Subgroups — #14413145

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#872	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×#681	C₂	1	2	yes	maximal
3	#11751025	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748440	C₂	1	2	yes	maximal, complemented
6	#11746993	C₂	1	8	yes	maximal, complemented
7	#10613935	—	3	1	no	maximal, Hall
8	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×#6710	C₂²	1	2	yes	—
11	C₂×C₅×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
12	C₂×C₅×Dic₃⋊C₄	C₄	1	2	yes	—
13	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₅×#872	C₄	1	4	yes	—
15	C₅×#872	C₂²	1	4	yes	—
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	C₂×#6650	C₂²	1	4	yes	—
18	D₃×#681	—	2	8	no	—
19	C₂×D₃×#168	C₂²	1	4	yes	—
20	C₄⋊(C₃⋊#168)	C₂²	1	4	yes	complemented
21	C₃×C₄⋊#168	C₂²	1	4	yes	complemented
22	Dic₃⋊#168	C₂²	1	8	yes	complemented
23	#79048	—	5	1	no	maximal, Hall
24	C₂×C₅×#117	—	3	1	no	—
25	C₂²×#681	—	3	2	no	—
26	#65288	—	3	8	no	—
27	#65213	—	3	2	no	—
28	#65213	—	3	1	no	—
29	#65213	D₃	1	1	yes	characteristic, complemented
30	C₂³×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
34	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×C₅×D₃	D₄	1	4	yes	—
37	C₂×C₄×C₅×D₃	Q₈	1	4	yes	—
38	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
39	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₃×C₄²×C₅	C₂³	1	2	yes	—
41	#6710	—	2	4	no	—
42	C₂×C₃⋊#168	C₂³	1	4	yes	—
43	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	—
44	C₅×Dic₃⋊C₄	C₂×C₄	1	8	yes	—
45	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
46	C₄×C₅×Dic₃	C₂³	1	2	yes	—
47	C₃×#681	—	2	4	no	—
48	C₂×C₃×#168	C₂³	1	4	yes	—
49	#6650	—	2	8	no	—
50	D₃×#168	—	2	8	no	—
51	C₂×D₃×M₁₆	—	5	2	no	—
52	C₂×#872	D₅	1	1	yes	characteristic
53	C₃×#423	—	5	1	no	—
54	#5118	—	5	2	no	—
55	#4924	—	5	1	no	—
56	#4839	—	5	8	no	—
57	C₂³×C₄×C₅	—	3	1	no	—
58	C₅×#117	—	3	4	no	—
59	C₅×#117	—	3	4	no	—
60	C₂×C₅×C₄⋊C₄	—	3	2	no	—
61	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
62	C₂×C₄²×C₅	—	3	1	no	—
63	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
64	C₂×#681	—	3	2	no	—
65	C₂×#681	—	3	4	no	—
66	C₂×#681	—	6	8	no	—
67	C₂×#681	C₂×D₃	1	2	yes	—
68	C₂²×#168	—	3	4	no	—
69	C₄⋊#168	—	3	4	no	—
70	C₄⋊#168	—	3	8	no	—
71	C₄⋊#168	C₂×D₃	1	4	yes	complemented
72	#2905	—	15	1	no	Sylow
73	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₅×D₃	—	4	2	no	—
75	C₂²×C₅×D₃	C₄⋊C₄	1	4	yes	—
76	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	—	4	2	no	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×C₅×Dic₃	—	4	2	no	—
84	C₂×C₅×Dic₃	C₄⋊C₄	1	4	yes	—
85	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
86	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
87	C₄×C₅×D₃	C₄⋊C₄	1	8	yes	—
88	C₄×C₅×D₃	C₂×D₄	1	4	yes	—
89	C₄×C₅×D₃	C₂×Q₈	1	4	yes	—
90	C₃⋊#168	—	2	4	no	—
91	C₃×#168	—	2	4	no	—
92	C₂²×C₄×D₃	C₂×D₅	1	1	yes	characteristic
93	C₂×C₃×M₁₆	—	5	2	no	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₂×(C₂×D₃)⋊C₄	Dic₅	1	2	yes	—
96	C₂×Dic₃⋊C₄	Dic₅	1	2	yes	—
97	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
98	C₂×#208	—	5	2	no	—
99	D₃×M₁₆	—	10	8	no	—
100	C₂×ΓL(1,𝔽₂₅)	—	5	4	no	—
101	C₂×D₃×C₈	—	5	4	no	—
102	#872	Dic₅	1	4	yes	—
103	#872	C₂×D₅	1	4	yes	—
104	C₃×C₄⋊C₈	—	5	4	no	—
105	Dic₃⋊C₈	—	5	8	no	—
106	C₄⋊(C₃⋊C₈)	—	5	4	no	—
107	C₂⁴×C₅	—	3	1	no	—
108	C₂²×C₄×C₅	—	3	1	no	—
109	C₂²×C₄×C₅	—	3	2	no	—
110	C₂²×C₄×C₅	—	3	4	no	—
111	C₂²×C₄×C₅	—	3	4	no	—
112	C₂²×C₄×C₅	—	3	2	no	—
113	C₂²×C₄×C₅	—	3	2	no	—
114	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
115	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₅×C₄⋊C₄	—	3	8	no	—
117	C₅×C₂²⋊C₄	—	3	8	no	—
118	C₄²×C₅	—	3	2	no	—
119	C₄²×C₅	—	3	2	no	—
120	C₄²×C₅	C₄×D₃	1	2	yes	—
121	C₄²×C₅	C₂²×D₃	1	2	yes	—
122	#681	—	2	4	no	—
123	#681	—	6	8	no	—
124	#681	—	6	4	no	—
125	C₂×#168	—	3	4	no	—
126	C₂×#168	—	6	8	no	—
127	C₂×#168	C₂²×D₃	1	4	yes	—
128	C₂²×M₁₆	—	15	2	no	—
129	C₂×#117	—	3	1	no	—
130	#424	—	15	8	no	—
131	#423	—	5	1	no	—
132	#423	—	15	2	no	—
133	#423	—	15	1	no	—
134	C₂²×C₃×C₅	—	2	2	no	—
135	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
136	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₅×D₃	—	2	4	no	—
138	C₂×C₅×D₃	—	4	4	no	—
139	C₂×C₅×D₃	C₂×C₄⋊C₄	1	4	yes	—
140	C₃×C₄×C₅	—	2	4	no	—
141	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
142	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
144	C₅×Dic₃	—	2	4	no	—
145	C₅×Dic₃	C₂×C₄⋊C₄	1	4	yes	—
146	C₂³×D₃	C₂×Dic₅	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
148	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
149	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
150	C₂²×Dic₃	C₂×Dic₅	1	1	yes	characteristic
151	C₂×C₄×D₃	Dic₁₀	1	4	yes	—
152	C₂×C₄×D₃	D₂₀	1	4	yes	—
153	C₂×C₄×D₃	C₂×Dic₅	1	2	yes	—
154	C₂×C₄×D₃	C₂²×D₅	1	2	yes	—
155	C₃×M₁₆	—	10	4	no	—
156	C₂×C₃×C₈	—	5	4	no	—
157	C₃×C₄²	C₂×Dic₅	1	2	yes	—
158	C₃×C₄²	C₂²×D₅	1	2	yes	—
159	(C₂×D₃)⋊C₄	C₂×Dic₅	1	8	yes	—
160	Dic₃⋊C₄	C₂×Dic₅	1	8	yes	—
161	C₄×Dic₃	C₂×Dic₅	1	2	yes	—
162	C₄×Dic₃	C₂²×D₅	1	2	yes	—
163	#208	—	10	4	no	—
164	C₂×C₃⋊C₈	—	5	4	no	—
165	ΓL(1,𝔽₂₅)	—	10	8	no	—
166	D₃×C₈	—	10	8	no	—
167	C₂³×C₅	—	3	2	no	—
168	C₂³×C₅	—	3	4	no	—
169	C₂³×C₅	—	12	2	no	—
170	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
171	C₂×C₄×C₅	—	3	4	no	—
172	C₂×C₄×C₅	—	3	2	no	—
173	C₂×C₄×C₅	—	3	8	no	—
174	C₂×C₄×C₅	—	3	4	no	—
175	C₂×C₄×C₅	—	3	4	no	—
176	C₂×C₄×C₅	—	3	4	no	—
177	C₂×C₄×C₅	—	4	2	no	—
178	C₂×C₄×C₅	—	12	2	no	—
179	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
180	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
182	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
183	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
184	#168	—	2	4	no	—
185	#168	—	6	4	no	—
186	C₂³×C₄	—	3	1	no	—
187	C₂×M₁₆	—	5	2	no	—
188	C₂×M₁₆	—	15	4	no	—
189	C₂×M₁₆	—	15	2	no	—
190	C₂×M₁₆	—	30	8	no	—
191	C₂²×C₈	—	15	4	no	—
192	#117	—	3	4	no	—
193	#117	—	3	4	no	—
194	C₂×C₄⋊C₄	—	3	2	no	—
195	C₂×C₂²⋊C₄	—	3	2	no	—
196	C₂×C₄²	—	3	1	no	—
197	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
198	C₄⋊C₈	—	5	4	no	—
199	C₄⋊C₈	—	15	8	no	—
200	C₄⋊C₈	—	15	4	no	—
201	C₂×C₃×C₅	—	2	2	no	—
202	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₃×C₅	#568	1	2	yes	—
204	C₅×D₃	—	2	4	no	—
205	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
206	C₂²×D₃	—	4	2	no	—
207	C₂²×D₃	C₄⋊Dic₅	1	4	yes	—
208	C₂²×D₃	C₂²×Dic₅	1	2	yes	—
209	C₂×C₃×C₄	—	4	2	no	—
210	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
211	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
212	C₂×C₃×C₄	C₂²×Dic₅	1	4	yes	—
213	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
215	C₂×Dic₃	—	4	2	no	—
216	C₂×Dic₃	C₄⋊Dic₅	1	4	yes	—
217	C₂×Dic₃	C₂²×Dic₅	1	4	yes	—
218	C₂×Dic₃	C₂²×Dic₅	1	2	yes	—
219	C₄×D₃	C₄⋊Dic₅	1	8	yes	—
220	C₄×D₃	C₂×Dic₁₀	1	4	yes	—
221	C₄×D₃	C₂×D₂₀	1	4	yes	—
222	C₃×C₈	—	10	4	no	—
223	C₃⋊C₈	—	10	4	no	—
224	C₂²×C₅	—	2	2	no	—
225	C₂²×C₅	—	3	4	no	—
226	C₂²×C₅	—	6	4	no	—
227	C₂²×C₅	—	12	4	no	—
228	C₂²×C₅	D₃×C₄⋊C₄	1	2	yes	—
229	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	—	3	4	no	—
232	C₄×C₅	—	6	4	no	—
233	C₄×C₅	D₃×C₄⋊C₄	1	2	yes	—
234	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
235	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
236	C₂⁴	—	3	1	no	—
237	C₂²×C₄	—	3	1	no	—
238	C₂²×C₄	—	3	4	no	—
239	C₂²×C₄	—	3	4	no	—
240	C₂²×C₄	—	3	2	no	—
241	C₂²×C₄	—	3	2	no	—
242	C₂²×C₄	—	3	2	no	—
243	C₂²×C₄	D₃×Dic₅	1	2	yes	—
244	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
245	M₁₆	—	10	4	no	—
246	M₁₆	—	30	8	no	—
247	M₁₆	—	30	4	no	—
248	C₂×C₈	—	5	4	no	—
249	C₂×C₈	—	15	4	no	—
250	C₂×C₈	—	30	8	no	—
251	C₄⋊C₄	—	3	8	no	—
252	C₂²⋊C₄	—	3	8	no	—
253	C₄²	—	3	2	no	—
254	C₄²	—	3	2	no	—
255	C₄²	D₃×Dic₅	1	2	yes	—
256	C₄²	C₂×D₃×D₅	1	2	yes	—
257	C₃×C₅	#2905	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	C₂×C₄⋊Dic₅	1	2	yes	—
260	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
261	C₂×D₃	—	2	4	no	—
262	C₂×D₃	—	4	4	no	—
263	C₂×D₃	C₂×C₄⋊Dic₅	1	4	yes	—
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
266	C₃×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
267	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
268	Dic₃	—	2	4	no	—
269	Dic₃	C₂×C₄⋊Dic₅	1	4	yes	—
270	C₂×C₅	—	2	2	no	—
271	C₂×C₅	—	6	4	no	—
272	C₂×C₅	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₅	#5745	1	2	yes	—
274	C₂³	—	3	2	no	—
275	C₂³	—	3	4	no	—
276	C₂³	—	12	2	no	—
277	C₂³	C₂×D₃×Dic₅	1	1	yes	characteristic
278	C₂×C₄	—	3	4	no	—
279	C₂×C₄	—	3	2	no	—
280	C₂×C₄	—	3	8	no	—
281	C₂×C₄	—	3	4	no	—
282	C₂×C₄	—	3	4	no	—
283	C₂×C₄	—	3	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	12	2	no	—
286	C₂×C₄	D₃×Dic₁₀	1	2	yes	—
287	C₂×C₄	D₃×D₂₀	1	2	yes	—
288	C₂×C₄	C₂×D₃×Dic₅	1	4	yes	—
289	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
291	C₈	—	10	4	no	—
292	C₈	—	30	4	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
295	C₂×C₃	#66153	1	2	yes	—
296	D₃	—	2	4	no	—
297	C₅	#79048	1	1	yes	characteristic, complemented, Sylow
298	C₂²	—	2	2	no	—
299	C₂²	—	3	4	no	—
300	C₂²	—	6	4	no	—
301	C₂²	—	12	4	no	—
302	C₂²	D₃×C₄⋊Dic₅	1	2	yes	—
303	C₂²	C₂²×D₃×Dic₅	1	1	yes	characteristic
304	C₄	—	2	4	no	—
305	C₄	—	3	4	no	—
306	C₄	—	6	4	no	—
307	C₄	D₃×C₄⋊Dic₅	1	2	yes	—
308	C₄	C₂×D₃×Dic₁₀	1	1	yes	characteristic
309	C₄	C₂×D₃×D₂₀	1	1	yes	characteristic
310	C₃	#10613935	1	1	yes	characteristic, complemented, Sylow
311	C₂	—	2	2	no	—
312	C₂	—	6	4	no	—
313	C₂	#11754125	1	2	yes	—
314	C₂	C₂×D₃×C₄⋊Dic₅	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×C₅×D₃	C₂×C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#2905
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
3. #11751025	C₂	semidirect
4. C₃×#65213	C₂	semidirect
5. #11748440	C₂	semidirect
6. #11746993	C₂	semidirect
20. C₄⋊(C₃⋊#168)	C₂²	semidirect
21. C₃×C₄⋊#168	C₂²	semidirect
22. Dic₃⋊#168	C₂²	semidirect
29. #65213	D₃	semidirect
71. C₄⋊#168	C₂×D₃	semidirect
257. C₃×C₅	#2905	semidirect
297. C₅	#79048	semidirect
310. C₃	#10613935	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#872	2
C₂×D₃×#681	2
#11751025	2
C₃×#65213	2
#11748440	2
#11746993	2
#10613935	3
#79048	5