Subgroups — C₃×#10598765 #14276692

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×#170	C₂	1	2	yes	maximal
2	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#10598765	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×#170	C₄	1	4	yes	—
5	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×#168	C₄	1	1	yes	characteristic
8	C₃×C₄×#170	—	2	2	no	—
9	C₃×C₄×#170	—	2	2	no	—
10	C₃×C₈×Dic₅	—	2	2	no	—
11	C₃×#1273	—	5	1	no	maximal, Hall
12	C₂×C₄×#170	C₂×C₃	1	2	yes	—
13	C₂×C₈×Dic₅	C₂×C₃	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
16	C₂×C₃×#170	—	2	4	no	—
17	C₂×C₃×#170	—	2	4	no	—
18	C₂×C₃×#170	C₈	1	8	yes	complemented
19	C₂×C₃×C₅×C₈	—	2	1	no	—
20	C₂×C₃×C₅×C₈	—	2	1	no	—
21	C₂×C₃×C₅×C₈	C₈	1	2	yes	complemented
22	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
23	C₃×C₄×Dic₅	D₄	1	2	yes	—
24	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
25	C₂×C₃×#168	—	2	1	no	—
26	C₂×C₃×#168	—	2	1	no	—
27	C₂×C₃×#168	C₈	1	2	yes	complemented
28	C₂×C₃×C₄×C₈	—	5	2	no	—
29	C₂×C₃×C₄×C₈	—	5	1	no	—
30	C₂²×#170	C₃×C₄	1	4	yes	—
31	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
33	C₂²×#168	C₃×C₄	1	1	yes	characteristic
34	C₄×#170	—	2	2	no	—
35	C₄×#170	—	2	2	no	—
36	C₈×Dic₅	—	2	2	no	—
37	#1273	—	5	1	no	Sylow
38	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
45	C₂×C₃×Dic₅	C₂²⋊C₄	1	2	yes	—
46	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
47	C₂×C₃×Dic₅	C₂×C₈	1	4	yes	—
48	C₂×C₃×Dic₅	M₁₆	1	4	yes	—
49	C₃×#170	—	2	8	no	—
50	C₃×C₅×C₈	—	2	2	no	—
51	C₃×#168	—	2	2	no	—
52	C₂²×C₃×C₈	—	5	4	no	—
53	C₂²×C₃×C₈	—	5	1	no	—
54	C₂²×C₃×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
55	C₂×C₃×C₄²	—	5	1	no	—
56	C₃×C₄×C₈	—	10	2	no	—
57	C₃×C₄×C₈	—	10	2	no	—
58	C₃×C₄×C₈	—	10	2	no	—
59	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
60	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
61	C₂×#170	—	2	4	no	—
62	C₂×#170	—	2	4	no	—
63	C₂×#170	C₃×C₈	1	8	yes	complemented
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	C₃×C₈	1	2	yes	complemented
67	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
68	C₄×Dic₅	C₃×D₄	1	2	yes	—
69	C₄×Dic₅	C₃×Q₈	1	1	yes	characteristic
70	C₂×#168	—	2	1	no	—
71	C₂×#168	—	2	1	no	—
72	C₂×#168	C₃×C₈	1	2	yes	complemented
73	C₂×C₄×C₈	—	5	2	no	—
74	C₂×C₄×C₈	—	5	1	no	—
75	C₂²×C₃×C₅	#95	1	1	yes	characteristic
76	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
77	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
78	C₂²×C₃×C₅	#97	1	2	yes	—
79	C₂²×C₃×C₅	#97	1	1	yes	characteristic
80	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
81	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
82	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
83	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
84	C₃×Dic₅	C₂²⋊C₈	1	4	yes	—
85	C₃×Dic₅	C₄⋊C₈	1	4	yes	—
86	C₂²×C₃×C₄	—	5	2	no	—
87	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	5	8	no	—
91	C₂×C₃×C₈	—	5	2	no	—
92	C₂×C₃×C₈	—	10	4	no	—
93	C₂×C₃×C₈	—	10	4	no	—
94	C₂×C₃×C₈	—	10	1	no	—
95	C₂×C₃×C₈	—	10	1	no	—
96	C₂×C₃×C₈	#170	1	2	yes	complemented
97	C₃×C₄²	—	5	1	no	—
98	C₃×C₄²	—	5	1	no	—
99	C₃×C₄²	—	5	2	no	—
100	C₂³×C₅	C₃×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
107	C₂×Dic₅	C₃×C₂²⋊C₄	1	2	yes	—
108	C₂×Dic₅	C₃×C₄⋊C₄	1	2	yes	—
109	C₂×Dic₅	C₂×C₃×C₈	1	4	yes	—
110	C₂×Dic₅	C₃×M₁₆	1	4	yes	—
111	#170	—	2	8	no	—
112	C₅×C₈	—	2	2	no	—
113	#168	—	2	2	no	—
114	C₂²×C₈	—	5	4	no	—
115	C₂²×C₈	—	5	1	no	—
116	C₂²×C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂×C₄²	—	5	1	no	—
118	C₄×C₈	—	10	2	no	—
119	C₄×C₈	—	10	2	no	—
120	C₄×C₈	—	10	2	no	—
121	C₂×C₃×C₅	C₈²	1	1	yes	characteristic
122	C₂×C₃×C₅	#322	1	1	yes	characteristic
123	C₂×C₃×C₅	#322	1	2	yes	—
124	C₂×C₃×C₅	#336	1	2	yes	—
125	C₂×C₃×C₅	#336	1	1	yes	characteristic
126	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₂×C₃×C₄	—	5	4	no	—
128	C₂×C₃×C₄	—	5	4	no	—
129	C₂×C₃×C₄	—	5	2	no	—
130	C₂×C₃×C₄	—	5	2	no	—
131	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
132	C₂×C₃×C₄	#700	1	1	yes	characteristic
133	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
135	C₂×C₃×C₄	#704	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₃×C₈	—	2	2	no	—
138	C₃×C₈	—	10	8	no	—
139	C₃×C₈	—	10	2	no	—
140	C₂²×C₅	C₃×#95	1	1	yes	characteristic
141	C₂²×C₅	C₃×C₄×C₈	1	2	yes	—
142	C₂²×C₅	C₃×C₄×C₈	1	1	yes	characteristic
143	C₂²×C₅	C₃×#97	1	2	yes	—
144	C₂²×C₅	C₃×#97	1	1	yes	characteristic
145	C₄×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
146	C₄×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
147	C₄×C₅	C₃×C₄⋊C₈	1	1	yes	characteristic
148	C₄×C₅	C₃×C₄⋊C₈	1	1	yes	characteristic
149	Dic₅	C₃×C₂²⋊C₈	1	4	yes	—
150	Dic₅	C₃×C₄⋊C₈	1	4	yes	—
151	C₂²×C₄	—	5	2	no	—
152	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	5	8	no	—
156	C₂×C₈	—	5	2	no	—
157	C₂×C₈	—	10	4	no	—
158	C₂×C₈	—	10	4	no	—
159	C₂×C₈	—	10	1	no	—
160	C₂×C₈	—	10	1	no	—
161	C₂×C₈	C₃×#170	1	2	yes	complemented
162	C₄²	—	5	1	no	—
163	C₄²	—	5	1	no	—
164	C₄²	—	5	2	no	—
165	C₃×C₅	#1273	1	1	yes	characteristic, complemented, Hall
166	C₂²×C₃	C₈×AGL(1,𝔽₅)	1	2	yes	—
167	C₂²×C₃	#3992	1	2	yes	—
168	C₂²×C₃	C₄×#170	1	1	yes	characteristic
169	C₂²×C₃	#4002	1	1	yes	characteristic
170	C₂²×C₃	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
171	C₃×C₄	—	5	4	no	—
172	C₃×C₄	—	5	4	no	—
173	C₃×C₄	C₄⋊#170	1	1	yes	characteristic
174	C₃×C₄	#4003	1	1	yes	characteristic
175	C₃×C₄	#4004	1	1	yes	characteristic
176	C₃×C₄	C₂²⋊#170	1	1	yes	characteristic
177	C₂×C₅	C₃×C₈²	1	1	yes	characteristic
178	C₂×C₅	C₃×#322	1	1	yes	characteristic
179	C₂×C₅	C₃×#322	1	2	yes	—
180	C₂×C₅	C₃×#336	1	2	yes	—
181	C₂×C₅	C₃×#336	1	1	yes	characteristic
182	C₂³	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₂×C₄	—	5	4	no	—
184	C₂×C₄	—	5	4	no	—
185	C₂×C₄	—	5	2	no	—
186	C₂×C₄	—	5	2	no	—
187	C₂×C₄	C₃×D₅⋊C₈	1	1	yes	characteristic
188	C₂×C₄	C₃×#700	1	1	yes	characteristic
189	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₂×C₄	C₂×C₃×#170	1	1	yes	characteristic
191	C₂×C₄	C₃×#704	1	1	yes	characteristic
192	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	10	8	no	—
195	C₈	—	10	2	no	—
196	C₂×C₃	C₈×#170	1	1	yes	characteristic
197	C₂×C₃	#64879	1	1	yes	characteristic
198	C₂×C₃	#170⋊C₈	1	2	yes	—
199	C₂×C₃	(C₂×AGL(1,𝔽₅))⋊C₈	1	2	yes	—
200	C₂×C₃	#64918	1	1	yes	characteristic
201	C₅	C₃×#1273	1	1	yes	characteristic, complemented, Sylow
202	C₂²	C₃×C₈×AGL(1,𝔽₅)	1	2	yes	—
203	C₂²	C₃×#3992	1	2	yes	—
204	C₂²	C₃×C₄×#170	1	1	yes	characteristic
205	C₂²	C₃×#4002	1	1	yes	characteristic
206	C₂²	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
207	C₄	—	5	4	no	—
208	C₄	—	5	4	no	—
209	C₄	C₃×C₄⋊#170	1	1	yes	characteristic
210	C₄	C₃×#4003	1	1	yes	characteristic
211	C₄	C₃×#4004	1	1	yes	characteristic
212	C₄	C₃×C₂²⋊#170	1	1	yes	characteristic
213	C₃	#10598765	1	1	yes	characteristic, direct product factor, Sylow
214	C₂	C₃×C₈×#170	1	1	yes	characteristic
215	C₂	C₃×#64879	1	1	yes	characteristic
216	C₂	C₃×#170⋊C₈	1	2	yes	—
217	C₂	C₃×(C₂×AGL(1,𝔽₅))⋊C₈	1	2	yes	—
218	C₂	C₃×#64918	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₃×C₈²
Frattini	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)
Fitting	C₂²×C₃×C₅×C₈	C₄
socle, prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₂×C₄×Dic₅	C₂²×C₃
hypercenter	C₂²×C₃×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1273
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. #10598765	C₃	direct (N × H)
18. C₂×C₃×#170	C₈	semidirect
21. C₂×C₃×C₅×C₈	C₈	semidirect
27. C₂×C₃×#168	C₈	semidirect
63. C₂×#170	C₃×C₈	semidirect
66. C₂×C₅×C₈	C₃×C₈	semidirect
72. C₂×#168	C₃×C₈	semidirect
96. C₂×C₃×C₈	#170	semidirect
161. C₂×C₈	C₃×#170	semidirect
165. C₃×C₅	#1273	semidirect
201. C₅	C₃×#1273	semidirect
213. C₃	#10598765	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×C₄×#170	2
C₂×C₃×C₈×Dic₅	2
#10598765	3
C₃×#1273	5

Subgroups — C3×#10598765 #14276692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10598765 #14276692