Finite Group Databasebeta

← C2×#11751970 #14416464

Subgroups — C2×#11751970 #14416464

Order 1920.

960 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×ΓL(1,𝔽16)C211yesmaximal, characteristic
2C2×ΓL(1,𝔽16)⋊Q8C211yesmaximal, characteristic
3C2×D5×#216C211yesmaximal, characteristic
4C2×C3×#4009C211yesmaximal, characteristic
5#11751970C218yesmaximal, direct product factor
6C2×#91159C211yesmaximal, characteristic
7C2×#91154C211yesmaximal, characteristic
8C2×#91150C211yesmaximal, characteristic
9C2×#65739—31nomaximal, Hall
10C2×C3×Q8×D5C2211yescharacteristic
11C2×D5×Dic6C2211yescharacteristic
12Q8×ΓL(1,𝔽16)—22no—
13Q8×ΓL(1,𝔽16)C2214yes—
14C2×C4⋊ΓL(1,𝔽16)—21no—
15C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
16C2×C4×ΓL(1,𝔽16)—21no—
17C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
18C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
19C2×C3×D5⋊C8C2211yescharacteristic
20C2×Dic3⋊AGL(1,𝔽5)—21no—
21C2×Dic3×AGL(1,𝔽5)—21no—
22ΓL(1,𝔽16)⋊Q8—22no—
23ΓL(1,𝔽16)⋊Q8C2214yes—
24C2×#6718C411yescharacteristic, complemented
25C2×C5×#216C411yescharacteristic, complemented
26D5×#216C414yescomplemented
27D5×#216C2214yes—
28C2×#6662C411yescharacteristic, complemented
29C2×#6661C411yescharacteristic, complemented
30C2×D5×C3⋊C8C2211yescharacteristic
31C3×#4009C2214yes—
32#91159C2214yes—
33#91154C2214yes—
34#91150C2214yes—
35C2×#4789—51nomaximal, Hall
36C2×Q8×AGL(1,𝔽5)—31no—
37C2×Q8×AGL(1,𝔽5)—31no—
38C2×D5×Q16—31no—
39C2×#4009—31no—
40C2×#4009D311yescharacteristic
41#65739—38no—
42C2×#3993—31no—
43C2×#3992—31no—
44C22×ΓL(1,𝔽16)—21no—
45C22×ΓL(1,𝔽16)D411yescharacteristic
46C22×ΓL(1,𝔽16)D411yescharacteristic
47C22×C3×AGL(1,𝔽5)—21no—
48C2×C3×C5×Q8C2×C411yescharacteristic
49C2×Dic30C2×C411yescharacteristic
50C2×C5×Dic6C2×C411yescharacteristic
51C3×Q8×D5—21no—
52C3×Q8×D5—21no—
53C3×Q8×D5C2×C412yes—
54C3×Q8×D5C2312yes—
55C2×C3×C4×D5—21no—
56C2×C3×C4×D5C2311yescharacteristic
57C2×C3×Dic10—21no—
58C2×C3×Dic10C2×C411yescharacteristic
59C2×#1204—21no—
60C2×D5×Dic3—21no—
61D5×Dic6—21no—
62D5×Dic6—21no—
63D5×Dic6C2×C412yes—
64D5×Dic6C2312yes—
65C4⋊ΓL(1,𝔽16)—24no—
66C4⋊ΓL(1,𝔽16)—21no—
67C4⋊ΓL(1,𝔽16)C2312yes—
68C4×ΓL(1,𝔽16)—24no—
69C4×ΓL(1,𝔽16)D412yes—
70C4×ΓL(1,𝔽16)C2312yes—
71C2×C3×#170—21no—
72C3×C4⋊AGL(1,𝔽5)—21no—
73C3×C4⋊AGL(1,𝔽5)C2312yes—
74C3×D5⋊C8—21no—
75C3×D5⋊C8C2312yes—
76Dic3⋊AGL(1,𝔽5)—24no—
77Dic3×AGL(1,𝔽5)—24no—
78#6718C2×C414yescomplemented
79C2×C3⋊#168C2×C411yescharacteristic
80C5×#216C2×C414yescomplemented
81C2×C5×C3⋊C8C2×C411yescharacteristic
82#6662C2×C414yescomplemented
83#6661C2×C414yescomplemented
84D5×C3⋊C8C2×C412yes—
85D5×C3⋊C8C2312yes—
86C2×Q8×Dic3—51no—
87C22×#216—51no—
88C2×#887—51no—
89C2×C3×#103—51no—
90C2×#816—51no—
91C2×#815—51no—
92C2×#808—51no—
93#4789—58no—
94C2×Q8×D5—31no—
95C2×Q8×D5C2×D311yescharacteristic
96Q8×AGL(1,𝔽5)—34no—
97Q8×AGL(1,𝔽5)—34no—
98Q8×AGL(1,𝔽5)—62no—
99Q8×AGL(1,𝔽5)—62no—
100C2×C4⋊AGL(1,𝔽5)—31no—
101C2×C4⋊AGL(1,𝔽5)—61no—
102C2×C4⋊AGL(1,𝔽5)—61no—
103C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
104C2×C4×AGL(1,𝔽5)—31no—
105C2×C4×AGL(1,𝔽5)—61no—
106C2×C4×AGL(1,𝔽5)—61no—
107C2×D5⋊C8C2×D311yescharacteristic
108C2×C5×Q16—31no—
109C2×#689—31no—
110C2×#689—31no—
111D5×Q16—34no—
112D5×Q16—34no—
113C2×Dic20—31no—
114C2×C8×D5—31no—
115#4009—34no—
116#4009C2×D314yes—
117#3993—34no—
118#3992—34no—
119C2×#441—151noSylow
120C22×C3×D5C2×D411yescharacteristic
121C2×ΓL(1,𝔽16)—22no—
122C2×ΓL(1,𝔽16)—22no—
123C2×ΓL(1,𝔽16)—22no—
124C2×ΓL(1,𝔽16)—42no—
125C2×ΓL(1,𝔽16)C2×D412yes—
126C2×ΓL(1,𝔽16)C2×D412yes—
127C2×C3×AGL(1,𝔽5)—22no—
128C2×C3×AGL(1,𝔽5)—42no—
129C3×C5×Q8—21no—
130C3×C5×Q8C22×C412yes—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5C22×C411yescharacteristic
133C2×Dic15—21no—
134Dic30—21no—
135Dic30C22×C412yes—
136C2×C5×Dic3—21no—
137C5×Dic6—21no—
138C5×Dic6C22×C412yes—
139C2×C3×Dic5—21no—
140C2×C3×Dic5#4111yescharacteristic
141C3×C4×D5—22no—
142C3×C4×D5—22no—
143C3×C4×D5C22×C411yescharacteristic
144C3×C4×D5C2×D411yescharacteristic
145C3×C4×D5#4111yescharacteristic
146C3×C4×D5C2411yescharacteristic
147C3×Dic10—21no—
148C3×Dic10—22no—
149C3×Dic10—22no—
150C3×Dic10C22×C412yes—
151#1204—22no—
152#1204—22no—
153D5×Dic3—22no—
154D5×Dic3—22no—
155C3×#170—22no—
156C3⋊#168C22×C412yes—
157C5×C3⋊C8C22×C412yes—
158C22×C3×Q8—51no—
159C22×Dic6—51no—
160C22×C3×C8—51no—
161C2×C3×C4⋊C4—51no—
162Q8×Dic3—54no—
163Q8×Dic3—102no—
164C2×#216—51no—
165C2×#216—51no—
166C2×#216—54no—
167C2×#216—54no—
168C2×#216—51no—
169C2×#216AGL(1,𝔽5)11yescharacteristic, complemented
170C2×C4⋊Dic3—51no—
171C2×C4⋊Dic3—101no—
172C2×Dic3⋊C4—101no—
173C2×C4×Dic3—51no—
174C2×C4×Dic3—101no—
175C2×C4×Dic3—101no—
176C22×C3⋊C8—51no—
177#887—54no—
178#887—102no—
179C3×#103—54no—
180#816—54no—
181#815—54no—
182#808—54no—
183C22×AGL(1,𝔽5)—21no—
184C22×AGL(1,𝔽5)—31no—
185C22×AGL(1,𝔽5)—31no—
186C22×AGL(1,𝔽5)—61no—
187C2×C5×Q8—31no—
188C2×C5×Q8C4×D311yescharacteristic
189Q8×D5—21no—
190Q8×D5—21no—
191Q8×D5—32no—
192Q8×D5—32no—
193Q8×D5—61no—
194Q8×D5—61no—
195Q8×D5C4×D312yes—
196Q8×D5C22×D312yes—
197C2×C4×D5—21no—
198C2×C4×D5—61no—
199C2×C4×D5C22×D311yescharacteristic
200C2×Dic10—21no—
201C2×Dic10—31no—
202C2×Dic10—61no—
203C2×Dic10C4×D311yescharacteristic
204C2×#170—21no—
205C4⋊AGL(1,𝔽5)—21no—
206C4⋊AGL(1,𝔽5)—32no—
207C4⋊AGL(1,𝔽5)—64no—
208C4⋊AGL(1,𝔽5)—64no—
209C4⋊AGL(1,𝔽5)—61no—
210C4⋊AGL(1,𝔽5)C22×D312yes—
211C4×AGL(1,𝔽5)—32no—
212C4×AGL(1,𝔽5)—32no—
213C4×AGL(1,𝔽5)—64no—
214C4×AGL(1,𝔽5)—64no—
215D5⋊C8—21no—
216D5⋊C8C22×D312yes—
217C5×Q16—34no—
218C2×C5×C8—31no—
219#689—34no—
220#689—34no—
221C2×#168—31no—
222Dic20—34no—
223C8×D5—32no—
224C8×D5—32no—
225C22×Q16—151no—
226C2×C4×Q8—151no—
227C2×C4×Q8—151no—
228#441—158no—
229C2×#106—151no—
230C2×#103—51no—
231C2×#103—151no—
232C2×#97—151no—
233C22×C3×C5C4×D411yescharacteristic
234C2×C3×D5C4×D411yescharacteristic
235C2×C3×D5#13712yes—
236C2×C3×D5#13712yes—
237C2×C3×D5C22×D411yescharacteristic
238ΓL(1,𝔽16)—22no—
239ΓL(1,𝔽16)—22no—
240ΓL(1,𝔽16)—42no—
241C3×AGL(1,𝔽5)—42no—
242C3×C4×C5—22no—
243C3×C4×C5C4×D411yescharacteristic
244C3×C4×C5C23×C411yescharacteristic
245Dic15—22no—
246C3×Dic5—22no—
247C3×Dic5C4×D411yescharacteristic
248C3×Dic5C2×#4111yescharacteristic
249C5×Dic3—22no—
250C2×C3×Q8—52no—
251C2×C3×Q8—52no—
252C2×C3×Q8—51no—
253C2×C3×Q8—101no—
254C2×C3×Q8—101no—
255C2×C3×Q8—101no—
256C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
257C22×C3×C4—51no—
258C22×C3×C4—101no—
259C22×C3×C4—101no—
260C22×Dic3—51no—
261C22×Dic3—51no—
262C22×Dic3—101no—
263C22×Dic3—101no—
264C2×Dic6—51no—
265C2×Dic6—52no—
266C2×Dic6—52no—
267C2×Dic6—101no—
268C2×Dic6—101no—
269C2×Dic6—101no—
270C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
271C2×C3×C8—52no—
272C2×C3×C8—101no—
273C2×C3×C8—101no—
274C3×C4⋊C4—52no—
275C3×C4⋊C4—101no—
276#216—54no—
277#216—54no—
278#216—54no—
279#216C2×AGL(1,𝔽5)14yescomplemented
280C4⋊Dic3—52no—
281C4⋊Dic3—104no—
282C4⋊Dic3—101no—
283Dic3⋊C4—104no—
284C4×Dic3—52no—
285C4×Dic3—52no—
286C4×Dic3—104no—
287C4×Dic3—104no—
288C2×C3⋊C8—51no—
289C2×C3⋊C8—52no—
290C2×C3⋊C8—52no—
291C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
292C22×D5D3×D411yescharacteristic
293C2×AGL(1,𝔽5)—22no—
294C2×AGL(1,𝔽5)—32no—
295C2×AGL(1,𝔽5)—32no—
296C2×AGL(1,𝔽5)—42no—
297C2×AGL(1,𝔽5)—62no—
298C2×AGL(1,𝔽5)—62no—
299C2×AGL(1,𝔽5)—62no—
300C2×AGL(1,𝔽5)—122no—
301C5×Q8—21no—
302C5×Q8—32no—
303C5×Q8—61no—
304C5×Q8C2×C4×D312yes—
305C2×C4×C5—21no—
306C2×C4×C5—61no—
307C2×C4×C5C2×C4×D311yescharacteristic
308C2×Dic5—21no—
309C2×Dic5—61no—
310C2×Dic5#23711yescharacteristic
311C4×D5—22no—
312C4×D5—22no—
313C4×D5—62no—
314C4×D5—62no—
315C4×D5C2×C4×D311yescharacteristic
316C4×D5D3×D411yescharacteristic
317C4×D5#23711yescharacteristic
318C4×D5C23×D311yescharacteristic
319Dic10—21no—
320Dic10—22no—
321Dic10—22no—
322Dic10—32no—
323Dic10—61no—
324Dic10—62no—
325Dic10—62no—
326Dic10C2×C4×D312yes—
327#170—22no—
328C5×C8—32no—
329#168—32no—
330C22×Q8—51no—
331C22×Q8—151no—
332C2×Q16—31no—
333C2×Q16—151no—
334C2×Q16—154no—
335C2×Q16—154no—
336C2×Q16—151no—
337C2×Q16—151no—
338C22×C8—51no—
339C22×C8—151no—
340C4×Q8—154no—
341C4×Q8—154no—
342C4×Q8—302no—
343C4×Q8—302no—
344C2×C4⋊C4—51no—
345C2×C4⋊C4—151no—
346C2×C4⋊C4—301no—
347C2×C4⋊C4—301no—
348C2×C42—151no—
349C2×C42—301no—
350C2×C42—301no—
351#106—154no—
352#103—54no—
353#103—154no—
354#97—154no—
355C2×C3×C5#44112yescomplemented
356C2×C3×C5C2×C4×D411yescharacteristic
357C3×D5#44112yescomplemented
358C3×D5C2×#13711yescharacteristic
359C3×D5C2×#13711yescharacteristic
360C23×C3—51no—
361C3×Q8—21no—
362C3×Q8—52no—
363C3×Q8—102no—
364C3×Q8—102no—
365C3×Q8—101no—
366C3×Q8C22×AGL(1,𝔽5)12yes—
367C2×C3×C4—21no—
368C2×C3×C4—51no—
369C2×C3×C4—51no—
370C2×C3×C4—51no—
371C2×C3×C4—51no—
372C2×C3×C4—51no—
373C2×C3×C4—102no—
374C2×C3×C4—102no—
375C2×C3×C4—101no—
376C2×C3×C4—102no—
377C2×C3×C4—202no—
378C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
379C2×Dic3—21no—
380C2×Dic3—52no—
381C2×Dic3—52no—
382C2×Dic3—102no—
383C2×Dic3—102no—
384C2×Dic3—102no—
385C2×Dic3—102no—
386C2×Dic3—101no—
387C2×Dic3—102no—
388C2×Dic3—202no—
389Dic6—21no—
390Dic6—52no—
391Dic6—102no—
392Dic6—102no—
393Dic6—101no—
394Dic6C22×AGL(1,𝔽5)12yes—
395C3×C8—102no—
396C3⋊C8—52no—
397C3⋊C8C22×AGL(1,𝔽5)12yes—
398C22×C5#67⋊C411yescharacteristic
399C2×D5#67⋊C411yescharacteristic
400C2×D5#91512yes—
401C2×D5#91812yes—
402C2×D5C2×D3×D411yescharacteristic
403AGL(1,𝔽5)—42no—
404AGL(1,𝔽5)—62no—
405AGL(1,𝔽5)—62no—
406AGL(1,𝔽5)—122no—
407C4×C5—22no—
408C4×C5—62no—
409C4×C5#67⋊C411yescharacteristic
410C4×C5C22×C4×D311yescharacteristic
411Dic5—22no—
412Dic5—62no—
413Dic5#67⋊C411yescharacteristic
414Dic5C2×#23711yescharacteristic
415C2×Q8—31no—
416C2×Q8—52no—
417C2×Q8—52no—
418C2×Q8—51no—
419C2×Q8—101no—
420C2×Q8—101no—
421C2×Q8—101no—
422C2×Q8—152no—
423C2×Q8—152no—
424C2×Q8—151no—
425C2×Q8—301no—
426C2×Q8—301no—
427C2×Q8—301no—
428C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
429C22×C4—51no—
430C22×C4—101no—
431C22×C4—101no—
432C22×C4—151no—
433C22×C4—151no—
434C22×C4—301no—
435C22×C4—301no—
436Q16—34no—
437Q16—154no—
438Q16—154no—
439Q16—154no—
440C2×C8—31no—
441C2×C8—52no—
442C2×C8—101no—
443C2×C8—101no—
444C2×C8—152no—
445C2×C8—152no—
446C2×C8—151no—
447C4⋊C4—52no—
448C4⋊C4—101no—
449C4⋊C4—152no—
450C4⋊C4—301no—
451C4⋊C4—304no—
452C4⋊C4—304no—
453C42—152no—
454C42—152no—
455C42—304no—
456C42—304no—
457C3×C5C2×#44111yescharacteristic, complemented, Hall
458C22×C3—52no—
459C22×C3—52no—
460C22×C3—51no—
461C22×C3—51no—
462C22×C3D4×AGL(1,𝔽5)11yescharacteristic
463C3×C4—22no—
464C3×C4—51no—
465C3×C4—51no—
466C3×C4—102no—
467C3×C4—202no—
468C3×C4D4×AGL(1,𝔽5)11yescharacteristic
469C3×C4C23×AGL(1,𝔽5)11yescharacteristic
470Dic3—22no—
471Dic3—102no—
472Dic3—102no—
473Dic3—102no—
474Dic3—202no—
475C2×C5#478912yescomplemented
476C2×C5C2×#67⋊C411yescharacteristic
477D5#478912yescomplemented
478D5C2×#91511yescharacteristic
479D5C2×#91811yescharacteristic
480C23—51no—
481Q8—21no—
482Q8—32no—
483Q8—52no—
484Q8—61no—
485Q8—102no—
486Q8—102no—
487Q8—101no—
488Q8—152no—
489Q8—302no—
490Q8—302no—
491Q8—301no—
492Q8C2×D3×AGL(1,𝔽5)12yes—
493C2×C4—21no—
494C2×C4—51no—
495C2×C4—51no—
496C2×C4—51no—
497C2×C4—51no—
498C2×C4—51no—
499C2×C4—61no—
500C2×C4—102no—
501C2×C4—102no—
502C2×C4—101no—
503C2×C4—102no—
504C2×C4—152no—
505C2×C4—152no—
506C2×C4—202no—
507C2×C4—302no—
508C2×C4—302no—
509C2×C4—302no—
510C2×C4—302no—
511C2×C4—301no—
512C2×C4—302no—
513C2×C4—602no—
514C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
515C8—32no—
516C8—102no—
517C8—152no—
518C2×C3—51no—
519C2×C3—51no—
520C2×C3—52no—
521C2×C3#6573912yescomplemented
522C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
523C5C2×#478911yescharacteristic, complemented, Sylow
524C22—52no—
525C22—52no—
526C22—51no—
527C22—51no—
528C22#67⋊AGL(1,𝔽5)11yescharacteristic
529C4—22no—
530C4—51no—
531C4—51no—
532C4—62no—
533C4—102no—
534C4—202no—
535C4—302no—
536C4—302no—
537C4—302no—
538C4—602no—
539C4#67⋊AGL(1,𝔽5)11yescharacteristic
540C4C22×D3×AGL(1,𝔽5)11yescharacteristic
541C3C2×#6573911yescharacteristic, complemented, Sylow
542C2—51no—
543C2—51no—
544C2—52no—
545C2#1175197012yesdirect product factor
546C2C2×#67⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22#67⋊AGL(1,𝔽5)
commutatorC3×C4×C5C23×C4
FrattiniC4C22×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C4×D4
prime radicalC22×C3×D5C2×D4
cosocleC3×C4×D5C24
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5C2×#441
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
5. #11751970C2direct (N × H)
24. C2×#6718C4semidirect
25. C2×C5×#216C4semidirect
26. D5×#216C4semidirect
28. C2×#6662C4semidirect
29. C2×#6661C4semidirect
78. #6718C2×C4semidirect
80. C5×#216C2×C4semidirect
82. #6662C2×C4semidirect
83. #6661C2×C4semidirect
169. C2×#216AGL(1,𝔽5)semidirect
279. #216C2×AGL(1,𝔽5)semidirect
355. C2×C3×C5#441semidirect
357. C3×D5#441semidirect
457. C3×C5C2×#441semidirect
475. C2×C5#4789semidirect
477. D5#4789semidirect
521. C2×C3#65739semidirect
523. C5C2×#4789semidirect
541. C3C2×#65739semidirect
545. C2#11751970direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×ΓL(1,𝔽16)2
C2×ΓL(1,𝔽16)⋊Q82
C2×D5×#2162
C2×C3×#40092
#117519702
C2×#911592
C2×#911542
C2×#911502
C2×#657393
C2×#47895