Subgroups — #14313953

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	#11751207	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3944	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745894	C₂	1	4	yes	maximal, complemented
5	#10602826	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	#91815	—	2	1	no	—
9	C₂×C₅×D₂₄	—	2	1	no	—
10	C₂×C₅×D₂₄	—	2	2	no	—
11	C₂×C₅×D₂₄	C₄	1	4	yes	complemented
12	C₂×C₅×D₂₄	C₂²	1	2	yes	—
13	#91102	C₂²	1	2	yes	complemented
14	#91101	—	2	1	no	—
15	C₃×#3944	—	2	2	no	—
16	C₂×C₃×#672	C₂²	1	2	yes	complemented
17	#71440	—	5	1	no	maximal, Hall
18	C₂²×C₅×D₈	—	3	1	no	—
19	#65395	—	3	1	no	—
20	C₂×#3944	D₃	1	1	yes	characteristic, complemented
21	#64782	—	3	4	no	—
22	C₂³×C₅×D₃	—	4	1	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×C₅×D₁₂	—	2	4	no	—
25	C₂×C₅×D₁₂	—	2	2	no	—
26	C₂×C₅×D₁₂	—	4	1	no	—
27	C₂×C₅×D₁₂	—	4	2	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
32	#6719	—	4	1	no	—
33	C₄⋊Dic₁₅	—	2	2	no	—
34	C₄×Dic₁₅	—	4	1	no	—
35	C₅×D₂₄	—	2	4	no	—
36	C₅×D₂₄	—	2	2	no	—
37	C₅×D₂₄	C₂×C₄	1	4	yes	—
38	C₃×#672	—	2	2	no	—
39	C₂²×D₂₄	D₅	1	1	yes	characteristic
40	C₂×C₃×M₃₂	—	5	1	no	—
41	#5106	—	5	1	no	—
42	#4509	—	5	4	no	—
43	C₂²×C₅×D₄	—	6	1	no	—
44	C₂×C₅×D₈	—	3	4	no	—
45	C₂×C₅×D₈	—	3	2	no	—
46	C₂×C₅×D₈	—	6	1	no	—
47	C₂×C₅×D₈	—	6	2	no	—
48	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	#4072	—	6	1	no	—
50	#3950	—	3	2	no	—
51	#3949	—	6	1	no	—
52	#3944	—	2	2	no	—
53	C₂×#672	C₂×D₃	1	2	yes	complemented
54	#2140	—	15	1	no	Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₂²×C₅×D₃	—	4	4	no	—
57	C₂²×C₅×D₃	—	4	2	no	—
58	C₂²×C₅×D₃	—	8	4	no	—
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₁₅	—	4	2	no	—
64	C₅×D₁₂	—	2	4	no	—
65	C₅×D₁₂	—	4	4	no	—
66	C₅×D₁₂	—	4	2	no	—
67	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×D₁₂	—	2	1	no	—
71	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
72	#926	—	10	1	no	—
73	C₂×D₂₄	—	2	2	no	—
74	C₂×D₂₄	—	2	1	no	—
75	C₂×D₂₄	Dic₅	1	4	yes	complemented
76	C₂×D₂₄	C₂×D₅	1	2	yes	—
77	C₃×M₃₂	—	10	2	no	—
78	C₂×C₃×C₁₆	—	5	2	no	—
79	#818	—	5	2	no	—
80	#817	—	10	1	no	—
81	C₂⁴×C₅	—	12	1	no	—
82	C₂×C₅×D₄	—	6	4	no	—
83	C₂×C₅×D₄	—	6	2	no	—
84	C₂×C₅×D₄	—	12	2	no	—
85	C₂×C₅×D₄	—	12	1	no	—
86	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
87	C₅×D₈	—	3	4	no	—
88	C₅×D₈	—	6	4	no	—
89	C₅×D₈	—	6	2	no	—
90	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
91	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
92	C₂×C₅×C₈	#67	1	2	yes	—
93	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
94	C₂²⋊Dic₅	—	12	1	no	—
95	C₄⋊Dic₅	—	6	2	no	—
96	C₄×Dic₅	—	12	1	no	—
97	#672	—	2	2	no	—
98	C₂²×D₈	—	3	1	no	—
99	C₂×M₃₂	—	5	1	no	—
100	#427	—	15	1	no	—
101	#357	—	15	4	no	—
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	#102	1	2	yes	—
104	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
105	C₂×C₅×D₃	—	4	4	no	—
106	C₂×C₅×D₃	—	8	8	no	—
107	C₂×C₅×D₃	—	8	4	no	—
108	C₃×C₄×C₅	#102	1	2	yes	—
109	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
111	Dic₁₅	—	8	2	no	—
112	C₂³×D₃	—	4	1	no	—
113	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
114	C₂×D₁₂	—	2	4	no	—
115	C₂×D₁₂	—	2	2	no	—
116	C₂×D₁₂	—	4	1	no	—
117	C₂×D₁₂	—	4	2	no	—
118	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
119	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
120	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
122	#217	—	20	1	no	—
123	C₄⋊Dic₃	—	10	2	no	—
124	C₄×Dic₃	—	20	1	no	—
125	D₂₄	—	2	4	no	—
126	D₂₄	—	2	2	no	—
127	D₂₄	C₂×Dic₅	1	4	yes	—
128	C₃×C₁₆	—	10	2	no	—
129	C₂³×C₅	—	12	4	no	—
130	C₂³×C₅	—	12	2	no	—
131	C₂³×C₅	—	24	4	no	—
132	C₂³×C₅	D₂₄	1	1	yes	characteristic
133	C₅×D₄	—	6	4	no	—
134	C₅×D₄	—	12	4	no	—
135	C₅×D₄	—	12	2	no	—
136	C₂×C₄×C₅	#204	1	2	yes	—
137	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
138	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
139	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
140	C₂×Dic₅	—	12	2	no	—
141	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
142	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
143	C₅×C₈	C₂×#67	1	1	yes	characteristic
144	C₂²×D₄	—	6	1	no	—
145	C₂×D₈	—	3	4	no	—
146	C₂×D₈	—	3	2	no	—
147	C₂×D₈	—	6	2	no	—
148	C₂×D₈	—	6	1	no	—
149	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
150	#117	—	30	1	no	—
151	M₃₂	—	10	2	no	—
152	C₂×C₁₆	—	5	2	no	—
153	#107	—	15	2	no	—
154	#106	—	30	1	no	—
155	C₂×C₃×C₅	—	2	2	no	—
156	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
157	C₂×C₃×C₅	#509	1	2	yes	—
158	C₅×D₃	—	8	4	no	—
159	C₂³×C₃	#686	1	1	yes	characteristic
160	C₂²×D₃	—	4	4	no	—
161	C₂²×D₃	—	4	2	no	—
162	C₂²×D₃	—	8	4	no	—
163	C₂×C₃×C₄	#686	1	1	yes	characteristic
164	C₂×C₃×C₄	#687	1	2	yes	—
165	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
167	C₂×Dic₃	—	20	2	no	—
168	D₁₂	—	2	4	no	—
169	D₁₂	—	4	4	no	—
170	D₁₂	—	4	2	no	—
171	C₃×C₈	C₂²⋊Dic₅	1	2	yes	—
172	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
173	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	12	4	no	—
176	C₂²×C₅	—	24	8	no	—
177	C₂²×C₅	—	24	4	no	—
178	C₂²×C₅	#821	1	2	yes	—
179	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
180	C₄×C₅	#821	1	2	yes	—
181	C₄×C₅	C₂×#204	1	1	yes	characteristic
182	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
183	Dic₅	—	24	2	no	—
184	C₂⁴	—	12	1	no	—
185	C₂×D₄	—	6	4	no	—
186	C₂×D₄	—	6	2	no	—
187	C₂×D₄	—	12	2	no	—
188	C₂×D₄	—	12	1	no	—
189	C₂²×C₄	#1203	1	1	yes	characteristic
190	D₈	—	3	4	no	—
191	D₈	—	6	4	no	—
192	D₈	—	6	2	no	—
193	C₂×C₈	D₃×Dic₅	1	2	yes	—
194	C₂×C₈	#1201	1	2	yes	—
195	C₂×C₈	#1203	1	1	yes	characteristic
196	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
197	C₄⋊C₄	—	30	2	no	—
198	C₂²⋊C₄	—	60	1	no	—
199	C₄²	—	60	1	no	—
200	C₁₆	—	10	2	no	—
201	C₃×C₅	#2140	1	1	yes	characteristic, complemented, Hall
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	#3964	1	2	yes	—
204	C₂²×C₃	C₂×#686	1	1	yes	characteristic
205	C₂×D₃	—	4	4	no	—
206	C₂×D₃	—	8	8	no	—
207	C₂×D₃	—	8	4	no	—
208	C₃×C₄	#3964	1	2	yes	—
209	C₃×C₄	C₂×#687	1	1	yes	characteristic
210	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
211	Dic₃	—	40	2	no	—
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	—	24	4	no	—
214	C₂×C₅	#4908	1	2	yes	—
215	C₂×C₅	C₂×#821	1	1	yes	characteristic
216	C₂³	—	12	4	no	—
217	C₂³	—	12	2	no	—
218	C₂³	—	24	4	no	—
219	C₂³	#6654	1	1	yes	characteristic
220	D₄	—	6	4	no	—
221	D₄	—	12	4	no	—
222	D₄	—	12	2	no	—
223	C₂×C₄	—	60	2	no	—
224	C₂×C₄	#6654	1	1	yes	characteristic
225	C₂×C₄	#6659	1	2	yes	—
226	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
227	C₂×C₄	C₂×#1203	1	1	yes	characteristic
228	C₈	(C₂×D₃)⋊Dic₅	1	2	yes	—
229	C₈	C₂×D₃×Dic₅	1	1	yes	characteristic
230	C₈	C₂×#1201	1	1	yes	characteristic
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	#65436	1	2	yes	—
233	C₂×C₃	C₂×#3964	1	1	yes	characteristic
234	D₃	—	8	4	no	—
235	C₅	#71440	1	1	yes	characteristic, complemented, Sylow
236	C₂²	—	2	2	no	—
237	C₂²	—	12	4	no	—
238	C₂²	—	24	8	no	—
239	C₂²	—	24	4	no	—
240	C₂²	#90967	1	2	yes	—
241	C₂²	C₂×#6654	1	1	yes	characteristic
242	C₄	—	120	2	no	—
243	C₄	#90967	1	2	yes	—
244	C₄	C₂×#6659	1	1	yes	characteristic
245	C₄	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
246	C₃	#10602826	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	2	2	no	—
248	C₂	—	24	4	no	—
249	C₂	#11747547	1	2	yes	—
250	C₂	C₂×#90967	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6654
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂²×C₅×D₂₄	C₂
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2140
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #11751207	C₂	semidirect
3. C₂×C₃×#3944	C₂	semidirect
4. #11745894	C₂	semidirect
11. C₂×C₅×D₂₄	C₄	semidirect
13. #91102	C₂²	semidirect
16. C₂×C₃×#672	C₂²	semidirect
20. C₂×#3944	D₃	semidirect
53. C₂×#672	C₂×D₃	semidirect
75. C₂×D₂₄	Dic₅	semidirect
201. C₃×C₅	#2140	semidirect
235. C₅	#71440	semidirect
246. C₃	#10602826	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₂₄	2
#11751207	2
C₂×C₃×#3944	2
#11745894	2
#10602826	3
#71440	5