	yes	—
175	C₂³	C₃×C₈×D₅	1	1	yes	characteristic
176	C₂×C₄	—	5	2	no	—
177	C₂×C₄	—	5	2	no	—
178	C₂×C₄	—	10	4	no	—
179	C₂×C₄	C₃×C₈×D₅	1	1	yes	characteristic
180	C₂×C₄	C₃×#676	1	2	yes	—
181	C₂×C₄	C₃×#683	1	2	yes	—
182	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
183	C₈	—	10	2	no	—
184	C₈	C₃×#683	1	2	yes	—
185	C₈	C₂×C₃×Dic₁₀	1	1	yes	characteristic
186	C₈	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	D₅×M₃₂	1	2	yes	—
189	C₂×C₃	C₂×#3946	1	1	yes	characteristic
190	C₅	C₃×#2150	1	1	yes	characteristic, complemented, Sylow
191	C₂²	—	2	2	no	—
192	C₂²	C₃×#3946	1	2	yes	—
193	C₂²	C₂×C₃×C₈×D₅	1	1	yes	characteristic
194	C₄	—	5	4	no	—
195	C₄	—	10	2	no	—
196	C₄	C₃×#3946	1	2	yes	—
197	C₄	C₂×C₃×#676	1	1	yes	characteristic
198	C₄	C₂×C₃×#683	1	1	yes	characteristic
199	C₃	#10602213	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	—	2	2	no	—
201	C₂	C₃×D₅×M₃₂	1	2	yes	—
202	C₂	C₂×C₃×#3946	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₂×C₃×M₃₂	D₅
hypocenter	C₅	C₃×#2150
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂×C₃×M₃₂

Complements

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

N	H ≅ G/N	type
4. C₃×#64721	C₂	semidirect
5. #10602213	C₃	direct (N × H)
19. #64721	C₂×C₃	semidirect
92. C₃×Dic₅	M₃₂	semidirect
149. Dic₅	C₃×M₃₂	semidirect
163. C₃×C₅	#2150	semidirect
190. C₅	C₃×#2150	semidirect
199. C₃	#10602213	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×M₃₂	2
C₂×C₃×C₈×Dic₅	2
C₂×C₃×#3944	2
C₃×#64721	2
#10602213	3
C₃×#2150	5

Subgroups — C3×#10602213 #14314326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602213 #14314326