Subgroups — C₃×#65128 #11749736

Order 960.

228 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#676	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#4018	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#3946	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#3946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#65128	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×#685	C₄	1	1	yes	characteristic
14	C₃×#685	C₄	1	1	yes	characteristic
15	C₃×#683	C₄	1	1	yes	characteristic
16	C₃×#683	C₄	1	1	yes	characteristic
17	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×C₃×#168	C₂²	1	1	yes	characteristic
19	C₂×C₃×#168	C₂²	1	1	yes	characteristic
20	C₃×#676	—	2	2	no	—
21	C₃×C₈×D₅	—	2	2	no	—
22	C₃×#424	—	5	1	no	maximal, Hall
23	C₂×#676	C₂×C₃	1	1	yes	characteristic
24	C₂×C₈×D₅	C₂×C₃	1	1	yes	characteristic
25	#4018	C₂×C₃	1	1	yes	characteristic
26	C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	#3946	C₂×C₃	1	1	yes	characteristic, complemented
28	#3946	C₂×C₃	1	1	yes	characteristic, complemented
29	C₄⋊#168	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×C₃×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×D₅	D₄	1	2	yes	—
38	C₃×C₄×D₅	Q₈	1	2	yes	—
39	C₃×C₅×C₈	—	2	1	no	—
40	C₃×C₅×C₈	—	2	1	no	—
41	C₃×#168	—	2	1	no	—
42	C₃×#168	—	2	1	no	—
43	C₂×C₃×M₁₆	—	5	1	no	—
44	C₂²×C₃×C₈	—	5	1	no	—
45	C₃×#117	—	5	1	no	—
46	C₃×C₄⋊C₈	—	5	1	no	—
47	C₃×C₄⋊C₈	—	5	1	no	—
48	C₃×C₄⋊C₈	—	5	1	no	—
49	C₃×C₄⋊C₈	D₅	1	1	yes	characteristic, complemented
50	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	#685	C₃×C₄	1	1	yes	characteristic
55	#685	C₃×C₄	1	1	yes	characteristic
56	#683	C₃×C₄	1	1	yes	characteristic
57	#683	C₃×C₄	1	1	yes	characteristic
58	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
59	C₂×#168	C₂²×C₃	1	1	yes	characteristic
60	C₂×#168	C₂²×C₃	1	1	yes	characteristic
61	#676	—	2	2	no	—
62	C₈×D₅	—	2	2	no	—
63	#424	—	5	1	no	Sylow
64	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×D₅	—	2	1	no	—
66	C₂×C₃×D₅	—	2	1	no	—
67	C₂×C₃×D₅	C₄⋊C₄	1	2	yes	—
68	C₃×C₄×C₅	—	2	1	no	—
69	C₃×C₄×C₅	—	2	1	no	—
70	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₃×Dic₅	—	2	1	no	—
73	C₃×Dic₅	—	2	1	no	—
74	C₃×Dic₅	C₄⋊C₄	1	2	yes	—
75	C₂²×C₃×C₄	—	5	1	no	—
76	C₃×M₁₆	—	10	2	no	—
77	C₂×C₃×C₈	—	5	1	no	—
78	C₂×C₃×C₈	—	5	1	no	—
79	C₂×C₃×C₈	—	10	2	no	—
80	C₂×C₃×C₈	C₂×D₅	1	1	yes	characteristic
81	C₂×C₃×C₈	C₂×D₅	1	1	yes	characteristic
82	C₃×C₄⋊C₄	—	5	1	no	—
83	C₃×C₄⋊C₄	—	5	1	no	—
84	C₃×C₂²⋊C₄	—	5	1	no	—
85	C₃×C₂²⋊C₄	—	5	1	no	—
86	C₃×C₄²	—	5	1	no	—
87	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
88	C₂²×D₅	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
92	C₂×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
95	C₄×D₅	C₃×D₄	1	2	yes	—
96	C₄×D₅	C₃×Q₈	1	2	yes	—
97	C₅×C₈	—	2	1	no	—
98	C₅×C₈	—	2	1	no	—
99	#168	—	2	1	no	—
100	#168	—	2	1	no	—
101	C₂×M₁₆	—	5	1	no	—
102	C₂²×C₈	—	5	1	no	—
103	#117	—	5	1	no	—
104	C₄⋊C₈	—	5	1	no	—
105	C₄⋊C₈	—	5	1	no	—
106	C₄⋊C₈	—	5	1	no	—
107	C₄⋊C₈	C₃×D₅	1	1	yes	characteristic, complemented
108	C₂×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₅	#131	1	1	yes	characteristic
110	C₂×C₃×C₅	#131	1	1	yes	characteristic
111	C₃×D₅	—	2	2	no	—
112	C₂³×C₃	—	5	1	no	—
113	C₂×C₃×C₄	—	5	1	no	—
114	C₂×C₃×C₄	—	5	2	no	—
115	C₂×C₃×C₄	—	5	2	no	—
116	C₂×C₃×C₄	—	5	1	no	—
117	C₂×C₃×C₄	—	5	1	no	—
118	C₂×C₃×C₄	C₄×D₅	1	1	yes	characteristic
119	C₂×C₃×C₄	C₄×D₅	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
121	C₃×C₈	—	2	1	no	—
122	C₃×C₈	—	2	1	no	—
123	C₃×C₈	—	10	1	no	—
124	C₃×C₈	—	10	1	no	—
125	C₂²×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₂×D₅	—	2	1	no	—
127	C₂×D₅	—	2	1	no	—
128	C₂×D₅	C₃×C₄⋊C₄	1	2	yes	—
129	C₄×C₅	—	2	1	no	—
130	C₄×C₅	—	2	1	no	—
131	C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
132	C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
133	Dic₅	—	2	1	no	—
134	Dic₅	—	2	1	no	—
135	Dic₅	C₃×C₄⋊C₄	1	2	yes	—
136	C₂²×C₄	—	5	1	no	—
137	M₁₆	—	10	2	no	—
138	C₂×C₈	—	5	1	no	—
139	C₂×C₈	—	5	1	no	—
140	C₂×C₈	—	10	2	no	—
141	C₂×C₈	C₂×C₃×D₅	1	1	yes	characteristic
142	C₂×C₈	C₂×C₃×D₅	1	1	yes	characteristic
143	C₄⋊C₄	—	5	1	no	—
144	C₄⋊C₄	—	5	1	no	—
145	C₂²⋊C₄	—	5	1	no	—
146	C₂²⋊C₄	—	5	1	no	—
147	C₄²	—	5	1	no	—
148	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
149	C₃×C₅	#424	1	1	yes	characteristic, complemented, Hall
150	C₂²×C₃	—	5	2	no	—
151	C₂²×C₃	—	10	1	no	—
152	C₂²×C₃	—	10	1	no	—
153	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
154	C₃×C₄	—	2	1	no	—
155	C₃×C₄	—	2	1	no	—
156	C₃×C₄	—	5	2	no	—
157	C₃×C₄	—	10	1	no	—
158	C₃×C₄	—	10	1	no	—
159	C₃×C₄	D₄×D₅	1	1	yes	characteristic
160	C₃×C₄	Q₈×D₅	1	1	yes	characteristic
161	C₂×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₅	C₃×#131	1	1	yes	characteristic
163	C₂×C₅	C₃×#131	1	1	yes	characteristic
164	D₅	—	2	2	no	—
165	C₂³	—	5	1	no	—
166	C₂×C₄	—	5	1	no	—
167	C₂×C₄	—	5	2	no	—
168	C₂×C₄	—	5	2	no	—
169	C₂×C₄	—	5	1	no	—
170	C₂×C₄	—	5	1	no	—
171	C₂×C₄	C₃×C₄×D₅	1	1	yes	characteristic
172	C₂×C₄	C₃×C₄×D₅	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	10	1	no	—
177	C₈	—	10	1	no	—
178	C₂×C₃	—	10	2	no	—
179	C₂×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₃	#4047	1	1	yes	characteristic
181	C₂×C₃	#4053	1	1	yes	characteristic
182	C₅	C₃×#424	1	1	yes	characteristic, complemented, Sylow
183	C₂²	—	5	2	no	—
184	C₂²	—	10	1	no	—
185	C₂²	—	10	1	no	—
186	C₂²	C₂×C₃×C₄×D₅	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	5	2	no	—
190	C₄	—	10	1	no	—
191	C₄	—	10	1	no	—
192	C₄	C₃×D₄×D₅	1	1	yes	characteristic
193	C₄	C₃×Q₈×D₅	1	1	yes	characteristic
194	C₃	#65128	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	—	10	2	no	—
196	C₂	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
197	C₂	C₃×#4047	1	1	yes	characteristic
198	C₂	C₃×#4053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂²×C₃×D₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	D₅
hypocenter	C₅	C₃×#424
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×C₄⋊C₈	C₂	semidirect
5. C₃×#3946	C₂	semidirect
6. C₃×#3946	C₂	semidirect
7. C₃×C₄⋊#168	C₂	semidirect
8. #65128	C₃	direct (N × H)
26. C₅×C₄⋊C₈	C₂×C₃	semidirect
27. #3946	C₂×C₃	semidirect
28. #3946	C₂×C₃	semidirect
29. C₄⋊#168	C₂×C₃	semidirect
49. C₃×C₄⋊C₈	D₅	semidirect
107. C₄⋊C₈	C₃×D₅	semidirect
149. C₃×C₅	#424	semidirect
182. C₅	C₃×#424	semidirect
194. C₃	#65128	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×#676	2
C₂×C₃×C₈×D₅	2
C₃×#4018	2
C₃×C₅×C₄⋊C₈	2
C₃×#3946	2
C₃×C₄⋊#168	2
#65128	3
C₃×#424	5

Subgroups — C3×#65128 #11749736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65128 #11749736