Finite Group Databasebeta

← #34316948

Subgroups — #34316948

Order 20224.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098606C211yesmaximal, characteristic, complemented
2C2×#19098590C211yesmaximal, characteristic, complemented
3C2×#19098562C211yesmaximal, characteristic
4C2×#19098557C211yesmaximal, characteristic
5#23765988C211yesmaximal, characteristic
6#23764792C212yesmaximal, complemented
7#23764785C212yesmaximal, complemented
8#23764778C212yesmaximal, complemented
9#23764771C212yesmaximal, complemented
10C2×#19098325C211yesmaximal, characteristic, complemented
11C79×#2317C211yesmaximal, characteristic, complemented
12C22×#11929624C2211yescharacteristic
13C22×#11929608C2211yescharacteristic
14C23×C79⋊C8C2211yescharacteristic
15C22×C4⋊Dic79C2211yescharacteristic
16#19098606—24no—
17C2×#14579094C2212yes—
18C2×#14579092C2212yes—
19#19098590—24no—
20C2×C4⋊(C79⋊C8)C2212yes—
21C2×C4⋊(C79⋊C8)C2212yes—
22#19098562—24no—
23#19098557—24no—
24C2×#14579084C2212yes—
25C2×#14579084C2211yescharacteristic
26C2×#14579084C2211yescharacteristic
27C2×#14579081C2212yes—
28C2×#14579081C2211yescharacteristic
29C2×#14579081C2211yescharacteristic
30#19098469C2212yes—
31#19098465C2212yes—
32#19098440—22no—
33#19098420—21no—
34#19098420—21no—
35#19098325C2212yes—
36#19098325C2212yes—
37C2×#117×C79C2211yescharacteristic
38C2×#117×C79C2211yescharacteristic
39C22×C4⋊C4×C79C2211yescharacteristic
40#389×C79C2212yes—
41#389×C79C2212yes—
42#385×C79—22no—
43#384×C79—21no—
44#382×C79—21no—
45C2×#11929624—22no—
46C2×#11929624—22no—
47C2×#11929624—22no—
48C2×#11929624—22no—
49C2×#11929624—22no—
50C2×#11929624Q814yes—
51C22×Dic158C2311yescharacteristic
52C22×D316C2311yescharacteristic
53C22×C22⋊D79—21no—
54C23×Dic79—21no—
55C22×C4×D79—21no—
56C2×#11929608—22no—
57C2×#11929608—22no—
58C2×#11929608D414yes—
59C22×C79⋊C8—22no—
60C22×C79⋊C8—22no—
61C22×C79⋊C8—22no—
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2312yes—
65C2×C4⋊Dic79—22no—
66C2×C4⋊Dic79—22no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79C2312yes—
69C2×(C2×D79)⋊C4—22no—
70C2×(C2×D79)⋊C4—22no—
71#14579094—24no—
72#14579092—24no—
73C4⋊(C79⋊C8)—24no—
74C4⋊(C79⋊C8)—24no—
75#14579084—22no—
76#14579084—22no—
77#14579084—24no—
78#14579081—22no—
79#14579081—22no—
80#14579081—24no—
81C79⋊#95—22no—
82C79⋊#95—22no—
83C23×C4×C79—21no—
84C23×C4×C79C2311yescharacteristic
85#117×C79—22no—
86#117×C79—22no—
87#117×C79—22no—
88#117×C79D414yes—
89C2×C4⋊C4×C79—22no—
90C2×C4⋊C4×C79—22no—
91C2×C4⋊C4×C79—21no—
92C2×C4⋊C4×C79—22no—
93C2×C4⋊C4×C79—22no—
94C2×C4⋊C4×C79D411yescharacteristic
95C2×C4⋊C4×C79D411yescharacteristic
96C2×C4⋊C4×C79C2312yes—
97C2×C4⋊C4×C79C2311yescharacteristic
98C2×C4⋊C4×C79C2311yescharacteristic
99C2×C22⋊C4×C79—21no—
100C2×C22⋊C4×C79D412yes—
101C2×C42×C79—21no—
102C2×C42×C79C2312yes—
103#95×C79—22no—
104#95×C79—22no—
105C23×D79—21no—
106#11929624—28no—
107#11929624—28no—
108#11929624—44no—
109#11929624—44no—
110C2×Dic158—21no—
111C2×Dic158—21no—
112C2×Dic158—21no—
113C2×Dic158—21no—
114C2×Dic158—21no—
115C2×Dic158C2×Q812yes—
116C2×D316—21no—
117C2×D316—21no—
118C2×D316—21no—
119C2×D316—21no—
120C2×D316—21no—
121C2×D316C2×Q812yes—
122C2×C22⋊D79—24no—
123C2×C22⋊D79—42no—
124C2×C22⋊D79—42no—
125C22×Dic79—21no—
126C22×Dic79—22no—
127C22×Dic79—42no—
128C22×Dic79—81no—
129C22×Dic79—81no—
130C2×C4×D79—24no—
131C2×C4×D79—42no—
132C2×C4×D79—42no—
133#11929608—28no—
134C2×C79⋊C8—22no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—22no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—44no—
143C2×C79⋊C8C2×D414yes—
144C4⋊Dic79—22no—
145C4⋊Dic79—22no—
146C4⋊Dic79—42no—
147(C2×D79)⋊C4—82no—
148(C2×D79)⋊C4—82no—
149C24×C79C2×D411yescharacteristic
150C22×C4×C79—21no—
151C22×C4×C79—21no—
152C22×C4×C79—22no—
153C22×C4×C79—42no—
154C22×C4×C79—81no—
155C22×C4×C79—81no—
156C22×C4×C79C2×D412yes—
157C22×C4×C79C2×D412yes—
158C22×C4×C79C2×D411yescharacteristic
159C22×C4×C79C2×D412yes—
160C22×C4×C79C2×D412yes—
161C22×C4×C79C2×Q812yes—
162C22×C4×C79#4112yes—
163C22×C4×C79#4112yes—
164C22×C4×C79#4112yes—
165C22×C4×C79C2411yescharacteristic
166C4⋊C4×C79—22no—
167C4⋊C4×C79—22no—
168C4⋊C4×C79—21no—
169C4⋊C4×C79—21no—
170C4⋊C4×C79—22no—
171C4⋊C4×C79—22no—
172C4⋊C4×C79—42no—
173C4⋊C4×C79—82no—
174C4⋊C4×C79—82no—
175C4⋊C4×C79C2×D412yes—
176C4⋊C4×C79C2×D412yes—
177C22⋊C4×C79—24no—
178C22⋊C4×C79—24no—
179C42×C79—24no—
180C42×C79—22no—
181C42×C79—22no—
182C23×C79—21no—
183C23×C79—21no—
184C23×C79—21no—
185C23×C79—21no—
186C23×C79#12112yes—
187C23×C79#12112yes—
188C23×C79C22⋊Q812yes—
189C23×C79C22×D411yescharacteristic
190C22×D79—22no—
191C22×D79—41no—
192C22×D79—41no—
193C22×D79—41no—
194C22×D79—41no—
195C22×D79—41no—
196C22×D79—41no—
197C2×C4×C79—22no—
198C2×C4×C79—22no—
199C2×C4×C79—22no—
200C2×C4×C79—22no—
201C2×C4×C79—22no—
202C2×C4×C79—22no—
203C2×C4×C79—22no—
204C2×C4×C79—22no—
205C2×C4×C79—41no—
206C2×C4×C79—41no—
207C2×C4×C79—41no—
208C2×C4×C79—41no—
209C2×C4×C79—42no—
210C2×C4×C79—42no—
211C2×C4×C79—42no—
212C2×C4×C79—84no—
213C2×C4×C79#12112yes—
214C2×C4×C79#12112yes—
215C2×C4×C79#12114yes—
216C2×C4×C79C22⋊Q812yes—
217C2×C4×C79C22⋊Q814yes—
218C2×C4×C79C22⋊Q814yes—
219C2×C4×C79#12614yes—
220C2×C4×C79C22×D411yescharacteristic
221C2×C4×C79C22×D411yescharacteristic
222C2×C4×C79C22×Q811yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C2×C4×C79C2×#4111yescharacteristic
225C2×C4×C79C2×#4111yescharacteristic
226C22⋊D79—48no—
227C2×Dic79—22no—
228C2×Dic79—41no—
229C2×Dic79—41no—
230C2×Dic79—42no—
231C2×Dic79—42no—
232C2×Dic79—42no—
233C2×Dic79—84no—
234D316—22no—
235D316—22no—
236D316—41no—
237D316—41no—
238C4×D79—48no—
239Dic158—22no—
240Dic158—22no—
241Dic158—41no—
242Dic158—41no—
243C79⋊C8—24no—
244C79⋊C8—41no—
245C79⋊C8—41no—
246C22×C79—22no—
247C22×C79—22no—
248C22×C79—22no—
249C22×C79—22no—
250C22×C79—22no—
251C22×C79—22no—
252C22×C79#39614yes—
253C22×C79C2×#12111yescharacteristic
254C22×C79C2×#12111yescharacteristic
255C22×C79C2×C22⋊Q811yescharacteristic
256C22×C79#57611yescharacteristic
257C22×C79#57711yescharacteristic
258C22×C79#57711yescharacteristic
259C22×C79#57811yescharacteristic
260C2×D79—42no—
261C2×D79—42no—
262C2×D79—42no—
263C2×D79—81no—
264C2×D79—81no—
265C2×D79—81no—
266C2×D79—81no—
267C4×C79—42no—
268C4×C79—42no—
269C4×C79—42no—
270C4×C79—42no—
271C4×C79—82no—
272C4×C79#39614yes—
273C4×C79C2×#12111yescharacteristic
274C4×C79C2×C22⋊Q811yescharacteristic
275C4×C79C2×C22⋊Q811yescharacteristic
276C4×C79C2×#12611yescharacteristic
277Dic79—42no—
278Dic79—82no—
279#23616—791nomaximal, Sylow
280C2×C79—24no—
281C2×C79C2×#39611yescharacteristic
282C2×C79#304511yescharacteristic
283C2×C79#304611yescharacteristic
284C2×C79#306211yescharacteristic
285C2×C79#306311yescharacteristic
286C2×C79#308311yescharacteristic
287C2×C79#308411yescharacteristic
288D79—82no—
289C2×#428—791no—
290C2×#424—791no—
291C2×#417—791no—
292C2×#416—791no—
293#2317D7911yescharacteristic, complemented
294#2311—791no—
295#2068—792no—
296#2067—792no—
297#2066—792no—
298#2065—792no—
299C2×#340—791no—
300C79#2361611yescharacteristic, complemented, Sylow
301C22×#41—791no—
302C22×M16—791no—
303C23×C8—791no—
304C2×#117C2×D7911yescharacteristic
305C2×#117C2×D7911yescharacteristic
306C22×C4⋊C4—791no—
307C22×C4⋊C4C2×D7911yescharacteristic
308#428—1584no—
309C2×#107—792no—
310C2×#106—792no—
311#424—1584no—
312C2×C4⋊C8—792no—
313C2×C4⋊C8—792no—
314#417—1584no—
315#416—1584no—
316C2×#103—791no—
317C2×#103—792no—
318C2×#103—791no—
319C2×#102—791no—
320C2×#102—792no—
321C2×#102—791no—
322#391—792no—
323#390—792no—
324#389C2×D7912yes—
325#389C2×D7912yes—
326#386—1582no—
327#385—22no—
328#384—21no—
329#382—21no—
330#380—1581no—
331#380—1581no—
332#340—792no—
333#340—792no—
334C2×#41—794no—
335C2×#41—1582no—
336C2×#41—1582no—
337C2×#41—1582no—
338C2×#41—1582no—
339C2×#41—1582no—
340C22×Q8—791no—
341C22×D4—791no—
342C22×D4—1581no—
343C23×C4—21no—
344C23×C4—1581no—
345C23×C4—1581no—
346C23×C4C22×D7911yescharacteristic
347C2×M16—794no—
348C2×M16—1582no—
349C2×M16—1582no—
350C22×C8—792no—
351C22×C8—791no—
352C22×C8—791no—
353C22×C8—1582no—
354C22×C8—1582no—
355C22×C8—1582no—
356#117—22no—
357#117—22no—
358#117—22no—
359#117D31614yes—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—792no—
366C2×C4⋊C4—1581no—
367C2×C4⋊C4—1582no—
368C2×C4⋊C4—1582no—
369C2×C4⋊C4C22⋊D7911yescharacteristic
370C2×C4⋊C4C22⋊D7911yescharacteristic
371C2×C4⋊C4C22×D7912yes—
372C2×C4⋊C4C22×D7911yescharacteristic
373C2×C4⋊C4C22×D7911yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1582no—
376C2×C22⋊C4—1582no—
377C2×C22⋊C4C22⋊D7912yes—
378C2×C42—21no—
379C2×C42C22×D7912yes—
380#107—1584no—
381#106—1584no—
382C4⋊C8—1584no—
383C4⋊C8—1584no—
384#103—1584no—
385#103—1582no—
386#103—1582no—
387#102—1584no—
388#102—1582no—
389#102—1582no—
390#95—22no—
391#95—22no—
392#95—1582no—
393#95—1582no—
394C24—1581no—
395C24C2×C22⋊D7911yescharacteristic
396#41—3164no—
397#41—3164no—
398#41—1588no—
399#41—1588no—
400C2×Q8—792no—
401C2×Q8—1581no—
402C2×Q8—1581no—
403C2×Q8—1581no—
404C2×Q8—1581no—
405C2×Q8—1581no—
406C2×D4—3162no—
407C2×D4—3162no—
408C2×D4—792no—
409C2×D4—1584no—
410C2×D4—1581no—
411C2×D4—1581no—
412C2×D4—1581no—
413C2×D4—1581no—
414C2×D4—1581no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—3162no—
422C22×C4—3162no—
423C22×C4—3162no—
424C22×C4—6321no—
425C22×C4—6321no—
426C22×C4—1582no—
427C22×C4—1581no—
428C22×C4—1584no—
429C22×C4D4×D7912yes—
430C22×C4C2×C22⋊D7912yes—
431C22×C4C2×C22⋊D7912yes—
432C22×C4C2×C22⋊D7911yescharacteristic
433C22×C4C2×D31612yes—
434C22×C4Q8×D7912yes—
435C22×C4#1192962312yes—
436C22×C4#1192962412yes—
437C22×C4#1192962512yes—
438C22×C4C23×D7911yescharacteristic
439M16—1588no—
440C2×C8—3164no—
441C2×C8—794no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8—1581no—
448C2×C8—1582no—
449C2×C8—1582no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—3162no—
460C4⋊C4—1582no—
461C4⋊C4—1582no—
462C4⋊C4C2×D31612yes—
463C4⋊C4C2×D31612yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—6322no—
467C22⋊C4—6322no—
468C42—22no—
469C42—24no—
470C42—22no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—3161no—
476C23—3161no—
477C23—3161no—
478C23—3161no—
479C23—3161no—
480C23—3161no—
481C23—1582no—
482C23#1457913312yes—
483C23#1457913512yes—
484C23#1457913712yes—
485C23C22×C22⋊D7911yescharacteristic
486Q8—3161no—
487Q8—3161no—
488Q8—1582no—
489Q8—1582no—
490D4—3168no—
491D4—3161no—
492D4—3161no—
493D4—1582no—
494D4—1582no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—3161no—
512C2×C4—3161no—
513C2×C4—3168no—
514C2×C4—3162no—
515C2×C4—3162no—
516C2×C4—3162no—
517C2×C4—6324no—
518C2×C4—1582no—
519C2×C4#1457913312yes—
520C2×C4#1457913512yes—
521C2×C4#1457913614yes—
522C2×C4#1457913712yes—
523C2×C4#1457913914yes—
524C2×C4#1457914114yes—
525C2×C4#1457915614yes—
526C2×C4C2×D4×D7911yescharacteristic
527C2×C4C22×D31611yescharacteristic
528C2×C4C2×Q8×D7911yescharacteristic
529C2×C4C2×#1192962311yescharacteristic
530C2×C4C2×#1192962411yescharacteristic
531C2×C4C2×#1192962511yescharacteristic
532C8—3161no—
533C8—3161no—
534C8—1584no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—3162no—
542C22—3162no—
543C22—3162no—
544C22—6321no—
545C22—6321no—
546C22—6321no—
547C22—6321no—
548C22#1909848514yes—
549C22C2×#1457913311yescharacteristic
550C22C2×#1457913511yescharacteristic
551C22C2×#1457913711yescharacteristic
552C22#1909936611yescharacteristic
553C22#1909937111yescharacteristic
554C22#1909937411yescharacteristic
555C22#1909937811yescharacteristic
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4—3162no—
562C4—6322no—
563C4#1909848514yes—
564C4C2×#1457913611yescharacteristic
565C4C2×#1457913911yescharacteristic
566C4C2×#1457914111yescharacteristic
567C4C2×#1457915611yescharacteristic
568C2—24no—
569C2—6322no—
570C2C2×#1909848511yescharacteristic
571C2#2377293911yescharacteristic
572C2#2377295011yescharacteristic
573C2#2377307611yescharacteristic
574C2#2377308711yescharacteristic
575C2#2377324711yescharacteristic
576C2#2377325811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2317C2
socleC23×C79C22×D4
prime radicalC22×#11929624C22
hypercenter#2317D79
hypocenterC79#23616
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
1. C2×#19098606C2semidirect
2. C2×#19098590C2semidirect
6. #23764792C2semidirect
7. #23764785C2semidirect
8. #23764778C2semidirect
9. #23764771C2semidirect
10. C2×#19098325C2semidirect
11. C79×#2317C2semidirect
293. #2317D79semidirect
300. C79#23616semidirect

Maximal subgroups

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

subgroupindex
C2×#190986062
C2×#190985902
C2×#190985622
C2×#190985572
#237659882
#237647922
#237647852
#237647782
#237647712
C2×#190983252
C79×#23172
#2361679