Subgroups — #11989613

Order 1280.

700 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	#10613780	C₂	1	1	yes	maximal, characteristic
3	C₂×#65213	C₂	1	1	yes	maximal, characteristic
4	#10598111	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#117	—	2	1	no	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×#681	—	2	1	no	—
8	C₄×C₅×C₄⋊C₄	—	2	1	no	—
9	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
11	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
12	C₅×#376	C₄	1	2	yes	—
13	C₄³×C₅	—	2	1	no	—
14	#65496	—	2	1	no	—
15	#65215	—	2	1	no	—
16	#65215	C₄	1	2	yes	complemented
17	#65214	—	2	1	no	—
18	#65214	C₂²	1	2	yes	—
19	#65213	—	2	1	no	—
20	#65213	—	2	1	no	—
21	#65213	C₄	1	2	yes	complemented
22	#65213	C₄	1	2	yes	complemented
23	#65213	C₂²	1	2	yes	—
24	C₂×C₄⋊#168	—	2	1	no	—
25	C₂×C₄⋊#168	C₄	1	2	yes	complemented
26	C₄×#681	—	2	1	no	—
27	#8097	—	5	1	no	maximal, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
31	C₅×#117	—	2	8	no	—
32	C₂×C₅×C₄⋊C₄	—	2	2	no	—
33	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
34	C₂×C₄²×C₅	—	2	2	no	—
35	C₂×C₄²×C₅	—	2	2	no	—
36	C₂×C₄²×C₅	—	2	2	no	—
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
47	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₂×#681	—	2	2	no	—
50	C₂×#681	—	2	2	no	—
51	C₂×#681	—	2	2	no	—
52	C₂×#681	—	4	4	no	—
53	C₂²×#168	—	2	2	no	—
54	C₅×#95	—	2	2	no	—
55	C₅×#95	C₂×C₄	1	4	yes	—
56	C₂²⋊#168	—	2	2	no	—
57	C₂²⋊#168	—	2	2	no	—
58	C₄⋊#168	—	2	2	no	—
59	C₄⋊#168	—	2	2	no	—
60	C₄⋊#168	—	2	2	no	—
61	C₄⋊#168	—	2	2	no	—
62	C₄⋊#168	C₂×C₄	1	4	yes	—
63	C₄⋊#168	C₂×C₄	1	4	yes	—
64	#3935	—	2	2	no	—
65	C₄×#168	—	2	2	no	—
66	#2920	—	5	1	no	—
67	C₂×#423	—	5	1	no	—
68	C₄×#117	D₅	1	1	yes	characteristic
69	#1303	—	5	4	no	—
70	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	—	2	4	no	—
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	—	4	2	no	—
76	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	2	8	no	—
95	C₅×C₂²⋊C₄	—	2	8	no	—
96	C₄²×C₅	—	2	8	no	—
97	C₄²×C₅	—	4	4	no	—
98	C₄²×C₅	—	4	2	no	—
99	C₄²×C₅	—	4	2	no	—
100	C₄²×C₅	C₄²	1	1	yes	characteristic
101	C₄²×C₅	C₄²	1	1	yes	characteristic
102	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
106	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
112	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
116	#681	—	4	4	no	—
117	#681	—	4	4	no	—
118	#681	—	4	4	no	—
119	C₂×#168	—	2	4	no	—
120	C₂×#168	—	2	4	no	—
121	C₂×#168	—	4	4	no	—
122	C₂²×M₁₆	—	10	1	no	—
123	C₂×#117	—	2	1	no	—
124	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
125	#432	—	5	2	no	—
126	#432	—	10	1	no	—
127	#431	—	5	2	no	—
128	#431	—	10	1	no	—
129	#423	—	5	2	no	—
130	#423	—	5	2	no	—
131	#423	—	5	2	no	—
132	#423	—	10	1	no	—
133	#423	—	10	1	no	—
134	C₂×C₄⋊C₈	—	5	2	no	—
135	C₂×C₄⋊C₈	—	10	1	no	—
136	#407	—	10	1	no	—
137	C₄×M₁₆	—	10	1	no	—
138	C₄×C₄⋊C₄	—	2	1	no	—
139	C₄×C₄⋊C₄	C₂×D₅	1	2	yes	—
140	C₄×C₂²⋊C₄	—	2	1	no	—
141	C₄×C₂²⋊C₄	Dic₅	1	2	yes	—
142	#376	Dic₅	1	2	yes	—
143	C₄³	—	2	1	no	—
144	C₂³×C₅	—	2	2	no	—
145	C₂³×C₅	—	2	2	no	—
146	C₂³×C₅	—	4	1	no	—
147	C₂³×C₅	#95	1	1	yes	characteristic
148	C₂³×C₅	#95	1	1	yes	characteristic
149	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	2	1	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	8	no	—
166	C₂×C₄×C₅	—	4	1	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	—	4	4	no	—
170	C₂×C₄×C₅	—	4	4	no	—
171	C₂×C₄×C₅	#95	1	1	yes	characteristic
172	C₂×C₄×C₅	#95	1	1	yes	characteristic
173	C₂×C₄×C₅	#95	1	1	yes	characteristic
174	C₂×C₄×C₅	#95	1	1	yes	characteristic
175	C₂×C₄×C₅	#95	1	1	yes	characteristic
176	C₂×C₄×C₅	#95	1	1	yes	characteristic
177	C₂×C₄×C₅	#104	1	4	yes	—
178	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
185	C₂×C₄×C₅	#137	1	2	yes	—
186	#168	—	4	4	no	—
187	#168	—	4	4	no	—
188	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
189	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
190	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
191	C₂×M₁₆	—	10	2	no	—
192	C₂×M₁₆	—	10	2	no	—
193	C₂×M₁₆	—	10	2	no	—
194	C₂×M₁₆	—	20	4	no	—
195	C₂²×C₈	—	10	2	no	—
196	#117	—	2	8	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
203	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
204	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
205	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
207	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
208	C₂×C₄²	D₂₀	1	1	yes	characteristic
209	C₂×C₄²	D₂₀	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
214	C₄⋊C₈	—	5	4	no	—
215	C₄⋊C₈	—	5	4	no	—
216	C₄⋊C₈	—	10	2	no	—
217	C₄⋊C₈	—	10	2	no	—
218	C₄⋊C₈	—	10	2	no	—
219	C₄⋊C₈	—	10	2	no	—
220	C₂²⋊C₈	—	10	2	no	—
221	C₂²⋊C₈	—	10	2	no	—
222	#97	—	10	2	no	—
223	C₄×C₈	—	10	2	no	—
224	#95	—	2	2	no	—
225	#95	C₂×Dic₅	1	4	yes	—
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	2	1	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	2	1	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	4	4	no	—
234	C₂²×C₅	#339	1	2	yes	—
235	C₂²×C₅	C₂×#95	1	1	yes	characteristic
236	C₂²×C₅	#417	1	2	yes	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	4	8	no	—
245	C₄×C₅	#339	1	2	yes	—
246	C₄×C₅	#344	1	2	yes	—
247	C₄×C₅	#418	1	1	yes	characteristic
248	C₄×C₅	#419	1	1	yes	characteristic
249	C₄×C₅	C₂×#104	1	2	yes	—
250	C₄×C₅	#428	1	2	yes	—
251	C₂⁴	C₂²⋊Dic₅	1	1	yes	characteristic
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
258	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
259	C₂²×C₄	#683	1	1	yes	characteristic
260	C₂²×C₄	#683	1	1	yes	characteristic
261	C₂²×C₄	#683	1	1	yes	characteristic
262	C₂²×C₄	#683	1	1	yes	characteristic
263	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
264	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
265	C₂²×C₄	#685	1	1	yes	characteristic
266	C₂²×C₄	#685	1	1	yes	characteristic
267	C₂²×C₄	#685	1	1	yes	characteristic
268	C₂²×C₄	#685	1	1	yes	characteristic
269	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
270	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
271	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
272	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
275	M₁₆	—	20	4	no	—
276	M₁₆	—	20	4	no	—
277	M₁₆	—	20	4	no	—
278	C₂×C₈	—	10	4	no	—
279	C₂×C₈	—	10	4	no	—
280	C₂×C₈	—	20	4	no	—
281	C₄⋊C₄	—	2	8	no	—
282	C₂²⋊C₄	—	2	8	no	—
283	C₄²	—	2	8	no	—
284	C₄²	—	4	4	no	—
285	C₄²	—	4	2	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₄×Dic₅	1	1	yes	characteristic
288	C₄²	C₄×Dic₅	1	1	yes	characteristic
289	C₄²	#683	1	1	yes	characteristic
290	C₄²	#683	1	1	yes	characteristic
291	C₄²	#683	1	1	yes	characteristic
292	C₄²	#683	1	1	yes	characteristic
293	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
294	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
295	C₄²	#685	1	1	yes	characteristic
296	C₄²	#685	1	1	yes	characteristic
297	C₄²	#685	1	1	yes	characteristic
298	C₄²	#685	1	1	yes	characteristic
299	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
300	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
301	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
302	C₄²	C₂×D₂₀	1	1	yes	characteristic
303	C₂×C₅	—	2	2	no	—
304	C₂×C₅	—	2	2	no	—
305	C₂×C₅	—	4	1	no	—
306	C₂×C₅	C₂×#339	1	1	yes	characteristic
307	C₂×C₅	#1736	1	1	yes	characteristic
308	C₂×C₅	C₂×#344	1	1	yes	characteristic
309	C₂³	—	2	2	no	—
310	C₂³	—	2	2	no	—
311	C₂³	—	4	1	no	—
312	C₂³	#3963	1	1	yes	characteristic
313	C₂³	#3963	1	1	yes	characteristic
314	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	8	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	—	4	4	no	—
336	C₂×C₄	#3937	1	4	yes	—
337	C₂×C₄	#3963	1	1	yes	characteristic
338	C₂×C₄	#3963	1	1	yes	characteristic
339	C₂×C₄	#3963	1	1	yes	characteristic
340	C₂×C₄	#3963	1	1	yes	characteristic
341	C₂×C₄	#3963	1	1	yes	characteristic
342	C₂×C₄	#3963	1	1	yes	characteristic
343	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
344	C₂×C₄	C₂×#683	1	1	yes	characteristic
345	C₂×C₄	C₂×#683	1	1	yes	characteristic
346	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
347	C₂×C₄	C₂×#685	1	1	yes	characteristic
348	C₂×C₄	C₂×#685	1	1	yes	characteristic
349	C₂×C₄	#4095	1	2	yes	—
350	C₂×C₄	#4097	1	2	yes	—
351	C₈	—	20	4	no	—
352	C₈	—	20	4	no	—
353	C₅	#8097	1	1	yes	characteristic, complemented, Sylow
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	4	4	no	—
362	C₂²	#64743	1	2	yes	—
363	C₂²	C₂×#3963	1	1	yes	characteristic
364	C₂²	#65521	1	2	yes	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	4	8	no	—
373	C₄	#64743	1	2	yes	—
374	C₄	#64753	1	2	yes	—
375	C₄	C₂×#3937	1	2	yes	—
376	C₄	#65284	1	2	yes	—
377	C₄	#65290	1	1	yes	characteristic
378	C₄	#65293	1	1	yes	characteristic
379	C₂	—	2	2	no	—
380	C₂	—	2	2	no	—
381	C₂	—	4	1	no	—
382	C₂	C₂×#64743	1	1	yes	characteristic
383	C₂	#10602374	1	1	yes	characteristic
384	C₂	C₂×#64753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#685
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×#117	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×#117	D₅
hypocenter	C₅	#8097
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
4. #10598111	C₂	semidirect
16. #65215	C₄	semidirect
21. #65213	C₄	semidirect
22. #65213	C₄	semidirect
25. C₂×C₄⋊#168	C₄	semidirect
353. C₅	#8097	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#117	2
#10613780	2
C₂×#65213	2
#10598111	2
#8097	5