Finite Group Databasebeta

← C2×#21447116 #29707886

Subgroups — C2×#21447116 #29707886

Order 15616.

1362 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519807C211yesmaximal, characteristic
2C22×#14519712C211yesmaximal, characteristic, complemented
3C22×#14519700C211yesmaximal, characteristic, complemented
4C22×#14519695C211yesmaximal, characteristic, complemented
5C2×#16839729C211yesmaximal, characteristic
6#21447116C218yesmaximal, direct product factor
7C2×#16839531C211yesmaximal, characteristic, complemented
8C2×C61×#417C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C2×#14519807—24no—
11C2×#14519807—22no—
12C2×#14519807—24no—
13C2×#14519807C2214yes—
14C22×#11758202—21no—
15C22×#11758200—21no—
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18C23×#92418C2211yescharacteristic
19C2×#14519712—24no—
20C2×#14519712C2214yescomplemented
21C2×#14519700—22no—
22C2×#14519700—22no—
23C2×#14519700C2214yescomplemented
24C2×#14519695—24no—
25C2×#14519695C2214yescomplemented
26#16839729—22no—
27#16839729C2214yes—
28C2×C22⋊#92418—22no—
29#16839531C2214yescomplemented
30C22×#41×C61C2211yescharacteristic
31C22×M16×C61C2211yescharacteristic
32C22×C4⋊C4×C61C2211yescharacteristic
33C61×#417—22no—
34C61×#417C2214yes—
35C2×#103×C61—21no—
36C2×#102×C61—21no—
37C22×Dic122—21no—
38C22×Dic122—21no—
39C22×Dic122—21no—
40C22×Dic122—22no—
41C22×Dic122—21no—
42C22×Dic122—22no—
43C22×Dic122—21no—
44C22×Dic122—22no—
45C22×Dic122D414yes—
46C22×Dic122C2312yes—
47C23×Dic61—21no—
48#14519807—216no—
49#14519807—28no—
50#14519807—28no—
51C2×#11758202—28no—
52C2×#11758202—24no—
53C2×#11758200—28no—
54C2×#11758200—24no—
55C2×#11758194—22no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194D414yes—
61C2×#11758194C2312yes—
62C2×#11758194C2312yes—
63C22×#92418—22no—
64C22×#92418—21no—
65C22×#92418—22no—
66C22×#92418—21no—
67C22×#92418D412yes—
68C22×#92418D412yes—
69C22×#92418C2312yes—
70#14519712—28no—
71#14519700—28no—
72#14519695—28no—
73C22⋊#92418—28no—
74C2×#41×C61—21no—
75C2×#41×C61—21no—
76C2×#41×C61—22no—
77C2×#41×C61—22no—
78C2×#41×C61—22no—
79C2×#41×C61—22no—
80C2×#41×C61D412yes—
81C2×#41×C61C2312yes—
82C22×Q8×C61D411yescharacteristic
83C22×D4×C61—21no—
84C22×D4×C61D411yescharacteristic
85C23×C4×C61—21no—
86C23×C4×C61—21no—
87C23×C4×C61C2311yescharacteristic
88C2×M16×C61—21no—
89C2×M16×C61—22no—
90C2×M16×C61—22no—
91C2×M16×C61C2312yes—
92C22×C8×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99#103×C61—24no—
100#102×C61—24no—
101C24×C61—21no—
102C24×C61#4111yescharacteristic
103#41×C61—24no—
104#41×C61—24no—
105#41×C61—22no—
106#41×C61—22no—
107#41×C61—22no—
108#41×C61—22no—
109#41×C61—42no—
110#41×C61—42no—
111#41×C61—42no—
112#41×C61—42no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—21no—
121C2×D4×C61—22no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61C2×D412yes—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61—41no—
132C22×C4×C61—41no—
133C22×C4×C61—82no—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61#4112yes—
141C22×C4×C61#4111yescharacteristic
142C22×C4×C61#4112yes—
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—24no—
145C22×Dic61—22no—
146C22×Dic61—41no—
147C22×Dic61—41no—
148C22×Dic61—42no—
149C2×Dic122—24no—
150C2×Dic122—24no—
151C2×Dic122—24no—
152C2×Dic122—24no—
153C2×Dic122—22no—
154C2×Dic122—22no—
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122—24no—
158C2×Dic122—22no—
159C2×Dic122—42no—
160C2×Dic122—42no—
161C2×Dic122—41no—
162C2×Dic122—41no—
163C2×Dic122—42no—
164C2×Dic122—42no—
165C2×Dic122—42no—
166C2×Dic122C2×D414yes—
167M16×C61—22no—
168M16×C61—22no—
169M16×C61—42no—
170C2×C8×C61—22no—
171C2×C8×C61—42no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176#11758202—216no—
177#11758200—216no—
178#11758194—24no—
179#11758194—22no—
180#11758194—22no—
181#11758194—22no—
182#11758194—42no—
183#11758194C2×D414yes—
184C2×#92418—24no—
185C2×#92418—24no—
186C2×#92418—24no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—42no—
192C2×#92418C2×D414yes—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61#12111yescharacteristic
200C23×C61#12111yescharacteristic
201C23×C61#12312yes—
202C23×C61#12412yes—
203C23×C61C2×#4111yescharacteristic
204Q8×C61—22no—
205Q8×C61—22no—
206Q8×C61—42no—
207D4×C61—22no—
208D4×C61—22no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—44no—
212D4×C61—42no—
213C2×C4×C61—22no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—44no—
218C2×C4×C61—42no—
219C2×C4×C61—82no—
220C2×C4×C61—82no—
221C2×C4×C61—82no—
222C2×C4×C61#12012yes—
223C2×C4×C61#12012yes—
224C2×C4×C61#12112yes—
225C2×C4×C61#12112yes—
226C2×C4×C61#12112yes—
227C2×C4×C61#12112yes—
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12312yes—
231C2×C4×C61#12312yes—
232C2×C4×C61#12312yes—
233C2×C4×C61#12412yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×Dic61—24no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—44no—
244C2×Dic61—44no—
245Dic122—24no—
246Dic122—24no—
247Dic122—42no—
248Dic122—42no—
249Dic122—44no—
250Dic122—44no—
251C8×C61—42no—
252#92418—24no—
253#92418—24no—
254#92418—42no—
255C2×#2052—611nomaximal, Sylow
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—84no—
266C22×C61#39412yes—
267C22×C61#39412yes—
268C22×C61#47112yes—
269C22×C61#47312yes—
270C22×C61C2×#12111yescharacteristic
271C22×C61C2×#12311yescharacteristic
272C22×C61C2×#12411yescharacteristic
273C4×C61—42no—
274C4×C61—82no—
275C4×C61#39412yes—
276C4×C61#39412yes—
277C4×C61C2×#12011yescharacteristic
278C4×C61C2×#12111yescharacteristic
279C4×C61C2×#12111yescharacteristic
280C4×C61C2×#12311yescharacteristic
281Dic61—44no—
282C22×#137—611no—
283C22×#108—611no—
284C2×#417D6111yescharacteristic
285C22×#103—611no—
286C22×#101—611no—
287C2×#408—611no—
288#2052—618no—
289C2×#343—611no—
290C2×C61—21no—
291C2×C61—21no—
292C2×C61—22no—
293C2×C61—82no—
294C2×C61#205214yescomplemented
295C2×C61C2×#39411yescharacteristic
296C2×C61C2×#47111yescharacteristic
297C2×C61C2×#47311yescharacteristic
298C22×#41C2×D6111yescharacteristic
299C23×Q8—611no—
300C2×#137—614no—
301C2×#137—1224no—
302C2×#137—1224no—
303C2×#137—1222no—
304C22×Q16—1221no—
305C22×SD16—1221no—
306C22×M16—611no—
307C22×M16—611no—
308C22×M16C2×D6111yescharacteristic
309C23×C8—611no—
310C22×C4⋊C4C2×D6111yescharacteristic
311C2×#108—614no—
312C2×#108—1224no—
313#417—22no—
314#417C2×D6114yes—
315C2×#103—21no—
316C2×#103—614no—
317C2×#103—1222no—
318C2×#103—1222no—
319C2×#102—21no—
320C2×#101—614no—
321C2×#101—1224no—
322#408—614no—
323#408—1222no—
324C2×C22⋊C8—1222no—
325#343—614no—
326C61C2×#205211yescharacteristic, complemented, Sylow
327C2×#41—21no—
328C2×#41—21no—
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41C22⋊D6112yes—
334C2×#41C22×D6112yes—
335C22×Q8—614no—
336C22×Q8—612no—
337C22×Q8—1222no—
338C22×Q8—1221no—
339C22×Q8—1222no—
340C22×Q8—1222no—
341C22×Q8—1221no—
342C22×Q8—1221no—
343C22×Q8—1221no—
344C22×Q8—1221no—
345C22×Q8C22⋊D6111yescharacteristic
346C22×D4—21no—
347C22×D4C22⋊D6111yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—1221no—
351C23×C4C22×D6111yescharacteristic
352#137—1228no—
353#137—1228no—
354#137—12216no—
355C2×Q16—1228no—
356C2×Q16—1224no—
357C2×SD16—1228no—
358C2×SD16—1224no—
359C2×M16—21no—
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—614no—
363C2×M16—612no—
364C2×M16—612no—
365C2×M16—1222no—
366C2×M16—1221no—
367C2×M16—1221no—
368C2×M16—1222no—
369C2×M16—1222no—
370C2×M16C22×D6112yes—
371C22×C8—21no—
372C22×C8—612no—
373C22×C8—612no—
374C22×C8—612no—
375C22×C8—1222no—
376C22×C8—1222no—
377C22×C8—1221no—
378C22×C8—1221no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4C22×D6112yes—
385#108—1228no—
386#103—24no—
387#103—1228no—
388#102—24no—
389#101—1228no—
390C22⋊C8—1228no—
391C24—21no—
392C24#1175822411yescharacteristic
393#41—22no—
394#41—22no—
395#41—22no—
396#41—22no—
397#41—24no—
398#41—24no—
399#41—42no—
400#41—42no—
401#41—42no—
402#41—42no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—614no—
407C2×Q8—1222no—
408C2×Q8—1224no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×Q8—1224no—
414C2×Q8—1224no—
415C2×Q8—1224no—
416C2×Q8—1224no—
417C2×Q8—2441no—
418C2×Q8—2441no—
419C2×Q8—2442no—
420C2×Q8—2442no—
421C2×Q8—2442no—
422C2×Q8—2442no—
423C2×Q8—2442no—
424C2×Q8C2×C22⋊D6112yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4C2×C22⋊D6112yes—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—82no—
442C22×C4—1222no—
443C22×C4—1224no—
444C22×C4—2441no—
445C22×C4—2441no—
446C22×C4—2442no—
447C22×C4#1175822212yes—
448C22×C4#1175822212yes—
449C22×C4D4×D6112yes—
450C22×C4D4×D6112yes—
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4#1175822411yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457Q16—12216no—
458SD16—12216no—
459M16—22no—
460M16—22no—
461M16—42no—
462M16—614no—
463M16—1222no—
464M16—1224no—
465M16—1222no—
466M16—1222no—
467M16—2442no—
468C2×C8—22no—
469C2×C8—42no—
470C2×C8—614no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1224no—
475C2×C8—1224no—
476C2×C8—1224no—
477C2×C8—1222no—
478C2×C8—2442no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23#1451974911yescharacteristic
490C23#1451974911yescharacteristic
491C23#1451975912yes—
492C23#1451976312yes—
493C23C2×#1175822411yescharacteristic
494Q8—22no—
495Q8—22no—
496Q8—42no—
497Q8—1224no—
498Q8—1224no—
499Q8—2442no—
500Q8—2442no—
501Q8—2444no—
502Q8—2444no—
503D4—22no—
504D4—22no—
505D4—42no—
506D4—42no—
507D4—44no—
508D4—42no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—1224no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2444no—
522C2×C4—2444no—
523C2×C4#1451974512yes—
524C2×C4#1451974512yes—
525C2×C4#1451974911yescharacteristic
526C2×C4#1451974911yescharacteristic
527C2×C4#1451975012yes—
528C2×C4#1451975012yes—
529C2×C4#1451975012yes—
530C2×C4#1451975012yes—
531C2×C4#1451975912yes—
532C2×C4#1451976012yes—
533C2×C4#1451976012yes—
534C2×C4#1451976312yes—
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C2×C4C22×C22⋊D6111yescharacteristic
539C2×C4C2×#1175822211yescharacteristic
540C2×C4C2×#1175822211yescharacteristic
541C8—42no—
542C8—1224no—
543C8—1224no—
544C8—2442no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—24no—
550C22—24no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—84no—
555C22#1683967712yes—
556C22#1683967712yes—
557C22#1684002312yes—
558C22#1684003512yes—
559C22C2×#1451974911yescharacteristic
560C22C2×#1451975911yescharacteristic
561C22C2×#1451976311yescharacteristic
562C4—42no—
563C4—82no—
564C4—2444no—
565C4#1683967712yes—
566C4#1683967712yes—
567C4C2×#1451974511yescharacteristic
568C4C2×#1451975011yescharacteristic
569C4C2×#1451975011yescharacteristic
570C4C2×#1451976011yescharacteristic
571C2—21no—
572C2—21no—
573C2—22no—
574C2—82no—
575C2#2144711614yesdirect product factor
576C2C2×#1683967711yescharacteristic
577C2C2×#1684002311yescharacteristic
578C2C2×#1684003511yescharacteristic
subgroupG/H
centerC23#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC23×C61#121
prime radicalC22×#41×C61C22
hypercenterC2×#417D61
hypocenterC61C2×#2052
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#14519712C2semidirect
3. C22×#14519700C2semidirect
4. C22×#14519695C2semidirect
6. #21447116C2direct (N × H)
7. C2×#16839531C2semidirect
20. C2×#14519712C22semidirect
23. C2×#14519700C22semidirect
25. C2×#14519695C22semidirect
29. #16839531C22semidirect
294. C2×C61#2052semidirect
326. C61C2×#2052semidirect
575. C2#21447116direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198072
C22×#145197122
C22×#145197002
C22×#145196952
C2×#168397292
#214471162
C2×#168395312
C2×C61×#4172
C2×#205261