Subgroups — C₂×#10603999 #13043548

Order 1280.

662 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	C₂×#65791	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#65725	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#65113	C₂	1	1	yes	maximal, characteristic
4	C₂×#64932	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64898	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#64861	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
8	#10603999	C₂	1	8	yes	maximal, direct product factor
9	C₂×#4096	C₄	1	1	yes	characteristic
10	C₂×#4094	C₂²	1	1	yes	characteristic
11	C₂×#4053	C₂²	1	1	yes	characteristic
12	#65791	—	2	2	no	—
13	#65791	C₂²	1	4	yes	complemented
14	C₂×#3998	—	2	1	no	—
15	C₂×#3998	C₂²	1	1	yes	characteristic
16	C₂×#3998	C₂²	1	1	yes	characteristic
17	C₂²×#674	—	2	1	no	—
18	C₂²×#674	C₂²	1	1	yes	characteristic
19	C₂²×#674	C₂²	1	1	yes	characteristic
20	#65725	—	2	2	no	—
21	#65725	C₂²	1	4	yes	complemented
22	C₂×#3990	—	2	1	no	—
23	C₂×D₅⋊C₁₆	—	2	1	no	—
24	C₂×C₅×#104	C₄	1	1	yes	characteristic
25	C₂×#3954	C₄	1	1	yes	characteristic
26	C₂×#3937	C₄	1	1	yes	characteristic
27	C₂×C₄×#168	C₂²	1	1	yes	characteristic
28	#65113	C₄	1	4	yes	—
29	#65113	C₂²	1	4	yes	—
30	#64932	C₂²	1	4	yes	complemented
31	#64898	C₂²	1	4	yes	complemented
32	#64861	C₂²	1	4	yes	complemented
33	C₄×#674	C₂²	1	4	yes	complemented
34	C₂×#2170	—	5	1	no	maximal, Sylow
35	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
36	C₂×#709	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	#4096	C₈	1	4	yes	—
40	#4096	C₂×C₄	1	4	yes	—
41	#4094	—	2	1	no	—
42	#4094	—	2	1	no	—
43	#4094	C₂×C₄	1	2	yes	—
44	#4094	C₂³	1	2	yes	—
45	C₂×#689	C₈	1	1	yes	characteristic
46	C₂×#688	C₈	1	1	yes	characteristic
47	C₂×#687	C₈	1	1	yes	characteristic
48	C₂×#686	C₈	1	1	yes	characteristic
49	C₂×#681	—	2	1	no	—
50	C₂×#681	C₂×C₄	1	1	yes	characteristic
51	C₂²×#168	—	2	1	no	—
52	C₂²×#168	C₂×C₄	1	1	yes	characteristic
53	C₂²×#168	C₂³	1	1	yes	characteristic
54	#4053	—	2	1	no	—
55	#4053	—	2	1	no	—
56	#4053	C₂×C₄	1	2	yes	—
57	#4053	C₂³	1	2	yes	—
58	C₂×#676	—	2	1	no	—
59	C₂×C₈×D₅	—	2	1	no	—
60	#3998	—	2	1	no	—
61	#3998	—	2	1	no	—
62	#3998	—	2	4	no	—
63	#3998	C₂³	1	2	yes	—
64	#3998	C₂³	1	2	yes	—
65	C₂×#674	—	2	1	no	—
66	C₂×#674	—	2	1	no	—
67	C₂×#674	—	2	1	no	—
68	C₂×#674	—	2	1	no	—
69	C₂×#674	—	2	4	no	—
70	C₂×#674	D₄	1	1	yes	characteristic
71	C₂×#674	D₄	1	1	yes	characteristic
72	C₂×#674	D₄	1	2	yes	—
73	C₂×#674	C₂³	1	2	yes	—
74	C₂×#674	C₂³	1	2	yes	—
75	#3990	—	2	4	no	—
76	D₅⋊C₁₆	—	2	4	no	—
77	C₅×#104	C₂×C₄	1	4	yes	—
78	#3954	C₂×C₄	1	4	yes	—
79	#3937	C₂×C₄	1	4	yes	—
80	C₄×#168	C₂×C₄	1	2	yes	—
81	C₄×#168	C₂³	1	2	yes	—
82	C₂×#504	—	5	1	no	—
83	C₂×#504	—	5	1	no	—
84	C₂×#443	—	5	1	no	—
85	#2170	—	5	8	no	—
86	C₂×#364	—	5	1	no	—
87	C₂×#350	—	5	1	no	—
88	C₂×#350	—	5	1	no	—
89	C₂×C₄×C₁₆	—	5	1	no	—
90	C₅×#41	—	2	1	no	—
91	C₅×#41	—	2	1	no	—
92	C₅×#41	C₂×C₈	1	2	yes	—
93	C₅×#41	C₂²×C₄	1	2	yes	—
94	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
95	C₂×C₅×D₄	—	2	1	no	—
96	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
97	C₂²×C₄×C₅	—	2	1	no	—
98	C₂²×C₄×C₅	—	2	1	no	—
99	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₂²⋊D₅	—	2	1	no	—
101	#709	—	2	1	no	—
102	#709	—	2	1	no	—
103	#709	C₂×C₈	1	2	yes	—
104	#709	C₂²×C₄	1	2	yes	—
105	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
106	C₂×C₄×D₅	—	2	1	no	—
107	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
108	C₅×M₁₆	—	2	1	no	—
109	C₅×M₁₆	C₂²×C₄	1	2	yes	—
110	C₂×C₅×C₈	—	2	1	no	—
111	C₄²×C₅	—	2	1	no	—
112	C₄²×C₅	C₂²×C₄	1	2	yes	—
113	#689	C₂×C₈	1	4	yes	—
114	#688	C₂×C₈	1	4	yes	—
115	#687	C₂×C₈	1	4	yes	—
116	#686	C₂×C₈	1	4	yes	—
117	#681	—	2	1	no	—
118	#681	—	2	2	no	—
119	#681	—	2	2	no	—
120	#681	C₂²×C₄	1	2	yes	—
121	C₂×#168	—	2	1	no	—
122	C₂×#168	—	2	2	no	—
123	C₂×#168	—	2	2	no	—
124	C₂×#168	C₂×C₈	1	2	yes	—
125	C₂×#168	C₂×C₈	1	1	yes	characteristic
126	C₂×#168	C₂×C₈	1	1	yes	characteristic
127	C₂×#168	C₂²×C₄	1	1	yes	characteristic
128	C₂×#168	C₂²×C₄	1	2	yes	—
129	C₂×#168	C₂×D₄	1	1	yes	characteristic
130	C₂×#168	C₂×D₄	1	1	yes	characteristic
131	C₂×#168	#41	1	1	yes	characteristic
132	C₂×#168	#41	1	1	yes	characteristic
133	C₂×#168	C₂⁴	1	1	yes	characteristic
134	#676	—	2	2	no	—
135	#676	—	2	2	no	—
136	C₈×D₅	—	2	2	no	—
137	C₈×D₅	—	2	2	no	—
138	#674	—	2	2	no	—
139	#674	—	2	2	no	—
140	#674	—	2	2	no	—
141	#674	C₂×D₄	1	2	yes	—
142	#674	C₂×D₄	1	2	yes	—
143	C₂×#135	—	5	1	no	—
144	C₂×#131	—	5	1	no	—
145	C₂×#131	—	5	1	no	—
146	#504	—	5	4	no	—
147	#504	—	5	4	no	—
148	#504	—	10	2	no	—
149	#504	—	10	2	no	—
150	C₂×M₃₂	—	5	1	no	—
151	C₂×M₃₂	—	5	1	no	—
152	C₂×M₃₂	—	10	1	no	—
153	C₂×M₃₂	—	10	1	no	—
154	C₂²×C₁₆	—	5	1	no	—
155	C₂²×C₁₆	—	5	1	no	—
156	C₂²×C₁₆	—	10	1	no	—
157	C₂²×C₁₆	—	10	1	no	—
158	#443	—	5	4	no	—
159	#443	—	5	4	no	—
160	C₂×#108	—	5	1	no	—
161	C₂×#104	—	5	1	no	—
162	C₂×#104	AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₂×C₄×C₈	—	5	1	no	—
164	#364	—	5	4	no	—
165	#350	—	5	4	no	—
166	#350	—	5	4	no	—
167	C₄×C₁₆	—	5	4	no	—
168	C₂³×C₅	—	2	1	no	—
169	C₂³×C₅	#131	1	1	yes	characteristic
170	C₂²×D₅	—	2	1	no	—
171	C₅×Q₈	—	2	1	no	—
172	C₅×Q₈	C₂²×C₈	1	2	yes	—
173	C₅×D₄	—	2	2	no	—
174	C₅×D₄	—	2	2	no	—
175	C₅×D₄	—	2	1	no	—
176	C₅×D₄	C₂²×C₈	1	2	yes	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	1	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
186	C₂×C₄×C₅	#131	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
188	C₂²⋊D₅	—	2	2	no	—
189	C₂²⋊D₅	—	2	2	no	—
190	C₂×Dic₅	—	2	1	no	—
191	D₂₀	—	2	1	no	—
192	D₂₀	C₂²×C₈	1	2	yes	—
193	C₄×D₅	—	2	2	no	—
194	C₄×D₅	—	2	2	no	—
195	Dic₁₀	—	2	1	no	—
196	Dic₁₀	C₂²×C₈	1	2	yes	—
197	C₅×C₈	—	2	2	no	—
198	#168	—	2	2	no	—
199	#168	C₄×D₄	1	1	yes	characteristic
200	#168	C₄×D₄	1	1	yes	characteristic
201	#168	C₂²×C₈	1	2	yes	—
202	#168	C₂²×C₈	1	2	yes	—
203	#168	C₂²×D₄	1	1	yes	characteristic
204	#168	C₂×#41	1	1	yes	characteristic
205	C₂×#41	—	5	1	no	—
206	C₂×#41	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
207	#135	—	5	4	no	—
208	#135	—	5	4	no	—
209	C₂×Q₁₆	—	5	1	no	—
210	C₂×SD₁₆	—	5	1	no	—
211	C₂×SD₁₆	—	5	1	no	—
212	C₂×D₈	—	5	1	no	—
213	#131	—	5	2	no	—
214	#131	—	5	2	no	—
215	#131	—	5	2	no	—
216	#131	—	5	2	no	—
217	#131	—	10	1	no	—
218	#131	—	10	1	no	—
219	#131	—	10	1	no	—
220	#131	—	10	1	no	—
221	C₂×M₁₆	—	5	1	no	—
222	C₂×M₁₆	—	10	1	no	—
223	C₂×M₁₆	—	10	1	no	—
224	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
225	C₂²×C₈	—	5	1	no	—
226	C₂²×C₈	—	5	1	no	—
227	C₂²×C₈	—	10	1	no	—
228	C₂²×C₈	—	10	1	no	—
229	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
230	M₃₂	—	5	2	no	—
231	M₃₂	—	5	2	no	—
232	M₃₂	—	10	4	no	—
233	M₃₂	—	10	4	no	—
234	M₃₂	—	10	1	no	—
235	M₃₂	—	10	1	no	—
236	C₂×C₁₆	—	5	2	no	—
237	C₂×C₁₆	—	5	1	no	—
238	C₂×C₁₆	—	5	1	no	—
239	C₂×C₁₆	—	5	2	no	—
240	C₂×C₁₆	—	5	2	no	—
241	C₂×C₁₆	—	10	4	no	—
242	C₂×C₁₆	—	10	4	no	—
243	C₂×C₁₆	—	10	1	no	—
244	C₂×C₁₆	—	10	1	no	—
245	C₂×C₁₆	—	10	1	no	—
246	C₂×C₁₆	—	10	1	no	—
247	#108	—	5	4	no	—
248	#104	—	5	4	no	—
249	#104	C₂×AGL(1,𝔽₅)	1	4	yes	—
250	C₄×C₈	—	5	2	no	—
251	C₄×C₈	—	5	2	no	—
252	C₂²×C₅	—	2	2	no	—
253	C₂²×C₅	—	2	2	no	—
254	C₂²×C₅	—	4	2	no	—
255	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
256	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
257	C₂²×C₅	C₂×#131	1	1	yes	characteristic
258	C₂×D₅	—	2	2	no	—
259	C₂×D₅	—	4	2	no	—
260	C₄×C₅	—	2	4	no	—
261	C₄×C₅	—	2	2	no	—
262	C₄×C₅	C₈×D₄	1	1	yes	characteristic
263	C₄×C₅	C₈×D₄	1	1	yes	characteristic
264	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
265	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
266	Dic₅	—	2	2	no	—
267	#41	—	2	1	no	—
268	#41	—	2	1	no	—
269	#41	—	5	2	no	—
270	#41	—	5	2	no	—
271	#41	—	10	1	no	—
272	#41	—	10	1	no	—
273	#41	D₅⋊C₈	1	2	yes	—
274	#41	C₂²×AGL(1,𝔽₅)	1	2	yes	—
275	C₂×Q₈	—	5	1	no	—
276	C₂×Q₈	D₅⋊C₈	1	1	yes	characteristic
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	5	1	no	—
279	C₂×D₄	—	10	1	no	—
280	C₂×D₄	D₅⋊C₈	1	1	yes	characteristic
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	10	1	no	—
284	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
285	Q₁₆	—	5	4	no	—
286	SD₁₆	—	5	4	no	—
287	SD₁₆	—	5	4	no	—
288	D₈	—	5	4	no	—
289	M₁₆	—	2	1	no	—
290	M₁₆	—	5	2	no	—
291	M₁₆	—	10	2	no	—
292	M₁₆	—	10	2	no	—
293	M₁₆	—	10	2	no	—
294	M₁₆	—	10	2	no	—
295	M₁₆	—	10	1	no	—
296	M₁₆	C₂²×AGL(1,𝔽₅)	1	2	yes	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	5	1	no	—
299	C₂×C₈	—	5	1	no	—
300	C₂×C₈	—	5	2	no	—
301	C₂×C₈	—	5	1	no	—
302	C₂×C₈	—	5	1	no	—
303	C₂×C₈	—	5	1	no	—
304	C₂×C₈	—	5	1	no	—
305	C₂×C₈	—	5	2	no	—
306	C₂×C₈	—	5	1	no	—
307	C₂×C₈	—	5	1	no	—
308	C₂×C₈	—	10	2	no	—
309	C₂×C₈	—	10	2	no	—
310	C₂×C₈	—	10	2	no	—
311	C₂×C₈	—	10	2	no	—
312	C₂×C₈	—	10	1	no	—
313	C₄²	—	2	1	no	—
314	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
315	C₁₆	—	5	2	no	—
316	C₁₆	—	5	2	no	—
317	C₁₆	—	10	2	no	—
318	C₁₆	—	10	2	no	—
319	C₁₆	—	10	2	no	—
320	C₂×C₅	—	2	1	no	—
321	C₂×C₅	—	2	1	no	—
322	C₂×C₅	—	4	2	no	—
323	C₂×C₅	#2170	1	2	yes	complemented
324	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
325	D₅	—	4	2	no	—
326	C₂³	—	2	1	no	—
327	C₂³	—	10	1	no	—
328	C₂³	#4131	1	1	yes	characteristic
329	Q₈	—	2	1	no	—
330	Q₈	—	5	2	no	—
331	Q₈	—	10	1	no	—
332	Q₈	C₂×D₅⋊C₈	1	2	yes	—
333	D₄	—	2	2	no	—
334	D₄	—	2	2	no	—
335	D₄	—	2	1	no	—
336	D₄	—	5	2	no	—
337	D₄	—	10	2	no	—
338	D₄	—	10	2	no	—
339	D₄	—	10	1	no	—
340	D₄	C₂×D₅⋊C₈	1	2	yes	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	10	1	no	—
347	C₂×C₄	—	10	2	no	—
348	C₂×C₄	—	10	2	no	—
349	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
350	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
351	C₂×C₄	#4131	1	1	yes	characteristic
352	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
353	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
354	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	5	2	no	—
357	C₈	—	5	2	no	—
358	C₈	—	5	1	no	—
359	C₈	—	5	1	no	—
360	C₈	—	5	1	no	—
361	C₈	—	5	1	no	—
362	C₈	—	10	2	no	—
363	C₅	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	10	2	no	—
368	C₂²	—	20	2	no	—
369	C₂²	#65692	1	1	yes	characteristic
370	C₂²	#65692	1	1	yes	characteristic
371	C₂²	C₂×#4131	1	1	yes	characteristic
372	C₄	—	2	4	no	—
373	C₄	—	2	2	no	—
374	C₄	—	10	2	no	—
375	C₄	#65692	1	1	yes	characteristic
376	C₄	#65692	1	1	yes	characteristic
377	C₄	C₂²×D₅⋊C₈	1	1	yes	characteristic
378	C₄	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	4	2	no	—
382	C₂	—	20	2	no	—
383	C₂	#10603999	1	2	yes	direct product factor
384	C₂	C₂×#65692	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂³×C₈
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₂×C₅×#104	C₄
socle	C₂²×C₅	C₈×D₄
prime radical	C₂×#4096	C₄
cosocle	C₂×#168	C₂⁴
hypercenter	C₂×#104	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#2170
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
1. C₂×#65791	C₂	semidirect
2. C₂×#65725	C₂	semidirect
4. C₂×#64932	C₂	semidirect
5. C₂×#64898	C₂	semidirect
6. C₂×#64861	C₂	semidirect
7. C₂×C₄×#674	C₂	semidirect
8. #10603999	C₂	direct (N × H)
13. #65791	C₂²	semidirect
21. #65725	C₂²	semidirect
30. #64932	C₂²	semidirect
31. #64898	C₂²	semidirect
32. #64861	C₂²	semidirect
33. C₄×#674	C₂²	semidirect
323. C₂×C₅	#2170	semidirect
363. C₅	C₂×#2170	semidirect
383. C₂	#10603999	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×#65791	2
C₂×#65725	2
C₂×#65113	2
C₂×#64932	2
C₂×#64898	2
C₂×#64861	2
C₂×C₄×#674	2
#10603999	2
C₂×#2170	5

Subgroups — C2×#10603999 #13043548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10603999 #13043548