Subgroups — C₃×#65417 #11750025

Order 960.

244 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×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#676	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#3952	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#3946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	#65417	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₂²⋊D₅	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×M₁₆	—	2	2	no	—
13	C₂×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₂²⋊Dic₅	C₄	1	1	yes	characteristic
16	C₃×#685	C₄	1	1	yes	characteristic
17	C₃×#683	C₄	1	1	yes	characteristic
18	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
19	C₂×C₃×#168	C₂²	1	1	yes	characteristic
20	C₂×C₃×#168	C₂²	1	1	yes	characteristic
21	C₃×#676	—	2	2	no	—
22	C₃×#436	—	5	1	no	maximal, Hall
23	C₂×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₄×C₂²⋊D₅	C₂×C₃	1	1	yes	characteristic
25	C₂×#676	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂²⋊#168	C₂×C₃	1	1	yes	characteristic, complemented
27	#3952	C₂×C₃	1	1	yes	characteristic, complemented
28	#3946	C₂×C₃	1	1	yes	characteristic, complemented
29	C₈×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂³×C₃×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×D₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₅	—	2	1	no	—
33	C₂×C₃×C₄×C₅	—	2	1	no	—
34	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₃×C₂²⋊D₅	—	2	2	no	—
37	C₂×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄×D₅	—	2	1	no	—
41	C₃×C₄×D₅	—	2	1	no	—
42	C₃×C₅×C₈	—	2	1	no	—
43	C₃×C₅×C₈	D₄	1	2	yes	complemented
44	C₃×#168	—	2	1	no	—
45	C₃×#168	—	2	1	no	—
46	C₂×C₃×M₁₆	—	5	1	no	—
47	C₂×C₃×M₁₆	D₅	1	1	yes	characteristic, complemented
48	C₃×C₄×D₄	—	5	1	no	—
49	C₃×C₄⋊C₈	—	5	1	no	—
50	C₃×C₂²⋊C₈	—	5	1	no	—
51	C₃×C₂²⋊C₈	—	5	1	no	—
52	C₃×C₄×C₈	—	5	1	no	—
53	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₂²⋊D₅	C₃×C₄	1	1	yes	characteristic
55	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
56	C₅×M₁₆	—	2	2	no	—
57	C₂×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂²⋊Dic₅	C₃×C₄	1	1	yes	characteristic
60	#685	C₃×C₄	1	1	yes	characteristic
61	#683	C₃×C₄	1	1	yes	characteristic
62	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
63	C₂×#168	C₂²×C₃	1	1	yes	characteristic
64	C₂×#168	C₂²×C₃	1	1	yes	characteristic
65	#676	—	2	2	no	—
66	#436	—	5	1	no	Sylow
67	C₂²×C₃×C₅	—	2	1	no	—
68	C₂²×C₃×C₅	—	2	1	no	—
69	C₂²×C₃×C₅	—	2	1	no	—
70	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×D₅	—	2	1	no	—
72	C₂×C₃×D₅	—	2	1	no	—
73	C₂×C₃×D₅	—	2	1	no	—
74	C₃×C₄×C₅	—	2	1	no	—
75	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄×C₅	#41	1	1	yes	characteristic
77	C₃×Dic₅	—	2	1	no	—
78	C₃×Dic₅	—	2	1	no	—
79	C₃×Dic₅	M₁₆	1	2	yes	complemented
80	C₂×C₃×D₄	—	5	1	no	—
81	C₂²×C₃×C₄	—	5	1	no	—
82	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
83	C₃×M₁₆	—	2	2	no	—
84	C₃×M₁₆	—	10	2	no	—
85	C₂×C₃×C₈	—	5	1	no	—
86	C₂×C₃×C₈	—	5	1	no	—
87	C₂×C₃×C₈	C₂×D₅	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂×D₅	1	1	yes	characteristic
89	C₃×C₄⋊C₄	—	5	1	no	—
90	C₃×C₂²⋊C₄	—	5	1	no	—
91	C₃×C₂²⋊C₄	—	5	1	no	—
92	C₃×C₄²	—	5	1	no	—
93	C₂³×C₅	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂²×D₅	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
99	C₂²⋊D₅	—	2	2	no	—
100	C₂×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
103	C₄×D₅	—	2	1	no	—
104	C₄×D₅	—	2	1	no	—
105	C₅×C₈	—	2	1	no	—
106	C₅×C₈	C₃×D₄	1	2	yes	complemented
107	#168	—	2	1	no	—
108	#168	—	2	1	no	—
109	C₂×M₁₆	—	5	1	no	—
110	C₂×M₁₆	C₃×D₅	1	1	yes	characteristic, complemented
111	C₄×D₄	—	5	1	no	—
112	C₄⋊C₈	—	5	1	no	—
113	C₂²⋊C₈	—	5	1	no	—
114	C₂²⋊C₈	—	5	1	no	—
115	C₄×C₈	—	5	1	no	—
116	C₂×C₃×C₅	—	4	1	no	—
117	C₂×C₃×C₅	C₄×D₄	1	1	yes	characteristic
118	C₂×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₃×C₅	#131	1	1	yes	characteristic
120	C₃×D₅	—	4	1	no	—
121	C₂³×C₃	—	5	1	no	—
122	C₂³×C₃	C₄×D₅	1	1	yes	characteristic
123	C₃×D₄	—	10	2	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	5	1	no	—
127	C₂×C₃×C₄	—	5	1	no	—
128	C₂×C₃×C₄	—	5	1	no	—
129	C₂×C₃×C₄	—	10	1	no	—
130	C₂×C₃×C₄	—	10	1	no	—
131	C₂×C₃×C₄	C₄×D₅	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
133	C₃×C₈	—	2	1	no	—
134	C₃×C₈	—	10	1	no	—
135	C₃×C₈	—	10	1	no	—
136	C₃×C₈	C₂²⋊D₅	1	2	yes	complemented
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	—	2	1	no	—
139	C₂²×C₅	—	2	1	no	—
140	C₂²×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₂×D₅	—	2	1	no	—
142	C₂×D₅	—	2	1	no	—
143	C₂×D₅	—	2	1	no	—
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
146	C₄×C₅	C₃×#41	1	1	yes	characteristic
147	Dic₅	—	2	1	no	—
148	Dic₅	—	2	1	no	—
149	Dic₅	C₃×M₁₆	1	2	yes	complemented
150	C₂×D₄	—	5	1	no	—
151	C₂²×C₄	—	5	1	no	—
152	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	10	2	no	—
155	C₂×C₈	—	5	1	no	—
156	C₂×C₈	—	5	1	no	—
157	C₂×C₈	C₂×C₃×D₅	1	1	yes	characteristic
158	C₂×C₈	C₂×C₃×D₅	1	1	yes	characteristic
159	C₄⋊C₄	—	5	1	no	—
160	C₂²⋊C₄	—	5	1	no	—
161	C₂²⋊C₄	—	5	1	no	—
162	C₄²	—	5	1	no	—
163	C₃×C₅	#436	1	1	yes	characteristic, complemented, Hall
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	10	1	no	—
168	C₂²×C₃	—	10	1	no	—
169	C₂²×C₃	—	10	1	no	—
170	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	5	2	no	—
173	C₃×C₄	—	10	1	no	—
174	C₃×C₄	—	10	1	no	—
175	C₃×C₄	#709	1	1	yes	characteristic
176	C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
177	C₂×C₅	—	4	1	no	—
178	C₂×C₅	C₃×C₄×D₄	1	1	yes	characteristic
179	C₂×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
180	C₂×C₅	C₃×#131	1	1	yes	characteristic
181	D₅	—	4	1	no	—
182	C₂³	—	5	1	no	—
183	C₂³	C₃×C₄×D₅	1	1	yes	characteristic
184	D₄	—	10	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	5	1	no	—
188	C₂×C₄	—	5	1	no	—
189	C₂×C₄	—	5	1	no	—
190	C₂×C₄	—	10	1	no	—
191	C₂×C₄	—	10	1	no	—
192	C₂×C₄	C₃×C₄×D₅	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	10	1	no	—
196	C₈	—	10	1	no	—
197	C₈	C₃×C₂²⋊D₅	1	2	yes	complemented
198	C₂×C₃	—	4	1	no	—
199	C₂×C₃	—	20	1	no	—
200	C₂×C₃	D₅×M₁₆	1	1	yes	characteristic
201	C₂×C₃	#4053	1	1	yes	characteristic
202	C₂×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
203	C₅	C₃×#436	1	1	yes	characteristic, complemented, Sylow
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	10	1	no	—
208	C₂²	—	10	1	no	—
209	C₂²	—	10	1	no	—
210	C₂²	C₂×C₃×C₄×D₅	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	5	2	no	—
213	C₄	—	10	1	no	—
214	C₄	—	10	1	no	—
215	C₄	C₃×#709	1	1	yes	characteristic
216	C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
217	C₃	#65417	1	1	yes	characteristic, direct product factor, Sylow
218	C₂	—	4	1	no	—
219	C₂	—	20	1	no	—
220	C₂	C₃×D₅×M₁₆	1	1	yes	characteristic
221	C₂	C₃×#4053	1	1	yes	characteristic
222	C₂	C₃×C₄×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂×C₃×C₂²⋊D₅	C₄
cosocle	C₂×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	D₅
hypocenter	C₅	C₃×#436
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×M₁₆	C₂	semidirect
3. C₂×C₃×#676	C₂	semidirect
4. C₃×C₂²⋊#168	C₂	semidirect
5. C₃×#3952	C₂	semidirect
6. C₃×#3946	C₂	semidirect
7. C₃×C₈×Dic₅	C₂	semidirect
8. #65417	C₃	direct (N × H)
23. C₂×C₅×M₁₆	C₂×C₃	semidirect
25. C₂×#676	C₂×C₃	semidirect
26. C₂²⋊#168	C₂×C₃	semidirect
27. #3952	C₂×C₃	semidirect
28. #3946	C₂×C₃	semidirect
29. C₈×Dic₅	C₂×C₃	semidirect
43. C₃×C₅×C₈	D₄	semidirect
47. C₂×C₃×M₁₆	D₅	semidirect
79. C₃×Dic₅	M₁₆	semidirect
106. C₅×C₈	C₃×D₄	semidirect
110. C₂×M₁₆	C₃×D₅	semidirect
136. C₃×C₈	C₂²⋊D₅	semidirect
149. Dic₅	C₃×M₁₆	semidirect
163. C₃×C₅	#436	semidirect
197. C₈	C₃×C₂²⋊D₅	semidirect
203. C₅	C₃×#436	semidirect
217. C₃	#65417	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×C₅×M₁₆	2
C₃×C₄×C₂²⋊D₅	2
C₂×C₃×#676	2
C₃×C₂²⋊#168	2
C₃×#3952	2
C₃×#3946	2
C₃×C₈×Dic₅	2
#65417	3
C₃×#436	5

Subgroups — C3×#65417 #11750025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65417 #11750025