Subgroups — #14299069

Order 1920.

560 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	#11751036	C₂	1	1	yes	maximal, characteristic
2	C₅×#4935	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
4	C₂×#90985	C₂	1	4	yes	maximal
5	#10600369	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₅	—	2	2	no	—
8	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
10	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂²×C₃×#168	C₂²	1	2	yes	—
12	C₃×C₄×#168	C₄	1	4	yes	complemented
13	#90985	—	2	8	no	—
14	#69764	—	5	1	no	maximal, Hall
15	C₅×#384	—	3	1	no	—
16	C₂×#3940	—	3	4	no	—
17	#65224	—	3	1	no	—
18	C₂×C₄×#168	D₃	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
20	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×Dic₁₅	—	2	2	no	—
22	C₂²×C₅×Dic₃	—	2	2	no	—
23	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
24	C₃×C₄²×C₅	Q₈	1	2	yes	—
25	C₄⋊Dic₁₅	—	2	4	no	—
26	C₄⋊Dic₁₅	—	4	4	no	—
27	C₅×C₄⋊Dic₃	—	2	4	no	—
28	C₅×C₄⋊Dic₃	—	4	4	no	—
29	C₂×C₃×#168	C₂×C₄	1	4	yes	—
30	C₂×C₃×#168	D₄	1	4	yes	—
31	C₂×C₃×#168	Q₈	1	4	yes	—
32	C₂×C₃×C₄×C₈	—	5	1	no	—
33	C₂×#817	—	5	4	no	—
34	#4935	—	5	1	no	—
35	#4935	D₅	1	1	yes	characteristic
36	C₂×C₅×C₄⋊C₄	—	3	2	no	—
37	C₂×C₅×C₄⋊C₄	—	6	2	no	—
38	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
39	C₂×C₄⋊Dic₅	—	3	2	no	—
40	C₂×C₄⋊Dic₅	—	6	2	no	—
41	C₂²×#168	C₂×D₃	1	2	yes	—
42	#3940	—	6	8	no	—
43	C₄×#168	Dic₃	1	4	yes	complemented
44	#1844	—	15	1	no	Sylow
45	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
47	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
48	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
49	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
53	C₂×Dic₁₅	—	4	2	no	—
54	C₂×Dic₁₅	—	4	4	no	—
55	C₂×C₅×Dic₃	—	4	2	no	—
56	C₂×C₅×Dic₃	—	4	4	no	—
57	C₃×#168	C₄⋊C₄	1	8	yes	complemented
58	C₂²×C₃×C₈	—	5	2	no	—
59	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
60	C₂×C₄⋊Dic₃	—	2	2	no	—
61	C₂×C₄⋊Dic₃	—	5	2	no	—
62	C₂×C₄⋊Dic₃	—	10	2	no	—
63	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
64	C₃×C₄×C₈	—	5	4	no	—
65	#817	—	10	8	no	—
66	C₂²×C₄×C₅	—	6	2	no	—
67	C₂²×C₄×C₅	D₁₂	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
69	C₂²×Dic₅	—	6	2	no	—
70	C₅×C₄⋊C₄	—	6	4	no	—
71	C₅×C₄⋊C₄	—	12	4	no	—
72	C₄²×C₅	Dic₆	1	2	yes	—
73	C₄²×C₅	C₂×Dic₃	1	2	yes	—
74	C₄⋊Dic₅	—	6	4	no	—
75	C₄⋊Dic₅	—	12	4	no	—
76	C₂×#168	Dic₆	1	4	yes	—
77	C₂×#168	D₁₂	1	4	yes	—
78	C₂×#168	C₂×Dic₃	1	4	yes	—
79	C₂×#106	—	15	4	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	#384	—	3	1	no	—
82	#384	—	15	1	no	—
83	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₃×C₅	#127	1	1	yes	characteristic
85	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
87	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
88	C₃×C₄×C₅	#106	1	8	yes	complemented
89	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
90	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
91	Dic₁₅	—	8	2	no	—
92	C₅×Dic₃	—	8	2	no	—
93	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
94	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
95	C₂²×Dic₃	—	2	2	no	—
96	C₂²×Dic₃	—	10	2	no	—
97	C₂×C₃×C₈	—	5	4	no	—
98	C₂×C₃×C₈	—	5	4	no	—
99	C₂×C₃×C₈	—	5	4	no	—
100	C₃×C₄²	Dic₁₀	1	2	yes	—
101	C₃×C₄²	C₄×D₅	1	2	yes	—
102	C₄⋊Dic₃	—	2	4	no	—
103	C₄⋊Dic₃	—	4	4	no	—
104	C₄⋊Dic₃	—	10	4	no	—
105	C₄⋊Dic₃	—	20	4	no	—
106	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
107	C₂×C₄×C₅	—	12	2	no	—
108	C₂×C₄×C₅	—	12	4	no	—
109	C₂×C₄×C₅	#204	1	4	yes	—
110	C₂×C₄×C₅	#204	1	4	yes	—
111	C₂×C₄×C₅	C₄⋊Dic₃	1	4	yes	—
112	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
113	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
115	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
116	C₂×Dic₅	—	12	2	no	—
117	C₂×Dic₅	—	12	4	no	—
118	#168	C₄⋊Dic₃	1	8	yes	complemented
119	C₂²×C₈	—	5	2	no	—
120	C₂×C₄⋊C₄	—	3	2	no	—
121	C₂×C₄⋊C₄	—	6	2	no	—
122	C₂×C₄⋊C₄	—	15	2	no	—
123	C₂×C₄⋊C₄	—	30	2	no	—
124	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
125	#106	—	30	8	no	—
126	C₄×C₈	—	5	4	no	—
127	C₂×C₃×C₅	#384	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂×#106	1	2	yes	—
129	C₂×C₃×C₅	#492	1	1	yes	characteristic
130	C₂×C₃×C₅	#492	1	1	yes	characteristic
131	C₂×C₃×C₅	#498	1	2	yes	—
132	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
133	C₂×C₃×C₄	#683	1	4	yes	—
134	C₂×C₃×C₄	#687	1	4	yes	—
135	C₂×C₃×C₄	#688	1	4	yes	—
136	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
138	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
139	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
140	C₂×Dic₃	—	4	4	no	—
141	C₂×Dic₃	—	4	2	no	—
142	C₂×Dic₃	—	20	2	no	—
143	C₂×Dic₃	—	20	4	no	—
144	C₃×C₈	—	5	8	no	—
145	C₂²×C₅	#869	1	1	yes	characteristic
146	C₂²×C₅	#874	1	1	yes	characteristic
147	C₂²×C₅	C₂×#204	1	2	yes	—
148	C₂²×C₅	C₂×#204	1	2	yes	—
149	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
150	C₄×C₅	—	24	2	no	—
151	C₄×C₅	#817	1	8	yes	complemented
152	C₄×C₅	#869	1	2	yes	—
153	C₄×C₅	C₂×C₄⋊Dic₃	1	2	yes	—
154	Dic₅	—	24	2	no	—
155	C₂²×C₄	—	6	2	no	—
156	C₂²×C₄	—	30	2	no	—
157	C₂²×C₄	#1203	1	2	yes	—
158	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
159	C₂×C₈	—	5	4	no	—
160	C₂×C₈	—	5	4	no	—
161	C₂×C₈	—	5	4	no	—
162	C₄⋊C₄	—	6	4	no	—
163	C₄⋊C₄	—	12	4	no	—
164	C₄⋊C₄	—	30	4	no	—
165	C₄⋊C₄	—	60	4	no	—
166	C₄²	D₅×Dic₃	1	2	yes	—
167	C₄²	#1204	1	2	yes	—
168	C₃×C₅	#1844	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	C₂×#683	1	1	yes	characteristic
170	C₂²×C₃	C₂×#687	1	2	yes	—
171	C₂²×C₃	#4086	1	1	yes	characteristic
172	C₂²×C₃	C₂×#688	1	2	yes	—
173	C₂²×C₃	#4090	1	1	yes	characteristic
174	C₃×C₄	#3940	1	8	yes	complemented
175	C₃×C₄	#4034	1	2	yes	—
176	C₃×C₄	D₅×C₄⋊C₄	1	2	yes	—
177	Dic₃	—	8	2	no	—
178	Dic₃	—	40	2	no	—
179	C₂×C₅	#4680	1	2	yes	—
180	C₂×C₅	#4693	1	1	yes	characteristic
181	C₂×C₅	#4693	1	1	yes	characteristic
182	C₂×C₅	#4935	1	1	yes	characteristic
183	C₂×C₅	C₂×#817	1	2	yes	—
184	C₂³	C₂×#1203	1	1	yes	characteristic
185	C₂×C₄	—	12	2	no	—
186	C₂×C₄	—	12	4	no	—
187	C₂×C₄	—	60	2	no	—
188	C₂×C₄	—	60	4	no	—
189	C₂×C₄	#6659	1	4	yes	—
190	C₂×C₄	#6660	1	4	yes	—
191	C₂×C₄	#6668	1	4	yes	—
192	C₂×C₄	D₅×Dic₆	1	2	yes	—
193	C₂×C₄	D₅×D₁₂	1	2	yes	—
194	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
195	C₂×C₄	C₂×#1204	1	1	yes	characteristic
196	C₈	—	5	8	no	—
197	C₂×C₃	C₂×#3940	1	2	yes	—
198	C₂×C₃	#65265	1	1	yes	characteristic
199	C₂×C₃	#65365	1	1	yes	characteristic
200	C₂×C₃	#65368	1	2	yes	—
201	C₂×C₃	#65374	1	1	yes	characteristic
202	C₅	#69764	1	1	yes	characteristic, complemented, Sylow
203	C₂²	C₂×#6659	1	2	yes	—
204	C₂²	C₂×#6660	1	2	yes	—
205	C₂²	#91376	1	1	yes	characteristic
206	C₂²	#91451	1	1	yes	characteristic
207	C₂²	C₂×#6668	1	1	yes	characteristic
208	C₄	—	24	2	no	—
209	C₄	—	120	2	no	—
210	C₄	#90985	1	8	yes	complemented
211	C₄	D₅×C₄⋊Dic₃	1	2	yes	—
212	C₄	#91468	1	2	yes	—
213	C₃	#10600369	1	1	yes	characteristic, complemented, Sylow
214	C₂	#11746941	1	2	yes	—
215	C₂	#11747066	1	1	yes	characteristic
216	C₂	#11747067	1	1	yes	characteristic
217	C₂	C₂×#90985	1	2	yes	—
218	C₂	#11748834	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1203
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1844
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
12. C₃×C₄×#168	C₄	semidirect
43. C₄×#168	Dic₃	semidirect
57. C₃×#168	C₄⋊C₄	semidirect
88. C₃×C₄×C₅	#106	semidirect
118. #168	C₄⋊Dic₃	semidirect
151. C₄×C₅	#817	semidirect
168. C₃×C₅	#1844	semidirect
174. C₃×C₄	#3940	semidirect
202. C₅	#69764	semidirect
210. C₄	#90985	semidirect
213. C₃	#10600369	semidirect

Maximal subgroups

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

subgroup	index
#11751036	2
C₅×#4935	2
C₂×C₃×C₄×#168	2
C₂×#90985	2
#10600369	3
#69764	5