Subgroups — #14299066

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91101	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#11748834	C₂	1	2	yes	maximal
5	C₂×#90985	C₂	1	2	yes	maximal
6	#10601347	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
10	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
12	C₂×#6668	—	2	4	no	—
13	#91101	—	2	2	no	—
14	C₅×#817	—	2	2	no	—
15	C₃×C₈×Dic₅	C₄	1	4	yes	complemented
16	#90985	—	2	4	no	—
17	#69764	—	5	1	no	maximal, Hall
18	C₂×C₅×#106	—	3	1	no	—
19	C₂×#3949	—	3	1	no	—
20	C₂×C₈×Dic₅	D₃	1	1	yes	characteristic
21	#65265	—	3	2	no	—
22	C₂×#3940	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₅	—	2	2	no	—
25	C₂²×C₅×Dic₃	—	2	2	no	—
26	C₂²×C₃×Dic₅	D₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
30	C₄⋊Dic₁₅	—	2	2	no	—
31	C₄⋊Dic₁₅	—	2	2	no	—
32	C₅×C₄⋊Dic₃	—	2	2	no	—
33	C₅×C₄⋊Dic₃	—	2	2	no	—
34	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
35	C₃×C₄×Dic₅	Q₈	1	2	yes	—
36	C₂×C₃×#168	C₂×C₄	1	2	yes	—
37	C₂×C₃×#168	D₄	1	2	yes	—
38	C₂×C₃×#168	Q₈	1	2	yes	—
39	#6668	—	4	8	no	—
40	C₂×C₃×C₄×C₈	—	5	1	no	—
41	C₂×#817	—	5	1	no	—
42	C₂×#817	—	5	2	no	—
43	C₂×#817	D₅	1	1	yes	characteristic
44	#4935	—	5	2	no	—
45	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
46	C₂×C₅×C₄⋊C₄	—	3	2	no	—
47	C₂×C₄⋊Dic₅	—	3	2	no	—
48	C₂×#683	—	6	4	no	—
49	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
50	C₂²×#168	C₂×D₃	1	1	yes	characteristic
51	C₅×#106	—	6	2	no	—
52	#3949	—	6	2	no	—
53	C₈×Dic₅	Dic₃	1	4	yes	complemented
54	#3940	—	6	4	no	—
55	#1844	—	15	1	no	Sylow
56	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂×Dic₁₅	—	4	2	no	—
64	C₂×Dic₁₅	—	4	4	no	—
65	C₂×C₅×Dic₃	—	4	2	no	—
66	C₂×C₅×Dic₃	—	4	4	no	—
67	C₂×C₃×Dic₅	C₄⋊C₄	1	4	yes	—
68	C₂×C₃×Dic₅	SD₁₆	1	8	yes	—
69	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	complemented
70	C₃×#168	C₄⋊C₄	1	4	yes	complemented
71	C₂²×C₃×C₈	—	5	1	no	—
72	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
73	C₂×C₃×C₄²	—	5	1	no	—
74	C₂×C₄⋊Dic₃	—	5	2	no	—
75	C₂×C₄⋊Dic₃	—	10	4	no	—
76	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
77	C₃×C₄×C₈	—	5	4	no	—
78	#817	—	2	2	no	—
79	#817	—	10	4	no	—
80	#817	—	10	2	no	—
81	C₂²×C₄×C₅	—	6	2	no	—
82	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
83	C₂²×Dic₅	—	6	2	no	—
84	C₂²×Dic₅	D₁₂	1	2	yes	—
85	C₂×C₅×C₈	Dic₆	1	2	yes	—
86	C₂×C₅×C₈	D₁₂	1	2	yes	—
87	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
88	C₅×C₄⋊C₄	—	6	2	no	—
89	C₅×C₄⋊C₄	—	6	2	no	—
90	C₄⋊Dic₅	—	6	2	no	—
91	C₄⋊Dic₅	—	6	2	no	—
92	#683	—	12	8	no	—
93	C₄×Dic₅	Dic₆	1	2	yes	—
94	C₄×Dic₅	C₂×Dic₃	1	2	yes	—
95	C₂×#168	Dic₆	1	2	yes	—
96	C₂×#168	D₁₂	1	2	yes	—
97	C₂×#168	C₂×Dic₃	1	2	yes	—
98	C₂×#106	—	3	1	no	—
99	C₂×#106	—	15	2	no	—
100	C₂×#106	—	15	1	no	—
101	C₂×C₄×C₈	—	5	1	no	—
102	#384	—	15	2	no	—
103	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	#127	1	1	yes	characteristic
105	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
106	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
107	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
110	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
111	Dic₁₅	—	8	2	no	—
112	C₃×Dic₅	#106	1	8	yes	complemented
113	C₅×Dic₃	—	8	2	no	—
114	C₂²×C₃×C₄	—	5	2	no	—
115	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
116	C₂²×Dic₃	—	2	2	no	—
117	C₂²×Dic₃	—	10	2	no	—
118	C₂×C₃×C₈	—	5	2	no	—
119	C₂×C₃×C₈	—	5	2	no	—
120	C₂×C₃×C₈	—	5	2	no	—
121	C₂×C₃×C₈	Dic₁₀	1	2	yes	—
122	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
123	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
124	C₃×C₄²	—	5	2	no	—
125	C₃×C₄²	—	5	2	no	—
126	C₄⋊Dic₃	—	2	2	no	—
127	C₄⋊Dic₃	—	2	2	no	—
128	C₄⋊Dic₃	—	10	2	no	—
129	C₄⋊Dic₃	—	10	2	no	—
130	C₄⋊Dic₃	—	20	8	no	—
131	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
132	C₂×C₄×C₅	—	12	2	no	—
133	C₂×C₄×C₅	—	12	4	no	—
134	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
140	C₂×Dic₅	—	12	2	no	—
141	C₂×Dic₅	—	12	4	no	—
142	C₂×Dic₅	#204	1	8	yes	—
143	C₂×Dic₅	C₄⋊Dic₃	1	4	yes	—
144	C₅×C₈	C₄⋊Dic₃	1	4	yes	complemented
145	#168	C₄⋊Dic₃	1	4	yes	complemented
146	C₂²×C₈	—	5	1	no	—
147	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
148	C₂×C₄⋊C₄	—	3	2	no	—
149	C₂×C₄⋊C₄	—	15	2	no	—
150	C₂×C₄⋊C₄	—	30	4	no	—
151	C₂×C₄²	—	5	1	no	—
152	#106	—	6	2	no	—
153	#106	—	30	4	no	—
154	#106	—	30	2	no	—
155	C₄×C₈	—	5	4	no	—
156	C₂×C₃×C₅	#384	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×#106	1	2	yes	—
158	C₂×C₃×C₅	#492	1	2	yes	—
159	C₂×C₃×C₅	#498	1	2	yes	—
160	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
161	C₂×C₃×C₄	—	5	4	no	—
162	C₂×C₃×C₄	—	5	8	no	—
163	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
165	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
166	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
167	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
169	C₂×Dic₃	—	4	4	no	—
170	C₂×Dic₃	—	4	2	no	—
171	C₂×Dic₃	—	20	4	no	—
172	C₂×Dic₃	—	20	2	no	—
173	C₃×C₈	—	5	4	no	—
174	C₃×C₈	#683	1	4	yes	complemented
175	C₂²×C₅	#869	1	1	yes	characteristic
176	C₂²×C₅	#874	1	1	yes	characteristic
177	C₂²×C₅	C₂×#204	1	4	yes	—
178	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
179	C₄×C₅	—	24	2	no	—
180	C₄×C₅	#869	1	1	yes	characteristic
181	C₄×C₅	#869	1	1	yes	characteristic
182	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
183	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
184	Dic₅	—	24	2	no	—
185	Dic₅	#817	1	8	yes	complemented
186	C₂²×C₄	—	5	2	no	—
187	C₂²×C₄	—	6	2	no	—
188	C₂²×C₄	—	30	2	no	—
189	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
190	C₂×C₈	—	5	2	no	—
191	C₂×C₈	—	5	2	no	—
192	C₂×C₈	—	5	2	no	—
193	C₂×C₈	D₅×Dic₃	1	2	yes	—
194	C₂×C₈	#1203	1	2	yes	—
195	C₂×C₈	#1204	1	2	yes	—
196	C₄⋊C₄	—	6	2	no	—
197	C₄⋊C₄	—	6	2	no	—
198	C₄⋊C₄	—	30	2	no	—
199	C₄⋊C₄	—	30	2	no	—
200	C₄⋊C₄	—	60	8	no	—
201	C₄²	—	5	2	no	—
202	C₄²	—	5	2	no	—
203	C₃×C₅	#1844	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	#4034	1	1	yes	characteristic
205	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
206	C₂²×C₃	D₅×SD₁₆	1	4	yes	—
207	C₂²×C₃	#4086	1	1	yes	characteristic
208	C₃×C₄	—	5	8	no	—
209	C₃×C₄	#4034	1	1	yes	characteristic
210	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
211	C₃×C₄	C₂×#683	1	1	yes	characteristic
212	C₃×C₄	#4090	1	1	yes	characteristic
213	Dic₃	—	8	2	no	—
214	Dic₃	—	40	2	no	—
215	C₂×C₅	#4680	1	2	yes	—
216	C₂×C₅	#4693	1	2	yes	—
217	C₂×C₅	#4935	1	1	yes	characteristic
218	C₂×C₅	C₂×#817	1	2	yes	—
219	C₂³	D₅×D₁₂	1	1	yes	characteristic
220	C₂×C₄	—	5	4	no	—
221	C₂×C₄	—	5	8	no	—
222	C₂×C₄	—	12	2	no	—
223	C₂×C₄	—	12	4	no	—
224	C₂×C₄	—	60	2	no	—
225	C₂×C₄	—	60	4	no	—
226	C₂×C₄	D₅×Dic₆	1	1	yes	characteristic
227	C₂×C₄	D₅×Dic₆	1	1	yes	characteristic
228	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
229	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
230	C₂×C₄	C₂×#1203	1	1	yes	characteristic
231	C₂×C₄	C₂×#1204	1	1	yes	characteristic
232	C₈	—	5	4	no	—
233	C₈	#6668	1	4	yes	complemented
234	C₂×C₃	#65144	1	2	yes	—
235	C₂×C₃	D₅×#106	1	2	yes	—
236	C₂×C₃	#65265	1	1	yes	characteristic
237	C₂×C₃	#65464	1	2	yes	—
238	C₅	#69764	1	1	yes	characteristic, complemented, Sylow
239	C₂²	D₅×#204	1	4	yes	—
240	C₂²	D₅×C₄⋊Dic₃	1	1	yes	characteristic
241	C₂²	#91451	1	1	yes	characteristic
242	C₂²	#91468	1	1	yes	characteristic
243	C₄	—	5	8	no	—
244	C₄	—	24	2	no	—
245	C₄	—	120	2	no	—
246	C₄	#91376	1	1	yes	characteristic
247	C₄	D₅×C₄⋊Dic₃	1	1	yes	characteristic
248	C₄	#91468	1	1	yes	characteristic
249	C₄	C₂×#6668	1	1	yes	characteristic
250	C₃	#10601347	1	1	yes	characteristic, complemented, Sylow
251	C₂	D₅×#817	1	2	yes	—
252	C₂	#11747994	1	2	yes	—
253	C₂	#11748016	1	2	yes	—
254	C₂	#11748834	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₅×D₁₂
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#1844
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
15. C₃×C₈×Dic₅	C₄	semidirect
53. C₈×Dic₅	Dic₃	semidirect
69. C₃×C₅×C₈	C₄⋊C₄	semidirect
70. C₃×#168	C₄⋊C₄	semidirect
112. C₃×Dic₅	#106	semidirect
144. C₅×C₈	C₄⋊Dic₃	semidirect
145. #168	C₄⋊Dic₃	semidirect
174. C₃×C₈	#683	semidirect
185. Dic₅	#817	semidirect
203. C₃×C₅	#1844	semidirect
233. C₈	#6668	semidirect
238. C₅	#69764	semidirect
250. C₃	#10601347	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91101	2
C₂×C₅×#817	2
C₂×C₃×C₈×Dic₅	2
#11748834	2
C₂×#90985	2
#10601347	3
#69764	5