Finite Group Databasebeta

← C2×#11751859 #14320760

Subgroups — C2×#11751859 #14320760

Order 1920.

1600 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
2C22×C3×C4×AGL(1,𝔽5)C211yesmaximal, characteristic
3C22×Dic3⋊AGL(1,𝔽5)C212yesmaximal
4C2×D5×C4⋊Dic3C211yesmaximal, characteristic
5C2×(C2×AGL(1,𝔽5))⋊Dic3C212yesmaximal
6#11751859C218yesmaximal, direct product factor
7C2×#65712—31nomaximal, Hall
8C23×ΓL(1,𝔽16)C2212yes—
9C23×C3×AGL(1,𝔽5)C2211yescharacteristic
10C23×C3×AGL(1,𝔽5)C2211yescharacteristic
11C22×C3×C4×D5C2211yescharacteristic
12C22×D5×Dic3C2212yes—
13C2×C4⋊ΓL(1,𝔽16)—24no—
14C2×C4⋊ΓL(1,𝔽16)C2214yes—
15C2×C3×C4×AGL(1,𝔽5)—24no—
16C2×C3×C4×AGL(1,𝔽5)C2214yes—
17C2×Dic3⋊AGL(1,𝔽5)—28no—
18C2×Dic3⋊AGL(1,𝔽5)C2218yes—
19C2×C4⋊Dic15C411yescharacteristic, complemented
20C2×C5×C4⋊Dic3C411yescharacteristic, complemented
21C2×#6668C412yescomplemented
22D5×C4⋊Dic3C414yescomplemented
23D5×C4⋊Dic3C2214yes—
24(C2×AGL(1,𝔽5))⋊Dic3C2218yes—
25C2×#4928—51nomaximal, Hall
26C22×C4⋊AGL(1,𝔽5)—32no—
27C22×C4⋊AGL(1,𝔽5)—31no—
28C22×C4×AGL(1,𝔽5)D311yescharacteristic
29C2×D5×C4⋊C4—31no—
30C2×(C2×AGL(1,𝔽5))⋊C4—32no—
31#65712—38no—
32C23×C3×D5C2311yescharacteristic
33C22×ΓL(1,𝔽16)—24no—
34C22×ΓL(1,𝔽16)—24no—
35C22×ΓL(1,𝔽16)—24no—
36C22×ΓL(1,𝔽16)C2314yes—
37C22×C3×AGL(1,𝔽5)—22no—
38C22×C3×AGL(1,𝔽5)—22no—
39C22×C3×AGL(1,𝔽5)—22no—
40C22×C3×AGL(1,𝔽5)—22no—
41C22×C3×AGL(1,𝔽5)D414yes—
42C22×C3×AGL(1,𝔽5)Q814yes—
43C22×C3×AGL(1,𝔽5)C2312yes—
44C22×C3×AGL(1,𝔽5)C2312yes—
45C22×C3×C4×C5C2×C411yescharacteristic
46C22×Dic15C2×C412yes—
47C22×C5×Dic3C2×C412yes—
48C22×C3×Dic5C2×C411yescharacteristic
49C2×C3×C4×D5C2×C412yes—
50C2×C3×C4×D5D414yes—
51C2×C3×C4×D5Q814yes—
52C2×C3×C4×D5C2312yes—
53C2×D5×Dic3—24no—
54C2×D5×Dic3—24no—
55C2×D5×Dic3C2×C414yes—
56C2×D5×Dic3C2314yes—
57C4⋊ΓL(1,𝔽16)—28no—
58C3×C4×AGL(1,𝔽5)—28no—
59Dic3⋊AGL(1,𝔽5)—216no—
60C4⋊Dic15C2×C414yescomplemented
61C5×C4⋊Dic3C2×C414yescomplemented
62#6668C2×C418yescomplemented
63C22×C3×C42—51no—
64C22×C4⋊Dic3—51no—
65C22×C4⋊Dic3—51no—
66C22×Dic3⋊C4—52no—
67C2×C3⋊#95—52no—
68#4928—58no—
69C23×AGL(1,𝔽5)—32no—
70C23×AGL(1,𝔽5)C2×D311yescharacteristic
71C23×AGL(1,𝔽5)C2×D311yescharacteristic
72C22×C4×D5—32no—
73C22×C4×D5C2×D311yescharacteristic
74C2×C4⋊AGL(1,𝔽5)—38no—
75C2×C4⋊AGL(1,𝔽5)—34no—
76C2×C4⋊AGL(1,𝔽5)—68no—
77C2×C4⋊AGL(1,𝔽5)—64no—
78C2×C4×AGL(1,𝔽5)—24no—
79C2×C4×AGL(1,𝔽5)C2×D314yes—
80C2×C5×C4⋊C4—31no—
81C2×C4⋊Dic5—31no—
82C2×#683—32no—
83D5×C4⋊C4—34no—
84D5×C4⋊C4—34no—
85(C2×AGL(1,𝔽5))⋊C4—38no—
86C2×#389—151noSylow
87C23×C3×C5C22×C411yescharacteristic
88C22×C3×D5C22×C411yescharacteristic
89C22×C3×D5C2×D412yes—
90C22×C3×D5C2×D412yes—
91C22×C3×D5C2×Q812yes—
92C22×C3×D5C2×Q812yes—
93C22×C3×D5#4112yes—
94C22×C3×D5#4112yes—
95C22×C3×D5C2411yescharacteristic
96C2×ΓL(1,𝔽16)—24no—
97C2×ΓL(1,𝔽16)—24no—
98C2×ΓL(1,𝔽16)—24no—
99C2×ΓL(1,𝔽16)—48no—
100C2×C3×AGL(1,𝔽5)—28no—
101C2×C3×AGL(1,𝔽5)—28no—
102C2×C3×AGL(1,𝔽5)—22no—
103C2×C3×AGL(1,𝔽5)—22no—
104C2×C3×AGL(1,𝔽5)—22no—
105C2×C3×AGL(1,𝔽5)—22no—
106C2×C3×AGL(1,𝔽5)C2×D414yes—
107C2×C3×AGL(1,𝔽5)C2×Q814yes—
108C2×C3×C4×C5C4⋊C414yescomplemented
109C2×C3×C4×C5C22×C412yes—
110C2×Dic15—24no—
111C2×Dic15C22×C414yes—
112C2×C5×Dic3—24no—
113C2×C5×Dic3C22×C414yes—
114C2×C3×Dic5C4⋊C414yescomplemented
115C2×C3×Dic5C22×C412yes—
116C3×C4×D5C4⋊C418yescomplemented
117C3×C4×D5C2×D414yes—
118C3×C4×D5C2×Q814yes—
119D5×Dic3—28no—
120D5×Dic3—24no—
121D5×Dic3—24no—
122C23×C3×C4—51no—
123C23×C3×C4—51no—
124C23×C3×C4—51no—
125C23×Dic3—52no—
126C23×Dic3—52no—
127C2×C3×C42—54no—
128C2×C3×C42—104no—
129C2×C4⋊Dic3—51no—
130C2×C4⋊Dic3—52no—
131C2×C4⋊Dic3—54no—
132C2×C4⋊Dic3—54no—
133C2×C4⋊Dic3—54no—
134C2×C4⋊Dic3—104no—
135C2×C4⋊Dic3AGL(1,𝔽5)11yescharacteristic, complemented
136C2×Dic3⋊C4—58no—
137C2×Dic3⋊C4—108no—
138C3⋊#95—58no—
139C23×D5C22×D311yescharacteristic
140C22×AGL(1,𝔽5)—22no—
141C22×AGL(1,𝔽5)—22no—
142C22×AGL(1,𝔽5)—22no—
143C22×AGL(1,𝔽5)—22no—
144C22×AGL(1,𝔽5)—34no—
145C22×AGL(1,𝔽5)—64no—
146C22×AGL(1,𝔽5)—64no—
147C22×AGL(1,𝔽5)—64no—
148C22×AGL(1,𝔽5)Dic614yes—
149C22×AGL(1,𝔽5)D1214yes—
150C22×AGL(1,𝔽5)C22×D312yes—
151C22×AGL(1,𝔽5)C22×D312yes—
152C22×C4×C5—32no—
153C22×C4×C5C4×D311yescharacteristic
154C22×Dic5—32no—
155C22×Dic5C4×D311yescharacteristic
156C2×C4×D5—34no—
157C2×C4×D5—34no—
158C2×C4×D5—64no—
159C2×C4×D5—64no—
160C2×C4×D5Dic614yes—
161C2×C4×D5C4×D312yes—
162C2×C4×D5#6714yes—
163C2×C4×D5C22×D312yes—
164C4⋊AGL(1,𝔽5)—616no—
165C4⋊AGL(1,𝔽5)—68no—
166C4×AGL(1,𝔽5)—28no—
167C5×C4⋊C4—34no—
168C4⋊Dic5—34no—
169#683—38no—
170C22×C4⋊C4—151no—
171C22×C4⋊C4—152no—
172C22×C4⋊C4—151no—
173C22×C42—51no—
174#389—158no—
175C2×#95—152no—
176C22×C3×C5C2×C4⋊C412yes—
177C22×C3×C5C4×D412yes—
178C22×C3×C5C4×Q812yes—
179C22×C3×C5C23×C411yescharacteristic
180C2×C3×D5C2×C4⋊C412yes—
181C2×C3×D5C4×D412yes—
182C2×C3×D5C4×Q812yes—
183C2×C3×D5#12114yes—
184C2×C3×D5C22⋊Q814yes—
185C2×C3×D5#12514yes—
186C2×C3×D5C4⋊Q814yes—
187C2×C3×D5C22×D411yescharacteristic
188C2×C3×D5C22×D411yescharacteristic
189C2×C3×D5C22×Q811yescharacteristic
190C2×C3×D5C22×Q811yescharacteristic
191C2×C3×D5C2×#4111yescharacteristic
192C2×C3×D5C2×#4111yescharacteristic
193ΓL(1,𝔽16)—44no—
194C3×AGL(1,𝔽5)—24no—
195C3×AGL(1,𝔽5)—24no—
196C3×C4×C5C2×C4⋊C414yescomplemented
197Dic15—24no—
198C3×Dic5C2×C4⋊C414yescomplemented
199C5×Dic3—24no—
200C24×C3—51no—
201C22×C3×C4—51no—
202C22×C3×C4—54no—
203C22×C3×C4—54no—
204C22×C3×C4—52no—
205C22×C3×C4—52no—
206C22×C3×C4—52no—
207C22×C3×C4—52no—
208C22×C3×C4—54no—
209C22×C3×C4—54no—
210C22×C3×C4—102no—
211C22×C3×C4—102no—
212C22×C3×C4—102no—
213C22×C3×C4—102no—
214C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
215C22×Dic3—52no—
216C22×Dic3—54no—
217C22×Dic3—54no—
218C22×Dic3—54no—
219C22×Dic3—104no—
220C22×Dic3—104no—
221C22×Dic3—104no—
222C22×Dic3—104no—
223C22×Dic3—104no—
224C22×Dic3C2×AGL(1,𝔽5)12yes—
225C3×C42—108no—
226C4⋊Dic3—54no—
227C4⋊Dic3—58no—
228C4⋊Dic3—108no—
229C4⋊Dic3C2×AGL(1,𝔽5)14yescomplemented
230Dic3⋊C4—1016no—
231C23×C5C2×C4×D311yescharacteristic
232C22×D5C2×Dic612yes—
233C22×D5C2×Dic612yes—
234C22×D5C2×C4×D311yescharacteristic
235C22×D5C2×D1212yes—
236C22×D5#23512yes—
237C22×D5#23512yes—
238C22×D5C2×#6712yes—
239C22×D5C23×D311yescharacteristic
240C2×AGL(1,𝔽5)—28no—
241C2×AGL(1,𝔽5)—28no—
242C2×AGL(1,𝔽5)—22no—
243C2×AGL(1,𝔽5)—22no—
244C2×AGL(1,𝔽5)—22no—
245C2×AGL(1,𝔽5)—22no—
246C2×AGL(1,𝔽5)—64no—
247C2×AGL(1,𝔽5)—64no—
248C2×AGL(1,𝔽5)—64no—
249C2×AGL(1,𝔽5)—128no—
250C2×AGL(1,𝔽5)C2×Dic614yes—
251C2×AGL(1,𝔽5)C2×D1214yes—
252C2×C4×C5—34no—
253C2×C4×C5—64no—
254C2×C4×C5Dic3⋊C414yescomplemented
255C2×C4×C5C2×C4×D312yes—
256C2×Dic5—34no—
257C2×Dic5—64no—
258C2×Dic5Dic3⋊C414yescomplemented
259C2×Dic5C2×C4×D312yes—
260C4×D5—68no—
261C4×D5—64no—
262C4×D5—64no—
263C4×D5Dic3⋊C418yescomplemented
264C4×D5C2×Dic614yes—
265C4×D5C2×#6714yes—
266C23×C4—51no—
267C23×C4—51no—
268C23×C4—51no—
269C23×C4—152no—
270C23×C4—152no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—154no—
273C2×C4⋊C4—154no—
274C2×C4⋊C4—152no—
275C2×C4⋊C4—151no—
276C2×C4⋊C4—158no—
277C2×C4⋊C4—154no—
278C2×C4⋊C4—308no—
279C2×C4⋊C4—304no—
280C2×C42—54no—
281C2×C42—104no—
282#95—158no—
283C2×C3×C5#38914yescomplemented
284C2×C3×C5C22×C4⋊C411yescharacteristic
285C2×C3×C5C2×C4×D411yescharacteristic
286C2×C3×C5C2×C4×Q811yescharacteristic
287C3×D5#38914yescomplemented
288C3×D5C2×#12111yescharacteristic
289C3×D5C2×C22⋊Q811yescharacteristic
290C3×D5C2×#12511yescharacteristic
291C3×D5C2×C4⋊Q811yescharacteristic
292C23×C3—52no—
293C23×C3—52no—
294C23×C3—52no—
295C23×C3—52no—
296C23×C3—52no—
297C23×C3—52no—
298C23×C3—51no—
299C23×C3—51no—
300C23×C3C22×AGL(1,𝔽5)11yescharacteristic
301C2×C3×C4—54no—
302C2×C3×C4—52no—
303C2×C3×C4—58no—
304C2×C3×C4—54no—
305C2×C3×C4—54no—
306C2×C3×C4—54no—
307C2×C3×C4—54no—
308C2×C3×C4—102no—
309C2×C3×C4—102no—
310C2×C3×C4—108no—
311C2×C3×C4—108no—
312C2×C3×C4—102no—
313C2×C3×C4—102no—
314C2×C3×C4C4⋊AGL(1,𝔽5)14yescomplemented
315C2×C3×C4C22×AGL(1,𝔽5)12yes—
316C2×Dic3—24no—
317C2×Dic3—54no—
318C2×Dic3—104no—
319C2×Dic3—108no—
320C2×Dic3—104no—
321C2×Dic3—104no—
322C2×Dic3—104no—
323C2×Dic3—104no—
324C2×Dic3—104no—
325C2×Dic3—208no—
326C2×Dic3C22×AGL(1,𝔽5)14yes—
327C22×C5C4×Dic612yes—
328C22×C5C4×D1212yes—
329C22×C5C2×Dic3⋊C412yes—
330C22×C5C22×C4×D311yescharacteristic
331C2×D5C4×Dic612yes—
332C2×D5#86914yes—
333C2×D5#87014yes—
334C2×D5C4×D1212yes—
335C2×D5C2×Dic3⋊C412yes—
336C2×D5#92414yes—
337C2×D5#93014yes—
338C2×D5C22×Dic611yescharacteristic
339C2×D5C22×Dic611yescharacteristic
340C2×D5C22×D1211yescharacteristic
341C2×D5C2×#23511yescharacteristic
342C2×D5C2×#23511yescharacteristic
343C2×D5C22×#6711yescharacteristic
344AGL(1,𝔽5)—24no—
345AGL(1,𝔽5)—24no—
346AGL(1,𝔽5)—124no—
347C4×C5—64no—
348C4×C5C2×Dic3⋊C414yescomplemented
349Dic5—64no—
350Dic5C2×Dic3⋊C414yescomplemented
351C24—51no—
352C22×C4—32no—
353C22×C4—51no—
354C22×C4—54no—
355C22×C4—54no—
356C22×C4—52no—
357C22×C4—52no—
358C22×C4—52no—
359C22×C4—52no—
360C22×C4—54no—
361C22×C4—54no—
362C22×C4—102no—
363C22×C4—102no—
364C22×C4—102no—
365C22×C4—102no—
366C22×C4—154no—
367C22×C4—154no—
368C22×C4—152no—
369C22×C4—154no—
370C22×C4—304no—
371C22×C4—304no—
372C22×C4—304no—
373C22×C4—304no—
374C22×C4—304no—
375C22×C4D3×AGL(1,𝔽5)11yescharacteristic
376C4⋊C4—34no—
377C4⋊C4—158no—
378C4⋊C4—154no—
379C4⋊C4—3016no—
380C4⋊C4—308no—
381C42—108no—
382C3×C5C2×#38911yescharacteristic, complemented, Hall
383C22×C3—54no—
384C22×C3—54no—
385C22×C3—54no—
386C22×C3—54no—
387C22×C3—52no—
388C22×C3—51no—
389C22×C3—51no—
390C22×C3—52no—
391C22×C3—51no—
392C22×C3—51no—
393C22×C3—52no—
394C22×C3—51no—
395C22×C3—51no—
396C22×C3C2×C4⋊AGL(1,𝔽5)12yes—
397C22×C3D4×AGL(1,𝔽5)12yes—
398C22×C3Q8×AGL(1,𝔽5)12yes—
399C22×C3C23×AGL(1,𝔽5)11yescharacteristic
400C3×C4—54no—
401C3×C4—104no—
402C3×C4—104no—
403C3×C4C2×C4⋊AGL(1,𝔽5)14yescomplemented
404Dic3—24no—
405Dic3—104no—
406Dic3—204no—
407C2×C5#492814yescomplemented
408C2×C5C2×C4×Dic611yescharacteristic
409C2×C5C2×C4×D1211yescharacteristic
410C2×C5C22×Dic3⋊C411yescharacteristic
411D5#492814yescomplemented
412D5C2×#86911yescharacteristic
413D5C2×#87011yescharacteristic
414D5C2×#92411yescharacteristic
415D5C2×#93011yescharacteristic
416C23—52no—
417C23—52no—
418C23—52no—
419C23—52no—
420C23—52no—
421C23—52no—
422C23—51no—
423C23—51no—
424C23C2×D3×AGL(1,𝔽5)11yescharacteristic
425C2×C4—34no—
426C2×C4—54no—
427C2×C4—52no—
428C2×C4—54no—
429C2×C4—58no—
430C2×C4—54no—
431C2×C4—54no—
432C2×C4—54no—
433C2×C4—64no—
434C2×C4—102no—
435C2×C4—102no—
436C2×C4—102no—
437C2×C4—102no—
438C2×C4—108no—
439C2×C4—108no—
440C2×C4—154no—
441C2×C4—304no—
442C2×C4—304no—
443C2×C4—308no—
444C2×C4—304no—
445C2×C4—304no—
446C2×C4—304no—
447C2×C4—304no—
448C2×C4—608no—
449C2×C4Dic3⋊AGL(1,𝔽5)14yescomplemented
450C2×C4C2×D3×AGL(1,𝔽5)12yes—
451C2×C3—51no—
452C2×C3—51no—
453C2×C3—51no—
454C2×C3—51no—
455C2×C3—54no—
456C2×C3#6571214yescomplemented
457C2×C3C22×C4⋊AGL(1,𝔽5)11yescharacteristic
458C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
459C2×C3C2×Q8×AGL(1,𝔽5)11yescharacteristic
460C5C2×#492811yescharacteristic, complemented, Sylow
461C22—51no—
462C22—51no—
463C22—52no—
464C22—51no—
465C22—51no—
466C22—52no—
467C22—51no—
468C22—52no—
469C22—51no—
470C22—54no—
471C22—54no—
472C22—54no—
473C22—54no—
474C22AGL(1,𝔽5)×Dic612yes—
475C22AGL(1,𝔽5)×D1212yes—
476C22C2×Dic3⋊AGL(1,𝔽5)12yes—
477C22C22×D3×AGL(1,𝔽5)11yescharacteristic
478C4—54no—
479C4—64no—
480C4—104no—
481C4—104no—
482C4—304no—
483C4—604no—
484C4C2×Dic3⋊AGL(1,𝔽5)14yescomplemented
485C3C2×#6571211yescharacteristic, complemented, Sylow
486C2—51no—
487C2—51no—
488C2—54no—
489C2—51no—
490C2—51no—
491C2#1175185914yesdirect product factor
492C2C2×AGL(1,𝔽5)×Dic611yescharacteristic
493C2C2×AGL(1,𝔽5)×D1211yescharacteristic
494C2C22×Dic3⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC23C2×D3×AGL(1,𝔽5)
commutatorC22×C3×C5C23×C4
FrattiniC22C22×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC23×C3×C5C22×C4
prime radicalC23×C3×D5C23
cosocleC22×C3×D5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5C2×#389
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11751859C2direct (N × H)
19. C2×C4⋊Dic15C4semidirect
20. C2×C5×C4⋊Dic3C4semidirect
21. C2×#6668C4semidirect
22. D5×C4⋊Dic3C4semidirect
60. C4⋊Dic15C2×C4semidirect
61. C5×C4⋊Dic3C2×C4semidirect
62. #6668C2×C4semidirect
108. C2×C3×C4×C5C4⋊C4semidirect
114. C2×C3×Dic5C4⋊C4semidirect
116. C3×C4×D5C4⋊C4semidirect
135. C2×C4⋊Dic3AGL(1,𝔽5)semidirect
196. C3×C4×C5C2×C4⋊C4semidirect
198. C3×Dic5C2×C4⋊C4semidirect
229. C4⋊Dic3C2×AGL(1,𝔽5)semidirect
254. C2×C4×C5Dic3⋊C4semidirect
258. C2×Dic5Dic3⋊C4semidirect
263. C4×D5Dic3⋊C4semidirect
283. C2×C3×C5#389semidirect
287. C3×D5#389semidirect
314. C2×C3×C4C4⋊AGL(1,𝔽5)semidirect
348. C4×C5C2×Dic3⋊C4semidirect
350. Dic5C2×Dic3⋊C4semidirect
382. C3×C5C2×#389semidirect
403. C3×C4C2×C4⋊AGL(1,𝔽5)semidirect
407. C2×C5#4928semidirect
411. D5#4928semidirect
449. C2×C4Dic3⋊AGL(1,𝔽5)semidirect
456. C2×C3#65712semidirect
460. C5C2×#4928semidirect
484. C4C2×Dic3⋊AGL(1,𝔽5)semidirect
485. C3C2×#65712semidirect
491. C2#11751859direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊ΓL(1,𝔽16)2
C22×C3×C4×AGL(1,𝔽5)2
C22×Dic3⋊AGL(1,𝔽5)2
C2×D5×C4⋊Dic32
C2×(C2×AGL(1,𝔽5))⋊Dic32
#117518592
C2×#657123
C2×#49285