Subgroups — C₃×#66064 #11755741

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4092	C₂	1	4	yes	maximal, complemented
3	C₃×#4043	C₂	1	2	yes	maximal, complemented
4	C₃×#4039	C₂	1	2	yes	maximal
5	C₃×D₅×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×#4036	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
9	#66064	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
11	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
13	C₂×C₃×C₄×D₅	C₂²	1	4	yes	—
14	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₃×#685	C₂²	1	2	yes	—
19	C₃×#685	C₂²	1	4	yes	—
20	C₃×C₄⋊Dic₅	C₂²	1	4	yes	—
21	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₃×#683	C₂²	1	2	yes	—
24	C₃×#683	C₂²	1	4	yes	—
25	C₃×C₄×Dic₅	C₂²	1	2	yes	—
26	C₃×#554	—	5	1	no	maximal, Hall
27	C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
28	#4092	C₂×C₃	1	4	yes	complemented
29	#4043	C₂×C₃	1	2	yes	complemented
30	#4039	C₂×C₃	1	2	yes	—
31	D₅×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
32	#4036	C₂×C₃	1	2	yes	complemented
33	C₄×D₂₀	C₂×C₃	1	1	yes	characteristic
34	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂²×C₃×D₅	C₂³	1	2	yes	—
36	C₃×C₅×Q₈	—	4	2	no	—
37	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
39	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×Dic₅	C₂³	1	2	yes	—
42	C₂×C₃×Dic₅	C₂³	1	2	yes	—
43	C₂×C₃×Dic₅	C₂³	1	2	yes	—
44	C₃×D₂₀	Q₈	1	4	yes	complemented
45	C₃×C₄×D₅	—	2	4	no	—
46	C₃×C₄×D₅	—	4	4	no	—
47	C₃×Dic₁₀	—	2	2	no	—
48	C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
49	C₃×#125	—	5	2	no	—
50	C₃×C₂²⋊Q₈	—	5	4	no	—
51	C₃×C₂²⋊Q₈	—	5	2	no	—
52	C₃×C₄×Q₈	—	5	1	no	—
53	C₃×C₄×D₄	—	5	2	no	—
54	C₃×C₄×D₄	—	5	1	no	—
55	C₂×C₃×C₄⋊C₄	—	5	2	no	—
56	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
57	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
59	C₂×C₄×D₅	C₂²×C₃	1	4	yes	—
60	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
61	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
63	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
64	#685	C₂²×C₃	1	2	yes	—
65	#685	C₂²×C₃	1	4	yes	—
66	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
67	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
68	C₄⋊Dic₅	C₂²×C₃	1	4	yes	—
69	#683	C₂²×C₃	1	2	yes	—
70	#683	C₂²×C₃	1	4	yes	—
71	C₄×Dic₅	C₂²×C₃	1	2	yes	—
72	#554	—	5	1	no	Sylow
73	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
74	C₂×C₃×D₅	—	2	2	no	—
75	C₂×C₃×D₅	—	2	2	no	—
76	C₂×C₃×D₅	C₂×Q₈	1	4	yes	—
77	C₃×C₄×C₅	—	2	4	no	—
78	C₃×C₄×C₅	—	2	1	no	—
79	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
80	C₃×C₄×C₅	#41	1	2	yes	—
81	C₃×Dic₅	—	2	2	no	—
82	C₃×Dic₅	—	2	2	no	—
83	C₃×Dic₅	—	2	2	no	—
84	C₂×C₃×Q₈	—	5	1	no	—
85	C₂×C₃×Q₈	C₂×D₅	1	2	yes	—
86	C₂×C₃×D₄	—	5	1	no	—
87	C₂²×C₃×C₄	—	5	2	no	—
88	C₂²×C₃×C₄	—	5	4	no	—
89	C₃×C₄⋊C₄	—	5	4	no	—
90	C₃×C₄⋊C₄	—	5	2	no	—
91	C₃×C₄⋊C₄	—	5	1	no	—
92	C₃×C₄⋊C₄	—	5	1	no	—
93	C₃×C₄⋊C₄	—	5	4	no	—
94	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
95	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
96	C₃×C₂²⋊C₄	—	5	4	no	—
97	C₃×C₂²⋊C₄	—	5	2	no	—
98	C₃×C₄²	—	5	2	no	—
99	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
100	C₂²×D₅	C₂³×C₃	1	2	yes	—
101	C₅×Q₈	—	4	2	no	—
102	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
104	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
106	C₂×Dic₅	C₂³×C₃	1	2	yes	—
107	C₂×Dic₅	C₂³×C₃	1	2	yes	—
108	C₂×Dic₅	C₂³×C₃	1	2	yes	—
109	D₂₀	C₃×Q₈	1	4	yes	complemented
110	C₄×D₅	—	2	4	no	—
111	C₄×D₅	—	4	4	no	—
112	Dic₁₀	—	2	2	no	—
113	C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
114	#125	—	5	2	no	—
115	C₂²⋊Q₈	—	5	4	no	—
116	C₂²⋊Q₈	—	5	2	no	—
117	C₄×Q₈	—	5	1	no	—
118	C₄×D₄	—	5	2	no	—
119	C₄×D₄	—	5	1	no	—
120	C₂×C₄⋊C₄	—	5	2	no	—
121	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
123	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
124	C₃×D₅	—	2	4	no	—
125	C₂³×C₃	—	5	2	no	—
126	C₃×Q₈	—	4	2	no	—
127	C₃×Q₈	—	10	2	no	—
128	C₃×D₄	—	5	4	no	—
129	C₂×C₃×C₄	—	5	2	no	—
130	C₂×C₃×C₄	—	5	2	no	—
131	C₂×C₃×C₄	—	5	2	no	—
132	C₂×C₃×C₄	—	10	4	no	—
133	C₂×C₃×C₄	—	20	4	no	—
134	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂²×D₅	1	4	yes	—
136	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
138	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
139	C₂×D₅	—	2	2	no	—
140	C₂×D₅	—	2	2	no	—
141	C₂×D₅	C₂×C₃×Q₈	1	4	yes	—
142	C₄×C₅	—	2	4	no	—
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
145	C₄×C₅	C₃×#41	1	2	yes	—
146	Dic₅	—	2	2	no	—
147	Dic₅	—	2	2	no	—
148	Dic₅	—	2	2	no	—
149	C₂×Q₈	—	5	1	no	—
150	C₂×Q₈	C₂×C₃×D₅	1	2	yes	—
151	C₂×D₄	—	5	1	no	—
152	C₂²×C₄	—	5	2	no	—
153	C₂²×C₄	—	5	4	no	—
154	C₄⋊C₄	—	5	1	no	—
155	C₄⋊C₄	—	5	4	no	—
156	C₄⋊C₄	—	5	1	no	—
157	C₄⋊C₄	—	5	2	no	—
158	C₄⋊C₄	—	5	4	no	—
159	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
160	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
161	C₂²⋊C₄	—	5	4	no	—
162	C₂²⋊C₄	—	5	2	no	—
163	C₄²	—	5	2	no	—
164	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
165	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
166	C₂²×C₃	—	5	4	no	—
167	C₂²×C₃	—	10	2	no	—
168	C₂²×C₃	—	10	2	no	—
169	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
170	C₃×C₄	—	2	4	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	10	2	no	—
173	C₃×C₄	—	10	2	no	—
174	C₃×C₄	—	10	2	no	—
175	C₃×C₄	Q₈×D₅	1	2	yes	—
176	C₃×C₄	#713	1	2	yes	—
177	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
178	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
179	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
180	D₅	—	2	4	no	—
181	C₂³	—	5	2	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	10	2	no	—
184	D₄	—	5	4	no	—
185	C₂×C₄	—	5	2	no	—
186	C₂×C₄	—	5	2	no	—
187	C₂×C₄	—	5	2	no	—
188	C₂×C₄	—	10	4	no	—
189	C₂×C₄	—	20	4	no	—
190	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₃×D₅	1	4	yes	—
192	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
194	C₂×C₃	—	10	4	no	—
195	C₂×C₃	#4144	1	1	yes	characteristic
196	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
197	C₂×C₃	C₂×#713	1	1	yes	characteristic
198	C₅	C₃×#554	1	1	yes	characteristic, complemented, Sylow
199	C₂²	—	5	4	no	—
200	C₂²	—	10	2	no	—
201	C₂²	—	10	2	no	—
202	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	1	no	—
205	C₄	—	10	2	no	—
206	C₄	—	10	2	no	—
207	C₄	—	10	2	no	—
208	C₄	C₃×Q₈×D₅	1	2	yes	—
209	C₄	C₃×#713	1	2	yes	—
210	C₃	#66064	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	10	4	no	—
212	C₂	C₃×#4144	1	1	yes	characteristic
213	C₂	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
214	C₂	C₂×C₃×#713	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×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×D₂₀	C₂²
hypercenter	C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₃×#554
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×C₅×C₄⋊Q₈	C₂	semidirect
2. C₃×#4092	C₂	semidirect
3. C₃×#4043	C₂	semidirect
5. C₃×D₅×C₄⋊C₄	C₂	semidirect
6. C₃×#4036	C₂	semidirect
8. C₃×C₄×Dic₁₀	C₂	semidirect
9. #66064	C₃	direct (N × H)
27. C₅×C₄⋊Q₈	C₂×C₃	semidirect
28. #4092	C₂×C₃	semidirect
29. #4043	C₂×C₃	semidirect
31. D₅×C₄⋊C₄	C₂×C₃	semidirect
32. #4036	C₂×C₃	semidirect
34. C₄×Dic₁₀	C₂×C₃	semidirect
44. C₃×D₂₀	Q₈	semidirect
48. C₃×C₄⋊Q₈	D₅	semidirect
109. D₂₀	C₃×Q₈	semidirect
113. C₄⋊Q₈	C₃×D₅	semidirect
165. C₃×C₅	#554	semidirect
198. C₅	C₃×#554	semidirect
210. C₃	#66064	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×C₅×C₄⋊Q₈	2
C₃×#4092	2
C₃×#4043	2
C₃×#4039	2
C₃×D₅×C₄⋊C₄	2
C₃×#4036	2
C₃×C₄×D₂₀	2
C₃×C₄×Dic₁₀	2
#66064	3
C₃×#554	5

Subgroups — C3×#66064 #11755741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66064 #11755741