Subgroups — C₃×#66066 #11755743

Order 960.

424 conjugacy classes of subgroups.

The subgroup lattice has 212 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×#4093	C₂	1	2	yes	maximal
3	C₃×Q₈×Dic₅	C₂	1	2	yes	maximal, complemented
4	C₃×#4044	C₂	1	4	yes	maximal, complemented
5	C₃×#4043	C₂	1	2	yes	maximal, complemented
6	C₃×#4038	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	#66066	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×Dic₁₀	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₃×#685	C₂²	1	4	yes	—
18	C₃×#685	C₂²	1	4	yes	—
19	C₃×#685	C₂²	1	2	yes	—
20	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊Dic₅	C₂²	1	4	yes	—
22	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₃×#683	C₂²	1	2	yes	—
24	C₃×C₄×Dic₅	C₂²	1	2	yes	—
25	C₃×C₄×Dic₅	C₂²	1	4	yes	—
26	C₃×#556	—	5	1	no	maximal, Hall
27	C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
28	#4093	C₂×C₃	1	2	yes	—
29	Q₈×Dic₅	C₂×C₃	1	2	yes	complemented
30	#4044	C₂×C₃	1	4	yes	complemented
31	#4043	C₂×C₃	1	2	yes	complemented
32	#4038	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	—	2	4	no	—
37	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
38	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
45	C₃×C₄×D₅	—	2	4	no	—
46	C₃×Dic₁₀	—	2	2	no	—
47	C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
48	C₃×#126	—	5	4	no	—
49	C₃×#124	—	5	2	no	—
50	C₃×C₂²⋊Q₈	—	5	2	no	—
51	C₃×C₄×Q₈	—	5	2	no	—
52	C₃×C₄×Q₈	—	5	1	no	—
53	C₃×C₄×D₄	—	5	1	no	—
54	C₃×#117	—	5	2	no	—
55	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
56	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
58	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
59	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
60	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
62	#685	C₂²×C₃	1	2	yes	—
63	#685	C₂²×C₃	1	4	yes	—
64	#685	C₂²×C₃	1	4	yes	—
65	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
66	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
67	C₄⋊Dic₅	C₂²×C₃	1	4	yes	—
68	#683	C₂²×C₃	1	2	yes	—
69	C₄×Dic₅	C₂²×C₃	1	2	yes	—
70	C₄×Dic₅	C₂²×C₃	1	4	yes	—
71	#556	—	5	1	no	Sylow
72	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
73	C₂×C₃×D₅	—	2	2	no	—
74	C₂×C₃×D₅	—	2	2	no	—
75	C₂×C₃×D₅	—	2	2	no	—
76	C₃×C₄×C₅	—	2	4	no	—
77	C₃×C₄×C₅	—	2	1	no	—
78	C₃×C₄×C₅	#41	1	2	yes	—
79	C₃×C₄×C₅	#41	1	2	yes	—
80	C₃×Dic₅	—	2	2	no	—
81	C₃×Dic₅	—	2	2	no	—
82	C₃×Dic₅	—	2	2	no	—
83	C₂×C₃×Q₈	—	5	1	no	—
84	C₂×C₃×Q₈	C₂×D₅	1	2	yes	—
85	C₂×C₃×D₄	—	5	1	no	—
86	C₂²×C₃×C₄	—	5	2	no	—
87	C₃×C₄⋊C₄	—	5	2	no	—
88	C₃×C₄⋊C₄	—	5	1	no	—
89	C₃×C₄⋊C₄	—	5	1	no	—
90	C₃×C₄⋊C₄	—	5	4	no	—
91	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
92	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
93	C₃×C₂²⋊C₄	—	5	4	no	—
94	C₃×C₂²⋊C₄	—	5	2	no	—
95	C₃×C₂²⋊C₄	—	5	4	no	—
96	C₃×C₄²	—	5	4	no	—
97	C₃×C₄²	—	5	2	no	—
98	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
99	C₂²×D₅	C₂³×C₃	1	2	yes	—
100	C₅×Q₈	—	2	4	no	—
101	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
102	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
105	C₂×Dic₅	C₂³×C₃	1	2	yes	—
106	C₂×Dic₅	C₂³×C₃	1	2	yes	—
107	C₂×Dic₅	C₂³×C₃	1	2	yes	—
108	D₂₀	—	2	2	no	—
109	C₄×D₅	—	2	4	no	—
110	Dic₁₀	—	2	2	no	—
111	C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
112	#126	—	5	4	no	—
113	#124	—	5	2	no	—
114	C₂²⋊Q₈	—	5	2	no	—
115	C₄×Q₈	—	5	2	no	—
116	C₄×Q₈	—	5	1	no	—
117	C₄×D₄	—	5	1	no	—
118	#117	—	5	2	no	—
119	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
120	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
121	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
122	C₃×D₅	—	4	2	no	—
123	C₂³×C₃	—	5	2	no	—
124	C₃×Q₈	—	2	4	no	—
125	C₃×Q₈	—	10	2	no	—
126	C₃×D₄	—	10	2	no	—
127	C₂×C₃×C₄	—	5	2	no	—
128	C₂×C₃×C₄	—	5	2	no	—
129	C₂×C₃×C₄	—	5	2	no	—
130	C₂×C₃×C₄	—	10	4	no	—
131	C₂×C₃×C₄	C₂²×D₅	1	4	yes	—
132	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
135	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
136	C₂×D₅	—	2	2	no	—
137	C₂×D₅	—	2	2	no	—
138	C₂×D₅	—	2	2	no	—
139	C₄×C₅	—	2	4	no	—
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	C₃×#41	1	2	yes	—
142	C₄×C₅	C₃×#41	1	2	yes	—
143	Dic₅	—	2	2	no	—
144	Dic₅	—	2	2	no	—
145	Dic₅	—	2	2	no	—
146	C₂×Q₈	—	5	1	no	—
147	C₂×Q₈	C₂×C₃×D₅	1	2	yes	—
148	C₂×D₄	—	5	1	no	—
149	C₂²×C₄	—	5	2	no	—
150	C₄⋊C₄	—	5	1	no	—
151	C₄⋊C₄	—	5	1	no	—
152	C₄⋊C₄	—	5	4	no	—
153	C₄⋊C₄	—	5	2	no	—
154	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
155	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
156	C₂²⋊C₄	—	5	4	no	—
157	C₂²⋊C₄	—	5	2	no	—
158	C₂²⋊C₄	—	5	4	no	—
159	C₄²	—	5	4	no	—
160	C₄²	—	5	2	no	—
161	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
162	C₃×C₅	#556	1	1	yes	characteristic, complemented, Hall
163	C₂²×C₃	—	10	2	no	—
164	C₂²×C₃	—	10	2	no	—
165	C₂²×C₃	—	10	2	no	—
166	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	10	2	no	—
170	C₃×C₄	—	10	2	no	—
171	C₃×C₄	—	10	2	no	—
172	C₃×C₄	#711	1	2	yes	—
173	C₃×C₄	#713	1	2	yes	—
174	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
175	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
176	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
177	D₅	—	4	2	no	—
178	C₂³	—	5	2	no	—
179	Q₈	—	2	4	no	—
180	Q₈	—	10	2	no	—
181	D₄	—	10	2	no	—
182	C₂×C₄	—	5	2	no	—
183	C₂×C₄	—	5	2	no	—
184	C₂×C₄	—	5	2	no	—
185	C₂×C₄	—	10	4	no	—
186	C₂×C₄	C₂²×C₃×D₅	1	4	yes	—
187	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
189	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
190	C₂×C₃	—	20	2	no	—
191	C₂×C₃	C₂×#711	1	1	yes	characteristic
192	C₂×C₃	C₂×#713	1	1	yes	characteristic
193	C₂×C₃	#4147	1	1	yes	characteristic
194	C₅	C₃×#556	1	1	yes	characteristic, complemented, Sylow
195	C₂²	—	10	2	no	—
196	C₂²	—	10	2	no	—
197	C₂²	—	10	2	no	—
198	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	1	no	—
201	C₄	—	10	2	no	—
202	C₄	—	10	2	no	—
203	C₄	—	10	2	no	—
204	C₄	C₃×#711	1	2	yes	—
205	C₄	C₃×#713	1	2	yes	—
206	C₃	#66066	1	1	yes	characteristic, direct product factor, Sylow
207	C₂	—	20	2	no	—
208	C₂	C₂×C₃×#711	1	1	yes	characteristic
209	C₂	C₂×C₃×#713	1	1	yes	characteristic
210	C₂	C₃×#4147	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×#556
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
3. C₃×Q₈×Dic₅	C₂	semidirect
4. C₃×#4044	C₂	semidirect
5. C₃×#4043	C₂	semidirect
6. C₃×#4038	C₂	semidirect
8. C₃×C₄×Dic₁₀	C₂	semidirect
9. #66066	C₃	direct (N × H)
27. C₅×C₄⋊Q₈	C₂×C₃	semidirect
29. Q₈×Dic₅	C₂×C₃	semidirect
30. #4044	C₂×C₃	semidirect
31. #4043	C₂×C₃	semidirect
32. #4038	C₂×C₃	semidirect
34. C₄×Dic₁₀	C₂×C₃	semidirect
47. C₃×C₄⋊Q₈	D₅	semidirect
111. C₄⋊Q₈	C₃×D₅	semidirect
162. C₃×C₅	#556	semidirect
194. C₅	C₃×#556	semidirect
206. C₃	#66066	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×#4093	2
C₃×Q₈×Dic₅	2
C₃×#4044	2
C₃×#4043	2
C₃×#4038	2
C₃×C₄×D₂₀	2
C₃×C₄×Dic₁₀	2
#66066	3
C₃×#556	5

Subgroups — C3×#66066 #11755743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66066 #11755743