Finite Group Databasebeta

← #34316842

Subgroups — #34316842

Order 20224.

1002 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098611C211yesmaximal, characteristic, complemented
2#23771964C211yesmaximal, characteristic, complemented
3#23771917C211yesmaximal, characteristic
4C2×#19098562C211yesmaximal, characteristic
5C2×#19098545C211yesmaximal, characteristic
6#23764764C214yesmaximal, complemented
7#23764757C214yesmaximal, complemented
8C2×#19098332C211yesmaximal, characteristic, complemented
9C2×#428×C79C211yesmaximal, characteristic, complemented
10C22×#11929624C2211yescharacteristic
11C22×#11929608C2211yescharacteristic
12C22×#11929608C2211yescharacteristic
13C22×#11929608C2211yescharacteristic
14#19098611—24no—
15C2×#14579096C2214yes—
16#19098590—22no—
17#19098590C2214yes—
18#19098587—22no—
19#19098571C2212yes—
20#19098567C2212yes—
21#19098562—22no—
22#19098562—22no—
23#19098562—21no—
24#19098562—21no—
25#19098557—22no—
26C2×#14579084C2212yes—
27C2×#14579081C2212yes—
28#19098545—24no—
29C2×#14579079C2212yes—
30C2×#14579077C2212yes—
31#19098332C2214yes—
32C22×M16×C79C2211yescharacteristic
33C2×#117×C79C2211yescharacteristic
34C22×C4⋊C4×C79C2211yescharacteristic
35#428×C79—22no—
36#428×C79C2214yes—
37C2×#107×C79—21no—
38C2×#106×C79—21no—
39C2×#11929624—22no—
40C2×#11929624—21no—
41C2×#11929624—21no—
42C2×#11929624—21no—
43C2×#11929624—21no—
44C2×#11929624—22no—
45C2×#11929624—22no—
46C2×#11929624Q814yes—
47C22×Dic158C2311yescharacteristic
48C22×D316C2311yescharacteristic
49C22×C22⋊D79—21no—
50C22×C4×D79—21no—
51C2×#11929608—22no—
52C2×#11929608—22no—
53C2×#11929608—21no—
54C2×#11929608—21no—
55C2×#11929608—22no—
56C2×#11929608—22no—
57C2×#11929608—22no—
58C2×#11929608—22no—
59C2×#11929608D412yes—
60C2×#11929608D412yes—
61C2×#11929608C2312yes—
62C2×#11929608C2312yes—
63C22×C79⋊C8—21no—
64C22×C79⋊C8—21no—
65C22×C79⋊C8C2312yes—
66#14579096—28no—
67C4⋊(C79⋊C8)—28no—
68#14579084—22no—
69#14579084—22no—
70#14579084—24no—
71#14579081—22no—
72#14579081—22no—
73#14579081—24no—
74#14579079—24no—
75#14579077—24no—
76C23×C4×C79—21no—
77C23×C4×C79C2311yescharacteristic
78C2×M16×C79—21no—
79C2×M16×C79—22no—
80C2×M16×C79D414yes—
81C2×M16×C79C2312yes—
82C22×C8×C79—21no—
83#117×C79—21no—
84#117×C79—21no—
85#117×C79D412yes—
86#117×C79C2312yes—
87C2×C4⋊C4×C79—21no—
88C2×C4⋊C4×C79—21no—
89C2×C4⋊C4×C79—21no—
90C2×C4⋊C4×C79—21no—
91C2×C4⋊C4×C79—21no—
92C2×C4⋊C4×C79D411yescharacteristic
93C2×C4⋊C4×C79D411yescharacteristic
94C2×C4⋊C4×C79C2312yes—
95C2×C22⋊C4×C79—21no—
96C2×C42×C79—21no—
97#107×C79—24no—
98#106×C79—24no—
99C23×D79—21no—
100#11929624—24no—
101#11929624—24no—
102#11929624—24no—
103#11929624—24no—
104#11929624—42no—
105#11929624—42no—
106#11929624—42no—
107#11929624—42no—
108C2×Dic158—21no—
109C2×Dic158—21no—
110C2×Dic158—21no—
111C2×Dic158—21no—
112C2×Dic158—21no—
113C2×Dic158C2×Q812yes—
114C2×D316—21no—
115C2×D316—21no—
116C2×D316—21no—
117C2×D316—21no—
118C2×D316—21no—
119C2×D316C2×Q812yes—
120C2×C22⋊D79—24no—
121C2×C22⋊D79—41no—
122C2×C22⋊D79—41no—
123C2×C22⋊D79—42no—
124C22×Dic79—21no—
125C2×C4×D79—24no—
126C2×C4×D79—41no—
127C2×C4×D79—41no—
128C2×C4×D79—42no—
129#11929608—24no—
130#11929608—24no—
131#11929608—22no—
132#11929608—22no—
133#11929608—22no—
134#11929608—22no—
135#11929608—42no—
136#11929608—42no—
137C2×C79⋊C8—22no—
138C2×C79⋊C8—22no—
139C2×C79⋊C8—22no—
140C2×C79⋊C8—22no—
141C2×C79⋊C8—42no—
142C2×C79⋊C8—42no—
143C2×C79⋊C8C2×D414yes—
144C24×C79#4111yescharacteristic
145C22×C4×C79—21no—
146C22×C4×C79—21no—
147C22×C4×C79—22no—
148C22×C4×C79—41no—
149C22×C4×C79—41no—
150C22×C4×C79—82no—
151C22×C4×C79C2×D412yes—
152C22×C4×C79C2×D411yescharacteristic
153C22×C4×C79C2×D411yescharacteristic
154C22×C4×C79C2×D411yescharacteristic
155C22×C4×C79C2×D411yescharacteristic
156C22×C4×C79C2×Q812yes—
157C22×C4×C79#4111yescharacteristic
158C22×C4×C79#4112yes—
159C22×C4×C79#4112yes—
160C22×C4×C79C2411yescharacteristic
161M16×C79—24no—
162M16×C79—22no—
163M16×C79C2×D414yes—
164C2×C8×C79—24no—
165C2×C8×C79—22no—
166C4⋊C4×C79—21no—
167C4⋊C4×C79—21no—
168C4⋊C4×C79—22no—
169C4⋊C4×C79—22no—
170C4⋊C4×C79—41no—
171C4⋊C4×C79—41no—
172C4⋊C4×C79C2×D412yes—
173C4⋊C4×C79C2×D412yes—
174C22⋊C4×C79—22no—
175C22⋊C4×C79—22no—
176C42×C79—22no—
177C42×C79—22no—
178C23×C79—41no—
179C23×C79—41no—
180C23×C79#12111yescharacteristic
181C23×C79#12111yescharacteristic
182C23×C79C22⋊Q812yes—
183C23×C79#12612yes—
184C23×C79C2×#4111yescharacteristic
185C22×D79—22no—
186C22×D79—41no—
187C22×D79—41no—
188C22×D79—81no—
189C22×D79—81no—
190C2×C4×C79—22no—
191C2×C4×C79—22no—
192C2×C4×C79—42no—
193C2×C4×C79—81no—
194C2×C4×C79—81no—
195C2×C4×C79—82no—
196C2×C4×C79—82no—
197C2×C4×C79—82no—
198C2×C4×C79#12111yescharacteristic
199C2×C4×C79#12111yescharacteristic
200C2×C4×C79#12112yes—
201C2×C4×C79#12112yes—
202C2×C4×C79#12112yes—
203C2×C4×C79#12112yes—
204C2×C4×C79C22⋊Q812yes—
205C2×C4×C79C22⋊Q812yes—
206C2×C4×C79C22⋊Q812yes—
207C2×C4×C79C22⋊Q812yes—
208C2×C4×C79C22⋊Q812yes—
209C2×C4×C79#12612yes—
210C2×C4×C79C22×D411yescharacteristic
211C2×C4×C79C22×D411yescharacteristic
212C2×C4×C79C22×D411yescharacteristic
213C2×C4×C79C22×Q811yescharacteristic
214C2×C4×C79C2×#4111yescharacteristic
215C2×C4×C79C2×#4111yescharacteristic
216C22⋊D79—44no—
217C22⋊D79—44no—
218C2×Dic79—22no—
219C2×Dic79—41no—
220C2×Dic79—41no—
221D316—22no—
222D316—22no—
223D316—41no—
224D316—41no—
225C4×D79—44no—
226C4×D79—44no—
227Dic158—22no—
228Dic158—22no—
229Dic158—41no—
230Dic158—41no—
231C8×C79—24no—
232C79⋊C8—24no—
233C79⋊C8—42no—
234C79⋊C8—42no—
235C22×C79—21no—
236C22×C79—21no—
237C22×C79—21no—
238C22×C79—21no—
239C22×C79—21no—
240C22×C79—21no—
241C22×C79—42no—
242C22×C79—42no—
243C22×C79#39612yes—
244C22×C79#39612yes—
245C22×C79C2×#12111yescharacteristic
246C22×C79C2×C22⋊Q811yescharacteristic
247C22×C79C2×#12611yescharacteristic
248C2×D79—42no—
249C2×D79—82no—
250C2×D79—81no—
251C2×D79—81no—
252C2×D79—82no—
253C4×C79—42no—
254C4×C79—42no—
255C4×C79—82no—
256C4×C79#39612yes—
257C4×C79#39612yes—
258C4×C79C2×#12111yescharacteristic
259C4×C79C2×#12111yescharacteristic
260C4×C79C2×C22⋊Q811yescharacteristic
261C4×C79C2×C22⋊Q811yescharacteristic
262Dic79—42no—
263#23604—791nomaximal, Sylow
264C2×C79—21no—
265C2×C79—21no—
266C2×C79—22no—
267C2×C79—22no—
268C2×C79C2×#39611yescharacteristic
269C2×C79#306811yescharacteristic
270C2×C79#306911yescharacteristic
271D79—82no—
272C2×#430—791no—
273C2×#428D7911yescharacteristic, complemented
274#2905—791no—
275#2896—791no—
276C2×#417—791no—
277C2×#413—791no—
278#2064—794no—
279#2063—794no—
280C2×#343—791no—
281C79#2360411yescharacteristic, complemented, Sylow
282C22×#41—791no—
283C22×M16—791no—
284C22×M16—791no—
285C22×M16—791no—
286C22×M16C2×D7911yescharacteristic
287C2×#117C2×D7911yescharacteristic
288C22×C4⋊C4C2×D7911yescharacteristic
289#430—1584no—
290C2×#108—794no—
291#428—22no—
292#428C2×D7914yes—
293C2×#107—21no—
294C2×#106—21no—
295#424—794no—
296#424—1582no—
297#423—1582no—
298#419—792no—
299#418—792no—
300#417—1582no—
301#417—1582no—
302#417—1581no—
303#417—1581no—
304#416—1582no—
305C2×#103—792no—
306C2×#102—792no—
307#413—1584no—
308C2×#101—792no—
309C2×#100—792no—
310#343—794no—
311C2×#41—794no—
312C2×#41—1581no—
313C2×#41—1581no—
314C2×#41—1581no—
315C2×#41—1581no—
316C2×#41—1582no—
317C2×#41—1582no—
318C2×#41—1582no—
319C22×Q8—791no—
320C22×D4—791no—
321C22×D4—1581no—
322C23×C4—21no—
323C23×C4—1581no—
324C23×C4C22×D7911yescharacteristic
325C2×M16—21no—
326C2×M16—22no—
327C2×M16—792no—
328C2×M16—792no—
329C2×M16—792no—
330C2×M16—792no—
331C2×M16—1582no—
332C2×M16—1582no—
333C2×M16—1582no—
334C2×M16—1582no—
335C2×M16—1581no—
336C2×M16—1581no—
337C2×M16—1582no—
338C2×M16—1582no—
339C2×M16D31614yes—
340C2×M16C22×D7912yes—
341C22×C8—21no—
342C22×C8—792no—
343C22×C8—1581no—
344C22×C8—1581no—
345#117—21no—
346#117—21no—
347#117C22⋊D7912yes—
348#117C22×D7912yes—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4C22⋊D7911yescharacteristic
355C2×C4⋊C4C22⋊D7911yescharacteristic
356C2×C4⋊C4C22×D7912yes—
357C2×C22⋊C4—21no—
358C2×C42—21no—
359#108—1588no—
360#107—24no—
361#106—24no—
362C4⋊C8—1588no—
363#103—1584no—
364#103—1582no—
365#103—1582no—
366#102—1584no—
367#102—1582no—
368#102—1582no—
369#101—1584no—
370#100—1584no—
371C24—1581no—
372C24#1192962311yescharacteristic
373#41—3162no—
374#41—3162no—
375#41—3162no—
376#41—3162no—
377#41—1584no—
378#41—1584no—
379#41—1584no—
380#41—1584no—
381C2×Q8—792no—
382C2×Q8—1581no—
383C2×Q8—1581no—
384C2×Q8—1581no—
385C2×Q8—1581no—
386C2×Q8—1581no—
387C2×D4—3161no—
388C2×D4—3161no—
389C2×D4—3162no—
390C2×D4—792no—
391C2×D4—1584no—
392C2×D4—1581no—
393C2×D4—1581no—
394C2×D4—1581no—
395C2×D4—1581no—
396C2×D4—1581no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—82no—
403C22×C4—3161no—
404C22×C4—3161no—
405C22×C4—3162no—
406C22×C4—1581no—
407C22×C4—1584no—
408C22×C4D4×D7911yescharacteristic
409C22×C4D4×D7911yescharacteristic
410C22×C4C2×C22⋊D7911yescharacteristic
411C22×C4C2×C22⋊D7911yescharacteristic
412C22×C4C2×D31612yes—
413C22×C4Q8×D7912yes—
414C22×C4#1192962311yescharacteristic
415C22×C4#1192962412yes—
416C22×C4#1192962512yes—
417C22×C4C23×D7911yescharacteristic
418M16—22no—
419M16—24no—
420M16—3162no—
421M16—3162no—
422M16—1582no—
423M16—1582no—
424M16—1582no—
425M16—1582no—
426M16—1584no—
427M16—1584no—
428M16C2×D31614yes—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—3162no—
432C2×C8—3162no—
433C2×C8—794no—
434C2×C8—1582no—
435C2×C8—1582no—
436C2×C8—1582no—
437C2×C8—1582no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4C2×C22⋊D7912yes—
445C4⋊C4C2×C22⋊D7912yes—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C42—22no—
449C42—22no—
450C23—41no—
451C23—41no—
452C23—3161no—
453C23—3161no—
454C23—6321no—
455C23—6321no—
456C23—1582no—
457C23#1457913311yescharacteristic
458C23#1457913311yescharacteristic
459C23#1457914112yes—
460C23#1457915612yes—
461C23C2×#1192962311yescharacteristic
462Q8—3161no—
463Q8—3161no—
464Q8—1582no—
465Q8—1582no—
466D4—3164no—
467D4—3164no—
468D4—3161no—
469D4—3161no—
470D4—1582no—
471D4—1582no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—3161no—
481C2×C4—3161no—
482C2×C4—3164no—
483C2×C4—3164no—
484C2×C4—1582no—
485C2×C4#1457913311yescharacteristic
486C2×C4#1457913311yescharacteristic
487C2×C4#1457913512yes—
488C2×C4#1457913512yes—
489C2×C4#1457913612yes—
490C2×C4#1457913612yes—
491C2×C4#1457913712yes—
492C2×C4#1457913712yes—
493C2×C4#1457913912yes—
494C2×C4#1457913912yes—
495C2×C4#1457914112yes—
496C2×C4#1457915612yes—
497C2×C4C2×D4×D7911yescharacteristic
498C2×C4C22×C22⋊D7911yescharacteristic
499C2×C4C22×D31611yescharacteristic
500C2×C4C2×Q8×D7911yescharacteristic
501C2×C4C2×#1192962411yescharacteristic
502C2×C4C2×#1192962511yescharacteristic
503C8—24no—
504C8—3162no—
505C8—3162no—
506C8—1584no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—42no—
515C22—3162no—
516C22—6322no—
517C22—6322no—
518C22—6321no—
519C22—6321no—
520C22#1909848512yes—
521C22#1909848512yes—
522C22C2×#1457913311yescharacteristic
523C22C2×#1457914111yescharacteristic
524C22C2×#1457915611yescharacteristic
525C4—42no—
526C4—42no—
527C4—82no—
528C4—3162no—
529C4#1909848512yes—
530C4#1909848512yes—
531C4C2×#1457913511yescharacteristic
532C4C2×#1457913611yescharacteristic
533C4C2×#1457913711yescharacteristic
534C4C2×#1457913911yescharacteristic
535C2—21no—
536C2—21no—
537C2—22no—
538C2—22no—
539C2—6322no—
540C2C2×#1909848511yescharacteristic
541C2#2377312511yescharacteristic
542C2#2377313411yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#428×C79C2
socleC22×C79C2×#121
prime radicalC22×#11929624C22
hypercenterC2×#428D79
hypocenterC79#23604
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#19098611C2semidirect
2. #23771964C2semidirect
6. #23764764C2semidirect
7. #23764757C2semidirect
8. C2×#19098332C2semidirect
9. C2×#428×C79C2semidirect
273. C2×#428D79semidirect
281. C79#23604semidirect

Maximal subgroups

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

subgroupindex
C2×#190986112
#237719642
#237719172
C2×#190985622
C2×#190985452
#237647642
#237647572
C2×#190983322
C2×#428×C792
#2360479