Subgroups — #12005102

Order 1280.

850 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	#10613793	C₂	1	1	yes	maximal, characteristic
3	C₄×#4072	C₂	1	1	yes	maximal, characteristic
4	C₂×#64777	C₂	1	4	yes	maximal
5	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×#4072	C₂²	1	2	yes	—
8	C₂×#4072	C₂²	1	2	yes	—
9	C₅×#423	—	2	2	no	—
10	C₅×#423	—	2	1	no	—
11	C₅×#423	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₄	1	4	yes	complemented
13	#65501	—	2	2	no	—
14	C₄×C₂²⋊Dic₅	—	2	2	no	—
15	#65225	—	2	1	no	—
16	#65224	—	2	1	no	—
17	C₄×C₄⋊Dic₅	—	2	2	no	—
18	C₄×C₄⋊Dic₅	C₄	1	4	yes	complemented
19	#65221	C₄	1	2	yes	complemented
20	C₄²×Dic₅	C₄	1	2	yes	complemented
21	#64777	—	2	8	no	—
22	#10836	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	D₄	1	2	yes	—
24	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	4	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	4	no	—
29	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	D₄	1	2	yes	—
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
39	C₂×C₂²⋊Dic₅	—	2	2	no	—
40	C₂×C₂²⋊Dic₅	—	2	2	no	—
41	#4072	—	2	2	no	—
42	#4072	—	2	2	no	—
43	#4072	—	2	4	no	—
44	#4072	—	2	4	no	—
45	#4072	D₄	1	4	yes	—
46	#4072	Q₈	1	4	yes	—
47	C₂×C₄⋊Dic₅	—	2	2	no	—
48	C₂×C₄⋊Dic₅	—	2	4	no	—
49	C₂×C₄⋊Dic₅	—	2	2	no	—
50	C₂×C₄×Dic₅	—	2	4	no	—
51	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
52	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
53	C₅×C₄⋊C₈	—	2	4	no	—
54	C₅×C₄⋊C₈	—	2	4	no	—
55	#3963	—	2	4	no	—
56	#3963	—	2	4	no	—
57	C₂×#423	D₅	1	1	yes	characteristic
58	#2285	—	5	1	no	—
59	C₄×#117	—	5	1	no	—
60	C₂×H(3,ℤ/4)	—	5	4	no	—
61	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	4	4	no	—
63	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₅	#41	1	2	yes	—
78	C₂²×C₄×C₅	#41	1	2	yes	—
79	C₂²×Dic₅	—	2	4	no	—
80	C₂²×Dic₅	—	2	4	no	—
81	C₅×M₁₆	—	4	4	no	—
82	C₅×M₁₆	—	4	2	no	—
83	C₅×M₁₆	—	4	2	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	8	no	—
86	C₄²×C₅	C₄²	1	2	yes	—
87	C₄²×C₅	C₄²	1	2	yes	—
88	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
91	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
92	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
93	C₄²×C₅	C₄⋊C₄	1	2	yes	—
94	C₄²×C₅	C₄⋊C₄	1	2	yes	—
95	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
96	C₂²⋊Dic₅	—	2	4	no	—
97	C₂²⋊Dic₅	—	2	4	no	—
98	C₂²⋊Dic₅	—	4	4	no	—
99	C₄⋊Dic₅	—	2	4	no	—
100	C₄⋊Dic₅	—	2	4	no	—
101	C₄⋊Dic₅	—	4	8	no	—
102	C₄⋊Dic₅	—	4	2	no	—
103	C₄⋊Dic₅	—	4	2	no	—
104	C₄×Dic₅	—	2	4	no	—
105	C₄×Dic₅	—	2	8	no	—
106	C₄×Dic₅	—	4	8	no	—
107	C₄×Dic₅	C₄⋊C₄	1	8	yes	complemented
108	C₂²×M₁₆	C₂×D₅	1	2	yes	—
109	C₂×#117	—	5	2	no	—
110	C₂×#117	—	5	2	no	—
111	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
112	#423	—	2	2	no	—
113	#423	—	2	1	no	—
114	#423	—	2	1	no	—
115	C₂×C₄⋊C₈	Dic₅	1	4	yes	complemented
116	#384	—	10	1	no	—
117	#382	—	10	1	no	—
118	#380	—	10	2	no	—
119	C₄×C₄⋊C₄	—	5	4	no	—
120	C₄×C₄⋊C₄	—	10	2	no	—
121	C₄×C₂²⋊C₄	—	10	2	no	—
122	#376	—	5	2	no	—
123	C₄³	—	5	2	no	—
124	H(3,ℤ/4)	—	10	8	no	—
125	C₂³×C₅	—	4	2	no	—
126	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂³×C₅	#121	1	2	yes	—
130	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	#95	1	2	yes	—
138	C₂×C₄×C₅	#95	1	2	yes	—
139	C₂×C₄×C₅	#95	1	2	yes	—
140	C₂×C₄×C₅	#95	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	#117	1	2	yes	—
147	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
150	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
151	C₂×C₄×C₅	#121	1	2	yes	—
152	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
153	C₂×C₄×C₅	#124	1	1	yes	characteristic
154	C₂×C₄×C₅	#124	1	1	yes	characteristic
155	C₂×C₄×C₅	#125	1	2	yes	—
156	C₂×C₄×C₅	#127	1	1	yes	characteristic
157	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
158	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
159	C₂×Dic₅	—	2	8	no	—
160	C₂×Dic₅	—	4	8	no	—
161	C₂×Dic₅	—	4	8	no	—
162	C₂×Dic₅	—	4	4	no	—
163	C₅×C₈	—	4	4	no	—
164	C₂³×C₄	D₂₀	1	2	yes	—
165	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂²×C₈	C₂×Dic₅	1	4	yes	—
171	#117	—	5	4	no	—
172	#117	—	5	4	no	—
173	#117	—	10	2	no	—
174	#117	—	10	2	no	—
175	#117	—	10	4	no	—
176	#117	—	10	4	no	—
177	C₂×C₄⋊C₄	—	10	2	no	—
178	C₂×C₄⋊C₄	—	10	4	no	—
179	C₂×C₄⋊C₄	—	10	2	no	—
180	C₂×C₂²⋊C₄	—	10	2	no	—
181	C₂×C₂²⋊C₄	—	10	2	no	—
182	C₂×C₄²	—	5	4	no	—
183	C₂×C₄²	—	5	4	no	—
184	C₂×C₄²	—	10	4	no	—
185	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
186	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
187	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
189	C₂×C₄²	C₄×D₅	1	2	yes	—
190	C₂×C₄²	C₂×Dic₅	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	2	4	no	—
196	#95	—	10	4	no	—
197	#95	—	10	4	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	—	4	2	no	—
202	C₂²×C₅	—	4	2	no	—
203	C₂²×C₅	C₂×#95	1	1	yes	characteristic
204	C₂²×C₅	#380	1	2	yes	—
205	C₂²×C₅	#389	1	2	yes	—
206	C₂²×C₅	#390	1	1	yes	characteristic
207	C₂²×C₅	#391	1	1	yes	characteristic
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	H(3,ℤ/4)	1	4	yes	complemented
212	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
213	C₄×C₅	#382	1	1	yes	characteristic
214	C₄×C₅	#384	1	1	yes	characteristic
215	C₄×C₅	#389	1	2	yes	—
216	C₄×C₅	#390	1	1	yes	characteristic
217	C₄×C₅	#391	1	1	yes	characteristic
218	Dic₅	—	4	8	no	—
219	Dic₅	—	8	4	no	—
220	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	10	4	no	—
223	C₂²×C₄	—	10	4	no	—
224	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
225	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
226	C₂²×C₄	#685	1	2	yes	—
227	C₂²×C₄	#685	1	2	yes	—
228	C₂²×C₄	#685	1	2	yes	—
229	C₂²×C₄	#685	1	2	yes	—
230	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₅	1	2	yes	—
234	C₂²×C₄	#711	1	2	yes	—
235	C₂²×C₄	Q₈×D₅	1	2	yes	—
236	C₂²×C₄	#713	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	4	no	—
242	M₁₆	—	4	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	8	no	—
245	C₄⋊C₄	—	10	4	no	—
246	C₄⋊C₄	—	10	4	no	—
247	C₄⋊C₄	—	20	8	no	—
248	C₄⋊C₄	—	20	2	no	—
249	C₄⋊C₄	—	20	2	no	—
250	C₂²⋊C₄	—	10	4	no	—
251	C₂²⋊C₄	—	10	4	no	—
252	C₂²⋊C₄	—	20	4	no	—
253	C₄²	—	5	8	no	—
254	C₄²	—	10	4	no	—
255	C₄²	—	10	8	no	—
256	C₄²	—	20	8	no	—
257	C₄²	C₄×Dic₅	1	2	yes	—
258	C₄²	C₄×Dic₅	1	2	yes	—
259	C₄²	#683	1	2	yes	—
260	C₄²	#683	1	1	yes	characteristic
261	C₄²	#683	1	1	yes	characteristic
262	C₄²	#683	1	1	yes	characteristic
263	C₄²	#683	1	2	yes	—
264	C₄²	#683	1	1	yes	characteristic
265	C₄²	C₂²⋊Dic₅	1	2	yes	—
266	C₄²	C₂²⋊Dic₅	1	2	yes	—
267	C₂×C₅	—	2	2	no	—
268	C₂×C₅	—	2	1	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
272	C₂×C₅	C₂×H(3,ℤ/4)	1	2	yes	—
273	C₂³	—	4	2	no	—
274	C₂³	#4041	1	2	yes	—
275	C₂³	#4043	1	2	yes	—
276	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
277	C₂³	C₂×#685	1	1	yes	characteristic
278	C₂³	C₂×#685	1	1	yes	characteristic
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	10	8	no	—
286	C₂×C₄	—	20	8	no	—
287	C₂×C₄	—	20	8	no	—
288	C₂×C₄	—	20	4	no	—
289	C₂×C₄	#3963	1	2	yes	—
290	C₂×C₄	#3963	1	2	yes	—
291	C₂×C₄	#3963	1	2	yes	—
292	C₂×C₄	#3963	1	2	yes	—
293	C₂×C₄	#4033	1	2	yes	—
294	C₂×C₄	#4034	1	2	yes	—
295	C₂×C₄	#4036	1	2	yes	—
296	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
297	C₂×C₄	#4038	1	2	yes	—
298	C₂×C₄	#4039	1	2	yes	—
299	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
300	C₂×C₄	C₂×#683	1	1	yes	characteristic
301	C₂×C₄	C₂×#683	1	1	yes	characteristic
302	C₂×C₄	D₄×Dic₅	1	2	yes	—
303	C₂×C₄	#4082	1	1	yes	characteristic
304	C₂×C₄	#4084	1	2	yes	—
305	C₂×C₄	#4086	1	1	yes	characteristic
306	C₂×C₄	#4090	1	1	yes	characteristic
307	C₂×C₄	Q₈×Dic₅	1	2	yes	—
308	C₂×C₄	#4092	1	2	yes	—
309	C₂×C₄	#4093	1	1	yes	characteristic
310	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
311	C₈	—	4	4	no	—
312	C₅	#10836	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	#65263	1	1	yes	characteristic
319	C₂²	#65274	1	2	yes	—
320	C₂²	#65276	1	2	yes	—
321	C₂²	#65277	1	1	yes	characteristic
322	C₂²	C₂×#3963	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	20	8	no	—
327	C₄	—	40	4	no	—
328	C₄	#64777	1	4	yes	complemented
329	C₄	#65262	1	2	yes	—
330	C₄	C₄⋊C₄×Dic₅	1	2	yes	—
331	C₄	#65265	1	1	yes	characteristic
332	C₄	#65266	1	1	yes	characteristic
333	C₄	#65510	1	1	yes	characteristic
334	C₄	#65519	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	2	1	no	—
337	C₂	—	2	1	no	—
338	C₂	—	2	2	no	—
339	C₂	#10602356	1	1	yes	characteristic
340	C₂	C₂×#64777	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂×C₅×#423	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂×D₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂×#423	D₅
hypocenter	C₅	#10836
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
12. C₂×C₅×C₄⋊C₈	C₄	semidirect
18. C₄×C₄⋊Dic₅	C₄	semidirect
19. #65221	C₄	semidirect
20. C₄²×Dic₅	C₄	semidirect
107. C₄×Dic₅	C₄⋊C₄	semidirect
115. C₂×C₄⋊C₈	Dic₅	semidirect
211. C₄×C₅	H(3,ℤ/4)	semidirect
312. C₅	#10836	semidirect
328. C₄	#64777	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#423	2
#10613793	2
C₄×#4072	2
C₂×#64777	2
#10836	5