Subgroups — #14313950

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×C₃×D₄₀	C₂	1	1	yes	maximal, characteristic
2	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5106	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745891	C₂	1	4	yes	maximal, complemented
5	#10601445	—	3	1	no	maximal, Hall
6	C₂²×C₃×D₂₀	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#926	—	2	1	no	—
9	C₂×C₃×D₄₀	—	2	1	no	—
10	C₂×C₃×D₄₀	—	2	2	no	—
11	C₂×C₃×D₄₀	C₄	1	4	yes	complemented
12	C₂×C₃×D₄₀	C₂²	1	2	yes	—
13	#91096	—	2	2	no	—
14	C₂×C₃⋊#672	C₂²	1	2	yes	complemented
15	C₅×#818	C₂²	1	2	yes	complemented
16	C₅×#817	—	2	1	no	—
17	#72033	—	5	1	no	maximal, Hall
18	C₂²×D₄₀	D₃	1	1	yes	characteristic
19	C₅×#427	—	3	1	no	—
20	C₂×#3944	—	3	1	no	—
21	#64711	—	3	4	no	—
22	C₂³×C₃×D₅	—	4	1	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×C₃×D₂₀	—	2	4	no	—
25	C₂×C₃×D₂₀	—	2	2	no	—
26	C₂×C₃×D₂₀	—	4	1	no	—
27	C₂×C₃×D₂₀	—	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	C₅×#217	—	4	1	no	—
33	C₅×C₄⋊Dic₃	—	2	2	no	—
34	C₄×C₅×Dic₃	—	4	1	no	—
35	C₃×D₄₀	—	2	4	no	—
36	C₃×D₄₀	—	2	2	no	—
37	C₃×D₄₀	C₂×C₄	1	4	yes	—
38	C₃⋊#672	—	2	2	no	—
39	C₂²×C₃×D₈	—	5	1	no	—
40	#5106	D₅	1	1	yes	characteristic, complemented
41	C₂×#812	—	5	1	no	—
42	#4562	—	5	4	no	—
43	C₂²×D₂₀	—	2	1	no	—
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₅×#117	—	6	1	no	—
46	C₂×D₄₀	—	2	1	no	—
47	C₂×D₄₀	—	2	2	no	—
48	C₂×D₄₀	Dic₃	1	4	yes	complemented
49	C₂×D₄₀	C₂×D₃	1	2	yes	—
50	C₅×#107	—	3	2	no	—
51	C₅×#106	—	6	1	no	—
52	#3944	—	6	2	no	—
53	C₂×#672	—	3	2	no	—
54	#2140	—	15	1	no	Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₂²×C₃×D₅	—	4	4	no	—
57	C₂²×C₃×D₅	—	4	2	no	—
58	C₂²×C₃×D₅	—	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×C₅×Dic₃	—	4	2	no	—
64	C₃×D₂₀	—	2	4	no	—
65	C₃×D₂₀	—	4	4	no	—
66	C₃×D₂₀	—	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×C₃×D₄	—	10	1	no	—
71	C₂×C₃×D₈	—	5	4	no	—
72	C₂×C₃×D₈	—	5	2	no	—
73	C₂×C₃×D₈	—	10	2	no	—
74	C₂×C₃×D₈	—	10	1	no	—
75	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
76	#926	—	2	1	no	—
77	#818	C₂×D₅	1	2	yes	complemented
78	#817	—	2	1	no	—
79	#812	—	10	2	no	—
80	C₂×C₃⋊C₁₆	—	5	2	no	—
81	C₂³×D₅	—	4	1	no	—
82	C₂²×C₄×C₅	#67	1	1	yes	characteristic
83	C₂×D₂₀	—	2	4	no	—
84	C₂×D₂₀	—	2	2	no	—
85	C₂×D₂₀	—	4	1	no	—
86	C₂×D₂₀	—	4	2	no	—
87	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
88	C₂×C₅×C₈	#67	1	1	yes	characteristic
89	C₂×C₅×C₈	#67	1	2	yes	—
90	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
91	C₅×C₄⋊C₄	—	6	2	no	—
92	C₅×C₂²⋊C₄	—	12	1	no	—
93	C₄²×C₅	—	12	1	no	—
94	D₄₀	—	2	4	no	—
95	D₄₀	—	2	2	no	—
96	D₄₀	C₂×Dic₃	1	4	yes	—
97	#672	—	6	2	no	—
98	C₂²×D₈	—	5	1	no	—
99	C₂×M₃₂	—	15	1	no	—
100	#427	—	3	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×C₃×D₅	—	4	4	no	—
106	C₂×C₃×D₅	—	8	8	no	—
107	C₂×C₃×D₅	—	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	C₅×Dic₃	—	8	2	no	—
112	C₂⁴×C₃	—	20	1	no	—
113	C₂×C₃×D₄	—	10	4	no	—
114	C₂×C₃×D₄	—	10	2	no	—
115	C₂×C₃×D₄	—	20	1	no	—
116	C₂×C₃×D₄	—	20	2	no	—
117	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
118	C₃×D₈	—	5	4	no	—
119	C₃×D₈	—	10	4	no	—
120	C₃×D₈	—	10	2	no	—
121	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
122	C₂×C₃×C₈	D₂₀	1	2	yes	—
123	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
125	#217	—	4	1	no	—
126	C₄⋊Dic₃	—	2	2	no	—
127	C₄×Dic₃	—	4	1	no	—
128	C₃⋊C₁₆	—	10	2	no	—
129	C₂³×C₅	#213	1	1	yes	characteristic
130	C₂²×D₅	—	4	4	no	—
131	C₂²×D₅	—	4	2	no	—
132	C₂²×D₅	—	8	4	no	—
133	C₂×C₄×C₅	—	12	2	no	—
134	C₂×C₄×C₅	#213	1	1	yes	characteristic
135	C₂×C₄×C₅	#214	1	2	yes	—
136	C₂×C₄×C₅	#217	1	2	yes	—
137	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
138	D₂₀	—	2	4	no	—
139	D₂₀	—	4	4	no	—
140	D₂₀	—	4	2	no	—
141	C₅×C₈	#217	1	2	yes	—
142	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
143	C₅×C₈	C₂×#67	1	1	yes	characteristic
144	C₂²×D₄	—	10	1	no	—
145	C₂×D₈	—	5	4	no	—
146	C₂×D₈	—	5	2	no	—
147	C₂×D₈	—	10	1	no	—
148	C₂×D₈	—	10	2	no	—
149	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
150	#117	—	6	1	no	—
151	M₃₂	—	30	2	no	—
152	C₂×C₁₆	—	15	2	no	—
153	#107	—	3	2	no	—
154	#106	—	6	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	C₃×D₅	—	8	4	no	—
159	C₂³×C₃	—	20	4	no	—
160	C₂³×C₃	—	20	2	no	—
161	C₂³×C₃	—	40	4	no	—
162	C₂³×C₃	#686	1	1	yes	characteristic
163	C₃×D₄	—	10	4	no	—
164	C₃×D₄	—	20	4	no	—
165	C₃×D₄	—	20	2	no	—
166	C₂×C₃×C₄	#685	1	2	yes	—
167	C₂×C₃×C₄	#686	1	1	yes	characteristic
168	C₂×C₃×C₄	#688	1	2	yes	—
169	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
170	C₂×Dic₃	—	4	2	no	—
171	C₃×C₈	#685	1	2	yes	—
172	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
173	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	#832	1	2	yes	—
176	C₂²×C₅	C₂×#213	1	1	yes	characteristic
177	C₂×D₅	—	4	4	no	—
178	C₂×D₅	—	8	8	no	—
179	C₂×D₅	—	8	4	no	—
180	C₄×C₅	—	24	2	no	—
181	C₄×C₅	#832	1	2	yes	—
182	C₄×C₅	C₂×#214	1	1	yes	characteristic
183	C₄×C₅	C₂×#217	1	1	yes	characteristic
184	C₂⁴	—	20	1	no	—
185	C₂×D₄	—	10	4	no	—
186	C₂×D₄	—	10	2	no	—
187	C₂×D₄	—	20	1	no	—
188	C₂×D₄	—	20	2	no	—
189	C₂²×C₄	#1201	1	1	yes	characteristic
190	D₈	—	5	4	no	—
191	D₈	—	10	4	no	—
192	D₈	—	10	2	no	—
193	C₂×C₈	D₅×Dic₃	1	2	yes	—
194	C₂×C₈	#1201	1	1	yes	characteristic
195	C₂×C₈	#1202	1	2	yes	—
196	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
197	C₄⋊C₄	—	6	2	no	—
198	C₂²⋊C₄	—	12	1	no	—
199	C₄²	—	12	1	no	—
200	C₁₆	—	30	2	no	—
201	C₃×C₅	#2140	1	1	yes	characteristic, complemented, Hall
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	—	20	4	no	—
204	C₂²×C₃	—	40	8	no	—
205	C₂²×C₃	—	40	4	no	—
206	C₂²×C₃	#3941	1	2	yes	—
207	C₂²×C₃	C₂×#686	1	1	yes	characteristic
208	C₃×C₄	#3941	1	2	yes	—
209	C₃×C₄	C₂×#685	1	1	yes	characteristic
210	C₃×C₄	C₂×#688	1	1	yes	characteristic
211	Dic₃	—	8	2	no	—
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	#5147	1	2	yes	—
214	C₂×C₅	C₂×#832	1	1	yes	characteristic
215	D₅	—	8	4	no	—
216	C₂³	—	20	4	no	—
217	C₂³	—	20	2	no	—
218	C₂³	—	40	4	no	—
219	C₂³	#6652	1	1	yes	characteristic
220	D₄	—	10	4	no	—
221	D₄	—	20	4	no	—
222	D₄	—	20	2	no	—
223	C₂×C₄	—	12	2	no	—
224	C₂×C₄	#6652	1	1	yes	characteristic
225	C₂×C₄	#6655	1	2	yes	—
226	C₂×C₄	#6665	1	2	yes	—
227	C₂×C₄	C₂×#1201	1	1	yes	characteristic
228	C₈	#6665	1	2	yes	—
229	C₈	C₂×D₅×Dic₃	1	1	yes	characteristic
230	C₈	C₂×#1202	1	1	yes	characteristic
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	—	40	4	no	—
233	C₂×C₃	C₂×#3941	1	1	yes	characteristic
234	C₂×C₃	#65482	1	2	yes	—
235	C₅	#72033	1	1	yes	characteristic, complemented, Sylow
236	C₂²	—	2	2	no	—
237	C₂²	—	20	4	no	—
238	C₂²	—	40	8	no	—
239	C₂²	—	40	4	no	—
240	C₂²	#90964	1	2	yes	—
241	C₂²	C₂×#6652	1	1	yes	characteristic
242	C₄	—	24	2	no	—
243	C₄	#90964	1	2	yes	—
244	C₄	C₂×#6655	1	1	yes	characteristic
245	C₄	C₂×#6665	1	1	yes	characteristic
246	C₃	#10601445	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	2	2	no	—
248	C₂	—	40	4	no	—
249	C₂	#11747539	1	2	yes	—
250	C₂	C₂×#90964	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6652
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×C₃×D₄₀	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. C₂×#91096	C₂	semidirect
3. C₅×#5106	C₂	semidirect
4. #11745891	C₂	semidirect
11. C₂×C₃×D₄₀	C₄	semidirect
14. C₂×C₃⋊#672	C₂²	semidirect
15. C₅×#818	C₂²	semidirect
40. #5106	D₅	semidirect
48. C₂×D₄₀	Dic₃	semidirect
77. #818	C₂×D₅	semidirect
201. C₃×C₅	#2140	semidirect
235. C₅	#72033	semidirect
246. C₃	#10601445	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₄₀	2
C₂×#91096	2
C₅×#5106	2
#11745891	2
#10601445	3
#72033	5