Subgroups — C₂×#10601722 #12252741

Order 1280.

710 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#444	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#65482	C₂	1	1	yes	maximal, characteristic
3	C₂×#65000	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#64786	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601722	C₂	1	8	yes	maximal, direct product factor
6	C₂×#64715	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64714	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#64686	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₂²	1	1	yes	characteristic
10	C₂×C₅×#131	—	2	1	no	—
11	C₂²×D₄₀	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₀	—	2	1	no	—
13	C₅×#444	—	2	2	no	—
14	C₅×#444	C₂²	1	4	yes	complemented
15	C₂×C₅×#108	C₂²	1	1	yes	characteristic
16	C₂×C₅×#104	—	2	1	no	—
17	C₂×C₅×#97	C₂²	1	1	yes	characteristic
18	#65482	C₄	1	4	yes	complemented
19	#65482	C₂²	1	4	yes	—
20	C₂×#3960	C₄	1	2	yes	complemented
21	C₂×#3958	C₄	1	2	yes	complemented
22	C₂×#3953	—	2	1	no	—
23	C₂×#3949	C₂²	1	1	yes	characteristic
24	C₂×#3944	C₂²	1	1	yes	characteristic
25	C₂×#3944	C₂²	1	1	yes	characteristic
26	#65000	—	2	2	no	—
27	#65000	C₂²	1	4	yes	complemented
28	#64786	C₂²	1	4	yes	complemented
29	#64715	C₂²	1	4	yes	complemented
30	#64714	C₂²	1	4	yes	complemented
31	#64686	C₂²	1	4	yes	complemented
32	C₂×AGL(1,ℤ/16)	—	5	1	no	maximal, Sylow
33	C₂×C₅×#41	—	2	1	no	—
34	C₂²×D₂₀	—	2	1	no	—
35	C₂²×C₄×D₅	—	4	1	no	—
36	C₅×#135	—	2	1	no	—
37	C₅×#135	—	2	1	no	—
38	C₅×#135	C₂×C₄	1	2	yes	complemented
39	C₅×#135	C₂³	1	2	yes	—
40	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
41	C₂×C₅×SD₁₆	—	2	1	no	—
42	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic, complemented
43	C₅×#131	—	2	4	no	—
44	C₅×#131	—	4	2	no	—
45	C₂×C₅×M₁₆	—	2	1	no	—
46	C₂×C₅×M₁₆	—	4	1	no	—
47	C₂²×C₅×C₈	—	4	1	no	—
48	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
51	C₂×#685	—	4	1	no	—
52	C₂×C₄⋊Dic₅	—	2	1	no	—
53	C₂×D₄₀	—	2	1	no	—
54	C₂×D₄₀	—	2	1	no	—
55	C₂×D₄₀	—	2	1	no	—
56	C₂×D₄₀	C₂×C₄	1	2	yes	—
57	C₂×D₄₀	C₂×C₄	1	2	yes	—
58	C₂×D₄₀	C₂³	1	2	yes	—
59	C₄×D₂₀	—	2	4	no	—
60	C₄×D₂₀	—	4	2	no	—
61	C₅×#108	—	2	1	no	—
62	C₅×#108	C₂³	1	2	yes	—
63	C₅×#104	—	2	4	no	—
64	C₅×#97	D₄	1	2	yes	—
65	C₅×#97	C₂³	1	2	yes	—
66	#3960	C₂×C₄	1	8	yes	complemented
67	#3958	C₂×C₄	1	8	yes	complemented
68	#3953	—	2	4	no	—
69	#3949	—	2	1	no	—
70	#3949	C₂³	1	2	yes	—
71	#3944	—	2	1	no	—
72	#3944	C₂×C₄	1	2	yes	complemented
73	#3944	C₂³	1	2	yes	—
74	#3944	C₂³	1	2	yes	—
75	C₂×#672	—	2	1	no	—
76	C₂×#672	C₂×C₄	1	2	yes	complemented
77	C₂×#509	—	5	1	no	—
78	C₂×#444	D₅	1	1	yes	characteristic, complemented
79	C₂×#442	—	5	1	no	—
80	AGL(1,ℤ/16)	—	5	8	no	—
81	C₂×#365	—	5	1	no	—
82	C₂×#361	—	5	1	no	—
83	C₂×#360	—	5	1	no	—
84	C₂×#347	—	5	1	no	—
85	C₂³×D₅	—	4	1	no	—
86	C₅×#41	—	2	2	no	—
87	C₅×#41	—	2	2	no	—
88	C₅×#41	—	4	1	no	—
89	C₅×#41	—	4	1	no	—
90	C₂×C₅×Q₈	—	2	1	no	—
91	C₂×C₅×D₄	—	2	1	no	—
92	C₂×C₅×D₄	—	4	1	no	—
93	C₂²×C₄×C₅	—	2	1	no	—
94	C₂²×C₄×C₅	—	4	1	no	—
95	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
96	C₂²×Dic₅	—	4	1	no	—
97	C₂×D₂₀	—	2	2	no	—
98	C₂×D₂₀	—	2	2	no	—
99	C₂×D₂₀	—	2	2	no	—
100	C₂×D₂₀	—	4	1	no	—
101	C₂×D₂₀	—	4	1	no	—
102	C₂×D₂₀	—	4	1	no	—
103	C₂×C₄×D₅	—	4	4	no	—
104	C₂×C₄×D₅	—	8	4	no	—
105	C₅×Q₁₆	—	2	1	no	—
106	C₅×Q₁₆	C₂²×C₄	1	2	yes	complemented
107	C₅×SD₁₆	—	2	2	no	—
108	C₅×SD₁₆	—	2	2	no	—
109	C₅×D₈	—	2	1	no	—
110	C₅×D₈	C₂²×C₄	1	2	yes	complemented
111	C₅×M₁₆	—	2	2	no	—
112	C₅×M₁₆	—	4	4	no	—
113	C₅×M₁₆	—	4	1	no	—
114	C₂×C₅×C₈	—	2	1	no	—
115	C₂×C₅×C₈	—	2	1	no	—
116	C₂×C₅×C₈	—	4	4	no	—
117	C₂×C₅×C₈	—	4	1	no	—
118	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
121	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₅×C₈	#41	1	1	yes	characteristic
123	C₂×C₅×C₈	#41	1	1	yes	characteristic
124	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
125	C₄²×C₅	—	2	1	no	—
126	C₄²×C₅	C₂×D₄	1	2	yes	—
127	#685	—	4	4	no	—
128	C₄⋊Dic₅	—	2	2	no	—
129	C₄⋊Dic₅	—	4	1	no	—
130	D₄₀	—	2	2	no	—
131	D₄₀	—	2	2	no	—
132	D₄₀	—	2	2	no	—
133	D₄₀	C₂²×C₄	1	4	yes	—
134	#672	—	2	2	no	—
135	#672	C₂²×C₄	1	4	yes	complemented
136	C₂×#135	C₂×D₅	1	1	yes	characteristic
137	C₂²×D₈	—	5	1	no	—
138	C₂×#131	—	2	1	no	—
139	C₂×C₄×D₄	—	10	1	no	—
140	#509	—	5	4	no	—
141	#509	—	5	4	no	—
142	C₂×SD₃₂	—	5	2	no	—
143	C₂×D₁₆	—	5	2	no	—
144	C₂×M₃₂	—	5	1	no	—
145	C₂×M₃₂	—	5	1	no	—
146	#444	—	2	2	no	—
147	#444	C₂×D₅	1	4	yes	complemented
148	#442	—	5	4	no	—
149	#442	—	10	2	no	—
150	C₂×#108	C₂×D₅	1	1	yes	characteristic
151	C₂×#106	—	5	1	no	—
152	C₂×#104	—	2	1	no	—
153	C₂×#102	—	10	1	no	—
154	C₂×#97	C₂×D₅	1	1	yes	characteristic
155	#365	—	5	4	no	—
156	#361	—	5	4	no	—
157	#360	—	5	4	no	—
158	#347	—	5	4	no	—
159	C₂³×C₅	—	4	1	no	—
160	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂²×D₅	—	4	2	no	—
162	C₂²×D₅	—	4	2	no	—
163	C₂²×D₅	—	4	2	no	—
164	C₂²×D₅	—	8	2	no	—
165	C₂²×D₅	—	8	2	no	—
166	C₅×Q₈	—	2	2	no	—
167	C₅×Q₈	—	4	1	no	—
168	C₅×D₄	—	2	2	no	—
169	C₅×D₄	—	4	2	no	—
170	C₅×D₄	—	4	2	no	—
171	C₅×D₄	—	4	1	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	4	1	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
183	C₂×Dic₅	—	4	2	no	—
184	C₂×Dic₅	—	8	2	no	—
185	D₂₀	—	2	4	no	—
186	D₂₀	—	4	2	no	—
187	D₂₀	—	4	2	no	—
188	D₂₀	—	4	2	no	—
189	C₄×D₅	—	8	8	no	—
190	C₅×C₈	—	2	2	no	—
191	C₅×C₈	—	4	2	no	—
192	C₅×C₈	C₄×D₄	1	1	yes	characteristic
193	C₅×C₈	C₄×D₄	1	1	yes	characteristic
194	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
195	C₅×C₈	C₂×#41	1	1	yes	characteristic
196	C₂×#41	—	2	1	no	—
197	C₂²×D₄	—	10	1	no	—
198	C₂³×C₄	—	20	1	no	—
199	#135	—	2	1	no	—
200	#135	—	2	1	no	—
201	#135	C₄×D₅	1	2	yes	complemented
202	#135	C₂²×D₅	1	2	yes	—
203	C₂×Q₁₆	C₄×D₅	1	1	yes	characteristic, complemented
204	C₂×SD₁₆	—	2	1	no	—
205	C₂×D₈	—	5	2	no	—
206	C₂×D₈	—	5	2	no	—
207	C₂×D₈	—	5	2	no	—
208	C₂×D₈	—	10	1	no	—
209	C₂×D₈	—	10	1	no	—
210	C₂×D₈	—	10	1	no	—
211	C₂×D₈	C₄×D₅	1	1	yes	characteristic, complemented
212	#131	—	2	4	no	—
213	#131	—	4	2	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	4	1	no	—
216	C₂²×C₈	—	4	1	no	—
217	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
219	C₄×D₄	—	10	4	no	—
220	C₄×D₄	—	20	2	no	—
221	C₂×C₄⋊C₄	—	10	1	no	—
222	C₂×C₂²⋊C₄	—	20	1	no	—
223	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
224	SD₃₂	—	5	8	no	—
225	D₁₆	—	5	8	no	—
226	M₃₂	—	5	2	no	—
227	M₃₂	—	5	2	no	—
228	M₃₂	—	5	2	no	—
229	M₃₂	—	10	1	no	—
230	C₂×C₁₆	—	5	2	no	—
231	C₂×C₁₆	—	10	1	no	—
232	#108	—	2	1	no	—
233	#108	C₂²×D₅	1	2	yes	—
234	#106	—	5	2	no	—
235	#106	—	10	1	no	—
236	#104	—	2	4	no	—
237	#102	—	10	4	no	—
238	#97	C₂²⋊D₅	1	2	yes	—
239	#97	C₂²×D₅	1	2	yes	—
240	C₂²×C₅	—	2	2	no	—
241	C₂²×C₅	—	4	2	no	—
242	C₂²×C₅	—	8	2	no	—
243	C₂²×C₅	#442	1	1	yes	characteristic
244	C₂²×C₅	#442	1	1	yes	characteristic
245	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
246	C₂×D₅	—	4	4	no	—
247	C₂×D₅	—	8	2	no	—
248	C₂×D₅	—	8	2	no	—
249	C₂×D₅	—	8	4	no	—
250	C₂×D₅	—	8	4	no	—
251	C₄×C₅	—	4	2	no	—
252	C₄×C₅	—	4	2	no	—
253	C₄×C₅	#442	1	1	yes	characteristic
254	C₄×C₅	#442	1	1	yes	characteristic
255	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
256	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
257	Dic₅	—	8	2	no	—
258	C₂⁴	—	20	1	no	—
259	#41	—	2	2	no	—
260	#41	—	2	2	no	—
261	#41	—	4	1	no	—
262	#41	—	4	1	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	10	2	no	—
267	C₂×D₄	—	10	2	no	—
268	C₂×D₄	—	10	2	no	—
269	C₂×D₄	—	20	1	no	—
270	C₂×D₄	—	20	1	no	—
271	C₂×D₄	—	20	1	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	20	4	no	—
275	C₂²×C₄	—	20	1	no	—
276	C₂²×C₄	—	40	4	no	—
277	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
278	Q₁₆	—	2	1	no	—
279	Q₁₆	C₂×C₄×D₅	1	2	yes	complemented
280	SD₁₆	—	2	2	no	—
281	SD₁₆	—	2	2	no	—
282	D₈	—	2	1	no	—
283	D₈	—	5	4	no	—
284	D₈	—	10	2	no	—
285	D₈	—	10	2	no	—
286	D₈	—	10	2	no	—
287	D₈	C₂×C₄×D₅	1	2	yes	complemented
288	M₁₆	—	2	2	no	—
289	M₁₆	—	4	4	no	—
290	M₁₆	—	4	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	4	4	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
297	C₂×C₈	#709	1	1	yes	characteristic
298	C₂×C₈	#709	1	1	yes	characteristic
299	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
300	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
301	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
302	C₄⋊C₄	—	10	2	no	—
303	C₄⋊C₄	—	20	1	no	—
304	C₂²⋊C₄	—	20	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
307	C₁₆	—	5	4	no	—
308	C₁₆	—	10	2	no	—
309	C₂×C₅	—	2	1	no	—
310	C₂×C₅	—	2	1	no	—
311	C₂×C₅	—	8	2	no	—
312	C₂×C₅	AGL(1,ℤ/16)	1	2	yes	complemented
313	C₂×C₅	C₂×#442	1	1	yes	characteristic
314	D₅	—	8	4	no	—
315	C₂³	—	4	1	no	—
316	C₂³	—	20	2	no	—
317	C₂³	—	20	2	no	—
318	C₂³	—	20	2	no	—
319	C₂³	—	40	2	no	—
320	C₂³	—	40	2	no	—
321	C₂³	#4095	1	1	yes	characteristic
322	Q₈	—	2	2	no	—
323	Q₈	—	4	1	no	—
324	D₄	—	2	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	10	4	no	—
329	D₄	—	20	2	no	—
330	D₄	—	20	2	no	—
331	D₄	—	20	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	20	2	no	—
338	C₂×C₄	—	40	8	no	—
339	C₂×C₄	—	40	2	no	—
340	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
341	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
342	C₂×C₄	#4078	1	1	yes	characteristic
343	C₂×C₄	#4078	1	1	yes	characteristic
344	C₂×C₄	#4095	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	4	2	no	—
348	C₈	C₄×C₂²⋊D₅	1	1	yes	characteristic
349	C₈	C₄×C₂²⋊D₅	1	1	yes	characteristic
350	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
351	C₈	C₂×#709	1	1	yes	characteristic
352	C₅	C₂×AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
353	C₂²	—	2	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	20	4	no	—
357	C₂²	—	40	2	no	—
358	C₂²	—	40	2	no	—
359	C₂²	—	40	4	no	—
360	C₂²	—	40	4	no	—
361	C₂²	#65303	1	1	yes	characteristic
362	C₂²	#65303	1	1	yes	characteristic
363	C₂²	C₂×#4095	1	1	yes	characteristic
364	C₄	—	4	2	no	—
365	C₄	—	4	2	no	—
366	C₄	—	40	2	no	—
367	C₄	#65303	1	1	yes	characteristic
368	C₄	#65303	1	1	yes	characteristic
369	C₄	C₂×C₄×C₂²⋊D₅	1	1	yes	characteristic
370	C₄	C₂×#4078	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	8	2	no	—
374	C₂	—	40	4	no	—
375	C₂	#10601722	1	2	yes	direct product factor
376	C₂	C₂×#65303	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65303
commutator	C₅×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#444	C₂
socle	C₂²×C₅	#442
prime radical	C₂×#65482	C₂
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#444	D₅
hypocenter	C₅	C₂×AGL(1,ℤ/16)
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#444

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#444	C₂	semidirect
3. C₂×#65000	C₂	semidirect
4. C₂×#64786	C₂	semidirect
5. #10601722	C₂	direct (N × H)
6. C₂×#64715	C₂	semidirect
7. C₂×#64714	C₂	semidirect
8. C₂×#64686	C₂	semidirect
14. C₅×#444	C₂²	semidirect
18. #65482	C₄	semidirect
20. C₂×#3960	C₄	semidirect
21. C₂×#3958	C₄	semidirect
27. #65000	C₂²	semidirect
28. #64786	C₂²	semidirect
29. #64715	C₂²	semidirect
30. #64714	C₂²	semidirect
31. #64686	C₂²	semidirect
38. C₅×#135	C₂×C₄	semidirect
40. C₂×C₅×Q₁₆	C₂×C₄	semidirect
42. C₂×C₅×D₈	C₂×C₄	semidirect
66. #3960	C₂×C₄	semidirect
67. #3958	C₂×C₄	semidirect
72. #3944	C₂×C₄	semidirect
76. C₂×#672	C₂×C₄	semidirect
78. C₂×#444	D₅	semidirect
106. C₅×Q₁₆	C₂²×C₄	semidirect
110. C₅×D₈	C₂²×C₄	semidirect
135. #672	C₂²×C₄	semidirect
147. #444	C₂×D₅	semidirect
201. #135	C₄×D₅	semidirect
203. C₂×Q₁₆	C₄×D₅	semidirect
211. C₂×D₈	C₄×D₅	semidirect
279. Q₁₆	C₂×C₄×D₅	semidirect
287. D₈	C₂×C₄×D₅	semidirect
312. C₂×C₅	AGL(1,ℤ/16)	semidirect
352. C₅	C₂×AGL(1,ℤ/16)	semidirect
375. C₂	#10601722	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#444	2
C₂×#65482	2
C₂×#65000	2
C₂×#64786	2
#10601722	2
C₂×#64715	2
C₂×#64714	2
C₂×#64686	2
C₂×AGL(1,ℤ/16)	5

Subgroups — C2×#10601722 #12252741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601722 #12252741