Subgroups — C₃×#66150 #11755827

Order 960.

540 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	C₂²×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#709	C₂	1	1	yes	maximal, characteristic
3	C₃×#4093	C₂	1	2	yes	maximal, complemented
4	C₃×#4092	C₂	1	4	yes	maximal, complemented
5	C₃×#4090	C₂	1	2	yes	maximal, complemented
6	C₃×#4075	C₂	1	4	yes	maximal, complemented
7	C₃×#4072	C₂	1	1	yes	maximal, characteristic, complemented
8	#66150	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₈	—	2	4	no	—
10	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
12	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₃×#709	—	2	4	no	—
16	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
18	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
19	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
20	C₃×#685	C₂²	1	8	yes	—
21	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
22	C₃×#683	C₂²	1	8	yes	—
23	C₃×C₄×Dic₅	C₂²	1	2	yes	—
24	C₃×#536	—	5	1	no	maximal, Hall
25	C₂²×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×#709	C₂×C₃	1	1	yes	characteristic
27	#4093	C₂×C₃	1	2	yes	complemented
28	#4092	C₂×C₃	1	4	yes	complemented
29	#4090	C₂×C₃	1	2	yes	complemented
30	#4075	C₂×C₃	1	4	yes	complemented
31	#4072	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×D₅	C₂³	1	2	yes	—
34	C₃×C₅×Q₈	—	2	4	no	—
35	C₃×C₅×Q₈	—	2	4	no	—
36	C₂×C₃×C₄×C₅	—	2	4	no	—
37	C₂×C₃×C₄×C₅	D₄	1	4	yes	—
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	C₃×C₂²⋊D₅	—	2	4	no	—
42	C₂×C₃×Dic₅	C₂³	1	4	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	C₂×C₃×#41	—	5	1	no	—
48	C₂²×C₃×Q₈	D₅	1	1	yes	characteristic, complemented
49	C₃×C₄⋊Q₈	—	5	2	no	—
50	C₃×#124	—	5	2	no	—
51	C₃×#123	—	5	4	no	—
52	C₃×C₂²⋊Q₈	—	5	4	no	—
53	C₃×#117	—	5	1	no	—
54	C₂×C₅×Q₈	—	2	4	no	—
55	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
56	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
58	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
60	#709	—	2	4	no	—
61	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
63	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
64	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
65	#685	C₂²×C₃	1	8	yes	—
66	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
67	#683	C₂²×C₃	1	8	yes	—
68	C₄×Dic₅	C₂²×C₃	1	2	yes	—
69	#536	—	5	1	no	Sylow
70	C₂²×C₃×C₅	—	2	2	no	—
71	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
72	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
73	C₂×C₃×D₅	—	2	2	no	—
74	C₂×C₃×D₅	—	2	4	no	—
75	C₃×C₄×C₅	—	2	4	no	—
76	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₃×Dic₅	—	2	2	no	—
79	C₃×Dic₅	—	2	4	no	—
80	C₃×#41	—	10	4	no	—
81	C₂×C₃×Q₈	—	2	4	no	—
82	C₂×C₃×Q₈	—	5	1	no	—
83	C₂×C₃×Q₈	C₂×D₅	1	2	yes	—
84	C₂×C₃×Q₈	C₂×D₅	1	2	yes	—
85	C₂×C₃×D₄	—	5	2	no	—
86	C₂×C₃×D₄	—	5	1	no	—
87	C₂²×C₃×C₄	—	5	2	no	—
88	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
89	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
90	C₃×C₄⋊C₄	—	5	8	no	—
91	C₃×C₄⋊C₄	—	5	2	no	—
92	C₃×C₂²⋊C₄	—	5	2	no	—
93	C₃×C₂²⋊C₄	—	5	8	no	—
94	C₃×C₄²	—	5	2	no	—
95	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
96	C₂²×D₅	C₂³×C₃	1	2	yes	—
97	C₅×Q₈	—	2	4	no	—
98	C₅×Q₈	—	2	4	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	C₃×D₄	1	4	yes	—
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	C₂²⋊D₅	—	2	4	no	—
105	C₂×Dic₅	C₂³×C₃	1	4	yes	—
106	C₂×Dic₅	C₂³×C₃	1	2	yes	—
107	D₂₀	—	2	2	no	—
108	C₄×D₅	—	2	4	no	—
109	Dic₁₀	—	2	2	no	—
110	C₂×#41	—	5	1	no	—
111	C₂²×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
112	C₄⋊Q₈	—	5	2	no	—
113	#124	—	5	2	no	—
114	#123	—	5	4	no	—
115	C₂²⋊Q₈	—	5	4	no	—
116	#117	—	5	1	no	—
117	C₂×C₃×C₅	—	2	2	no	—
118	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
120	C₃×D₅	—	4	2	no	—
121	C₂³×C₃	—	5	2	no	—
122	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
123	C₃×Q₈	—	2	4	no	—
124	C₃×Q₈	—	2	4	no	—
125	C₃×Q₈	—	10	2	no	—
126	C₃×D₄	—	10	4	no	—
127	C₃×D₄	—	10	2	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	5	4	no	—
130	C₂×C₃×C₄	—	5	2	no	—
131	C₂×C₃×C₄	—	10	4	no	—
132	C₂×C₃×C₄	C₂²⋊D₅	1	4	yes	—
133	C₂×C₃×C₄	C₂²×D₅	1	4	yes	—
134	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
138	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
139	C₂×D₅	—	2	2	no	—
140	C₂×D₅	—	2	4	no	—
141	C₄×C₅	—	2	4	no	—
142	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
143	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
144	Dic₅	—	2	2	no	—
145	Dic₅	—	2	4	no	—
146	#41	—	10	4	no	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	5	1	no	—
149	C₂×Q₈	C₂×C₃×D₅	1	2	yes	—
150	C₂×Q₈	C₂×C₃×D₅	1	2	yes	—
151	C₂×D₄	—	5	2	no	—
152	C₂×D₄	—	5	1	no	—
153	C₂²×C₄	—	5	2	no	—
154	C₂²×C₄	C₂×C₃×D₅	1	2	yes	—
155	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
156	C₄⋊C₄	—	5	2	no	—
157	C₄⋊C₄	—	5	8	no	—
158	C₂²⋊C₄	—	5	2	no	—
159	C₂²⋊C₄	—	5	8	no	—
160	C₄²	—	5	2	no	—
161	C₃×C₅	#536	1	1	yes	characteristic, complemented, Hall
162	C₂²×C₃	—	2	2	no	—
163	C₂²×C₃	—	10	4	no	—
164	C₂²×C₃	—	10	2	no	—
165	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
166	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	—	10	4	no	—
169	C₃×C₄	—	10	2	no	—
170	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
171	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
172	C₂×C₅	—	2	2	no	—
173	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
174	C₂×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
175	D₅	—	4	2	no	—
176	C₂³	—	5	2	no	—
177	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
178	Q₈	—	2	4	no	—
179	Q₈	—	2	4	no	—
180	Q₈	—	10	2	no	—
181	D₄	—	10	4	no	—
182	D₄	—	10	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	5	4	no	—
185	C₂×C₄	—	5	2	no	—
186	C₂×C₄	—	10	4	no	—
187	C₂×C₄	C₃×C₂²⋊D₅	1	4	yes	—
188	C₂×C₄	C₂²×C₃×D₅	1	4	yes	—
189	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
191	C₂×C₃	—	2	2	no	—
192	C₂×C₃	—	20	2	no	—
193	C₂×C₃	#4147	1	2	yes	—
194	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
195	C₅	C₃×#536	1	1	yes	characteristic, complemented, Sylow
196	C₂²	—	2	2	no	—
197	C₂²	—	10	4	no	—
198	C₂²	—	10	2	no	—
199	C₂²	C₂×C₃×C₂²⋊D₅	1	2	yes	—
200	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	—	10	4	no	—
203	C₄	—	10	2	no	—
204	C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
205	C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
206	C₃	#66150	1	1	yes	characteristic, direct product factor, Sylow
207	C₂	—	2	2	no	—
208	C₂	—	20	2	no	—
209	C₂	C₃×#4147	1	2	yes	—
210	C₂	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₂²×C₃×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×#709	C₂
hypercenter	C₂²×C₃×Q₈	D₅
hypocenter	C₅	C₃×#536
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×Q₈	C₂	semidirect
3. C₃×#4093	C₂	semidirect
4. C₃×#4092	C₂	semidirect
5. C₃×#4090	C₂	semidirect
6. C₃×#4075	C₂	semidirect
7. C₃×#4072	C₂	semidirect
8. #66150	C₃	direct (N × H)
25. C₂²×C₅×Q₈	C₂×C₃	semidirect
27. #4093	C₂×C₃	semidirect
28. #4092	C₂×C₃	semidirect
29. #4090	C₂×C₃	semidirect
30. #4075	C₂×C₃	semidirect
31. #4072	C₂×C₃	semidirect
48. C₂²×C₃×Q₈	D₅	semidirect
111. C₂²×Q₈	C₃×D₅	semidirect
161. C₃×C₅	#536	semidirect
195. C₅	C₃×#536	semidirect
206. C₃	#66150	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×Q₈	2
C₂×C₃×#709	2
C₃×#4093	2
C₃×#4092	2
C₃×#4090	2
C₃×#4075	2
C₃×#4072	2
#66150	3
C₃×#536	5

Subgroups — C3×#66150 #11755827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66150 #11755827