Subgroups — C₃×#65906 #11755583

Order 960.

532 conjugacy classes of subgroups.

The subgroup lattice has 184 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	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×#4038	C₂	1	3	yes	maximal
5	C₃×#4033	C₂	1	3	yes	maximal, complemented
6	C₃×#4018	C₂	1	3	yes	maximal
7	C₃×C₄×Dic₁₀	C₂	1	3	yes	maximal, complemented
8	#65906	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₃×Q₈×D₅	C₄	1	8	yes	complemented
11	C₂×C₃×C₄×D₅	C₂²	1	3	yes	—
12	C₂×C₃×Dic₁₀	C₂²	1	3	yes	—
13	C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
14	C₃×C₄²×C₅	C₂²	1	3	yes	—
15	C₃×#685	C₂²	1	1	yes	characteristic
16	C₃×#685	C₂²	1	3	yes	—
17	C₃×C₄⋊Dic₅	C₂²	1	3	yes	—
18	C₃×#683	C₂²	1	3	yes	—
19	C₃×#683	C₂²	1	3	yes	—
20	C₃×C₄×Dic₅	C₂²	1	3	yes	—
21	C₃×C₄×Dic₅	C₂²	1	6	yes	—
22	C₃×#519	—	5	1	no	maximal, Hall
23	C₂×Q₈×D₅	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	#4038	C₂×C₃	1	3	yes	—
27	#4033	C₂×C₃	1	3	yes	complemented
28	#4018	C₂×C₃	1	3	yes	—
29	C₄×Dic₁₀	C₂×C₃	1	3	yes	complemented
30	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
31	C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
33	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
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	3	yes	—
38	C₃×C₄×D₅	C₂×C₄	1	12	yes	—
39	C₃×Dic₁₀	C₂×C₄	1	12	yes	—
40	C₂²×C₃×Q₈	—	5	1	no	—
41	C₃×C₄×Q₈	—	5	1	no	—
42	C₃×C₄×Q₈	—	5	3	no	—
43	C₃×C₄×Q₈	—	5	3	no	—
44	C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
45	C₃×#117	—	5	3	no	—
46	C₃×#117	—	5	3	no	—
47	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
48	Q₈×D₅	C₃×C₄	1	8	yes	complemented
49	C₂×C₄×D₅	C₂²×C₃	1	3	yes	—
50	C₂×Dic₁₀	C₂²×C₃	1	3	yes	—
51	C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
52	C₄²×C₅	C₂²×C₃	1	3	yes	—
53	#685	C₂²×C₃	1	1	yes	characteristic
54	#685	C₂²×C₃	1	3	yes	—
55	C₄⋊Dic₅	C₂²×C₃	1	3	yes	—
56	#683	C₂²×C₃	1	3	yes	—
57	#683	C₂²×C₃	1	3	yes	—
58	C₄×Dic₅	C₂²×C₃	1	6	yes	—
59	C₄×Dic₅	C₂²×C₃	1	3	yes	—
60	#519	—	5	1	no	Sylow
61	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
62	C₂×C₃×D₅	—	2	1	no	—
63	C₂×C₃×D₅	—	2	1	no	—
64	C₂×C₃×D₅	C₂²×C₄	1	2	yes	—
65	C₃×C₄×C₅	—	2	1	no	—
66	C₃×C₄×C₅	—	2	3	no	—
67	C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
68	C₃×Dic₅	—	2	3	no	—
69	C₃×Dic₅	—	2	1	no	—
70	C₃×Dic₅	C₂²×C₄	1	6	yes	—
71	C₂×C₃×Q₈	—	5	8	no	—
72	C₂×C₃×Q₈	—	5	3	no	—
73	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
74	C₂²×C₃×C₄	—	5	3	no	—
75	C₃×C₄⋊C₄	—	5	3	no	—
76	C₃×C₄⋊C₄	—	5	3	no	—
77	C₃×C₄⋊C₄	—	5	3	no	—
78	C₃×C₄⋊C₄	C₂×D₅	1	3	yes	—
79	C₃×C₂²⋊C₄	—	5	3	no	—
80	C₃×C₂²⋊C₄	—	5	1	no	—
81	C₃×C₄²	—	5	6	no	—
82	C₃×C₄²	—	5	3	no	—
83	C₃×C₄²	C₂×D₅	1	3	yes	—
84	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
85	C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
86	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
87	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
89	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
90	C₂×Dic₅	C₂³×C₃	1	3	yes	—
91	C₂×Dic₅	C₂³×C₃	1	3	yes	—
92	C₄×D₅	C₂×C₃×C₄	1	12	yes	—
93	Dic₁₀	C₂×C₃×C₄	1	12	yes	—
94	C₂²×Q₈	—	5	1	no	—
95	C₄×Q₈	—	5	1	no	—
96	C₄×Q₈	—	5	3	no	—
97	C₄×Q₈	—	5	3	no	—
98	C₄×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
99	#117	—	5	3	no	—
100	#117	—	5	3	no	—
101	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
104	C₃×D₅	—	2	2	no	—
105	C₂³×C₃	—	5	1	no	—
106	C₃×Q₈	—	5	12	no	—
107	C₃×Q₈	C₄×D₅	1	4	yes	—
108	C₂×C₃×C₄	—	5	3	no	—
109	C₂×C₃×C₄	—	5	12	no	—
110	C₂×C₃×C₄	—	5	3	no	—
111	C₂×C₃×C₄	—	5	1	no	—
112	C₂×C₃×C₄	C₂²×D₅	1	3	yes	—
113	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂²×D₅	1	3	yes	—
115	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
116	C₂×D₅	—	2	1	no	—
117	C₂×D₅	—	2	1	no	—
118	C₂×D₅	C₂²×C₃×C₄	1	2	yes	—
119	C₄×C₅	—	2	1	no	—
120	C₄×C₅	—	2	3	no	—
121	C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
122	Dic₅	—	2	3	no	—
123	Dic₅	—	2	1	no	—
124	Dic₅	C₂²×C₃×C₄	1	6	yes	—
125	C₂×Q₈	—	5	8	no	—
126	C₂×Q₈	—	5	3	no	—
127	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
128	C₂²×C₄	—	5	3	no	—
129	C₄⋊C₄	—	5	3	no	—
130	C₄⋊C₄	—	5	3	no	—
131	C₄⋊C₄	—	5	3	no	—
132	C₄⋊C₄	C₂×C₃×D₅	1	3	yes	—
133	C₂²⋊C₄	—	5	3	no	—
134	C₂²⋊C₄	—	5	1	no	—
135	C₄²	—	5	6	no	—
136	C₄²	—	5	3	no	—
137	C₄²	C₂×C₃×D₅	1	3	yes	—
138	C₃×C₅	#519	1	1	yes	characteristic, complemented, Hall
139	C₂²×C₃	—	5	2	no	—
140	C₂²×C₃	—	10	1	no	—
141	C₂²×C₃	—	10	1	no	—
142	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
143	C₃×C₄	—	2	1	no	—
144	C₃×C₄	—	2	3	no	—
145	C₃×C₄	—	5	6	no	—
146	C₃×C₄	—	10	3	no	—
147	C₃×C₄	—	10	1	no	—
148	C₃×C₄	C₂×C₄×D₅	1	6	yes	—
149	C₂×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
150	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
152	D₅	—	2	2	no	—
153	C₂³	—	5	1	no	—
154	Q₈	—	5	12	no	—
155	Q₈	C₃×C₄×D₅	1	4	yes	—
156	C₂×C₄	—	5	3	no	—
157	C₂×C₄	—	5	12	no	—
158	C₂×C₄	—	5	3	no	—
159	C₂×C₄	—	5	1	no	—
160	C₂×C₄	C₂²×C₃×D₅	1	3	yes	—
161	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
162	C₂×C₄	C₂²×C₃×D₅	1	3	yes	—
163	C₂×C₃	—	10	2	no	—
164	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
165	C₂×C₃	#4147	1	1	yes	characteristic
166	C₂×C₃	#4150	1	1	yes	characteristic
167	C₅	C₃×#519	1	1	yes	characteristic, complemented, Sylow
168	C₂²	—	5	2	no	—
169	C₂²	—	10	1	no	—
170	C₂²	—	10	1	no	—
171	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	3	no	—
174	C₄	—	5	6	no	—
175	C₄	—	10	3	no	—
176	C₄	—	10	1	no	—
177	C₄	C₂×C₃×C₄×D₅	1	6	yes	—
178	C₃	#65906	1	1	yes	characteristic, direct product factor, Sylow
179	C₂	—	10	2	no	—
180	C₂	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
181	C₂	C₃×#4147	1	1	yes	characteristic
182	C₂	C₃×#4150	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	C₂²×C₃×D₅	C₂³
cosocle	C₂²×C₅	C₂⁴×C₃
hypercenter	C₃×C₄×Q₈	D₅
hypocenter	C₅	C₃×#519
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
5. C₃×#4033	C₂	semidirect
7. C₃×C₄×Dic₁₀	C₂	semidirect
8. #65906	C₃	direct (N × H)
10. C₃×Q₈×D₅	C₄	semidirect
24. C₄×C₅×Q₈	C₂×C₃	semidirect
25. Q₈×Dic₅	C₂×C₃	semidirect
27. #4033	C₂×C₃	semidirect
29. C₄×Dic₁₀	C₂×C₃	semidirect
44. C₃×C₄×Q₈	D₅	semidirect
48. Q₈×D₅	C₃×C₄	semidirect
98. C₄×Q₈	C₃×D₅	semidirect
138. C₃×C₅	#519	semidirect
167. C₅	C₃×#519	semidirect
178. C₃	#65906	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×C₄×C₅×Q₈	2
C₃×Q₈×Dic₅	2
C₃×#4038	2
C₃×#4033	2
C₃×#4018	2
C₃×C₄×Dic₁₀	2
#65906	3
C₃×#519	5

Subgroups — C3×#65906 #11755583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65906 #11755583