Subgroups — #14285309

Order 1920.

492 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5716	C₂	1	1	yes	maximal, characteristic
2	#11747011	C₂	1	1	yes	maximal, characteristic, complemented
3	#11746875	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64764	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745858	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745857	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602602	—	3	1	no	maximal, Hall
9	C₂×C₅×#237	—	2	1	no	—
10	C₃×C₅×#127	C₂²	1	1	yes	characteristic
11	C₅×#938	—	2	1	no	—
12	C₅×#938	C₄	1	2	yes	—
13	C₅×#936	—	2	1	no	—
14	C₅×D₄×Dic₃	—	2	1	no	—
15	C₅×D₄×Dic₃	C₄	1	2	yes	—
16	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₅×#869	C₂²	1	1	yes	characteristic
18	C₂×#6650	—	2	1	no	—
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
23	Dic₃⋊#168	—	2	1	no	—
24	(C₂×D₃)⋊#168	—	2	1	no	—
25	#90953	—	2	1	no	—
26	#68388	—	5	1	no	maximal, Hall
27	C₅×#532	—	3	1	no	—
28	#65215	—	3	1	no	—
29	#65213	—	3	1	no	—
30	#64767	—	3	1	no	—
31	#64767	—	3	1	no	—
32	#64764	—	3	1	no	—
33	#64764	D₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₃×C₅×D₄	—	2	1	no	—
36	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
37	C₂×C₅×#67	—	2	2	no	—
38	C₂²×C₅×Dic₃	—	2	2	no	—
39	C₅×#237	—	4	2	no	—
40	C₅×#237	—	4	2	no	—
41	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
43	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
44	C₂×C₅×Dic₆	—	2	1	no	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₅×#217	—	2	2	no	—
49	C₅×#217	—	2	2	no	—
50	C₅×C₄⋊Dic₃	—	2	1	no	—
51	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
52	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
54	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
55	C₂×C₃×#168	—	2	1	no	—
56	C₂×C₃×#168	—	2	1	no	—
57	#6650	—	4	2	no	—
58	#5716	D₅	1	1	yes	characteristic
59	#4837	—	5	1	no	—
60	#4835	—	5	1	no	—
61	C₃×#331	—	5	1	no	—
62	#4544	—	5	1	no	—
63	#4486	—	5	1	no	—
64	#4482	—	5	1	no	—
65	C₂×C₅×#41	—	6	1	no	—
66	C₅×C₄⋊Q₈	—	3	1	no	—
67	C₅×#127	C₂×D₃	1	1	yes	characteristic
68	C₅×#124	—	6	1	no	—
69	C₅×#121	—	3	2	no	—
70	C₅×#121	—	6	1	no	—
71	C₄×C₅×D₄	—	3	2	no	—
72	C₄×C₅×D₄	—	6	1	no	—
73	C₂×C₄²×C₅	—	3	1	no	—
74	C₂×#681	—	6	1	no	—
75	C₂²⋊#168	—	6	1	no	—
76	C₄⋊#168	—	3	1	no	—
77	C₄⋊#168	—	3	1	no	—
78	C₄⋊#168	—	6	1	no	—
79	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
80	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
81	#3935	—	6	1	no	—
82	#1557	—	15	1	no	Sylow
83	C₂³×C₃×C₅	—	2	2	no	—
84	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₃×C₅×D₄	—	4	2	no	—
86	C₃×C₅×D₄	—	4	1	no	—
87	C₃×C₅×D₄	—	4	1	no	—
88	C₃×C₅×D₄	—	4	1	no	—
89	C₃×C₅×D₄	—	4	1	no	—
90	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₅×#67	—	4	4	no	—
94	C₂×C₅×Dic₃	—	2	2	no	—
95	C₂×C₅×Dic₃	—	4	4	no	—
96	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₅×D₃	—	2	1	no	—
100	C₄×C₅×D₃	—	2	1	no	—
101	C₄×C₅×D₃	—	2	1	no	—
102	C₄×C₅×D₃	—	2	1	no	—
103	C₄×C₅×D₃	—	4	1	no	—
104	C₅×Dic₆	—	4	1	no	—
105	C₅×Dic₆	—	4	1	no	—
106	C₃⋊#168	—	4	1	no	—
107	C₃⋊#168	—	4	1	no	—
108	C₃×#168	—	4	1	no	—
109	C₃×#168	—	4	1	no	—
110	C₂×#237	—	2	1	no	—
111	C₃×#127	C₂×D₅	1	1	yes	characteristic
112	#938	—	2	1	no	—
113	#938	Dic₅	1	2	yes	—
114	#936	—	2	1	no	—
115	D₄×Dic₃	—	2	1	no	—
116	D₄×Dic₃	Dic₅	1	2	yes	—
117	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
118	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
119	#869	C₂×D₅	1	1	yes	characteristic
120	C₃×C₄⋊C₈	—	5	1	no	—
121	C₃×C₄⋊C₈	—	5	1	no	—
122	(C₂×D₃)⋊C₈	—	10	1	no	—
123	#815	—	10	1	no	—
124	Dic₃⋊C₈	—	10	1	no	—
125	C₄⋊(C₃⋊C₈)	—	5	1	no	—
126	C₄⋊(C₃⋊C₈)	—	5	1	no	—
127	C₅×#41	—	12	2	no	—
128	C₅×#41	—	12	2	no	—
129	C₂×C₅×Q₈	—	6	1	no	—
130	C₂×C₅×D₄	—	2	1	no	—
131	C₂×C₅×D₄	—	2	1	no	—
132	C₂×C₅×D₄	—	6	2	no	—
133	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
134	C₂²×C₄×C₅	—	3	1	no	—
135	C₂²×C₄×C₅	—	3	1	no	—
136	C₂²×C₄×C₅	—	3	1	no	—
137	C₂²×C₄×C₅	—	6	2	no	—
138	C₅×C₄⋊C₄	—	3	2	no	—
139	C₅×C₄⋊C₄	—	6	1	no	—
140	C₅×C₂²⋊C₄	—	6	2	no	—
141	C₅×C₂²⋊C₄	—	6	2	no	—
142	C₄²×C₅	—	3	1	no	—
143	C₄²×C₅	—	3	1	no	—
144	C₄²×C₅	—	3	1	no	—
145	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
146	#681	—	12	2	no	—
147	C₂×#168	—	2	1	no	—
148	C₂×#168	—	2	1	no	—
149	C₂×#168	—	6	1	no	—
150	C₂×#168	—	6	1	no	—
151	#532	—	3	1	no	—
152	#432	—	15	1	no	—
153	#423	—	15	1	no	—
154	#332	—	15	1	no	—
155	#332	—	15	1	no	—
156	#331	—	5	1	no	—
157	#331	—	15	1	no	—
158	C₂²×C₃×C₅	—	4	2	no	—
159	C₂²×C₃×C₅	—	4	2	no	—
160	C₂²×C₃×C₅	—	4	2	no	—
161	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₅×D₃	—	2	1	no	—
163	C₂×C₅×D₃	—	2	1	no	—
164	C₂×C₅×D₃	—	2	1	no	—
165	C₃×C₄×C₅	—	2	1	no	—
166	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₃×C₄×C₅	#137	1	1	yes	characteristic
170	C₅×Dic₃	—	2	1	no	—
171	C₅×Dic₃	—	2	1	no	—
172	C₅×Dic₃	—	2	1	no	—
173	C₅×Dic₃	—	4	2	no	—
174	C₂×C₃×D₄	—	2	1	no	—
175	C₂×C₃×D₄	—	2	1	no	—
176	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
177	C₂×#67	—	2	2	no	—
178	C₂²×Dic₃	—	2	2	no	—
179	#237	—	4	2	no	—
180	#237	—	4	2	no	—
181	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
182	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
183	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
184	C₂×Dic₆	—	2	1	no	—
185	C₂×C₃×C₈	—	10	1	no	—
186	C₂×C₃×C₈	—	10	1	no	—
187	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
188	#217	—	2	2	no	—
189	#217	—	2	2	no	—
190	C₄⋊Dic₃	—	2	1	no	—
191	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
192	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
193	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
194	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
195	C₂×C₃⋊C₈	—	10	1	no	—
196	C₂×C₃⋊C₈	—	10	1	no	—
197	ΓL(1,𝔽₂₅)	—	20	2	no	—
198	C₂³×C₅	—	2	2	no	—
199	C₂³×C₅	—	3	1	no	—
200	C₅×Q₈	—	12	1	no	—
201	C₅×Q₈	—	12	1	no	—
202	C₅×D₄	—	4	2	no	—
203	C₅×D₄	—	4	1	no	—
204	C₅×D₄	—	4	1	no	—
205	C₅×D₄	—	4	1	no	—
206	C₅×D₄	—	4	1	no	—
207	C₅×D₄	—	12	4	no	—
208	C₂×C₄×C₅	—	3	1	no	—
209	C₂×C₄×C₅	—	3	1	no	—
210	C₂×C₄×C₅	—	3	1	no	—
211	C₂×C₄×C₅	—	6	1	no	—
212	C₂×C₄×C₅	—	6	1	no	—
213	C₂×C₄×C₅	—	6	1	no	—
214	C₂×C₄×C₅	—	6	1	no	—
215	C₂×C₄×C₅	—	6	2	no	—
216	C₂×C₄×C₅	—	12	1	no	—
217	C₂×C₄×C₅	—	12	4	no	—
218	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
219	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
220	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
221	#168	—	4	1	no	—
222	#168	—	4	1	no	—
223	#168	—	12	1	no	—
224	#168	—	12	1	no	—
225	C₂×#41	—	6	1	no	—
226	C₂×M₁₆	—	30	1	no	—
227	C₄⋊Q₈	—	3	1	no	—
228	#127	D₃×D₅	1	1	yes	characteristic
229	#124	—	6	1	no	—
230	#121	—	3	2	no	—
231	#121	—	6	1	no	—
232	C₄×D₄	—	3	2	no	—
233	C₄×D₄	—	6	1	no	—
234	C₂×C₄²	—	3	1	no	—
235	C₄⋊C₈	—	5	1	no	—
236	C₄⋊C₈	—	5	1	no	—
237	C₄⋊C₈	—	15	1	no	—
238	C₄⋊C₈	—	15	1	no	—
239	C₄⋊C₈	—	30	1	no	—
240	C₂²⋊C₈	—	30	1	no	—
241	#97	—	30	1	no	—
242	C₂×C₃×C₅	—	8	2	no	—
243	C₂×C₃×C₅	#413	1	1	yes	characteristic
244	C₂×C₃×C₅	#417	1	1	yes	characteristic
245	C₂×C₃×C₅	#418	1	1	yes	characteristic
246	C₅×D₃	—	4	1	no	—
247	C₂³×C₃	—	2	2	no	—
248	C₂²×D₃	C₂²⋊Dic₅	1	1	yes	characteristic
249	C₃×D₄	—	4	2	no	—
250	C₃×D₄	—	4	1	no	—
251	C₃×D₄	—	4	1	no	—
252	C₃×D₄	—	4	1	no	—
253	C₃×D₄	—	4	1	no	—
254	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
255	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
256	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
257	#67	—	4	4	no	—
258	C₂×Dic₃	—	2	2	no	—
259	C₂×Dic₃	—	4	4	no	—
260	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
261	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
262	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
263	C₄×D₃	—	2	1	no	—
264	C₄×D₃	—	2	1	no	—
265	C₄×D₃	—	2	1	no	—
266	C₄×D₃	—	2	1	no	—
267	C₄×D₃	—	4	1	no	—
268	Dic₆	—	4	1	no	—
269	Dic₆	—	4	1	no	—
270	C₃×C₈	—	20	1	no	—
271	C₃×C₈	—	20	1	no	—
272	C₃⋊C₈	—	20	1	no	—
273	C₃⋊C₈	—	20	1	no	—
274	C₂²×C₅	—	4	2	no	—
275	C₂²×C₅	—	4	2	no	—
276	C₂²×C₅	—	4	2	no	—
277	C₂²×C₅	—	6	1	no	—
278	C₂²×C₅	—	6	1	no	—
279	C₂²×C₅	—	6	1	no	—
280	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
281	C₄×C₅	—	2	1	no	—
282	C₄×C₅	—	6	1	no	—
283	C₄×C₅	—	6	1	no	—
284	C₄×C₅	—	6	1	no	—
285	C₄×C₅	—	12	2	no	—
286	C₄×C₅	#911	1	1	yes	characteristic
287	C₄×C₅	#911	1	1	yes	characteristic
288	C₄×C₅	#914	1	1	yes	characteristic
289	C₄×C₅	#915	1	1	yes	characteristic
290	#41	—	12	2	no	—
291	#41	—	12	2	no	—
292	C₂×Q₈	—	6	1	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	6	2	no	—
296	C₂×D₄	D₃×Dic₅	1	2	yes	—
297	C₂²×C₄	—	3	1	no	—
298	C₂²×C₄	—	3	1	no	—
299	C₂²×C₄	—	3	1	no	—
300	C₂²×C₄	—	6	2	no	—
301	M₁₆	—	60	2	no	—
302	C₂×C₈	—	10	1	no	—
303	C₂×C₈	—	10	1	no	—
304	C₂×C₈	—	30	1	no	—
305	C₂×C₈	—	30	1	no	—
306	C₄⋊C₄	—	3	2	no	—
307	C₄⋊C₄	—	6	1	no	—
308	C₂²⋊C₄	—	6	2	no	—
309	C₂²⋊C₄	—	6	2	no	—
310	C₄²	—	3	1	no	—
311	C₄²	—	3	1	no	—
312	C₄²	—	3	1	no	—
313	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
314	C₃×C₅	#1557	1	1	yes	characteristic, complemented, Hall
315	C₂²×C₃	—	4	2	no	—
316	C₂²×C₃	—	4	2	no	—
317	C₂²×C₃	—	4	2	no	—
318	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
319	C₂×D₃	—	2	1	no	—
320	C₂×D₃	—	2	1	no	—
321	C₂×D₃	—	2	1	no	—
322	C₃×C₄	—	2	1	no	—
323	C₃×C₄	#4078	1	1	yes	characteristic
324	C₃×C₄	#4078	1	1	yes	characteristic
325	C₃×C₄	#4095	1	1	yes	characteristic
326	C₃×C₄	#4097	1	1	yes	characteristic
327	Dic₃	—	2	1	no	—
328	Dic₃	—	2	1	no	—
329	Dic₃	—	2	1	no	—
330	Dic₃	—	4	2	no	—
331	C₂×C₅	—	8	2	no	—
332	C₂×C₅	—	12	1	no	—
333	C₂×C₅	#4745	1	1	yes	characteristic
334	C₂×C₅	#4770	1	1	yes	characteristic
335	C₂×C₅	#4771	1	1	yes	characteristic
336	C₂³	—	2	2	no	—
337	C₂³	—	3	1	no	—
338	Q₈	—	12	1	no	—
339	Q₈	—	12	1	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	1	no	—
342	D₄	—	4	1	no	—
343	D₄	—	4	1	no	—
344	D₄	—	4	1	no	—
345	D₄	—	12	4	no	—
346	C₂×C₄	—	3	1	no	—
347	C₂×C₄	—	3	1	no	—
348	C₂×C₄	—	3	1	no	—
349	C₂×C₄	—	6	1	no	—
350	C₂×C₄	—	6	1	no	—
351	C₂×C₄	—	6	1	no	—
352	C₂×C₄	—	6	1	no	—
353	C₂×C₄	—	6	2	no	—
354	C₂×C₄	—	12	1	no	—
355	C₂×C₄	—	12	4	no	—
356	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
357	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
358	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
359	C₈	—	20	1	no	—
360	C₈	—	20	1	no	—
361	C₈	—	60	1	no	—
362	C₈	—	60	1	no	—
363	C₂×C₃	—	8	2	no	—
364	C₂×C₃	#65504	1	1	yes	characteristic
365	C₂×C₃	#65521	1	1	yes	characteristic
366	C₂×C₃	#65526	1	1	yes	characteristic
367	D₃	—	4	1	no	—
368	C₅	#68388	1	1	yes	characteristic, complemented, Sylow
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	6	1	no	—
373	C₂²	—	6	1	no	—
374	C₂²	—	6	1	no	—
375	C₂²	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	6	1	no	—
378	C₄	—	6	1	no	—
379	C₄	—	6	1	no	—
380	C₄	—	12	2	no	—
381	C₄	#91480	1	1	yes	characteristic
382	C₄	#91486	1	1	yes	characteristic
383	C₄	#91494	1	1	yes	characteristic
384	C₄	#91496	1	1	yes	characteristic
385	C₃	#10602602	1	1	yes	characteristic, complemented, Sylow
386	C₂	—	8	2	no	—
387	C₂	—	12	1	no	—
388	C₂	#11748955	1	1	yes	characteristic
389	C₂	#11748967	1	1	yes	characteristic
390	C₂	#11749126	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×C₂²⋊Dic₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×#127	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5716	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#127	D₃×D₅
hypocenter	C₃×C₅	#1557
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. #11747011	C₂	semidirect
3. #11746875	C₂	semidirect
4. #11746365	C₂	semidirect
5. C₃×#64764	C₂	semidirect
6. #11745858	C₂	semidirect
7. #11745857	C₂	semidirect
33. #64764	D₃	semidirect
314. C₃×C₅	#1557	semidirect
368. C₅	#68388	semidirect
385. C₃	#10602602	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5716	2
#11747011	2
#11746875	2
#11746365	2
C₃×#64764	2
#11745858	2
#11745857	2
#10602602	3
#68388	5