Subgroups — #14307148

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5470	C₂	1	1	yes	maximal, characteristic
2	C₄×#6710	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	#11745977	C₂	1	4	yes	maximal
5	#10602634	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×#6710	C₂²	1	2	yes	—
8	C₅×#926	C₂²	1	2	yes	—
9	C₅×#924	—	2	1	no	—
10	C₅×#924	—	2	1	no	—
11	C₅×#870	C₄	1	2	yes	complemented
12	C₅×#869	C₄	1	1	yes	characteristic, complemented
13	C₅×#869	C₄	1	1	yes	characteristic, complemented
14	C₄×C₅×Dic₆	—	2	1	no	—
15	C₄×C₅×Dic₆	—	2	1	no	—
16	C₂×C₃×C₄⋊Dic₅	—	2	1	no	—
17	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
18	#91087	—	2	1	no	—
19	C₄×C₃⋊#168	—	2	1	no	—
20	C₃×#3963	—	2	1	no	—
21	#69493	—	5	1	no	maximal, Hall
22	C₅×#533	—	3	1	no	—
23	C₄×C₄⋊Dic₅	D₃	1	1	yes	characteristic
24	C₄×#681	—	3	1	no	—
25	#64752	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₅×Dic₆	—	2	1	no	—
30	C₂×C₅×Dic₆	—	2	1	no	—
31	C₂²×C₃×Dic₅	—	2	2	no	—
32	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
36	#6710	—	2	2	no	—
37	#6710	Q₈	1	4	yes	complemented
38	C₂×C₃⋊#168	—	2	2	no	—
39	C₅×#217	—	2	2	no	—
40	C₅×C₄⋊Dic₃	—	2	1	no	—
41	C₅×C₄⋊Dic₃	—	2	1	no	—
42	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
43	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
44	C₅×Dic₃⋊C₄	—	2	2	no	—
45	C₅×Dic₃⋊C₄	—	2	2	no	—
46	C₄×C₅×Dic₃	—	2	2	no	—
47	C₃×C₄⋊Dic₅	—	2	4	no	—
48	C₃×C₄×Dic₅	—	2	4	no	—
49	#5470	D₅	1	1	yes	characteristic
50	C₃×C₄×C₄⋊C₄	—	5	1	no	—
51	C₄×#208	—	5	1	no	—
52	#4523	—	5	4	no	—
53	C₅×C₄⋊Q₈	—	3	1	no	—
54	C₅×C₄⋊Q₈	—	3	1	no	—
55	C₅×#125	—	3	2	no	—
56	C₅×C₂²⋊Q₈	—	6	1	no	—
57	C₅×C₂²⋊Q₈	—	6	1	no	—
58	C₄×C₅×Q₈	—	6	1	no	—
59	C₄×C₅×Q₈	—	6	1	no	—
60	C₅×#117	—	3	2	no	—
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	C₂×C₄⋊Dic₅	—	2	1	no	—
63	C₂×C₄×Dic₅	C₂×D₃	1	2	yes	—
64	C₂×#681	—	3	2	no	—
65	#3963	—	2	1	no	—
66	#3935	—	6	1	no	—
67	C₄×#168	—	6	1	no	—
68	#1986	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	4	1	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	C₂×C₅×Dic₃	—	2	2	no	—
81	C₂×C₅×Dic₃	—	2	2	no	—
82	C₅×Dic₆	—	4	2	no	—
83	C₅×Dic₆	—	4	2	no	—
84	C₂×C₃×Dic₅	—	2	4	no	—
85	C₂×C₃×Dic₅	—	4	2	no	—
86	C₂×C₃×Dic₅	—	4	2	no	—
87	C₃⋊#168	—	2	4	no	—
88	C₂×C₃×C₄⋊C₄	—	10	1	no	—
89	C₂×C₃×C₄²	—	5	2	no	—
90	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
91	#926	C₂×D₅	1	2	yes	—
92	#924	—	2	1	no	—
93	#924	—	2	1	no	—
94	C₂×#208	—	5	2	no	—
95	#870	Dic₅	1	2	yes	complemented
96	#869	Dic₅	1	1	yes	characteristic, complemented
97	#869	Dic₅	1	1	yes	characteristic, complemented
98	C₄×Dic₆	—	2	1	no	—
99	C₄×Dic₆	—	2	1	no	—
100	C₃×#95	—	10	1	no	—
101	#803	—	10	1	no	—
102	C₄×C₃⋊C₈	—	10	1	no	—
103	C₂×C₅×Q₈	—	6	1	no	—
104	C₂×C₅×Q₈	—	6	1	no	—
105	C₂²×C₄×C₅	#67	1	1	yes	characteristic
106	C₂²×C₄×C₅	#67	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
108	C₂²×Dic₅	—	2	2	no	—
109	C₅×C₄⋊C₄	—	3	2	no	—
110	C₅×C₄⋊C₄	—	3	2	no	—
111	C₅×C₄⋊C₄	—	6	2	no	—
112	C₅×C₄⋊C₄	—	6	2	no	—
113	C₅×C₄⋊C₄	—	6	1	no	—
114	C₅×C₄⋊C₄	—	6	1	no	—
115	C₅×C₂²⋊C₄	—	6	2	no	—
116	C₄²×C₅	—	6	2	no	—
117	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
118	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
119	C₄²×C₅	D₁₂	1	1	yes	characteristic
120	C₄²×C₅	D₁₂	1	1	yes	characteristic
121	C₄⋊Dic₅	—	2	4	no	—
122	C₄×Dic₅	—	2	4	no	—
123	#681	—	3	4	no	—
124	#681	—	6	2	no	—
125	C₂×#168	—	6	2	no	—
126	#533	—	3	1	no	—
127	C₄×M₁₆	—	15	1	no	—
128	C₄×C₄⋊C₄	—	5	1	no	—
129	#339	—	15	4	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₃×C₅	#124	1	1	yes	characteristic
134	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	#104	1	2	yes	—
139	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
140	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
141	C₃×Dic₅	—	4	4	no	—
142	C₅×Dic₃	—	4	2	no	—
143	C₅×Dic₃	—	4	2	no	—
144	C₂²×C₃×C₄	—	10	2	no	—
145	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
146	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
148	C₂×Dic₆	—	2	1	no	—
149	C₂×Dic₆	—	2	1	no	—
150	C₃×C₄⋊C₄	—	10	4	no	—
151	C₃×C₄²	—	10	4	no	—
152	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
153	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
154	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
155	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
156	#217	—	2	2	no	—
157	C₄⋊Dic₃	—	2	1	no	—
158	C₄⋊Dic₃	—	2	1	no	—
159	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
160	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
161	Dic₃⋊C₄	—	2	2	no	—
162	Dic₃⋊C₄	—	2	2	no	—
163	C₄×Dic₃	—	2	2	no	—
164	#208	—	5	4	no	—
165	#208	—	10	2	no	—
166	C₂×C₃⋊C₈	—	10	2	no	—
167	C₂³×C₅	C₂×#67	1	1	yes	characteristic
168	C₅×Q₈	—	12	2	no	—
169	C₅×Q₈	—	12	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	6	2	no	—
173	C₂×C₄×C₅	—	6	2	no	—
174	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
180	C₂×C₄×C₅	#237	1	2	yes	—
181	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
182	C₂×Dic₅	—	2	4	no	—
183	C₂×Dic₅	—	4	2	no	—
184	C₂×Dic₅	—	4	2	no	—
185	#168	—	6	4	no	—
186	C₂×M₁₆	—	15	2	no	—
187	C₄⋊Q₈	—	3	1	no	—
188	C₄⋊Q₈	—	3	1	no	—
189	#125	—	3	2	no	—
190	C₂²⋊Q₈	—	6	1	no	—
191	C₂²⋊Q₈	—	6	1	no	—
192	C₄×Q₈	—	6	1	no	—
193	C₄×Q₈	—	6	1	no	—
194	#117	—	3	2	no	—
195	C₂×C₄⋊C₄	—	10	1	no	—
196	C₂×C₄²	—	5	2	no	—
197	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
198	#97	—	30	1	no	—
199	C₄×C₈	—	30	1	no	—
200	#95	—	10	1	no	—
201	C₂×C₃×C₅	—	2	1	no	—
202	C₂×C₃×C₅	—	2	1	no	—
203	C₂×C₃×C₅	#391	1	1	yes	characteristic
204	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
205	C₂×C₃×C₅	#421	1	1	yes	characteristic
206	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	4	1	no	—
209	C₂×C₃×C₄	—	10	4	no	—
210	C₂×C₃×C₄	—	20	2	no	—
211	C₂×C₃×C₄	—	20	2	no	—
212	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
216	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
217	C₂×C₃×C₄	#713	1	2	yes	—
218	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
220	C₂×Dic₃	—	2	2	no	—
221	C₂×Dic₃	—	2	2	no	—
222	Dic₆	—	4	2	no	—
223	Dic₆	—	4	2	no	—
224	C₃⋊C₈	—	10	4	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
228	C₂²×C₅	#936	1	1	yes	characteristic
229	C₂²×C₅	#944	1	1	yes	characteristic
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	12	2	no	—
234	C₄×C₅	—	12	2	no	—
235	C₄×C₅	#805	1	2	yes	—
236	C₄×C₅	#887	1	2	yes	—
237	C₄×C₅	#897	1	2	yes	—
238	Dic₅	—	4	4	no	—
239	C₂×Q₈	—	6	1	no	—
240	C₂×Q₈	—	6	1	no	—
241	C₂²×C₄	—	10	2	no	—
242	C₂²×C₄	#1201	1	1	yes	characteristic
243	C₂²×C₄	#1201	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
245	M₁₆	—	15	4	no	—
246	M₁₆	—	30	2	no	—
247	C₂×C₈	—	30	2	no	—
248	C₄⋊C₄	—	3	2	no	—
249	C₄⋊C₄	—	3	2	no	—
250	C₄⋊C₄	—	6	2	no	—
251	C₄⋊C₄	—	6	2	no	—
252	C₄⋊C₄	—	6	1	no	—
253	C₄⋊C₄	—	6	1	no	—
254	C₄⋊C₄	—	10	4	no	—
255	C₂²⋊C₄	—	6	2	no	—
256	C₄²	—	6	2	no	—
257	C₄²	—	10	4	no	—
258	C₄²	D₃×Dic₅	1	1	yes	characteristic
259	C₄²	D₃×Dic₅	1	1	yes	characteristic
260	C₄²	#1203	1	1	yes	characteristic
261	C₄²	#1203	1	1	yes	characteristic
262	C₃×C₅	#1986	1	1	yes	characteristic, complemented, Hall
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	#4090	1	1	yes	characteristic
266	C₂²×C₃	#4093	1	1	yes	characteristic
267	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	20	4	no	—
272	C₃×C₄	#3969	1	2	yes	—
273	C₃×C₄	Q₈×Dic₅	1	2	yes	—
274	C₃×C₄	#4092	1	2	yes	—
275	Dic₃	—	4	2	no	—
276	Dic₃	—	4	2	no	—
277	C₂×C₅	—	2	1	no	—
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	C₂×#805	1	1	yes	characteristic
280	C₂×C₅	#4973	1	1	yes	characteristic
281	C₂×C₅	#5005	1	1	yes	characteristic
282	C₂³	C₂×#1201	1	1	yes	characteristic
283	Q₈	—	12	2	no	—
284	Q₈	—	12	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	6	2	no	—
288	C₂×C₄	—	6	2	no	—
289	C₂×C₄	—	10	4	no	—
290	C₂×C₄	—	20	2	no	—
291	C₂×C₄	—	20	2	no	—
292	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
293	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
294	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
295	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
296	C₂×C₄	#6766	1	2	yes	—
297	C₂×C₄	D₁₅⋊Q₈	1	2	yes	—
298	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
299	C₂×C₄	C₂×#1203	1	1	yes	characteristic
300	C₈	—	30	4	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	#65519	1	1	yes	characteristic
304	C₂×C₃	C₂×#3969	1	1	yes	characteristic
305	C₂×C₃	#65525	1	1	yes	characteristic
306	C₅	#69493	1	1	yes	characteristic, complemented, Sylow
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	#91368	1	1	yes	characteristic
310	C₂²	#91385	1	1	yes	characteristic
311	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	12	2	no	—
316	C₄	—	12	2	no	—
317	C₄	—	20	4	no	—
318	C₄	#90979	1	2	yes	—
319	C₄	#91344	1	2	yes	—
320	C₄	#91383	1	2	yes	—
321	C₃	#10602634	1	1	yes	characteristic, complemented, Sylow
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	C₂×#90979	1	1	yes	characteristic
325	C₂	#11748425	1	1	yes	characteristic
326	C₂	#11748607	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1203
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1986
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×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
11. C₅×#870	C₄	semidirect
12. C₅×#869	C₄	semidirect
13. C₅×#869	C₄	semidirect
37. #6710	Q₈	semidirect
95. #870	Dic₅	semidirect
96. #869	Dic₅	semidirect
97. #869	Dic₅	semidirect
262. C₃×C₅	#1986	semidirect
306. C₅	#69493	semidirect
321. C₃	#10602634	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5470	2
C₄×#6710	2
C₃×C₄×C₄⋊Dic₅	2
#11745977	2
#10602634	3
#69493	5