Subgroups — C₃×#66061 #11755738

Order 960.

560 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×D₅	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4092	C₂	1	2	yes	maximal, complemented
4	C₃×#4042	C₂	1	4	yes	maximal, complemented
5	C₃×#4039	C₂	1	2	yes	maximal
6	C₃×#4034	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	#66061	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
11	C₃×Q₈×D₅	—	2	8	no	—
12	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
14	C₂×C₃×C₄×D₅	C₂²	1	4	yes	—
15	C₂×C₃×Dic₁₀	C₂²	1	4	yes	—
16	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₃×#685	C₂²	1	4	yes	—
21	C₃×#685	C₂²	1	2	yes	—
22	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₃×#683	C₂²	1	4	yes	—
25	C₃×#683	C₂²	1	2	yes	—
26	C₃×C₄×Dic₅	C₂²	1	2	yes	—
27	C₃×D₄×Q₈	—	5	1	no	maximal, Hall
28	C₂×Q₈×D₅	C₂×C₃	1	2	yes	complemented
29	C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
30	#4092	C₂×C₃	1	2	yes	complemented
31	#4042	C₂×C₃	1	4	yes	complemented
32	#4039	C₂×C₃	1	2	yes	—
33	#4034	C₂×C₃	1	2	yes	complemented
34	C₄×D₂₀	C₂×C₃	1	1	yes	characteristic
35	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂²×C₃×D₅	C₂³	1	2	yes	—
37	C₃×C₅×Q₈	—	2	4	no	—
38	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₂³	1	4	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	C₂×C₃×Dic₅	C₂³	1	2	yes	—
45	C₃×D₂₀	Q₈	1	4	yes	complemented
46	C₃×C₄×D₅	—	2	4	no	—
47	C₃×C₄×D₅	—	2	8	no	—
48	C₃×Dic₁₀	—	2	8	no	—
49	C₃×Dic₁₀	D₄	1	4	yes	complemented
50	C₂²×C₃×Q₈	—	5	2	no	—
51	C₃×C₄⋊Q₈	—	5	2	no	—
52	C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
53	C₃×C₂²⋊Q₈	—	5	2	no	—
54	C₃×C₂²⋊Q₈	—	5	4	no	—
55	C₃×C₄×Q₈	—	5	1	no	—
56	C₃×C₄×D₄	—	5	2	no	—
57	C₃×C₄×D₄	—	5	1	no	—
58	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
59	Q₈×D₅	—	2	8	no	—
60	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
62	C₂×C₄×D₅	C₂²×C₃	1	4	yes	—
63	C₂×Dic₁₀	C₂²×C₃	1	4	yes	—
64	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
65	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
68	#685	C₂²×C₃	1	2	yes	—
69	#685	C₂²×C₃	1	4	yes	—
70	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
71	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
72	#683	C₂²×C₃	1	2	yes	—
73	#683	C₂²×C₃	1	4	yes	—
74	C₄×Dic₅	C₂²×C₃	1	2	yes	—
75	D₄×Q₈	—	5	1	no	Sylow
76	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
77	C₂×C₃×D₅	—	2	2	no	—
78	C₂×C₃×D₅	—	2	2	no	—
79	C₂×C₃×D₅	C₂×Q₈	1	4	yes	complemented
80	C₃×C₄×C₅	—	2	4	no	—
81	C₃×C₄×C₅	—	2	1	no	—
82	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₃×C₄×C₅	C₂×Q₈	1	2	yes	complemented
84	C₃×Dic₅	—	2	2	no	—
85	C₃×Dic₅	—	2	2	no	—
86	C₃×Dic₅	C₂×D₄	1	4	yes	—
87	C₂×C₃×Q₈	—	5	1	no	—
88	C₂×C₃×Q₈	—	5	4	no	—
89	C₂×C₃×Q₈	—	10	8	no	—
90	C₂×C₃×Q₈	C₂×D₅	1	2	yes	—
91	C₂×C₃×D₄	—	5	1	no	—
92	C₂²×C₃×C₄	—	5	4	no	—
93	C₂²×C₃×C₄	—	5	2	no	—
94	C₃×C₄⋊C₄	—	5	2	no	—
95	C₃×C₄⋊C₄	—	5	4	no	—
96	C₃×C₄⋊C₄	—	5	1	no	—
97	C₃×C₄⋊C₄	—	5	1	no	—
98	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
99	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
100	C₃×C₂²⋊C₄	—	5	4	no	—
101	C₃×C₂²⋊C₄	—	5	2	no	—
102	C₃×C₄²	—	5	2	no	—
103	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
104	C₂²×D₅	C₂³×C₃	1	2	yes	—
105	C₅×Q₈	—	2	4	no	—
106	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
108	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
110	C₂×Dic₅	C₂³×C₃	1	2	yes	—
111	C₂×Dic₅	C₂³×C₃	1	2	yes	—
112	C₂×Dic₅	C₂³×C₃	1	2	yes	—
113	D₂₀	C₃×Q₈	1	4	yes	complemented
114	C₄×D₅	—	2	4	no	—
115	C₄×D₅	—	2	8	no	—
116	Dic₁₀	—	2	8	no	—
117	Dic₁₀	C₃×D₄	1	4	yes	complemented
118	C₂²×Q₈	—	5	2	no	—
119	C₄⋊Q₈	—	5	2	no	—
120	C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
121	C₂²⋊Q₈	—	5	2	no	—
122	C₂²⋊Q₈	—	5	4	no	—
123	C₄×Q₈	—	5	1	no	—
124	C₄×D₄	—	5	2	no	—
125	C₄×D₄	—	5	1	no	—
126	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
129	C₃×D₅	—	2	4	no	—
130	C₂³×C₃	—	5	2	no	—
131	C₃×Q₈	—	2	4	no	—
132	C₃×Q₈	—	5	4	no	—
133	C₃×Q₈	—	10	8	no	—
134	C₃×D₄	—	5	4	no	—
135	C₂×C₃×C₄	—	5	2	no	—
136	C₂×C₃×C₄	—	5	2	no	—
137	C₂×C₃×C₄	—	5	2	no	—
138	C₂×C₃×C₄	—	10	8	no	—
139	C₂×C₃×C₄	—	10	4	no	—
140	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂²×D₅	1	4	yes	—
142	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
144	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
145	C₂×D₅	—	2	2	no	—
146	C₂×D₅	—	2	2	no	—
147	C₂×D₅	C₂×C₃×Q₈	1	4	yes	complemented
148	C₄×C₅	—	2	4	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
151	C₄×C₅	C₂×C₃×Q₈	1	2	yes	complemented
152	Dic₅	—	2	2	no	—
153	Dic₅	—	2	2	no	—
154	Dic₅	C₂×C₃×D₄	1	4	yes	—
155	C₂×Q₈	—	5	1	no	—
156	C₂×Q₈	—	5	4	no	—
157	C₂×Q₈	—	10	8	no	—
158	C₂×Q₈	C₂×C₃×D₅	1	2	yes	—
159	C₂×D₄	—	5	1	no	—
160	C₂²×C₄	—	5	4	no	—
161	C₂²×C₄	—	5	2	no	—
162	C₄⋊C₄	—	5	1	no	—
163	C₄⋊C₄	—	5	2	no	—
164	C₄⋊C₄	—	5	1	no	—
165	C₄⋊C₄	—	5	4	no	—
166	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
167	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
168	C₂²⋊C₄	—	5	4	no	—
169	C₂²⋊C₄	—	5	2	no	—
170	C₄²	—	5	2	no	—
171	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
172	C₃×C₅	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
173	C₂²×C₃	—	5	4	no	—
174	C₂²×C₃	—	10	2	no	—
175	C₂²×C₃	—	10	2	no	—
176	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	5	4	no	—
180	C₃×C₄	—	10	2	no	—
181	C₃×C₄	—	10	2	no	—
182	C₃×C₄	D₄×D₅	1	2	yes	—
183	C₃×C₄	Q₈×D₅	1	2	yes	complemented
184	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
185	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
186	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
187	D₅	—	2	4	no	—
188	C₂³	—	5	2	no	—
189	Q₈	—	2	4	no	—
190	Q₈	—	5	4	no	—
191	Q₈	—	10	8	no	—
192	D₄	—	5	4	no	—
193	C₂×C₄	—	5	2	no	—
194	C₂×C₄	—	5	2	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	10	8	no	—
197	C₂×C₄	—	10	4	no	—
198	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₃×D₅	1	4	yes	—
200	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
201	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
202	C₂×C₃	—	10	4	no	—
203	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
204	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
205	C₂×C₃	#4147	1	1	yes	characteristic
206	C₅	C₃×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
207	C₂²	—	5	4	no	—
208	C₂²	—	10	2	no	—
209	C₂²	—	10	2	no	—
210	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	1	no	—
213	C₄	—	5	4	no	—
214	C₄	—	10	2	no	—
215	C₄	—	10	2	no	—
216	C₄	C₃×D₄×D₅	1	2	yes	—
217	C₄	C₃×Q₈×D₅	1	2	yes	complemented
218	C₃	#66061	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	10	4	no	—
220	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
221	C₂	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
222	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×D₄×Q₈
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. C₂×C₃×Q₈×D₅	C₂	semidirect
2. C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₃×#4092	C₂	semidirect
4. C₃×#4042	C₂	semidirect
6. C₃×#4034	C₂	semidirect
8. C₃×C₄×Dic₁₀	C₂	semidirect
9. #66061	C₃	direct (N × H)
28. C₂×Q₈×D₅	C₂×C₃	semidirect
29. C₅×C₄⋊Q₈	C₂×C₃	semidirect
30. #4092	C₂×C₃	semidirect
31. #4042	C₂×C₃	semidirect
33. #4034	C₂×C₃	semidirect
35. C₄×Dic₁₀	C₂×C₃	semidirect
45. C₃×D₂₀	Q₈	semidirect
49. C₃×Dic₁₀	D₄	semidirect
52. C₃×C₄⋊Q₈	D₅	semidirect
79. C₂×C₃×D₅	C₂×Q₈	semidirect
83. C₃×C₄×C₅	C₂×Q₈	semidirect
113. D₂₀	C₃×Q₈	semidirect
117. Dic₁₀	C₃×D₄	semidirect
120. C₄⋊Q₈	C₃×D₅	semidirect
147. C₂×D₅	C₂×C₃×Q₈	semidirect
151. C₄×C₅	C₂×C₃×Q₈	semidirect
172. C₃×C₅	D₄×Q₈	semidirect
183. C₃×C₄	Q₈×D₅	semidirect
206. C₅	C₃×D₄×Q₈	semidirect
217. C₄	C₃×Q₈×D₅	semidirect
218. C₃	#66061	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×Q₈×D₅	2
C₃×C₅×C₄⋊Q₈	2
C₃×#4092	2
C₃×#4042	2
C₃×#4039	2
C₃×#4034	2
C₃×C₄×D₂₀	2
C₃×C₄×Dic₁₀	2
#66061	3
C₃×D₄×Q₈	5

Subgroups — C3×#66061 #11755738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66061 #11755738