Subgroups — #14307102

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₂×#91087	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11745966	C₂	1	4	yes	maximal
5	#10601266	—	3	1	no	maximal, Hall
6	C₂²×C₃×Dic₁₀	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊#168	C₂²	1	2	yes	—
9	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₃×#683	C₄	1	2	yes	—
12	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
13	C₃×C₄×Dic₁₀	—	2	2	no	—
14	C₃×C₄×Dic₁₀	—	2	2	no	—
15	#91087	—	2	2	no	—
16	C₅×C₄⋊(C₃⋊C₈)	—	2	2	no	—
17	#71635	—	5	1	no	maximal, Hall
18	C₂×C₄×Dic₁₀	D₃	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₈	—	3	1	no	—
20	C₂×#3935	—	3	1	no	—
21	#64770	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
26	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
27	C₂×C₃×Dic₁₀	—	2	2	no	—
28	C₂×C₃×Dic₁₀	—	2	4	no	—
29	C₃×C₄²×C₅	D₄	1	2	yes	—
30	C₃×C₄²×C₅	D₄	1	2	yes	—
31	C₂×C₃⋊#168	—	2	2	no	—
32	C₂×C₃⋊#168	—	2	2	no	—
33	C₂×C₃⋊#168	Q₈	1	4	yes	complemented
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	—	2	4	no	—
36	C₃×C₄⋊Dic₅	—	2	2	no	—
37	C₃×#683	—	2	4	no	—
38	C₃×C₄×Dic₅	—	2	2	no	—
39	C₃×C₄×Dic₅	—	2	2	no	—
40	C₂×C₃×C₄×Q₈	—	5	1	no	—
41	C₂×C₄⋊(C₃⋊C₈)	D₅	1	1	yes	characteristic
42	C₂×#803	—	5	1	no	—
43	#4524	—	5	4	no	—
44	C₂²×Dic₁₀	Dic₃	1	1	yes	characteristic
45	C₂²×C₅×C₈	—	3	2	no	—
46	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
47	C₂×C₄⋊Dic₅	Dic₃	1	1	yes	characteristic
48	C₂×#683	Dic₃	1	2	yes	—
49	C₂×C₄×Dic₅	C₂×D₃	1	2	yes	—
50	C₂²×#168	—	3	2	no	—
51	C₄×Dic₁₀	—	2	2	no	—
52	C₄×Dic₁₀	—	2	2	no	—
53	C₅×C₄⋊C₈	—	6	2	no	—
54	#3935	—	6	2	no	—
55	#1985	—	15	1	no	Sylow
56	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
64	C₂×C₃×C₄×C₅	#41	1	2	yes	—
65	C₂×C₃×Dic₅	—	2	2	no	—
66	C₂×C₃×Dic₅	—	2	2	no	—
67	C₂×C₃×Dic₅	—	2	2	no	—
68	C₂×C₃×Dic₅	—	2	4	no	—
69	C₂×C₃×Dic₅	M₁₆	1	4	yes	—
70	C₃×Dic₁₀	—	2	8	no	—
71	C₃⋊#168	—	2	4	no	—
72	C₅×C₃⋊C₈	—	4	2	no	—
73	C₂²×C₃×Q₈	—	5	1	no	—
74	C₃×C₄×Q₈	—	10	2	no	—
75	C₃×C₄×Q₈	—	10	2	no	—
76	C₂×C₃×C₄⋊C₄	—	5	2	no	—
77	C₂×C₃×C₄⋊C₄	—	5	1	no	—
78	C₂×C₃×C₄²	—	5	2	no	—
79	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
80	C₂²×C₃⋊C₈	—	5	2	no	—
81	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
82	C₄⋊(C₃⋊C₈)	—	2	2	no	—
83	#803	—	10	2	no	—
84	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
88	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
89	C₂×Dic₁₀	—	2	2	no	—
90	C₂×Dic₁₀	—	2	4	no	—
91	C₂×C₅×C₈	—	6	4	no	—
92	C₂×C₅×C₈	—	6	2	no	—
93	C₄²×C₅	#67	1	2	yes	—
94	C₄²×C₅	#67	1	2	yes	—
95	C₄⋊Dic₅	—	2	2	no	—
96	#683	—	2	4	no	—
97	C₄×Dic₅	—	2	2	no	—
98	C₄×Dic₅	—	2	2	no	—
99	C₂×#168	—	3	4	no	—
100	C₂×#168	—	6	2	no	—
101	C₂×#168	—	6	2	no	—
102	C₂×C₄×Q₈	—	5	1	no	—
103	C₂×C₄⋊C₈	—	3	1	no	—
104	C₂×#97	—	15	1	no	—
105	#336	—	15	4	no	—
106	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₄×Q₈	1	2	yes	—
108	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
109	C₂²×C₃×C₅	#131	1	2	yes	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	2	2	no	—
112	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
113	C₃×C₄×C₅	#124	1	1	yes	characteristic
114	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
115	C₃×Dic₅	—	2	4	no	—
116	C₃×Dic₅	—	4	2	no	—
117	C₂×C₃×Q₈	—	10	4	no	—
118	C₂×C₃×Q₈	—	10	2	no	—
119	C₂²×C₃×C₄	—	5	2	no	—
120	C₂²×C₃×C₄	—	5	2	no	—
121	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	—	10	4	no	—
125	C₃×C₄⋊C₄	—	10	2	no	—
126	C₃×C₄²	—	10	2	no	—
127	C₃×C₄²	—	10	2	no	—
128	C₃×C₄²	D₂₀	1	2	yes	—
129	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
130	C₂×C₃⋊C₈	—	2	4	no	—
131	C₂×C₃⋊C₈	—	2	2	no	—
132	C₂×C₃⋊C₈	—	5	4	no	—
133	C₂×C₃⋊C₈	—	10	2	no	—
134	C₂×C₃⋊C₈	—	10	2	no	—
135	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	#217	1	2	yes	—
139	C₂×C₄×C₅	#217	1	2	yes	—
140	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
141	C₂×C₄×C₅	#239	1	2	yes	—
142	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
144	C₂×Dic₅	—	2	2	no	—
145	C₂×Dic₅	—	2	2	no	—
146	C₂×Dic₅	—	2	4	no	—
147	C₂×Dic₅	—	2	2	no	—
148	C₂×Dic₅	#208	1	4	yes	—
149	Dic₁₀	—	2	8	no	—
150	C₅×C₈	—	12	2	no	—
151	#168	—	6	4	no	—
152	C₂²×Q₈	—	5	1	no	—
153	C₂²×C₈	—	3	2	no	—
154	C₂²×C₈	—	15	2	no	—
155	C₄×Q₈	—	10	2	no	—
156	C₄×Q₈	—	10	2	no	—
157	C₂×C₄⋊C₄	—	5	2	no	—
158	C₂×C₄⋊C₄	—	5	1	no	—
159	C₂×C₄²	—	5	2	no	—
160	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
161	C₄⋊C₈	—	6	2	no	—
162	#97	—	30	2	no	—
163	C₂×C₃×C₅	#391	1	1	yes	characteristic
164	C₂×C₃×C₅	#407	1	1	yes	characteristic
165	C₂×C₃×C₅	#408	1	1	yes	characteristic
166	C₂×C₃×C₅	#446	1	4	yes	—
167	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
168	C₃×Q₈	—	10	8	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	5	4	no	—
172	C₂×C₃×C₄	—	10	2	no	—
173	C₂×C₃×C₄	—	10	2	no	—
174	C₂×C₃×C₄	—	10	4	no	—
175	C₂×C₃×C₄	—	10	2	no	—
176	C₂×C₃×C₄	#685	1	2	yes	—
177	C₂×C₃×C₄	#685	1	2	yes	—
178	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
179	C₂×C₃×C₄	#711	1	2	yes	—
180	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
181	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
182	C₃⋊C₈	—	4	2	no	—
183	C₃⋊C₈	—	10	4	no	—
184	C₂²×C₅	C₂×#208	1	2	yes	—
185	C₂²×C₅	Q₈×Dic₃	1	2	yes	—
186	C₂²×C₅	#948	1	2	yes	—
187	C₂²×C₅	C₂×#217	1	1	yes	characteristic
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	#944	1	1	yes	characteristic
191	C₄×C₅	#946	1	2	yes	—
192	C₄×C₅	#947	1	1	yes	characteristic
193	Dic₅	—	2	4	no	—
194	Dic₅	—	4	2	no	—
195	C₂×Q₈	—	10	4	no	—
196	C₂×Q₈	—	10	2	no	—
197	C₂²×C₄	—	5	2	no	—
198	C₂²×C₄	—	5	2	no	—
199	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
200	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
202	C₂×C₈	—	6	4	no	—
203	C₂×C₈	—	6	2	no	—
204	C₂×C₈	—	15	4	no	—
205	C₂×C₈	—	30	2	no	—
206	C₂×C₈	—	30	2	no	—
207	C₄⋊C₄	—	10	4	no	—
208	C₄⋊C₄	—	10	2	no	—
209	C₄²	—	10	2	no	—
210	C₄²	—	10	2	no	—
211	C₄²	#1201	1	2	yes	—
212	C₄²	#1202	1	2	yes	—
213	C₃×C₅	#1985	1	1	yes	characteristic, complemented, Hall
214	C₂²×C₃	#4033	1	2	yes	—
215	C₂²×C₃	#4047	1	2	yes	—
216	C₂²×C₃	D₅×M₁₆	1	2	yes	—
217	C₂²×C₃	C₂×#685	1	1	yes	characteristic
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	10	4	no	—
221	C₃×C₄	—	20	2	no	—
222	C₃×C₄	#4043	1	2	yes	—
223	C₃×C₄	#4082	1	1	yes	characteristic
224	C₃×C₄	#4090	1	1	yes	characteristic
225	C₂×C₅	#5024	1	4	yes	—
226	C₂×C₅	#5207	1	1	yes	characteristic
227	C₂×C₅	#5229	1	1	yes	characteristic
228	C₂×C₅	#5234	1	1	yes	characteristic
229	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
230	Q₈	—	10	8	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	5	4	no	—
234	C₂×C₄	—	10	2	no	—
235	C₂×C₄	—	10	2	no	—
236	C₂×C₄	—	10	2	no	—
237	C₂×C₄	—	10	4	no	—
238	C₂×C₄	#6665	1	2	yes	—
239	C₂×C₄	#6665	1	2	yes	—
240	C₂×C₄	#6768	1	2	yes	—
241	C₂×C₄	D₁₅⋊Q₈	1	2	yes	—
242	C₂×C₄	C₂×#1201	1	1	yes	characteristic
243	C₂×C₄	C₂×#1202	1	1	yes	characteristic
244	C₈	—	12	2	no	—
245	C₈	—	30	4	no	—
246	C₂×C₃	#65117	1	4	yes	—
247	C₂×C₃	#65263	1	1	yes	characteristic
248	C₂×C₃	#65399	1	1	yes	characteristic
249	C₂×C₃	#65415	1	1	yes	characteristic
250	C₅	#71635	1	1	yes	characteristic, complemented, Sylow
251	C₂²	D₅×#208	1	2	yes	—
252	C₂²	#91283	1	2	yes	—
253	C₂²	#91331	1	2	yes	—
254	C₂²	C₂×#6665	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	10	4	no	—
258	C₄	—	20	2	no	—
259	C₄	#91370	1	1	yes	characteristic
260	C₄	#91385	1	1	yes	characteristic
261	C₄	#91390	1	2	yes	—
262	C₃	#10601266	1	1	yes	characteristic, complemented, Sylow
263	C₂	#11746848	1	4	yes	—
264	C₂	#11748341	1	1	yes	characteristic
265	C₂	#11748342	1	1	yes	characteristic
266	C₂	#11748610	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃×D₅
commutator, socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
Fitting	C₂×C₃×C₄²×C₅	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1985
LCS	G ▷ C₂³×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
33. C₂×C₃⋊#168	Q₈	semidirect
213. C₃×C₅	#1985	semidirect
250. C₅	#71635	semidirect
262. C₃	#10601266	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×Dic₁₀	2
C₂×#91087	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
#11745966	2
#10601266	3
#71635	5