Subgroups — #14418513

Order 1920.

728 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756229	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#65213	C₂	1	1	yes	maximal, characteristic
4	#11748436	C₂	1	2	yes	maximal
5	#11748326	C₂	1	2	yes	maximal
6	#11746865	C₂	1	4	yes	maximal, complemented
7	#11746855	C₂	1	4	yes	maximal, complemented
8	#10610176	—	3	1	no	maximal, Hall
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	#91815	C₂²	1	2	yes	—
11	#91813	C₄	1	4	yes	—
12	#91759	C₄	1	2	yes	—
13	#91758	C₄	1	2	yes	—
14	C₄×Dic₃₀	C₂²	1	4	yes	—
15	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×#681	C₂²	1	2	yes	—
18	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
19	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
20	C₃×C₄⋊#168	C₂²	1	4	yes	—
21	#90983	C₂²	1	8	yes	—
22	#90954	C₂²	1	4	yes	—
23	#90951	C₂²	1	4	yes	—
24	#79044	—	5	1	no	maximal, Hall
25	#65804	—	3	1	no	—
26	C₂×C₄×C₅×C₈	—	3	1	no	—
27	#65391	—	3	2	no	—
28	#65390	—	3	2	no	—
29	#65213	D₃	1	1	yes	characteristic
30	#64968	—	3	4	no	—
31	C₈×Dic₁₀	—	3	4	no	—
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×Dic₃₀	C₂×C₄	1	4	yes	—
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	C₂³	1	2	yes	—
38	#6719	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
41	#6712	C₂×C₄	1	8	yes	—
42	C₄×Dic₁₅	C₂³	1	4	yes	—
43	C₂×C₅×C₃⋊C₈	—	2	2	no	—
44	C₂×C₅×C₃⋊C₈	Q₈	1	4	yes	complemented
45	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
46	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
48	C₃×#681	—	2	4	no	—
49	C₂×C₃×#168	C₂³	1	4	yes	—
50	#5470	—	5	1	no	—
51	C₃×#423	—	5	1	no	—
52	#5122	—	5	2	no	—
53	#5119	—	5	2	no	—
54	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic
55	#4831	—	5	4	no	—
56	#4830	—	5	4	no	—
57	C₂²×C₅×C₈	—	3	1	no	—
58	C₂²×C₅×C₈	—	3	1	no	—
59	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
60	#4072	—	3	2	no	—
61	#4070	—	3	4	no	—
62	C₂×#681	C₂×D₃	1	2	yes	—
63	#4016	—	3	2	no	—
64	#4015	—	3	2	no	—
65	C₄×Dic₁₀	—	3	4	no	—
66	C₄×C₅×C₈	—	3	2	no	—
67	C₄×C₅×C₈	—	3	2	no	—
68	#3947	—	3	4	no	—
69	#3946	—	3	8	no	—
70	C₈×Dic₅	—	3	4	no	—
71	C₄⋊#168	C₂×D₃	1	4	yes	—
72	#2960	—	15	1	no	Sylow
73	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₅	C₂²×C₄	1	8	yes	—
82	Dic₃₀	—	2	8	no	—
83	C₃×#168	—	2	4	no	—
84	C₅×C₃⋊C₈	—	2	2	no	—
85	C₅×C₃⋊C₈	C₂×Q₈	1	2	yes	—
86	C₅×C₃⋊C₈	C₂×Q₈	1	2	yes	—
87	C₂×C₃×M₁₆	—	5	2	no	—
88	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
89	#926	—	5	2	no	—
90	#924	—	5	4	no	—
91	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
93	#870	—	5	2	no	—
94	#869	—	5	2	no	—
95	C₄×Dic₆	—	5	4	no	—
96	C₃×C₄⋊C₈	—	5	4	no	—
97	#815	—	5	4	no	—
98	Dic₃⋊C₈	—	5	8	no	—
99	C₈×Dic₃	—	5	4	no	—
100	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
101	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
102	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
103	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
104	C₂×Dic₁₀	—	3	4	no	—
105	C₂×C₅×C₈	—	3	4	no	—
106	C₂×C₅×C₈	—	3	1	no	—
107	C₂×C₅×C₈	—	3	1	no	—
108	C₂×C₅×C₈	—	3	2	no	—
109	C₂×C₅×C₈	—	6	2	no	—
110	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
111	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
112	C₄²×C₅	C₂²×D₃	1	2	yes	—
113	C₂²⋊Dic₅	—	3	4	no	—
114	C₄⋊Dic₅	—	3	4	no	—
115	C₄⋊Dic₅	—	3	4	no	—
116	#683	—	3	8	no	—
117	C₄×Dic₅	—	3	4	no	—
118	#681	—	2	4	no	—
119	C₂×#168	C₂²×D₃	1	4	yes	—
120	#533	—	15	1	no	—
121	#446	—	15	4	no	—
122	C₈×Q₈	—	15	4	no	—
123	#424	—	15	2	no	—
124	#423	—	5	1	no	—
125	#405	—	15	2	no	—
126	C₂×C₄×C₈	—	3	1	no	—
127	C₂²×C₃×C₅	—	2	2	no	—
128	C₂²×C₃×C₅	C₄×Q₈	1	2	yes	—
129	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
130	C₃×C₄×C₅	—	2	2	no	—
131	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
132	C₃×C₄×C₅	#131	1	4	yes	—
133	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
134	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
135	Dic₁₅	—	2	8	no	—
136	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
137	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
138	C₂×Dic₆	—	5	4	no	—
139	C₃×M₁₆	—	10	4	no	—
140	C₂×C₃×C₈	—	5	4	no	—
141	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
142	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
143	C₃×C₄²	C₂²×D₅	1	2	yes	—
144	#217	—	5	4	no	—
145	C₄⋊Dic₃	—	5	4	no	—
146	C₄⋊Dic₃	—	5	4	no	—
147	Dic₃⋊C₄	—	5	8	no	—
148	C₄×Dic₃	—	5	4	no	—
149	C₂×C₃⋊C₈	—	2	2	no	—
150	C₂×C₃⋊C₈	Dic₁₀	1	4	yes	complemented
151	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
152	C₂×C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
153	C₂×C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
154	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
157	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
159	C₂×C₄×C₅	#237	1	2	yes	—
160	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
161	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
162	C₂×Dic₅	—	3	8	no	—
163	Dic₁₀	—	6	8	no	—
164	C₅×C₈	—	3	2	no	—
165	C₅×C₈	—	3	2	no	—
166	C₅×C₈	—	6	2	no	—
167	#168	—	2	4	no	—
168	C₂×M₁₆	—	5	2	no	—
169	C₂²×C₈	—	3	1	no	—
170	C₂²×C₈	—	3	1	no	—
171	C₄⋊Q₈	—	15	2	no	—
172	#125	—	15	2	no	—
173	C₂²⋊Q₈	—	15	4	no	—
174	C₄×Q₈	—	15	4	no	—
175	#117	—	15	2	no	—
176	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
177	C₄⋊C₈	—	5	4	no	—
178	C₄⋊C₈	—	15	8	no	—
179	#97	—	15	4	no	—
180	C₄×C₈	—	3	2	no	—
181	C₄×C₈	—	3	2	no	—
182	C₄×C₈	—	15	4	no	—
183	C₂×C₃×C₅	—	2	2	no	—
184	C₂×C₃×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂×#131	1	2	yes	—
186	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
189	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
190	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
192	C₂×C₃×C₄	#709	1	2	yes	—
193	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
194	C₂×Dic₃	—	5	8	no	—
195	Dic₆	—	10	8	no	—
196	C₃×C₈	—	10	4	no	—
197	C₃⋊C₈	—	2	2	no	—
198	C₃⋊C₈	C₂×Dic₁₀	1	2	yes	—
199	C₃⋊C₈	C₂×Dic₁₀	1	2	yes	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	#887	1	2	yes	—
202	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	#887	1	2	yes	—
205	C₄×C₅	#907	1	4	yes	—
206	C₄×C₅	C₂×#237	1	1	yes	characteristic
207	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
208	Dic₅	—	6	8	no	—
209	C₂×Q₈	—	15	4	no	—
210	C₂²×C₄	#1200	1	2	yes	—
211	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
212	M₁₆	—	10	4	no	—
213	C₂×C₈	—	3	4	no	—
214	C₂×C₈	—	3	1	no	—
215	C₂×C₈	—	3	1	no	—
216	C₂×C₈	—	3	2	no	—
217	C₂×C₈	—	5	4	no	—
218	C₂×C₈	—	6	2	no	—
219	C₄⋊C₄	—	15	4	no	—
220	C₄⋊C₄	—	15	8	no	—
221	C₄⋊C₄	—	15	4	no	—
222	C₂²⋊C₄	—	15	4	no	—
223	C₄²	—	15	4	no	—
224	C₄²	#1200	1	1	yes	characteristic
225	C₄²	#1200	1	1	yes	characteristic
226	C₄²	C₂×D₃×D₅	1	2	yes	—
227	C₃×C₅	#2960	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	C₄×Dic₁₀	1	2	yes	—
230	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	C₄×Dic₁₀	1	2	yes	—
233	C₃×C₄	#4047	1	4	yes	—
234	C₃×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
235	C₃×C₄	C₂×#709	1	1	yes	characteristic
236	Dic₃	—	10	8	no	—
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	C₂×#887	1	1	yes	characteristic
239	C₂×C₅	C₂×#907	1	2	yes	—
240	C₂³	C₂×#1200	1	1	yes	characteristic
241	Q₈	—	30	8	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	15	8	no	—
244	C₂×C₄	#6765	1	2	yes	—
245	C₂×C₄	D₃×Dic₁₀	1	2	yes	—
246	C₂×C₄	C₂×#1200	1	2	yes	—
247	C₂×C₄	C₂×#1200	1	2	yes	—
248	C₂×C₄	C₂×#1200	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
250	C₈	—	3	2	no	—
251	C₈	—	3	2	no	—
252	C₈	—	6	2	no	—
253	C₈	—	10	4	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	C₂×C₄×Dic₁₀	1	1	yes	characteristic
256	C₂×C₃	C₂×#4047	1	2	yes	—
257	C₅	#79044	1	1	yes	characteristic, complemented, Sylow
258	C₂²	—	2	2	no	—
259	C₂²	#91340	1	2	yes	—
260	C₂²	C₂²×#1200	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	30	8	no	—
263	C₄	#91291	1	4	yes	—
264	C₄	#91340	1	2	yes	—
265	C₄	C₂×#6765	1	1	yes	characteristic
266	C₄	C₂×D₃×Dic₁₀	1	1	yes	characteristic
267	C₃	#10610176	1	1	yes	characteristic, complemented, Sylow
268	C₂	—	2	2	no	—
269	C₂	C₂×#91291	1	2	yes	—
270	C₂	C₂×#91340	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₂×C₃×C₄²×C₅	C₂²
prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#2960
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. #11746865	C₂	semidirect
7. #11746855	C₂	semidirect
44. C₂×C₅×C₃⋊C₈	Q₈	semidirect
150. C₂×C₃⋊C₈	Dic₁₀	semidirect
227. C₃×C₅	#2960	semidirect
257. C₅	#79044	semidirect
267. C₃	#10610176	semidirect

Maximal subgroups

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

subgroup	index
#11756229	2
C₂×C₄×C₅×C₃⋊C₈	2
C₃×#65213	2
#11748436	2
#11748326	2
#11746865	2
#11746855	2
#10610176	3
#79044	5