Subgroups — C₃×#65908 #11755585

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#713	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4039	C₂	1	3	yes	maximal, complemented
5	C₃×D₅×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₃×C₄×D₂₀	C₂	1	3	yes	maximal, complemented
7	C₃×#4018	C₂	1	3	yes	maximal, complemented
8	#65908	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₃×#713	C₄	1	8	yes	complemented
11	C₂×C₃×D₂₀	C₂²	1	3	yes	—
12	C₂×C₃×C₄×D₅	C₂²	1	6	yes	—
13	C₂×C₃×C₄×D₅	C₂²	1	3	yes	—
14	C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
15	C₃×C₄²×C₅	C₂²	1	3	yes	—
16	C₃×#685	C₂²	1	3	yes	—
17	C₃×#685	C₂²	1	3	yes	—
18	C₃×C₄⋊Dic₅	C₂²	1	3	yes	—
19	C₃×#683	C₂²	1	1	yes	characteristic
20	C₃×#683	C₂²	1	3	yes	—
21	C₃×C₄×Dic₅	C₂²	1	3	yes	—
22	C₃×#520	—	5	1	no	maximal, Hall
23	C₂×#713	C₂×C₃	1	1	yes	characteristic
24	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
25	Q₈×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
26	#4039	C₂×C₃	1	3	yes	complemented
27	D₅×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
28	C₄×D₂₀	C₂×C₃	1	3	yes	complemented
29	#4018	C₂×C₃	1	3	yes	complemented
30	C₂²×C₃×D₅	C₂³	1	3	yes	—
31	C₃×C₅×Q₈	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
34	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
35	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
36	C₂×C₃×Dic₅	C₂³	1	3	yes	—
37	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
38	C₃×D₂₀	C₂×C₄	1	12	yes	complemented
39	C₃×C₄×D₅	—	4	6	no	—
40	C₃×C₄×D₅	C₂×C₄	1	12	yes	complemented
41	C₂×C₃×#41	—	5	1	no	—
42	C₃×C₄×Q₈	—	5	1	no	—
43	C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
44	C₃×C₄×D₄	—	5	3	no	—
45	C₃×C₄×D₄	—	5	3	no	—
46	C₃×#117	—	5	3	no	—
47	C₂×C₃×C₄⋊C₄	—	5	3	no	—
48	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
49	#713	C₃×C₄	1	8	yes	complemented
50	C₂×D₂₀	C₂²×C₃	1	3	yes	—
51	C₂×C₄×D₅	C₂²×C₃	1	6	yes	—
52	C₂×C₄×D₅	C₂²×C₃	1	3	yes	—
53	C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
54	C₄²×C₅	C₂²×C₃	1	3	yes	—
55	#685	C₂²×C₃	1	3	yes	—
56	#685	C₂²×C₃	1	3	yes	—
57	C₄⋊Dic₅	C₂²×C₃	1	3	yes	—
58	#683	C₂²×C₃	1	1	yes	characteristic
59	#683	C₂²×C₃	1	3	yes	—
60	C₄×Dic₅	C₂²×C₃	1	3	yes	—
61	#520	—	5	1	no	Sylow
62	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
63	C₂×C₃×D₅	—	2	3	no	—
64	C₂×C₃×D₅	—	2	3	no	—
65	C₂×C₃×D₅	C₂²×C₄	1	6	yes	—
66	C₃×C₄×C₅	—	2	3	no	—
67	C₃×C₄×C₅	—	2	1	no	—
68	C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
69	C₃×Dic₅	—	2	3	no	—
70	C₃×Dic₅	—	2	1	no	—
71	C₃×Dic₅	C₂²×C₄	1	2	yes	—
72	C₃×#41	—	5	8	no	—
73	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
74	C₂×C₃×D₄	—	5	3	no	—
75	C₂²×C₃×C₄	—	5	3	no	—
76	C₂²×C₃×C₄	—	5	6	no	—
77	C₃×C₄⋊C₄	—	5	3	no	—
78	C₃×C₄⋊C₄	—	5	1	no	—
79	C₃×C₄⋊C₄	—	5	3	no	—
80	C₃×C₄⋊C₄	C₂×D₅	1	3	yes	—
81	C₃×C₂²⋊C₄	—	5	3	no	—
82	C₃×C₂²⋊C₄	—	5	3	no	—
83	C₃×C₄²	—	5	3	no	—
84	C₃×C₄²	C₂×D₅	1	3	yes	—
85	C₂²×D₅	C₂³×C₃	1	3	yes	—
86	C₅×Q₈	C₂×C₃×C₄	1	4	yes	complemented
87	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
89	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
90	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
91	C₂×Dic₅	C₂³×C₃	1	3	yes	—
92	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
93	D₂₀	C₂×C₃×C₄	1	12	yes	complemented
94	C₄×D₅	—	4	6	no	—
95	C₄×D₅	C₂×C₃×C₄	1	12	yes	complemented
96	C₂×#41	—	5	1	no	—
97	C₄×Q₈	—	5	1	no	—
98	C₄×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
99	C₄×D₄	—	5	3	no	—
100	C₄×D₄	—	5	3	no	—
101	#117	—	5	3	no	—
102	C₂×C₄⋊C₄	—	5	3	no	—
103	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
106	C₃×D₅	—	2	6	no	—
107	C₂³×C₃	—	5	3	no	—
108	C₃×Q₈	C₄×D₅	1	4	yes	complemented
109	C₃×D₄	—	5	12	no	—
110	C₂×C₃×C₄	—	5	1	no	—
111	C₂×C₃×C₄	—	5	12	no	—
112	C₂×C₃×C₄	—	5	1	no	—
113	C₂×C₃×C₄	—	5	3	no	—
114	C₂×C₃×C₄	—	20	6	no	—
115	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂²×D₅	1	3	yes	—
117	C₂×C₃×C₄	C₂²×D₅	1	3	yes	—
118	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
119	C₂×D₅	—	2	3	no	—
120	C₂×D₅	—	2	3	no	—
121	C₂×D₅	C₂²×C₃×C₄	1	6	yes	—
122	C₄×C₅	—	2	3	no	—
123	C₄×C₅	—	2	1	no	—
124	C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
125	Dic₅	—	2	3	no	—
126	Dic₅	—	2	1	no	—
127	Dic₅	C₂²×C₃×C₄	1	2	yes	—
128	#41	—	5	8	no	—
129	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
130	C₂×D₄	—	5	3	no	—
131	C₂²×C₄	—	5	3	no	—
132	C₂²×C₄	—	5	6	no	—
133	C₄⋊C₄	—	5	3	no	—
134	C₄⋊C₄	—	5	3	no	—
135	C₄⋊C₄	—	5	1	no	—
136	C₄⋊C₄	C₂×C₃×D₅	1	3	yes	—
137	C₂²⋊C₄	—	5	3	no	—
138	C₂²⋊C₄	—	5	3	no	—
139	C₄²	—	5	3	no	—
140	C₄²	C₂×C₃×D₅	1	3	yes	—
141	C₃×C₅	#520	1	1	yes	characteristic, complemented, Hall
142	C₂²×C₃	—	5	6	no	—
143	C₂²×C₃	—	10	3	no	—
144	C₂²×C₃	—	10	3	no	—
145	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
146	C₃×C₄	—	2	1	no	—
147	C₃×C₄	—	2	3	no	—
148	C₃×C₄	—	5	2	no	—
149	C₃×C₄	—	10	1	no	—
150	C₃×C₄	—	10	3	no	—
151	C₃×C₄	C₂×C₄×D₅	1	6	yes	—
152	C₂×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
153	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
155	D₅	—	2	6	no	—
156	C₂³	—	5	3	no	—
157	Q₈	C₃×C₄×D₅	1	4	yes	complemented
158	D₄	—	5	12	no	—
159	C₂×C₄	—	5	1	no	—
160	C₂×C₄	—	5	12	no	—
161	C₂×C₄	—	5	1	no	—
162	C₂×C₄	—	5	3	no	—
163	C₂×C₄	—	20	6	no	—
164	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
165	C₂×C₄	C₂²×C₃×D₅	1	3	yes	—
166	C₂×C₄	C₂²×C₃×D₅	1	3	yes	—
167	C₂×C₃	—	10	6	no	—
168	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
169	C₂×C₃	#4147	1	1	yes	characteristic
170	C₂×C₃	#4149	1	1	yes	characteristic
171	C₅	C₃×#520	1	1	yes	characteristic, complemented, Sylow
172	C₂²	—	5	6	no	—
173	C₂²	—	10	3	no	—
174	C₂²	—	10	3	no	—
175	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	3	no	—
178	C₄	—	5	2	no	—
179	C₄	—	10	1	no	—
180	C₄	—	10	3	no	—
181	C₄	C₂×C₃×C₄×D₅	1	6	yes	—
182	C₃	#65908	1	1	yes	characteristic, direct product factor, Sylow
183	C₂	—	10	6	no	—
184	C₂	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
185	C₂	C₃×#4147	1	1	yes	characteristic
186	C₂	C₃×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator	C₂×C₅	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×#713	C₂
cosocle	C₂²×C₅	C₂⁴×C₃
hypercenter	C₃×C₄×Q₈	D₅
hypocenter	C₅	C₃×#520
LCS	G ▷ C₂×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
2. C₃×C₄×C₅×Q₈	C₂	semidirect
3. C₃×Q₈×Dic₅	C₂	semidirect
4. C₃×#4039	C₂	semidirect
5. C₃×D₅×C₄⋊C₄	C₂	semidirect
6. C₃×C₄×D₂₀	C₂	semidirect
7. C₃×#4018	C₂	semidirect
8. #65908	C₃	direct (N × H)
10. C₃×#713	C₄	semidirect
24. C₄×C₅×Q₈	C₂×C₃	semidirect
25. Q₈×Dic₅	C₂×C₃	semidirect
26. #4039	C₂×C₃	semidirect
27. D₅×C₄⋊C₄	C₂×C₃	semidirect
28. C₄×D₂₀	C₂×C₃	semidirect
29. #4018	C₂×C₃	semidirect
31. C₃×C₅×Q₈	C₂×C₄	semidirect
38. C₃×D₂₀	C₂×C₄	semidirect
40. C₃×C₄×D₅	C₂×C₄	semidirect
43. C₃×C₄×Q₈	D₅	semidirect
49. #713	C₃×C₄	semidirect
86. C₅×Q₈	C₂×C₃×C₄	semidirect
93. D₂₀	C₂×C₃×C₄	semidirect
95. C₄×D₅	C₂×C₃×C₄	semidirect
98. C₄×Q₈	C₃×D₅	semidirect
108. C₃×Q₈	C₄×D₅	semidirect
141. C₃×C₅	#520	semidirect
157. Q₈	C₃×C₄×D₅	semidirect
171. C₅	C₃×#520	semidirect
182. C₃	#65908	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×#713	2
C₃×C₄×C₅×Q₈	2
C₃×Q₈×Dic₅	2
C₃×#4039	2
C₃×D₅×C₄⋊C₄	2
C₃×C₄×D₂₀	2
C₃×#4018	2
#65908	3
C₃×#520	5

Subgroups — C3×#65908 #11755585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65908 #11755585