Finite Group Databasebeta

← #34215488

Subgroups — #34215488

Order 20224.

1542 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic158C211yesmaximal, characteristic
2C2×#19098430C211yesmaximal, characteristic
3C2×#19098427C211yesmaximal, characteristic
4#23761670C212yesmaximal
5#23761666C212yesmaximal
6#23761652C212yesmaximal
7#23761651C212yesmaximal
8#23761627C211yesmaximal, characteristic
9#23761624C211yesmaximal, characteristic
10#23761624C211yesmaximal, characteristic
11C79×#1434C211yesmaximal, characteristic
12C23×Dic158C2211yescharacteristic
13C2×C4×Dic158—24no—
14C2×C4×Dic158—24no—
15C2×C4×Dic158C418yescomplemented
16C22×C4⋊Dic79C2211yescharacteristic
17C22×Dic79⋊C4C2211yescharacteristic
18C22×Dic79⋊C4C2211yescharacteristic
19C22×Dic79⋊C4C2212yes—
20C22×C4×Dic79C2211yescharacteristic
21C22×C4×Dic79C2211yescharacteristic
22#19098430—24no—
23#19098427—24no—
24C2×C79⋊#95C2212yes—
25C2×C79⋊#95C2212yes—
26C2×C79⋊#95C2212yes—
27C2×C79⋊#95C2212yes—
28C2×C79⋊#95C2212yes—
29C2×C79⋊#95C2211yescharacteristic
30C2×C79⋊#95C2211yescharacteristic
31C2×C79⋊#95C2212yes—
32C2×C79⋊#95C2212yes—
33C2×C79⋊#95C2212yes—
34C2×C79⋊#95C2212yes—
35C22×C42×C79C2211yescharacteristic
36C2×#95×C79C2212yes—
37C2×#95×C79C2211yescharacteristic
38C2×#95×C79C2211yescharacteristic
39C2×#95×C79C2212yes—
40C22×Dic158—24no—
41C22×Dic158—22no—
42C22×Dic158—22no—
43C22×Dic158—24no—
44C22×Dic158C2×C414yes—
45C23×Dic79C2312yes—
46C23×Dic79C2312yes—
47C23×Dic79C2311yescharacteristic
48C23×Dic79C2311yescharacteristic
49C23×Dic79C2311yescharacteristic
50C23×Dic79C2311yescharacteristic
51C4×Dic158—232no—
52C2×C4⋊Dic79—22no—
53C2×C4⋊Dic79—22no—
54C2×C4⋊Dic79C2×C414yes—
55C2×Dic79⋊C4—22no—
56C2×Dic79⋊C4—22no—
57C2×Dic79⋊C4—22no—
58C2×Dic79⋊C4—22no—
59C2×Dic79⋊C4—24no—
60C2×Dic79⋊C4—42no—
61C2×Dic79⋊C4—42no—
62C2×Dic79⋊C4C2×C414yes—
63C2×Dic79⋊C4C2×C414yes—
64C2×C4×Dic79—22no—
65C2×C4×Dic79—22no—
66C2×C4×Dic79C2×C414yes—
67C2×C4×Dic79C2×C414yes—
68C2×C4×Dic79D414yes—
69C2×C4×Dic79D414yes—
70C79⋊#95—22no—
71C79⋊#95—22no—
72C79⋊#95—24no—
73C79⋊#95—24no—
74C79⋊#95—24no—
75C79⋊#95—24no—
76C79⋊#95—24no—
77C79⋊#95—24no—
78C79⋊#95—42no—
79C79⋊#95—42no—
80C79⋊#95—42no—
81C23×C4×C79C2312yes—
82C23×C4×C79C2312yes—
83C23×C4×C79C2311yescharacteristic
84C23×C4×C79C2311yescharacteristic
85C23×C4×C79C2311yescharacteristic
86C2×C42×C79—22no—
87C2×C42×C79—22no—
88C2×C42×C79C2×C414yes—
89#95×C79—22no—
90#95×C79—22no—
91#95×C79—24no—
92#95×C79—42no—
93C2×Dic158—216no—
94C2×Dic158—216no—
95C2×Dic158—28no—
96C2×Dic158—28no—
97C2×Dic158—28no—
98C22×Dic79—21no—
99C22×Dic79—21no—
100C22×Dic79—21no—
101C22×Dic79—21no—
102C22×Dic79—22no—
103C22×Dic79—22no—
104C22×Dic79—22no—
105C22×Dic79—22no—
106C22×Dic79—22no—
107C22×Dic79—22no—
108C22×Dic79—22no—
109C22×Dic79—22no—
110C22×Dic79—22no—
111C22×Dic79—21no—
112C22×Dic79—21no—
113C22×Dic79—21no—
114C22×Dic79—21no—
115C22×Dic79—22no—
116C22×Dic79—22no—
117C22×Dic79—22no—
118C22×Dic79—21no—
119C22×Dic79—21no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—22no—
123C22×Dic79—22no—
124C22×Dic79—22no—
125C22×Dic79—22no—
126C22×Dic79—22no—
127C22×Dic79C22×C412yes—
128C22×Dic79C22×C412yes—
129C22×Dic79C22×C412yes—
130C22×Dic79C22×C412yes—
131C22×Dic79C2×D412yes—
132C22×Dic79C2×D412yes—
133C22×Dic79C2×D412yes—
134C22×Dic79C2×D412yes—
135C22×Dic79#4114yes—
136C4⋊Dic79—28no—
137Dic79⋊C4—28no—
138Dic79⋊C4—28no—
139Dic79⋊C4—48no—
140C4×Dic79—28no—
141C4×Dic79C22⋊C4116yes—
142C24×C79C2411yescharacteristic
143C22×C4×C79—21no—
144C22×C4×C79—21no—
145C22×C4×C79—22no—
146C22×C4×C79—22no—
147C22×C4×C79—22no—
148C22×C4×C79—21no—
149C22×C4×C79—21no—
150C22×C4×C79—21no—
151C22×C4×C79—21no—
152C22×C4×C79—21no—
153C22×C4×C79—21no—
154C22×C4×C79—22no—
155C22×C4×C79—22no—
156C22×C4×C79—22no—
157C22×C4×C79—22no—
158C22×C4×C79—22no—
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—22no—
166C22×C4×C79—22no—
167C22×C4×C79—22no—
168C22×C4×C79C22×C412yes—
169C22×C4×C79C22×C412yes—
170C22×C4×C79C22×C412yes—
171C22×C4×C79C2×D412yes—
172C22×C4×C79C2×D412yes—
173C22×C4×C79#4114yes—
174C42×C79—28no—
175C23×C79C23×C411yescharacteristic
176C23×C79C22×D411yescharacteristic
177C23×C79C22×D411yescharacteristic
178C23×C79C2×#4112yes—
179C23×C79C2×#4112yes—
180C23×C79H(5,𝔽2)11yescharacteristic
181C23×C79H(5,𝔽2)11yescharacteristic
182C23×C792−1+412yes—
183C23×C792−1+412yes—
184C23×C792−1+411yescharacteristic
185C23×C792−1+411yescharacteristic
186C2×C4×C79—22no—
187C2×C4×C79—22no—
188C2×C4×C79—22no—
189C2×C4×C79—22no—
190C2×C4×C79—24no—
191C2×C4×C79—24no—
192C2×C4×C79—24no—
193C2×C4×C79—24no—
194C2×C4×C79—22no—
195C2×C4×C79—22no—
196C2×C4×C79—22no—
197C2×C4×C79—41no—
198C2×C4×C79—41no—
199C2×C4×C79—41no—
200C2×C4×C79—41no—
201C2×C4×C79—42no—
202C2×C4×C79—42no—
203C2×C4×C79—42no—
204C2×C4×C79—42no—
205C2×C4×C79—42no—
206C2×C4×C79—42no—
207C2×C4×C79—42no—
208C2×C4×C79—42no—
209C2×C4×C79—42no—
210C2×C4×C79—42no—
211C2×C4×C79—42no—
212C2×C4×C79—42no—
213C2×C4×C79—42no—
214C2×C4×C79—42no—
215C2×C4×C79C2×C22⋊C414yes—
216C2×C4×C79#11714yes—
217C2×Dic79—22no—
218C2×Dic79—22no—
219C2×Dic79—22no—
220C2×Dic79—22no—
221C2×Dic79—22no—
222C2×Dic79—22no—
223C2×Dic79—24no—
224C2×Dic79—24no—
225C2×Dic79—24no—
226C2×Dic79—24no—
227C2×Dic79—22no—
228C2×Dic79—22no—
229C2×Dic79—22no—
230C2×Dic79—22no—
231C2×Dic79—22no—
232C2×Dic79—22no—
233C2×Dic79—22no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—42no—
239C2×Dic79—42no—
240C2×Dic79—42no—
241C2×Dic79—42no—
242C2×Dic79—42no—
243C2×Dic79—42no—
244C2×Dic79—42no—
245C2×Dic79—42no—
246C2×Dic79—42no—
247C2×Dic79—42no—
248C2×Dic79—42no—
249C2×Dic79—42no—
250C2×Dic79—42no—
251C2×Dic79—42no—
252C2×Dic79C2×C22⋊C414yes—
253C2×Dic79C2×C22⋊C414yes—
254C2×Dic79#11714yes—
255Dic158—232no—
256C22×C79C22×C22⋊C411yescharacteristic
257C22×C79C2×#11711yescharacteristic
258C22×C79C2×#11711yescharacteristic
259C22×C79#51911yescharacteristic
260C22×C79#51911yescharacteristic
261C22×C79#52011yescharacteristic
262C22×C79#52011yescharacteristic
263C22×C79#52512yes—
264C22×C79#52512yes—
265C22×C79#53511yescharacteristic
266C22×C79#53611yescharacteristic
267C22×C79#53612yes—
268C22×C79#53612yes—
269C22×C79#53711yescharacteristic
270C22×C79#53711yescharacteristic
271C22×C79#54112yes—
272C22×C79#54112yes—
273C22×C79#54112yes—
274C22×C79#54112yes—
275C22×C79#54212yes—
276C22×C79#54212yes—
277C22×C79#54212yes—
278C22×C79#54212yes—
279C4×C79—24no—
280C4×C79—24no—
281C4×C79—42no—
282C4×C79—82no—
283C4×C79—82no—
284Dic79—24no—
285Dic79—24no—
286Dic79—24no—
287Dic79—42no—
288Dic79—82no—
289Dic79—82no—
290#14346—791nomaximal, Sylow
291C2×C79#228011yescharacteristic
292C2×C79#231011yescharacteristic
293C2×C79#231111yescharacteristic
294C2×C79#232011yescharacteristic
295C2×C79#232011yescharacteristic
296C2×C79#232011yescharacteristic
297C2×C79#232011yescharacteristic
298C2×C79#263212yes—
299C2×C79#263412yes—
300C2×C79#263712yes—
301C2×C79#263712yes—
302C22×C4×Q8—791no—
303C2×#383—791no—
304C2×#382—791no—
305#1446—792no—
306#1445—792no—
307#1441—792no—
308#1441—792no—
309#1434—791no—
310#1434—791no—
311#1434—791no—
312#1434D7911yescharacteristic
313C79#1434611yescharacteristic, complemented, Sylow
314C23×Q8—791no—
315C2×C4×Q8—798no—
316C2×C4×Q8—1584no—
317C2×C4×Q8—1584no—
318C22×C4⋊C4—792no—
319C22×C4⋊C4—791no—
320C22×C4⋊C4—791no—
321C22×C4⋊C4—791no—
322C22×C42—791no—
323C22×C42—791no—
324C22×C42C2×D7911yescharacteristic
325#383—1584no—
326#382—1584no—
327C2×#95—792no—
328C2×#95—792no—
329C2×#95—792no—
330C2×#95—791no—
331C2×#95—791no—
332C2×#95—792no—
333C2×#95—792no—
334C2×#95—792no—
335C2×#95—792no—
336C2×#95—792no—
337C2×#95—792no—
338C2×#95C2×D7911yescharacteristic
339C2×#95C2×D7911yescharacteristic
340C2×#95C2×D7912yes—
341C2×#95C2×D7912yes—
342C22×Q8—794no—
343C22×Q8—1582no—
344C22×Q8—1582no—
345C22×Q8—1584no—
346C22×Q8—1584no—
347C23×C4—791no—
348C23×C4—791no—
349C23×C4—791no—
350C23×C4—791no—
351C23×C4—792no—
352C23×C4—792no—
353C23×C4C22×D7912yes—
354C23×C4C22×D7912yes—
355C23×C4C22×D7911yescharacteristic
356C23×C4C22×D7911yescharacteristic
357C23×C4C22×D7911yescharacteristic
358C4×Q8—15832no—
359C2×C4⋊C4—3162no—
360C2×C4⋊C4—3162no—
361C2×C4⋊C4—794no—
362C2×C4⋊C4—794no—
363C2×C4⋊C4—794no—
364C2×C4⋊C4—1584no—
365C2×C4⋊C4—1582no—
366C2×C4⋊C4—1582no—
367C2×C4⋊C4—1582no—
368C2×C4⋊C4—1582no—
369C2×C4⋊C4—1582no—
370C2×C4⋊C4—1582no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—794no—
374C2×C42—794no—
375C2×C42—794no—
376C2×C42—794no—
377C2×C42—1582no—
378C2×C42—1582no—
379C2×C42C4×D7914yes—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—42no—
384#95—3162no—
385#95—3162no—
386#95—3162no—
387#95—1584no—
388#95—1584no—
389#95—1584no—
390#95—1584no—
391#95—1584no—
392#95—1582no—
393#95—1582no—
394#95—1584no—
395C24C23×D7911yescharacteristic
396C2×Q8—1588no—
397C2×Q8—1588no—
398C2×Q8—15816no—
399C2×Q8—15816no—
400C2×Q8—1588no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—794no—
427C22×C4—792no—
428C22×C4—792no—
429C22×C4—792no—
430C22×C4—792no—
431C22×C4—792no—
432C22×C4—792no—
433C22×C4—792no—
434C22×C4—792no—
435C22×C4—1582no—
436C22×C4—1582no—
437C22×C4—1582no—
438C22×C4—1582no—
439C22×C4—1582no—
440C22×C4—1582no—
441C22×C4—1582no—
442C22×C4—1582no—
443C22×C4—1582no—
444C22×C4—1582no—
445C22×C4—1582no—
446C22×C4—1582no—
447C22×C4—1582no—
448C22×C4—1582no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4—1582no—
456C22×C4—1582no—
457C22×C4—1582no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4—1581no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4C2×C4×D7912yes—
465C22×C4C2×C4×D7912yes—
466C22×C4C2×C4×D7912yes—
467C22×C4D4×D7912yes—
468C22×C4D4×D7912yes—
469C22×C4#1192962414yes—
470C4⋊C4—3168no—
471C4⋊C4—1588no—
472C4⋊C4—1588no—
473C4⋊C4—1588no—
474C42—28no—
475C42—7916no—
476C42—1588no—
477C23C22×C4×D7911yescharacteristic
478C23C2×D4×D7911yescharacteristic
479C23C2×D4×D7911yescharacteristic
480C23#41×D7912yes—
481C23C2×#1192962412yes—
482C23#1457920611yescharacteristic
483C23#1457920711yescharacteristic
484C23#1457920811yescharacteristic
485C23#1457920812yes—
486C23#1457920812yes—
487C23#1457920911yescharacteristic
488Q8—15832no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3162no—
523C2×C4—3162no—
524C2×C4—3162no—
525C2×C4—3162no—
526C2×C4—3162no—
527C2×C4—3162no—
528C2×C4—3162no—
529C2×C4—3162no—
530C2×C4—3162no—
531C2×C4—3162no—
532C2×C4—3162no—
533C2×C4—3162no—
534C2×C4—3162no—
535C2×C4—3162no—
536C2×C4—794no—
537C2×C4—794no—
538C2×C4—794no—
539C2×C4—1582no—
540C2×C4—1582no—
541C2×C4—1582no—
542C2×C4—1582no—
543C2×C4—1582no—
544C2×C4—1582no—
545C2×C4—1582no—
546C2×C4—1582no—
547C2×C4—1582no—
548C2×C4—1582no—
549C2×C4—1582no—
550C2×C4—1582no—
551C2×C4—1584no—
552C2×C4—1584no—
553C2×C4—1584no—
554C2×C4—1584no—
555C2×C4—1582no—
556C2×C4C22⋊C4×D7914yes—
557C2×C4#1457911814yes—
558C22C2×C22⋊C4×D7911yescharacteristic
559C22#117×D7911yescharacteristic
560C22C2×#1457911811yescharacteristic
561C22#1909904011yescharacteristic
562C22#1909904011yescharacteristic
563C22#1909904311yescharacteristic
564C22#1909904411yescharacteristic
565C22#1909907212yes—
566C22#1909907212yes—
567C22#1909912911yescharacteristic
568C22#1909913512yes—
569C22#1909913512yes—
570C22#1909913611yescharacteristic
571C22#1909914211yescharacteristic
572C22#1909914311yescharacteristic
573C22#1909917012yes—
574C22#1909917112yes—
575C22#1909917212yes—
576C22#1909917512yes—
577C22#1909917712yes—
578C22#1909917812yes—
579C22#1909917912yes—
580C22#1909918412yes—
581C4—24no—
582C4—24no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—3162no—
587C4—6322no—
588C4—6322no—
589C4—1584no—
590C4—1584no—
591C4—1584no—
592C2#2376578111yescharacteristic
593C2#2376598511yescharacteristic
594C2#2376598911yescharacteristic
595C2#2376604611yescharacteristic
596C2#2376604811yescharacteristic
597C2#2376605011yescharacteristic
598C2#2376605211yescharacteristic
599C2#2376926412yes—
600C2#2376928012yes—
601C2#2376931212yes—
602C2#2376931412yes—
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC79×#1434C2
socle, prime radical, cosocleC24×C79C24
hypercenter#1434D79
hypocenterC79#14346
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
15. C2×C4×Dic158C4semidirect
313. C79#14346semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1582
C2×#190984302
C2×#190984272
#237616702
#237616662
#237616522
#237616512
#237616272
#237616242
C79×#14342
#1434679