Finite Group Databasebeta

← #34322200

Subgroups — #34322200

Order 20224.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098845C211yesmaximal, characteristic, complemented
2C2×C4×#11929612C211yesmaximal, characteristic, complemented
3#23768322C211yesmaximal, characteristic, complemented
4#23764586C212yesmaximal, complemented
5#23764403C212yesmaximal, complemented
6#23764362C211yesmaximal, characteristic, complemented
7#23764195C211yesmaximal, characteristic, complemented
8#23763707C212yesmaximal, complemented
9#23763593C212yesmaximal, complemented
10#23763400C211yesmaximal, characteristic, complemented
11C2×#547×C79C211yesmaximal, characteristic
12C22×#11929612C2211yescharacteristic
13C2×#14579140C2212yes—
14C2×C4×Dic158C2211yescharacteristic
15#19098845—24no—
16C4×#11929612—24no—
17C2×#14579092C2212yes—
18C2×#14579092C2211yescharacteristic
19C2×C4⋊(C79⋊C8)C2211yescharacteristic
20C2×#14579084C2212yes—
21C2×#14579084C2211yescharacteristic
22C2×#14579082C2211yescharacteristic
23C2×#14579082C2212yes—
24C2×C22⋊(C79⋊C8)C2212yes—
25C2×C22⋊(C79⋊C8)C2212yes—
26C2×C4×C79⋊C8C2211yescharacteristic
27#19098486—22no—
28#19098466—21no—
29#19098459—21no—
30#19098433C2211yescharacteristic
31#19098423C2212yes—
32#19098325C2212yes—
33#19098325C2212yes—
34#19098325C2212yes—
35C22×#41×C79—21no—
36#547×C79—24no—
37#547×C79—24no—
38C2×C4⋊Q8×C79C2211yescharacteristic
39C2×#123×C79—22no—
40C2×C22⋊Q8×C79—21no—
41C2×#121×C79C2212yes—
42C2×C4×D4×C79C2211yescharacteristic
43C2×C4×D4×C79C2211yescharacteristic
44C22×C4⋊C4×C79C2212yes—
45C22×Dic158C2311yescharacteristic
46C2×#11929612—22no—
47C2×#11929612—22no—
48C2×#11929612—22no—
49C22×C79⋊C8C2311yescharacteristic
50C22×C79⋊C8C2312yes—
51C22×C79⋊C8C2311yescharacteristic
52#14579140—24no—
53#14579140—42no—
54C4×Dic158—22no—
55C4×Dic158—22no—
56C2×C4⋊Dic79—21no—
57C2×C4⋊Dic79—21no—
58C2×C4⋊Dic79C2311yescharacteristic
59C2×C4⋊Dic79C2312yes—
60C2×Dic79⋊C4—22no—
61C2×Dic79⋊C4—21no—
62C2×C22⋊Dic79—22no—
63C2×C22⋊Dic79—22no—
64C2×C4×Dic79—21no—
65#14579092—22no—
66#14579092—24no—
67C4⋊(C79⋊C8)—22no—
68#14579084—22no—
69#14579084—24no—
70#14579082—22no—
71#14579082—42no—
72C22⋊(C79⋊C8)—24no—
73C22⋊(C79⋊C8)—42no—
74C4×C79⋊C8—22no—
75C79⋊#95—22no—
76C2×#41×C79—28no—
77C2×#41×C79—44no—
78C2×#41×C79—44no—
79C22×Q8×C79—21no—
80C22×D4×C79—22no—
81C22×D4×C79D411yescharacteristic
82C22×D4×C79C2311yescharacteristic
83C23×C4×C79—22no—
84C23×C4×C79—21no—
85C23×C4×C79C2312yes—
86C4⋊Q8×C79—22no—
87C4⋊Q8×C79D414yes—
88#123×C79—216no—
89C22⋊Q8×C79—28no—
90#121×C79—24no—
91#121×C79—24no—
92C4×D4×C79—22no—
93C4×D4×C79—22no—
94C4×D4×C79—22no—
95C4×D4×C79D414yes—
96C2×C4⋊C4×C79—22no—
97C2×C4⋊C4×C79—21no—
98C2×C4⋊C4×C79—24no—
99C2×C4⋊C4×C79—24no—
100C2×C4⋊C4×C79D411yescharacteristic
101C2×C4⋊C4×C79C2311yescharacteristic
102C2×C4⋊C4×C79C2312yes—
103C2×C22⋊C4×C79—21no—
104C2×C22⋊C4×C79—22no—
105C2×C22⋊C4×C79D412yes—
106C2×C42×C79C2311yescharacteristic
107C2×Dic158—21no—
108C2×Dic158—21no—
109C2×Dic158—21no—
110C2×Dic158—21no—
111C2×Dic158—21no—
112C2×Dic158#4112yescomplemented
113C22×Dic79—21no—
114C22×Dic79—21no—
115C22×Dic79—22no—
116#11929612—28no—
117C2×C79⋊C8—21no—
118C2×C79⋊C8—21no—
119C2×C79⋊C8—22no—
120C2×C79⋊C8—21no—
121C2×C79⋊C8—22no—
122C2×C79⋊C8—21no—
123C2×C79⋊C8—22no—
124C2×C79⋊C8—21no—
125C2×C79⋊C8#4112yescomplemented
126C4⋊Dic79—21no—
127C4⋊Dic79—21no—
128C4⋊Dic79—22no—
129C4⋊Dic79—22no—
130C4⋊Dic79—42no—
131C4⋊Dic79—81no—
132Dic79⋊C4—44no—
133Dic79⋊C4—42no—
134C22⋊Dic79—44no—
135C22⋊Dic79—82no—
136C4×Dic79—42no—
137C24×C79—21no—
138C24×C79C2×D412yes—
139#41×C79—432no—
140C2×Q8×C79—24no—
141C2×Q8×C79—42no—
142C2×Q8×C79—42no—
143C2×D4×C79—28no—
144C2×D4×C79—22no—
145C2×D4×C79—22no—
146C2×D4×C79—21no—
147C2×D4×C79—21no—
148C2×D4×C79—21no—
149C2×D4×C79—21no—
150C2×D4×C79—21no—
151C2×D4×C79—44no—
152C2×D4×C79—44no—
153C2×D4×C79SD1614yescomplemented
154C2×D4×C79C2×D412yes—
155C22×C4×C79—28no—
156C22×C4×C79—21no—
157C22×C4×C79—22no—
158C22×C4×C79—24no—
159C22×C4×C79—22no—
160C22×C4×C79—22no—
161C22×C4×C79—22no—
162C22×C4×C79—22no—
163C22×C4×C79—22no—
164C22×C4×C79—22no—
165C22×C4×C79—44no—
166C22×C4×C79—44no—
167C22×C4×C79—42no—
168C22×C4×C79—42no—
169C22×C4×C79C2×D412yes—
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2411yescharacteristic
173C4⋊C4×C79—28no—
174C4⋊C4×C79—24no—
175C4⋊C4×C79—22no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—22no—
178C4⋊C4×C79C2×D412yes—
179C4⋊C4×C79C2×D414yes—
180C22⋊C4×C79—28no—
181C22⋊C4×C79—24no—
182C22⋊C4×C79—24no—
183C42×C79C2×D412yes—
184C42×C79C2×D412yes—
185C23×C79—22no—
186C23×C79—22no—
187C23×C79—22no—
188C23×C79—22no—
189C23×C79—22no—
190C23×C79—22no—
191C23×C79—22no—
192C23×C79—41no—
193C23×C79—41no—
194C23×C79—41no—
195C23×C79—41no—
196C23×C79—41no—
197C23×C79—41no—
198C23×C79C2×SD1614yes—
199C23×C79C22×D411yescharacteristic
200Q8×C79—48no—
201D4×C79—22no—
202D4×C79—22no—
203D4×C79—28no—
204D4×C79—416no—
205D4×C79—41no—
206D4×C79—41no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—22no—
210C2×C4×C79—22no—
211C2×C4×C79—24no—
212C2×C4×C79—22no—
213C2×C4×C79—22no—
214C2×C4×C79—416no—
215C2×C4×C79—41no—
216C2×C4×C79—42no—
217C2×C4×C79—41no—
218C2×C4×C79—42no—
219C2×C4×C79—42no—
220C2×C4×C79—42no—
221C2×C4×C79—42no—
222C2×C4×C79—42no—
223C2×C4×C79—48no—
224C2×C4×C79#12012yes—
225C2×C4×C79#12012yes—
226C2×C4×C79C2×SD1612yes—
227C2×C4×C79AGL(1,ℤ/8)12yes—
228C2×C4×C79C22×D411yescharacteristic
229C2×C4×C79C22×D411yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79H(5,𝔽2)11yescharacteristic
233C2×C4×C792−1+411yescharacteristic
234C2×Dic79—22no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—42no—
239C2×Dic79—42no—
240C2×Dic79—41no—
241C2×Dic79—42no—
242C2×Dic79—41no—
243Dic158—22no—
244Dic158—22no—
245Dic158—41no—
246Dic158—41no—
247C79⋊C8—22no—
248C79⋊C8—41no—
249C79⋊C8—42no—
250C22×C79—24no—
251C22×C79—24no—
252C22×C79—24no—
253C22×C79—42no—
254C22×C79—42no—
255C22×C79—42no—
256C22×C79—42no—
257C22×C79—42no—
258C22×C79—42no—
259C22×C79—42no—
260C22×C79—81no—
261C22×C79—81no—
262C22×C79—81no—
263C22×C79—81no—
264C22×C79C2×#12011yescharacteristic
265C22×C79#54611yescharacteristic
266C22×C79#54711yescharacteristic
267C22×C79C22×SD1611yescharacteristic
268C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C79#57711yescharacteristic
270C22×C79#57811yescharacteristic
271C4×C79—22no—
272C4×C79—42no—
273C4×C79—42no—
274C4×C79—44no—
275C4×C79#45014yes—
276C4×C79#52511yescharacteristic
277C4×C79#53711yescharacteristic
278C4×C79#54811yescharacteristic
279C4×C79#55011yescharacteristic
280Dic79—42no—
281Dic79—82no—
282Dic79—81no—
283#24120—791nomaximal, Sylow
284C2×C79—44no—
285C2×C79—82no—
286C2×C79#240711yescharacteristic
287C2×C79C2×#45011yescharacteristic
288C2×C79#301311yescharacteristic
289C2×C79#329811yescharacteristic
290C2×C79#333211yescharacteristic
291C2×C79#333511yescharacteristic
292C2×C79#335311yescharacteristic
293C2×#547D7911yescharacteristic
294C2×#475—791no—
295C2×C4×SD16—791no—
296#2538—791no—
297#2034—792no—
298#2000—792no—
299#1990—791no—
300#1954—791no—
301#1872—792no—
302#1853—792no—
303#1802—791no—
304C79#2412011yescharacteristic, complemented, Sylow
305C22×#41—21no—
306C22×SD16—791no—
307#547—24no—
308#547—24no—
309C2×C4⋊Q8C2×D7911yescharacteristic
310C2×#123—22no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8—792no—
313C2×#121C2×D7912yes—
314C2×C4×Q8—791no—
315C2×C4×D4C2×D7911yescharacteristic
316C2×C4×D4C2×D7911yescharacteristic
317C22×C4⋊C4C2×D7912yes—
318#475—1584no—
319C4×SD16—1584no—
320C2×#106—792no—
321C2×#106—791no—
322C2×C4⋊C8—791no—
323C2×#103—792no—
324C2×#103—791no—
325C2×#102—792no—
326C2×#102—791no—
327C2×C22⋊C8—792no—
328C2×C22⋊C8—792no—
329C2×C4×C8—791no—
330#396—1582no—
331#390—1581no—
332#389—1581no—
333#384—791no—
334#380—792no—
335#340—792no—
336#340—792no—
337#340—792no—
338C2×#41—28no—
339C2×#41—44no—
340C2×#41—44no—
341C22×Q8—21no—
342C22×Q8—791no—
343C22×D4—22no—
344C22×D4C22⋊D7911yescharacteristic
345C22×D4C22×D7911yescharacteristic
346C23×C4—22no—
347C23×C4—21no—
348C23×C4C22×D7912yes—
349C2×SD16—1582no—
350C2×SD16—1582no—
351C2×SD16—1582no—
352C22×C8—791no—
353C22×C8—791no—
354C22×C8—792no—
355C4⋊Q8—22no—
356C4⋊Q8C22⋊D7914yes—
357#123—216no—
358C22⋊Q8—28no—
359C22⋊Q8—3162no—
360C22⋊Q8—1584no—
361#121—24no—
362#121—24no—
363C4×Q8—1582no—
364C4×Q8—1582no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4C22⋊D7914yes—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4—792no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4—1582no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4C22⋊D7911yescharacteristic
380C2×C4⋊C4C22×D7911yescharacteristic
381C2×C4⋊C4C22×D7912yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1582no—
385C2×C22⋊C4—1582no—
386C2×C22⋊C4C22⋊D7912yes—
387C2×C42—1581no—
388C2×C42C22×D7911yescharacteristic
389#106—1582no—
390#106—1584no—
391C4⋊C8—1582no—
392#103—1582no—
393#103—1584no—
394#102—3162no—
395#102—1582no—
396C22⋊C8—3162no—
397C22⋊C8—1584no—
398C4×C8—1582no—
399#95—1582no—
400C24—21no—
401C24C2×C22⋊D7912yes—
402#41—432no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—792no—
407C2×Q8—1581no—
408C2×Q8—1581no—
409C2×Q8—1581no—
410C2×Q8—1581no—
411C2×Q8—1581no—
412C2×D4—28no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4#1192961214yescomplemented
423C2×D4C2×C22⋊D7912yes—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—1581no—
439C22×C4—1582no—
440C22×C4—1581no—
441C22×C4C2×C22⋊D7912yes—
442C22×C4C2×C22⋊D7911yescharacteristic
443C22×C4C2×C22⋊D7911yescharacteristic
444C22×C4C23×D7911yescharacteristic
445SD16—1588no—
446C2×C8—792no—
447C2×C8—1582no—
448C2×C8—1581no—
449C2×C8—1582no—
450C2×C8—1581no—
451C2×C8—1581no—
452C2×C8—1582no—
453C2×C8—1581no—
454C2×C8—1581no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—3164no—
461C4⋊C4—3162no—
462C4⋊C4—3162no—
463C4⋊C4—6321no—
464C4⋊C4—1582no—
465C4⋊C4—1582no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4C2×C22⋊D7912yes—
469C4⋊C4C2×C22⋊D7914yes—
470C22⋊C4—28no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—3164no—
474C22⋊C4—6322no—
475C42—3162no—
476C42C2×C22⋊D7912yes—
477C42C2×C22⋊D7912yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×#1192961214yes—
492C23C22×C22⋊D7911yescharacteristic
493Q8—48no—
494Q8—3161no—
495Q8—3161no—
496Q8—1582no—
497Q8—1582no—
498D4—22no—
499D4—28no—
500D4—22no—
501D4—416no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—416no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—3162no—
524C2×C4—3161no—
525C2×C4—3162no—
526C2×C4—3161no—
527C2×C4—3162no—
528C2×C4—3161no—
529C2×C4—1582no—
530C2×C4#1457913012yes—
531C2×C4#1457913012yes—
532C2×C4C2×#1192961212yes—
533C2×C4#1457918712yes—
534C2×C4C22×C22⋊D7911yescharacteristic
535C2×C4C22×C22⋊D7911yescharacteristic
536C2×C4#41×D7911yescharacteristic
537C2×C4#41×D7911yescharacteristic
538C2×C4#1457920711yescharacteristic
539C2×C4#1457920811yescharacteristic
540C8—3161no—
541C8—3162no—
542C8—1582no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1457913011yescharacteristic
558C22#1909920111yescharacteristic
559C22#1909921211yescharacteristic
560C22C22×#1192961211yescharacteristic
561C22C2×#1457918711yescharacteristic
562C22#1909937111yescharacteristic
563C22#1909937811yescharacteristic
564C4—22no—
565C4—42no—
566C4—44no—
567C4—42no—
568C4—3162no—
569C4—6321no—
570C4—6322no—
571C4#1909871414yes—
572C4#1909907011yescharacteristic
573C4#1909914111yescharacteristic
574C4#1909922011yescharacteristic
575C4#1909923211yescharacteristic
576C2—44no—
577C2—82no—
578C2#2376673911yescharacteristic
579C2C2×#1909871411yescharacteristic
580C2#2377271511yescharacteristic
581C2#2377525911yescharacteristic
582C2#2377572111yescharacteristic
583C2#2377575711yescharacteristic
584C2#2377598411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
Fitting, prime radicalC2×#547×C79C2
socleC23×C79C22×D4
hypercenterC2×#547D79
hypocenterC79#24120
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#19098845C2semidirect
2. C2×C4×#11929612C2semidirect
3. #23768322C2semidirect
4. #23764586C2semidirect
5. #23764403C2semidirect
6. #23764362C2semidirect
7. #23764195C2semidirect
8. #23763707C2semidirect
9. #23763593C2semidirect
10. #23763400C2semidirect
112. C2×Dic158#41semidirect
125. C2×C79⋊C8#41semidirect
153. C2×D4×C79SD16semidirect
304. C79#24120semidirect
422. C2×D4#11929612semidirect

Maximal subgroups

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

subgroupindex
C2×#190988452
C2×C4×#119296122
#237683222
#237645862
#237644032
#237643622
#237641952
#237637072
#237635932
#237634002
C2×#547×C792
#2412079