Subgroups — #14415396

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65811	C₂	1	1	yes	maximal, characteristic
2	C₂×#91284	C₂	1	2	yes	maximal, complemented
3	C₂×#91087	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4924	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748343	C₂	1	2	yes	maximal, complemented
6	#11746997	C₂	1	8	yes	maximal, complemented
7	#10610773	—	3	1	no	maximal, Hall
8	C₂×C₃×#709	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
12	C₂×C₅×#208	C₂²	1	2	yes	—
13	C₃×#4076	C₄	1	4	yes	—
14	C₃×#4021	C₄	1	2	yes	—
15	C₃×#4020	C₄	1	1	yes	characteristic
16	C₃×C₄×D₂₀	C₂²	1	4	yes	—
17	C₃×#4015	C₄	1	1	yes	characteristic
18	C₂×#6649	C₂²	1	4	yes	—
19	#91284	—	2	8	no	—
20	C₂×D₅×C₃⋊C₈	C₂²	1	4	yes	—
21	#91087	C₂²	1	4	yes	—
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
23	#90955	C₂²	1	8	yes	—
24	#83276	—	5	1	no	maximal, Hall
25	C₂×#4053	—	3	2	no	—
26	#65811	D₃	1	1	yes	characteristic
27	C₅×#423	—	3	1	no	—
28	#65418	—	3	2	no	—
29	C₂×#3935	—	3	1	no	—
30	#65125	—	3	8	no	—
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	4	yes	—
34	C₃×#709	—	2	4	no	—
35	C₃×#709	—	2	4	no	—
36	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
37	C₂×C₃×D₂₀	C₂×C₄	1	4	yes	—
38	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
39	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	2	yes	—
43	C₂×C₃⋊#168	—	2	2	no	—
44	C₂×C₃⋊#168	D₄	1	4	yes	complemented
45	C₂×C₃⋊#168	C₂³	1	2	yes	—
46	C₂×C₃⋊#168	C₂³	1	2	yes	—
47	C₅×#208	—	2	4	no	—
48	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
49	C₃×#685	C₂×C₄	1	8	yes	—
50	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
51	#6649	—	2	8	no	—
52	D₅×C₃⋊C₈	—	2	8	no	—
53	C₃×#532	—	5	1	no	—
54	C₂×#948	—	5	2	no	—
55	#5234	—	5	2	no	—
56	#5011	—	5	8	no	—
57	#4924	D₅	1	1	yes	characteristic, complemented
58	C₂×#803	—	5	1	no	—
59	C₂×#709	C₂×D₃	1	2	yes	—
60	C₂×C₅×M₁₆	—	3	2	no	—
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	#4076	Dic₃	1	4	yes	—
63	C₂²×#168	—	3	1	no	—
64	C₂²×#168	—	3	1	no	—
65	#4053	—	6	8	no	—
66	C₂×#676	—	3	4	no	—
67	C₂×C₈×D₅	—	3	4	no	—
68	#4021	Dic₃	1	2	yes	—
69	#4020	Dic₃	1	1	yes	characteristic
70	C₄×D₂₀	C₂×D₃	1	4	yes	—
71	#4015	Dic₃	1	1	yes	characteristic
72	C₅×C₄⋊C₈	—	3	4	no	—
73	#3952	—	3	8	no	—
74	#3935	—	3	4	no	—
75	#2932	—	15	1	no	Sylow
76	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₃×D₅	C₂²×C₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	—	4	2	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	#41	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₃×C₂²⋊D₅	—	2	8	no	—
86	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
87	C₃×D₂₀	—	2	4	no	—
88	C₃×D₂₀	—	4	4	no	—
89	C₃×C₄×D₅	—	2	4	no	—
90	C₃×C₄×D₅	—	2	4	no	—
91	C₃×Dic₁₀	—	2	4	no	—
92	C₃⋊#168	—	2	2	no	—
93	C₃⋊#168	C₂×D₄	1	4	yes	—
94	C₅×C₃⋊C₈	—	2	4	no	—
95	C₂×C₃×#41	—	5	2	no	—
96	C₃×C₄⋊Q₈	—	5	1	no	—
97	C₃×#127	—	5	1	no	—
98	C₃×#124	—	5	2	no	—
99	C₃×#121	—	5	4	no	—
100	C₃×C₄×D₄	—	5	4	no	—
101	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
102	#948	—	10	8	no	—
103	C₂×#208	—	5	4	no	—
104	C₂×#208	C₂×D₅	1	2	yes	—
105	C₂²×C₃⋊C₈	—	5	1	no	—
106	C₂²×C₃⋊C₈	—	5	1	no	—
107	C₂²×C₃⋊C₈	—	5	4	no	—
108	#830	—	5	8	no	—
109	C₄⋊(C₃⋊C₈)	C₂×D₅	1	4	yes	—
110	#803	—	5	4	no	—
111	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
112	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₂²⋊D₅	C₂×Dic₃	1	4	yes	—
114	#709	—	2	4	no	—
115	#709	—	2	4	no	—
116	C₂×D₂₀	C₂×Dic₃	1	2	yes	—
117	C₂×D₂₀	C₂×Dic₃	1	4	yes	—
118	C₂×C₄×D₅	C₂²×D₃	1	4	yes	—
119	C₂×Dic₁₀	C₂×Dic₃	1	2	yes	—
120	C₅×M₁₆	—	6	4	no	—
121	C₂×C₅×C₈	—	3	4	no	—
122	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
123	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
124	C₄²×C₅	C₂²×D₃	1	2	yes	—
125	#685	C₂×Dic₃	1	8	yes	—
126	C₄⋊Dic₅	C₂×Dic₃	1	4	yes	—
127	C₂×#168	—	3	4	no	—
128	C₂×#168	—	3	2	no	—
129	C₂×#168	—	3	2	no	—
130	C₂×#168	—	6	2	no	—
131	#676	—	6	8	no	—
132	C₈×D₅	—	6	8	no	—
133	C₂×#131	—	15	2	no	—
134	#532	—	5	1	no	—
135	#435	—	15	8	no	—
136	#423	—	3	1	no	—
137	#408	—	15	2	no	—
138	C₂×#97	—	15	1	no	—
139	C₂²×C₃×C₅	—	2	2	no	—
140	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
141	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₃×D₅	—	2	4	no	—
143	C₂×C₃×D₅	—	2	8	no	—
144	C₃×C₄×C₅	—	2	2	no	—
145	C₃×C₄×C₅	—	2	2	no	—
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
149	C₃×Dic₅	—	2	4	no	—
150	C₃×#41	—	10	4	no	—
151	C₃×#41	—	10	4	no	—
152	C₂×C₃×Q₈	—	5	2	no	—
153	C₂×C₃×D₄	—	5	4	no	—
154	C₂×C₃×D₄	—	5	2	no	—
155	C₂×C₃×D₄	—	5	4	no	—
156	C₂²×C₃×C₄	—	5	4	no	—
157	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
158	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
159	C₃×C₄⋊C₄	—	5	4	no	—
160	C₃×C₂²⋊C₄	—	5	8	no	—
161	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
162	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
163	C₃×C₄²	C₂²×D₅	1	2	yes	—
164	#208	—	2	4	no	—
165	#208	—	10	8	no	—
166	C₂×C₃⋊C₈	—	5	4	no	—
167	C₂×C₃⋊C₈	—	5	2	no	—
168	C₂×C₃⋊C₈	—	5	2	no	—
169	C₂×C₃⋊C₈	—	10	8	no	—
170	C₂×C₃⋊C₈	—	10	2	no	—
171	C₂×C₃⋊C₈	C₂²×D₅	1	4	yes	—
172	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
173	C₂²×D₅	C₂²×Dic₃	1	4	yes	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
176	C₂×C₄×C₅	#237	1	2	yes	—
177	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
178	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
179	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
181	C₂²⋊D₅	—	2	8	no	—
182	C₂×Dic₅	C₂²×Dic₃	1	4	yes	—
183	D₂₀	—	2	4	no	—
184	D₂₀	—	4	4	no	—
185	C₄×D₅	—	2	4	no	—
186	C₄×D₅	—	2	4	no	—
187	Dic₁₀	—	2	4	no	—
188	C₅×C₈	—	6	4	no	—
189	#168	—	3	4	no	—
190	#168	—	6	2	no	—
191	C₂×#41	—	5	2	no	—
192	#131	—	30	8	no	—
193	C₂×M₁₆	—	3	2	no	—
194	C₂×M₁₆	—	15	4	no	—
195	C₂²×C₈	—	15	4	no	—
196	C₂²×C₈	—	15	1	no	—
197	C₂²×C₈	—	15	1	no	—
198	C₄⋊Q₈	—	5	1	no	—
199	#127	—	5	1	no	—
200	#124	—	5	2	no	—
201	#121	—	5	4	no	—
202	C₄×D₄	—	5	4	no	—
203	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
204	C₄⋊C₈	—	3	4	no	—
205	C₂²⋊C₈	—	15	8	no	—
206	#97	—	15	4	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
209	C₂×C₃×C₅	#568	1	2	yes	—
210	C₃×D₅	—	4	4	no	—
211	C₂³×C₃	—	5	4	no	—
212	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
213	C₃×Q₈	—	10	4	no	—
214	C₃×D₄	—	10	8	no	—
215	C₃×D₄	—	10	4	no	—
216	C₃×D₄	—	20	4	no	—
217	C₂×C₃×C₄	—	4	2	no	—
218	C₂×C₃×C₄	—	5	4	no	—
219	C₂×C₃×C₄	—	10	4	no	—
220	C₂×C₃×C₄	—	10	4	no	—
221	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
222	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
223	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
224	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
225	C₂×C₃×C₄	#713	1	2	yes	—
226	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
227	C₃⋊C₈	—	2	4	no	—
228	C₃⋊C₈	—	5	4	no	—
229	C₃⋊C₈	—	10	2	no	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	D₄×Dic₃	1	2	yes	—
232	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
233	C₂×D₅	—	2	4	no	—
234	C₂×D₅	—	2	8	no	—
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	D₄×Dic₃	1	2	yes	—
238	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
239	C₄×C₅	C₂×#237	1	1	yes	characteristic
240	Dic₅	—	2	4	no	—
241	#41	—	10	4	no	—
242	#41	—	10	4	no	—
243	C₂×Q₈	—	5	2	no	—
244	C₂×D₄	—	5	4	no	—
245	C₂×D₄	—	5	2	no	—
246	C₂×D₄	—	5	4	no	—
247	C₂²×C₄	—	5	4	no	—
248	C₂²×C₄	D₅×Dic₃	1	2	yes	—
249	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
250	M₁₆	—	6	4	no	—
251	M₁₆	—	30	8	no	—
252	C₂×C₈	—	3	4	no	—
253	C₂×C₈	—	15	4	no	—
254	C₂×C₈	—	15	2	no	—
255	C₂×C₈	—	15	2	no	—
256	C₂×C₈	—	30	8	no	—
257	C₂×C₈	—	30	2	no	—
258	C₄⋊C₄	—	5	4	no	—
259	C₂²⋊C₄	—	5	8	no	—
260	C₄²	D₅×Dic₃	1	1	yes	characteristic
261	C₄²	D₅×Dic₃	1	1	yes	characteristic
262	C₄²	C₂×D₃×D₅	1	2	yes	—
263	C₃×C₅	#2932	1	1	yes	characteristic, complemented, Hall
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	10	8	no	—
266	C₂²×C₃	—	10	4	no	—
267	C₂²×C₃	#4039	1	2	yes	—
268	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	10	4	no	—
272	C₃×C₄	#4039	1	2	yes	—
273	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
274	C₃×C₄	C₂×#713	1	1	yes	characteristic
275	C₂×C₅	—	2	2	no	—
276	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
277	C₂×C₅	#5819	1	2	yes	—
278	D₅	—	4	4	no	—
279	C₂³	—	5	4	no	—
280	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
281	Q₈	—	10	4	no	—
282	D₄	—	10	8	no	—
283	D₄	—	10	4	no	—
284	D₄	—	20	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	5	4	no	—
287	C₂×C₄	—	10	4	no	—
288	C₂×C₄	—	10	4	no	—
289	C₂×C₄	#6766	1	2	yes	—
290	C₂×C₄	#6777	1	2	yes	—
291	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
292	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
293	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
295	C₈	—	6	4	no	—
296	C₈	—	15	4	no	—
297	C₈	—	30	2	no	—
298	C₂×C₃	—	2	2	no	—
299	C₂×C₃	—	20	4	no	—
300	C₂×C₃	C₂×#4039	1	1	yes	characteristic
301	C₂×C₃	#66079	1	2	yes	—
302	C₅	#83276	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	2	2	no	—
304	C₂²	—	10	8	no	—
305	C₂²	—	10	4	no	—
306	C₂²	#91434	1	2	yes	—
307	C₂²	C₂²×D₅×Dic₃	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	10	4	no	—
311	C₄	#91434	1	2	yes	—
312	C₄	C₂×#6766	1	1	yes	characteristic
313	C₄	C₂×#6777	1	1	yes	characteristic
314	C₃	#10610773	1	1	yes	characteristic, complemented, Sylow
315	C₂	—	2	2	no	—
316	C₂	—	20	4	no	—
317	C₂	#11754120	1	2	yes	—
318	C₂	C₂×#91434	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	C₃×#65811	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#2932
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
2. C₂×#91284	C₂	semidirect
3. C₂×#91087	C₂	semidirect
4. C₅×#4924	C₂	semidirect
5. #11748343	C₂	semidirect
6. #11746997	C₂	semidirect
44. C₂×C₃⋊#168	D₄	semidirect
57. #4924	D₅	semidirect
263. C₃×C₅	#2932	semidirect
302. C₅	#83276	semidirect
314. C₃	#10610773	semidirect

Maximal subgroups

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

subgroup	index
C₃×#65811	2
C₂×#91284	2
C₂×#91087	2
C₅×#4924	2
#11748343	2
#11746997	2
#10610773	3
#83276	5