Subgroups — C₃×D₈×#170 #14286080

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₄×#170	C₂	1	2	yes	maximal, complemented
2	C₃×D₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₃×D₄⋊#170	C₂	1	2	yes	maximal, complemented
4	C₃×#64882	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×#170	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×#170	—	2	2	no	—
8	C₂×C₃×C₅×D₈	C₄	1	1	yes	characteristic
9	C₃×D₄×Dic₅	C₂²	1	2	yes	—
10	C₃×C₂²⋊#170	—	2	2	no	—
11	C₃×C₄⋊#170	C₂²	1	2	yes	—
12	C₃×C₄×#170	C₂²	1	1	yes	characteristic
13	C₃×C₄×#170	C₂²	1	1	yes	characteristic
14	C₃×#3964	C₄	1	2	yes	—
15	C₃×#3950	C₄	1	1	yes	characteristic
16	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₃×C₈×D₈	—	5	1	no	maximal, Hall
18	D₄×#170	C₂×C₃	1	2	yes	complemented
19	D₈×Dic₅	C₂×C₃	1	1	yes	characteristic
20	D₄⋊#170	C₂×C₃	1	2	yes	complemented
21	#64882	C₂×C₃	1	1	yes	characteristic, complemented
22	C₈×#170	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×Dic₅	—	2	2	no	—
25	C₂×C₃×#170	—	2	1	no	—
26	C₂×C₃×#170	—	2	2	no	—
27	C₂×C₃×#170	—	4	4	no	—
28	C₂×C₃×#170	D₄	1	1	yes	characteristic
29	C₂×C₃×#170	D₄	1	1	yes	characteristic
30	C₃×C₅×D₈	C₈	1	4	yes	complemented
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₃×C₂²⋊Dic₅	—	2	2	no	—
33	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
34	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
35	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×D₈	—	5	1	no	—
37	C₃×C₈×D₄	—	5	2	no	—
38	C₃×#335	—	5	1	no	—
39	C₃×D₄⋊C₈	—	5	2	no	—
40	C₃×C₈²	—	5	1	no	—
41	C₂²×#170	—	2	2	no	—
42	C₂×C₅×D₈	C₃×C₄	1	1	yes	characteristic
43	D₄×Dic₅	C₂²×C₃	1	2	yes	—
44	C₂²⋊#170	—	2	2	no	—
45	C₄⋊#170	C₂²×C₃	1	2	yes	—
46	C₄×#170	C₂²×C₃	1	1	yes	characteristic
47	C₄×#170	C₂²×C₃	1	1	yes	characteristic
48	#3964	C₃×C₄	1	2	yes	—
49	#3950	C₃×C₄	1	1	yes	characteristic
50	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
51	C₈×D₈	—	5	1	no	Sylow
52	C₂³×C₃×C₅	—	2	2	no	—
53	C₃×C₅×D₄	—	2	2	no	—
54	C₃×C₅×D₄	C₂×C₈	1	4	yes	complemented
55	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃×Dic₅	—	2	2	no	—
57	C₂×C₃×Dic₅	—	4	2	no	—
58	C₂×C₃×Dic₅	—	4	2	no	—
59	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
61	C₃×#170	—	2	2	no	—
62	C₃×#170	—	2	1	no	—
63	C₃×#170	—	4	2	no	—
64	C₃×#170	D₈	1	2	yes	direct product factor
65	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
66	C₃×#168	—	2	1	no	—
67	C₂×C₃×D₈	AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₂²×C₃×C₈	—	10	2	no	—
69	C₃×C₄×D₄	—	5	2	no	—
70	C₃×#107	—	5	1	no	—
71	C₃×C₄⋊C₈	—	5	2	no	—
72	C₃×#102	—	5	2	no	—
73	C₃×C₂²⋊C₈	—	10	2	no	—
74	C₃×C₄×C₈	—	5	1	no	—
75	C₃×C₄×C₈	—	5	1	no	—
76	C₃×C₄×C₈	—	5	1	no	—
77	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
78	C₂²×Dic₅	—	2	2	no	—
79	C₂×#170	—	2	1	no	—
80	C₂×#170	—	2	2	no	—
81	C₂×#170	—	4	4	no	—
82	C₂×#170	C₃×D₄	1	1	yes	characteristic
83	C₂×#170	C₃×D₄	1	1	yes	characteristic
84	C₅×D₈	C₃×C₈	1	4	yes	complemented
85	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂²⋊Dic₅	—	2	2	no	—
87	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
88	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
89	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
90	C₄×D₈	—	5	1	no	—
91	C₈×D₄	—	5	2	no	—
92	#335	—	5	1	no	—
93	D₄⋊C₈	—	5	2	no	—
94	C₈²	—	5	1	no	—
95	C₂²×C₃×C₅	—	2	4	no	—
96	C₂²×C₃×C₅	—	4	2	no	—
97	C₂²×C₃×C₅	—	4	2	no	—
98	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
99	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
100	C₃×C₄×C₅	#131	1	1	yes	characteristic
101	C₃×Dic₅	—	2	1	no	—
102	C₃×Dic₅	—	4	2	no	—
103	C₃×Dic₅	C₂×D₈	1	1	yes	characteristic
104	C₃×Dic₅	#135	1	1	yes	characteristic
105	C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
106	C₂²×C₃×C₄	—	10	2	no	—
107	C₃×D₈	#170	1	4	yes	direct product factor
108	C₂×C₃×C₈	—	5	1	no	—
109	C₂×C₃×C₈	—	5	1	no	—
110	C₂×C₃×C₈	—	5	1	no	—
111	C₂×C₃×C₈	—	10	1	no	—
112	C₂×C₃×C₈	—	10	2	no	—
113	C₂×C₃×C₈	—	20	4	no	—
114	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₃×C₄⋊C₄	—	5	2	no	—
116	C₃×C₂²⋊C₄	—	10	2	no	—
117	C₃×C₄²	—	5	1	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₅×D₄	—	2	2	no	—
120	C₅×D₄	C₂×C₃×C₈	1	4	yes	complemented
121	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₂×Dic₅	—	2	2	no	—
123	C₂×Dic₅	—	4	2	no	—
124	C₂×Dic₅	—	4	2	no	—
125	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
126	C₂×Dic₅	C₃×#41	1	1	yes	characteristic
127	#170	—	2	2	no	—
128	#170	—	2	1	no	—
129	#170	—	4	2	no	—
130	#170	C₃×D₈	1	2	yes	direct product factor
131	C₅×C₈	C₂×C₃×C₈	1	2	yes	complemented
132	#168	—	2	1	no	—
133	C₂×D₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₂²×C₈	—	10	2	no	—
135	C₄×D₄	—	5	2	no	—
136	#107	—	5	1	no	—
137	C₄⋊C₈	—	5	2	no	—
138	#102	—	5	2	no	—
139	C₂²⋊C₈	—	10	2	no	—
140	C₄×C₈	—	5	1	no	—
141	C₄×C₈	—	5	1	no	—
142	C₄×C₈	—	5	1	no	—
143	C₂×C₃×C₅	—	4	4	no	—
144	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
145	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
146	C₂×C₃×C₅	#443	1	1	yes	characteristic
147	C₂³×C₃	—	2	2	no	—
148	C₃×D₄	—	2	2	no	—
149	C₃×D₄	C₂×#170	1	4	yes	complemented
150	C₂×C₃×C₄	—	5	1	no	—
151	C₂×C₃×C₄	—	5	1	no	—
152	C₂×C₃×C₄	—	10	2	no	—
153	C₂×C₃×C₄	—	20	2	no	—
154	C₂×C₃×C₄	—	20	2	no	—
155	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₃×C₈	—	5	2	no	—
157	C₃×C₈	—	10	1	no	—
158	C₃×C₈	—	10	2	no	—
159	C₃×C₈	—	10	1	no	—
160	C₃×C₈	—	20	2	no	—
161	C₃×C₈	C₂×#170	1	2	yes	complemented
162	C₂²×C₅	—	2	4	no	—
163	C₂²×C₅	—	4	2	no	—
164	C₂²×C₅	—	4	2	no	—
165	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
166	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
167	C₄×C₅	C₃×#131	1	1	yes	characteristic
168	Dic₅	—	2	1	no	—
169	Dic₅	—	4	2	no	—
170	Dic₅	C₂×C₃×D₈	1	1	yes	characteristic
171	Dic₅	C₃×#135	1	1	yes	characteristic
172	C₂×D₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
173	C₂²×C₄	—	10	2	no	—
174	D₈	C₃×#170	1	4	yes	direct product factor
175	C₂×C₈	—	5	1	no	—
176	C₂×C₈	—	5	1	no	—
177	C₂×C₈	—	5	1	no	—
178	C₂×C₈	—	10	1	no	—
179	C₂×C₈	—	10	2	no	—
180	C₂×C₈	—	20	4	no	—
181	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₄⋊C₄	—	5	2	no	—
183	C₂²⋊C₄	—	10	2	no	—
184	C₄²	—	5	1	no	—
185	C₃×C₅	C₈×D₈	1	1	yes	characteristic, complemented, Hall
186	C₂²×C₃	—	2	4	no	—
187	C₂²×C₃	—	4	2	no	—
188	C₂²×C₃	—	4	2	no	—
189	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₃×C₄	—	5	1	no	—
191	C₃×C₄	—	5	1	no	—
192	C₃×C₄	—	10	1	no	—
193	C₃×C₄	—	20	2	no	—
194	C₃×C₄	#4131	1	1	yes	characteristic
195	C₃×C₄	C₂²×#170	1	1	yes	characteristic
196	C₂×C₅	—	4	4	no	—
197	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
198	C₂×C₅	C₃×C₄×D₈	1	1	yes	characteristic
199	C₂×C₅	C₃×#443	1	1	yes	characteristic
200	C₂³	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	C₂×C₃×#170	1	4	yes	complemented
203	C₂×C₄	—	5	1	no	—
204	C₂×C₄	—	5	1	no	—
205	C₂×C₄	—	10	2	no	—
206	C₂×C₄	—	20	2	no	—
207	C₂×C₄	—	20	2	no	—
208	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₈	—	5	2	no	—
210	C₈	—	10	1	no	—
211	C₈	—	10	2	no	—
212	C₈	—	10	1	no	—
213	C₈	—	20	2	no	—
214	C₈	C₂×C₃×#170	1	2	yes	complemented
215	C₂×C₃	—	4	4	no	—
216	C₂×C₃	D₈×AGL(1,𝔽₅)	1	1	yes	characteristic
217	C₂×C₃	#65732	1	1	yes	characteristic
218	C₂×C₃	D₄×#170	1	1	yes	characteristic
219	C₅	C₃×C₈×D₈	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	4	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
224	C₄	—	5	1	no	—
225	C₄	—	5	1	no	—
226	C₄	—	10	1	no	—
227	C₄	—	20	2	no	—
228	C₄	C₃×#4131	1	1	yes	characteristic
229	C₄	C₂²×C₃×#170	1	1	yes	characteristic
230	C₃	D₈×#170	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	—	4	4	no	—
232	C₂	C₃×D₈×AGL(1,𝔽₅)	1	1	yes	characteristic
233	C₂	C₃×#65732	1	1	yes	characteristic
234	C₂	C₃×D₄×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting, prime radical	C₂×C₃×C₅×D₈	C₄
socle	C₂²×C₃×C₅	C₄×D₄
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₂×C₃×D₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₈×D₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×D₈

Complements

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

N	H ≅ G/N	type
1. C₃×D₄×#170	C₂	semidirect
3. C₃×D₄⋊#170	C₂	semidirect
4. C₃×#64882	C₂	semidirect
5. C₃×C₈×#170	C₂	semidirect
6. D₈×#170	C₃	direct (N × H)
18. D₄×#170	C₂×C₃	semidirect
20. D₄⋊#170	C₂×C₃	semidirect
21. #64882	C₂×C₃	semidirect
22. C₈×#170	C₂×C₃	semidirect
30. C₃×C₅×D₈	C₈	semidirect
54. C₃×C₅×D₄	C₂×C₈	semidirect
64. C₃×#170	D₈	direct (N × H)
65. C₃×C₅×C₈	C₂×C₈	semidirect
84. C₅×D₈	C₃×C₈	semidirect
107. C₃×D₈	#170	direct (N × H)
120. C₅×D₄	C₂×C₃×C₈	semidirect
130. #170	C₃×D₈	direct (N × H)
131. C₅×C₈	C₂×C₃×C₈	semidirect
149. C₃×D₄	C₂×#170	semidirect
161. C₃×C₈	C₂×#170	semidirect
174. D₈	C₃×#170	direct (N × H)
185. C₃×C₅	C₈×D₈	semidirect
202. D₄	C₂×C₃×#170	semidirect
214. C₈	C₂×C₃×#170	semidirect
219. C₅	C₃×C₈×D₈	semidirect
230. C₃	D₈×#170	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×D₄×#170	2
C₃×D₈×Dic₅	2
C₃×D₄⋊#170	2
C₃×#64882	2
C₃×C₈×#170	2
D₈×#170	3
C₃×C₈×D₈	5

Subgroups — C3×D8×#170 #14286080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₈×#170 #14286080