Subgroups — #14299032

Order 1920.

600 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×C₄⋊(C₃⋊#168)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#90982	C₂	1	4	yes	maximal
5	#10601238	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊#168	C₂²	1	2	yes	—
8	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₅	C₄	1	2	yes	—
11	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
13	C₄⋊(C₃⋊#168)	—	2	2	no	—
14	C₅×C₄⋊(C₃⋊C₈)	—	2	2	no	—
15	#90982	—	2	8	no	—
16	#69394	—	5	1	no	maximal, Hall
17	C₂×C₅×C₄⋊C₈	—	3	1	no	—
18	C₂×#3946	—	3	4	no	—
19	C₄²×Dic₅	D₃	1	1	yes	characteristic
20	C₂×C₄⋊#168	—	3	1	no	—
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
26	C₃×C₄²×C₅	D₄	1	2	yes	—
27	C₃×C₄²×C₅	Q₈	1	2	yes	—
28	C₂×C₃⋊#168	—	2	4	no	—
29	C₂×C₃⋊#168	—	2	2	no	—
30	C₂×C₅×C₃⋊C₈	—	2	4	no	—
31	C₂×C₅×C₃⋊C₈	—	2	2	no	—
32	C₃×C₄×Dic₅	—	2	2	no	—
33	C₃×C₄×Dic₅	—	2	4	no	—
34	C₃×C₄×Dic₅	C₈	1	4	yes	complemented
35	C₃×C₄×Dic₅	D₄	1	4	yes	—
36	C₃×C₄×Dic₅	Q₈	1	4	yes	—
37	C₃×C₄³	—	5	1	no	—
38	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
39	C₂×C₄⋊(C₃⋊C₈)	—	5	4	no	—
40	C₂×C₄⋊(C₃⋊C₈)	D₅	1	1	yes	characteristic
41	C₂²×C₅×C₈	—	3	2	no	—
42	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄×Dic₅	Dic₃	1	1	yes	characteristic
44	C₂×C₄×Dic₅	Dic₃	1	2	yes	—
45	C₂×C₄×Dic₅	Dic₃	1	1	yes	characteristic
46	C₂×C₄×Dic₅	C₂×D₃	1	2	yes	—
47	C₂²×#168	—	3	2	no	—
48	C₅×C₄⋊C₈	—	6	2	no	—
49	#3946	—	6	8	no	—
50	C₄⋊#168	—	6	2	no	—
51	#1842	—	15	1	no	Sylow
52	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
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	2	yes	—
62	C₂×C₃×Dic₅	—	2	2	no	—
63	C₂×C₃×Dic₅	—	2	2	no	—
64	C₂×C₃×Dic₅	C₄⋊C₄	1	4	yes	—
65	C₂×C₃×Dic₅	C₄⋊C₄	1	4	yes	—
66	C₂×C₃×Dic₅	C₂×C₈	1	4	yes	—
67	C₂×C₃×Dic₅	M₁₆	1	4	yes	—
68	C₃⋊#168	—	4	2	no	—
69	C₅×C₃⋊C₈	—	4	2	no	—
70	C₂×C₃×C₄²	—	5	1	no	—
71	C₂×C₃×C₄²	—	5	2	no	—
72	C₂×C₃×C₄²	—	5	2	no	—
73	C₂×C₃×C₄²	—	5	1	no	—
74	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
75	C₂²×C₃⋊C₈	—	5	2	no	—
76	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
77	C₄⋊(C₃⋊C₈)	—	2	2	no	—
78	C₄⋊(C₃⋊C₈)	—	10	8	no	—
79	C₄⋊(C₃⋊C₈)	—	10	2	no	—
80	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
83	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
84	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
85	C₂×C₅×C₈	—	6	4	no	—
86	C₂×C₅×C₈	—	6	2	no	—
87	C₄²×C₅	Dic₆	1	2	yes	—
88	C₄²×C₅	D₁₂	1	2	yes	—
89	C₄×Dic₅	—	2	2	no	—
90	C₄×Dic₅	—	2	4	no	—
91	C₄×Dic₅	C₃⋊C₈	1	4	yes	complemented
92	C₄×Dic₅	Dic₆	1	4	yes	—
93	C₄×Dic₅	D₁₂	1	4	yes	—
94	C₂×#168	—	6	2	no	—
95	C₂×#168	—	6	4	no	—
96	C₂×C₄⋊C₈	—	3	1	no	—
97	C₂×C₄⋊C₈	—	15	4	no	—
98	C₂×C₄⋊C₈	—	15	1	no	—
99	C₄³	—	5	1	no	—
100	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
101	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
104	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	C₄⋊C₈	1	4	yes	complemented
107	C₃×C₄×C₅	#127	1	1	yes	characteristic
108	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
109	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
110	C₃×Dic₅	—	2	4	no	—
111	C₃×Dic₅	C₄⋊C₈	1	8	yes	complemented
112	C₂²×C₃×C₄	—	5	2	no	—
113	C₂²×C₃×C₄	—	5	2	no	—
114	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
117	C₃×C₄²	—	5	4	no	—
118	C₃×C₄²	—	5	4	no	—
119	C₃×C₄²	—	5	4	no	—
120	C₃×C₄²	—	10	4	no	—
121	C₃×C₄²	—	10	2	no	—
122	C₃×C₄²	Dic₁₀	1	2	yes	—
123	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
124	C₂×C₃⋊C₈	—	2	4	no	—
125	C₂×C₃⋊C₈	—	2	2	no	—
126	C₂×C₃⋊C₈	—	10	4	no	—
127	C₂×C₃⋊C₈	—	10	2	no	—
128	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
131	C₂×C₄×C₅	#208	1	2	yes	—
132	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
133	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
134	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
135	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
137	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
138	C₂×Dic₅	—	2	2	no	—
139	C₂×Dic₅	—	2	2	no	—
140	C₂×Dic₅	C₂×C₃⋊C₈	1	4	yes	—
141	C₂×Dic₅	#208	1	4	yes	—
142	C₂×Dic₅	C₄⋊Dic₃	1	4	yes	—
143	C₂×Dic₅	C₄⋊Dic₃	1	4	yes	—
144	C₅×C₈	—	12	2	no	—
145	#168	—	12	2	no	—
146	C₂²×C₈	—	3	2	no	—
147	C₂²×C₈	—	15	2	no	—
148	C₂×C₄²	—	5	1	no	—
149	C₂×C₄²	—	5	2	no	—
150	C₂×C₄²	—	5	2	no	—
151	C₂×C₄²	—	5	1	no	—
152	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
153	C₄⋊C₈	—	6	2	no	—
154	C₄⋊C₈	—	30	8	no	—
155	C₄⋊C₈	—	30	2	no	—
156	C₂×C₃×C₅	#384	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×C₄⋊C₈	1	2	yes	—
158	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
159	C₂×C₃×C₅	#423	1	2	yes	—
160	C₂×C₃×C₅	#423	1	1	yes	characteristic
161	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	5	4	no	—
164	C₂×C₃×C₄	—	5	4	no	—
165	C₂×C₃×C₄	—	5	4	no	—
166	C₂×C₃×C₄	—	5	4	no	—
167	C₂×C₃×C₄	—	10	2	no	—
168	C₂×C₃×C₄	—	10	2	no	—
169	C₂×C₃×C₄	C₈×D₅	1	2	yes	—
170	C₂×C₃×C₄	#676	1	2	yes	—
171	C₂×C₃×C₄	#683	1	2	yes	—
172	C₂×C₃×C₄	#683	1	2	yes	—
173	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
174	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
175	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
177	C₃⋊C₈	—	4	2	no	—
178	C₃⋊C₈	—	20	2	no	—
179	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
180	C₂²×C₅	C₂×#208	1	2	yes	—
181	C₂²×C₅	C₂×#208	1	1	yes	characteristic
182	C₂²×C₅	C₂×C₄⋊Dic₃	1	2	yes	—
183	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	C₄⋊(C₃⋊C₈)	1	4	yes	complemented
186	C₄×C₅	#869	1	2	yes	—
187	C₄×C₅	#869	1	1	yes	characteristic
188	C₄×C₅	#874	1	1	yes	characteristic
189	Dic₅	—	2	4	no	—
190	Dic₅	C₄⋊(C₃⋊C₈)	1	8	yes	complemented
191	C₂²×C₄	—	5	2	no	—
192	C₂²×C₄	—	5	2	no	—
193	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
194	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
196	C₂×C₈	—	6	4	no	—
197	C₂×C₈	—	6	2	no	—
198	C₂×C₈	—	30	4	no	—
199	C₂×C₈	—	30	2	no	—
200	C₄²	—	5	4	no	—
201	C₄²	—	5	4	no	—
202	C₄²	—	5	4	no	—
203	C₄²	—	10	4	no	—
204	C₄²	—	10	2	no	—
205	C₄²	#1203	1	2	yes	—
206	C₄²	#1204	1	2	yes	—
207	C₃×C₅	#1842	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	D₅×C₄⋊C₄	1	2	yes	—
209	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
210	C₂²×C₃	C₂×#676	1	1	yes	characteristic
211	C₂²×C₃	D₅×M₁₆	1	2	yes	—
212	C₂²×C₃	C₂×#683	1	1	yes	characteristic
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	5	8	no	—
215	C₃×C₄	—	10	4	no	—
216	C₃×C₄	#3946	1	4	yes	complemented
217	C₃×C₄	#4034	1	2	yes	—
218	C₃×C₄	#4086	1	1	yes	characteristic
219	C₃×C₄	#4090	1	1	yes	characteristic
220	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
221	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
222	C₂×C₅	#4924	1	1	yes	characteristic
223	C₂×C₅	#4924	1	2	yes	—
224	C₂×C₅	#4935	1	1	yes	characteristic
225	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	5	4	no	—
228	C₂×C₄	—	5	4	no	—
229	C₂×C₄	—	5	4	no	—
230	C₂×C₄	—	5	4	no	—
231	C₂×C₄	—	10	2	no	—
232	C₂×C₄	—	10	2	no	—
233	C₂×C₄	D₅×C₃⋊C₈	1	2	yes	—
234	C₂×C₄	#6649	1	2	yes	—
235	C₂×C₄	#6668	1	2	yes	—
236	C₂×C₄	#6668	1	2	yes	—
237	C₂×C₄	D₅×Dic₆	1	2	yes	—
238	C₂×C₄	D₅×D₁₂	1	2	yes	—
239	C₂×C₄	C₂×#1203	1	1	yes	characteristic
240	C₂×C₄	C₂×#1204	1	1	yes	characteristic
241	C₈	—	12	2	no	—
242	C₈	—	60	2	no	—
243	C₂×C₃	D₅×C₄⋊C₈	1	2	yes	—
244	C₂×C₃	#65126	1	2	yes	—
245	C₂×C₃	#65265	1	1	yes	characteristic
246	C₂×C₃	C₂×#3946	1	1	yes	characteristic
247	C₂×C₃	#65407	1	1	yes	characteristic
248	C₅	#69394	1	1	yes	characteristic, complemented, Sylow
249	C₂²	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
250	C₂²	D₅×#208	1	2	yes	—
251	C₂²	C₂×#6649	1	1	yes	characteristic
252	C₂²	D₅×C₄⋊Dic₃	1	2	yes	—
253	C₂²	C₂×#6668	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	5	8	no	—
256	C₄	—	10	4	no	—
257	C₄	#90982	1	4	yes	complemented
258	C₄	#91376	1	1	yes	characteristic
259	C₄	#91451	1	1	yes	characteristic
260	C₄	#91468	1	2	yes	—
261	C₃	#10601238	1	1	yes	characteristic, complemented, Sylow
262	C₂	D₅×C₄⋊(C₃⋊C₈)	1	2	yes	—
263	C₂	#11747007	1	2	yes	—
264	C₂	C₂×#90982	1	1	yes	characteristic
265	C₂	#11748431	1	1	yes	characteristic
266	C₂	#11748834	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃×D₅
commutator	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1842
LCS	G ▷ C₂²×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
34. C₃×C₄×Dic₅	C₈	semidirect
91. C₄×Dic₅	C₃⋊C₈	semidirect
106. C₃×C₄×C₅	C₄⋊C₈	semidirect
111. C₃×Dic₅	C₄⋊C₈	semidirect
185. C₄×C₅	C₄⋊(C₃⋊C₈)	semidirect
190. Dic₅	C₄⋊(C₃⋊C₈)	semidirect
207. C₃×C₅	#1842	semidirect
216. C₃×C₄	#3946	semidirect
248. C₅	#69394	semidirect
257. C₄	#90982	semidirect
261. C₃	#10601238	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊#168)	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₃×C₄²×Dic₅	2
C₂×#90982	2
#10601238	3
#69394	5