Subgroups — C₃×#65919 #11755596

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4093	C₂	1	2	yes	maximal, complemented
3	C₃×#4040	C₂	1	4	yes	maximal, complemented
4	C₃×#4039	C₂	1	2	yes	maximal, complemented
5	C₃×#4021	C₂	1	2	yes	maximal, complemented
6	C₃×#4020	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×D₂₀	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
9	#65919	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₂₀	C₂²	1	2	yes	—
12	C₂×C₃×D₂₀	C₂²	1	2	yes	—
13	C₂×C₃×D₂₀	C₂²	1	2	yes	—
14	C₂×C₃×C₄×D₅	C₂²	1	4	yes	—
15	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₃×C₄²×C₅	C₂²	1	2	yes	—
20	C₃×#685	C₂²	1	4	yes	—
21	C₃×#685	C₂²	1	4	yes	—
22	C₃×#685	C₂²	1	4	yes	—
23	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₃×#683	C₂²	1	2	yes	—
25	C₃×C₄×Dic₅	C₂²	1	2	yes	—
26	C₃×#559	—	5	1	no	maximal, Hall
27	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
28	#4093	C₂×C₃	1	2	yes	complemented
29	#4040	C₂×C₃	1	4	yes	complemented
30	#4039	C₂×C₃	1	2	yes	complemented
31	#4021	C₂×C₃	1	2	yes	complemented
32	#4020	C₂×C₃	1	1	yes	characteristic
33	C₄×D₂₀	C₂×C₃	1	2	yes	complemented
34	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂²×C₃×D₅	C₂³	1	4	yes	—
36	C₃×C₅×Q₈	—	2	2	no	—
37	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
39	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×Dic₅	C₂³	1	2	yes	—
43	C₂×C₃×Dic₅	C₂³	1	2	yes	—
44	C₃×D₂₀	—	2	4	no	—
45	C₃×D₂₀	—	2	4	no	—
46	C₃×D₂₀	—	4	2	no	—
47	C₃×C₄×D₅	—	4	4	no	—
48	C₃×Dic₁₀	—	2	2	no	—
49	C₃×#127	—	5	1	no	—
50	C₃×#124	—	5	2	no	—
51	C₃×#124	—	5	2	no	—
52	C₃×#123	—	5	4	no	—
53	C₃×C₄×Q₈	—	5	1	no	—
54	C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
55	C₃×C₄×D₄	—	5	2	no	—
56	C₃×C₄×D₄	—	5	2	no	—
57	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₂×D₂₀	C₂²×C₃	1	2	yes	—
59	C₂×D₂₀	C₂²×C₃	1	2	yes	—
60	C₂×D₂₀	C₂²×C₃	1	2	yes	—
61	C₂×C₄×D₅	C₂²×C₃	1	4	yes	—
62	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
63	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
66	C₄²×C₅	C₂²×C₃	1	2	yes	—
67	#685	C₂²×C₃	1	4	yes	—
68	#685	C₂²×C₃	1	4	yes	—
69	#685	C₂²×C₃	1	4	yes	—
70	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
71	#683	C₂²×C₃	1	2	yes	—
72	C₄×Dic₅	C₂²×C₃	1	2	yes	—
73	#559	—	5	1	no	Sylow
74	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×C₃×D₅	—	2	4	no	—
76	C₂×C₃×D₅	—	2	4	no	—
77	C₂×C₃×D₅	—	2	4	no	—
78	C₃×C₄×C₅	—	2	2	no	—
79	C₃×C₄×C₅	—	2	2	no	—
80	C₃×C₄×C₅	—	2	1	no	—
81	C₃×C₄×C₅	#41	1	2	yes	—
82	C₃×C₄×C₅	#41	1	2	yes	—
83	C₃×Dic₅	—	2	2	no	—
84	C₃×Dic₅	—	2	2	no	—
85	C₂×C₃×Q₈	—	5	1	no	—
86	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
87	C₂×C₃×D₄	—	5	2	no	—
88	C₂×C₃×D₄	—	5	2	no	—
89	C₂×C₃×D₄	—	5	2	no	—
90	C₂²×C₃×C₄	—	5	4	no	—
91	C₃×C₄⋊C₄	—	5	2	no	—
92	C₃×C₄⋊C₄	—	5	1	no	—
93	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
94	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
95	C₃×C₂²⋊C₄	—	5	4	no	—
96	C₃×C₂²⋊C₄	—	5	4	no	—
97	C₃×C₂²⋊C₄	—	5	4	no	—
98	C₃×C₄²	—	5	2	no	—
99	C₃×C₄²	C₂×D₅	1	2	yes	—
100	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
101	C₂²×D₅	C₂³×C₃	1	4	yes	—
102	C₅×Q₈	—	2	2	no	—
103	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
105	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
106	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₂×Dic₅	C₂³×C₃	1	2	yes	—
109	C₂×Dic₅	C₂³×C₃	1	2	yes	—
110	D₂₀	—	2	4	no	—
111	D₂₀	—	2	4	no	—
112	D₂₀	—	4	2	no	—
113	C₄×D₅	—	4	4	no	—
114	Dic₁₀	—	2	2	no	—
115	#127	—	5	1	no	—
116	#124	—	5	2	no	—
117	#124	—	5	2	no	—
118	#123	—	5	4	no	—
119	C₄×Q₈	—	5	1	no	—
120	C₄×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
121	C₄×D₄	—	5	2	no	—
122	C₄×D₄	—	5	2	no	—
123	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
124	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
125	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
126	C₃×D₅	—	4	4	no	—
127	C₂³×C₃	—	5	4	no	—
128	C₃×Q₈	—	2	2	no	—
129	C₃×Q₈	—	10	2	no	—
130	C₃×D₄	—	10	4	no	—
131	C₃×D₄	—	10	4	no	—
132	C₃×D₄	—	20	2	no	—
133	C₂×C₃×C₄	—	5	2	no	—
134	C₂×C₃×C₄	—	5	2	no	—
135	C₂×C₃×C₄	—	20	4	no	—
136	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
137	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
138	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
141	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
142	C₂×D₅	—	2	4	no	—
143	C₂×D₅	—	2	4	no	—
144	C₂×D₅	—	2	4	no	—
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	C₃×#41	1	2	yes	—
149	C₄×C₅	C₃×#41	1	2	yes	—
150	Dic₅	—	2	2	no	—
151	Dic₅	—	2	2	no	—
152	C₂×Q₈	—	5	1	no	—
153	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
154	C₂×D₄	—	5	2	no	—
155	C₂×D₄	—	5	2	no	—
156	C₂×D₄	—	5	2	no	—
157	C₂²×C₄	—	5	4	no	—
158	C₄⋊C₄	—	5	1	no	—
159	C₄⋊C₄	—	5	2	no	—
160	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
161	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
162	C₂²⋊C₄	—	5	4	no	—
163	C₂²⋊C₄	—	5	4	no	—
164	C₂²⋊C₄	—	5	4	no	—
165	C₄²	—	5	2	no	—
166	C₄²	C₂×C₃×D₅	1	2	yes	—
167	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
168	C₃×C₅	#559	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	—	10	4	no	—
170	C₂²×C₃	—	10	4	no	—
171	C₂²×C₃	—	10	4	no	—
172	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	10	2	no	—
177	C₃×C₄	—	10	2	no	—
178	C₃×C₄	#709	1	2	yes	—
179	C₃×C₄	#713	1	2	yes	—
180	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
181	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
182	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
183	D₅	—	4	4	no	—
184	C₂³	—	5	4	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	10	2	no	—
187	D₄	—	10	4	no	—
188	D₄	—	10	4	no	—
189	D₄	—	20	2	no	—
190	C₂×C₄	—	5	2	no	—
191	C₂×C₄	—	5	2	no	—
192	C₂×C₄	—	20	4	no	—
193	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
194	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
195	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
198	C₂×C₃	—	20	4	no	—
199	C₂×C₃	C₂×#709	1	1	yes	characteristic
200	C₂×C₃	C₂×#713	1	1	yes	characteristic
201	C₂×C₃	#4149	1	1	yes	characteristic
202	C₅	C₃×#559	1	1	yes	characteristic, complemented, Sylow
203	C₂²	—	10	4	no	—
204	C₂²	—	10	4	no	—
205	C₂²	—	10	4	no	—
206	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	10	2	no	—
211	C₄	—	10	2	no	—
212	C₄	C₃×#709	1	2	yes	—
213	C₄	C₃×#713	1	2	yes	—
214	C₃	#65919	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	—	20	4	no	—
216	C₂	C₂×C₃×#709	1	1	yes	characteristic
217	C₂	C₂×C₃×#713	1	1	yes	characteristic
218	C₂	C₃×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₃×#4020	C₂
hypercenter	C₃×C₄×Q₈	D₅
hypocenter	C₅	C₃×#559
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×Q₈	C₂	semidirect
2. C₃×#4093	C₂	semidirect
3. C₃×#4040	C₂	semidirect
4. C₃×#4039	C₂	semidirect
5. C₃×#4021	C₂	semidirect
7. C₃×C₄×D₂₀	C₂	semidirect
8. C₃×C₄×Dic₁₀	C₂	semidirect
9. #65919	C₃	direct (N × H)
27. C₄×C₅×Q₈	C₂×C₃	semidirect
28. #4093	C₂×C₃	semidirect
29. #4040	C₂×C₃	semidirect
30. #4039	C₂×C₃	semidirect
31. #4021	C₂×C₃	semidirect
33. C₄×D₂₀	C₂×C₃	semidirect
34. C₄×Dic₁₀	C₂×C₃	semidirect
54. C₃×C₄×Q₈	D₅	semidirect
120. C₄×Q₈	C₃×D₅	semidirect
168. C₃×C₅	#559	semidirect
202. C₅	C₃×#559	semidirect
214. C₃	#65919	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₄×C₅×Q₈	2
C₃×#4093	2
C₃×#4040	2
C₃×#4039	2
C₃×#4021	2
C₃×#4020	2
C₃×C₄×D₂₀	2
C₃×C₄×Dic₁₀	2
#65919	3
C₃×#559	5

Subgroups — C3×#65919 #11755596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65919 #11755596