Finite Group Databasebeta

← #34442844

Subgroups — #34442844

Order 20224.

642 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098993C211yesmaximal, characteristic, complemented
2C2×#19098980C211yesmaximal, characteristic, complemented
3#23773120C211yesmaximal, characteristic, complemented
4#23765532C212yesmaximal, complemented
5#23765528C212yesmaximal, complemented
6#23765523C212yesmaximal, complemented
7#23765518C212yesmaximal, complemented
8C2×#19098392C211yesmaximal, characteristic, complemented
9#23765149C211yesmaximal, characteristic, complemented
10#23765147C211yesmaximal, characteristic, complemented
11C79×#3067C211yesmaximal, characteristic, complemented
12#19099361—21no—
13C2×#14579183C2211yescharacteristic
14#19098993—21no—
15#19098993—21no—
16#19098993—21no—
17#19098993—21no—
18C2×#14579107C2211yescharacteristic, complemented
19C2×#14579105C2211yescharacteristic, complemented
20C2×#14579104C2211yescharacteristic, complemented
21C2×#14579102C2211yescharacteristic, complemented
22#19098980—21no—
23#19098980—21no—
24#19098980C2212yes—
25#19098980C2212yes—
26C2×#14579101—21no—
27C2×#14579101C2211yescharacteristic
28C22×C79⋊C16—21no—
29C22×C79⋊C16C2211yescharacteristic
30#19098838C2212yes—
31#19098833C2212yes—
32#19098826—22no—
33C2×#14579097C2211yescharacteristic
34#19098545—21no—
35#19098392C2212yescomplemented
36#19098392C2212yescomplemented
37#19098379C2212yescomplemented
38#19098378C2212yescomplemented
39#19098376C2212yescomplemented
40#19098375C2212yescomplemented
41#575×C79—21no—
42C2×#135×C79C2211yescharacteristic
43C2×#131×C79C2211yescharacteristic
44#473×C79C2212yes—
45#472×C79C2212yes—
46#471×C79—22no—
47C2×#108×C79C2211yescharacteristic
48#413×C79—21no—
49C2×#11929624—21no—
50#14579192—41no—
51#14579191—22no—
52#14579187—41no—
53#14579186—22no—
54#14579183—21no—
55#14579183—21no—
56#14579183—21no—
57#14579183—21no—
58#14579181—42no—
59C2×Dic316C2311yescharacteristic
60C2×#11929613—21no—
61C2×D632C2311yescharacteristic
62C2×#11929608—21no—
63#14579107—21no—
64#14579107—21no—
65#14579105—21no—
66#14579105—21no—
67#14579104—21no—
68#14579104—21no—
69#14579102—21no—
70#14579102—21no—
71#14579101—22no—
72#14579101—22no—
73#14579101—21no—
74#14579101C2312yes—
75C2×C79⋊C16—22no—
76C2×C79⋊C16—22no—
77C2×C79⋊C16—21no—
78C2×C79⋊C16D411yescharacteristic
79C2×C79⋊C16D411yescharacteristic
80C2×C79⋊C16D411yescharacteristic
81C2×C79⋊C16D411yescharacteristic
82C2×C79⋊C16C2311yescharacteristic
83C2×C79⋊C16C2311yescharacteristic
84#14579097—21no—
85#14579097C2312yes—
86#14579079—22no—
87#14579077—22no—
88C2×#41×C79—21no—
89C2×#41×C79D411yescharacteristic
90#137×C79—22no—
91#137×C79—41no—
92AGL(1,ℤ/8)×C79—22no—
93AGL(1,ℤ/8)×C79—41no—
94#135×C79—21no—
95#135×C79—21no—
96#135×C79—21no—
97#135×C79—21no—
98#135×C79—42no—
99C2×Q16×C79C2311yescharacteristic
100C2×SD16×C79—21no—
101C2×D8×C79C2311yescharacteristic
102#131×C79—21no—
103#131×C79—21no—
104#131×C79D412yes—
105#131×C79C2312yes—
106C2×M16×C79—21no—
107C2×M16×C79—21no—
108C2×M16×C79D411yescharacteristic
109C22×C8×C79—21no—
110C22×C8×C79C2311yescharacteristic
111#108×C79—21no—
112#108×C79C2312yes—
113#101×C79—22no—
114#100×C79—22no—
115#11929624—41no—
116#11929624—41no—
117#11929624—41no—
118#11929624—41no—
119C2×Dic158—21no—
120C2×D316—21no—
121C2×C22⋊D79—41no—
122C2×C4×D79—41no—
123Dic316—21no—
124Dic316—21no—
125#11929614—42no—
126#11929613—41no—
127#11929613—41no—
128#11929612—42no—
129#11929611—42no—
130D632—21no—
131D632—21no—
132#11929609—42no—
133#11929608—22no—
134#11929608—41no—
135C2×C79⋊C8—41no—
136C79⋊C16—22no—
137C79⋊C16C2×D411yescharacteristic
138C79⋊C16C2×D411yescharacteristic
139C79⋊C16C2×D411yescharacteristic
140C79⋊C16C2×D411yescharacteristic
141#41×C79—21no—
142#41×C79—21no—
143#41×C79—41no—
144#41×C79—41no—
145#41×C79—41no—
146#41×C79—41no—
147#41×C79D812yes—
148#41×C79C2×D412yes—
149C2×Q8×C79—21no—
150C2×Q8×C79D811yescharacteristic
151C2×D4×C79—21no—
152C2×D4×C79—21no—
153C2×D4×C79—41no—
154C2×D4×C79D811yescharacteristic
155C22×C4×C79—21no—
156C22×C4×C79—41no—
157C22×C4×C79C2×D411yescharacteristic
158Q16×C79—21no—
159Q16×C79—21no—
160Q16×C79—42no—
161SD16×C79—41no—
162SD16×C79—41no—
163SD16×C79—42no—
164SD16×C79—42no—
165D8×C79—21no—
166D8×C79—21no—
167D8×C79—42no—
168M16×C79—22no—
169M16×C79—22no—
170M16×C79—21no—
171M16×C79—22no—
172M16×C79—41no—
173M16×C79C2×D412yes—
174C2×C8×C79—22no—
175C2×C8×C79—22no—
176C2×C8×C79—21no—
177C2×C8×C79—41no—
178C2×C8×C79C2×D411yescharacteristic
179C2×C8×C79C2×D411yescharacteristic
180C2×C8×C79C2×D411yescharacteristic
181C2×C8×C79#4111yescharacteristic
182C2×C8×C79#4111yescharacteristic
183C2×C8×C79C2411yescharacteristic
184C23×C79—21no—
185C23×C79—41no—
186C23×C79#13511yescharacteristic
187C22×D79—41no—
188Q8×C79—21no—
189Q8×C79—41no—
190Q8×C79—41no—
191Q8×C79C2×D812yes—
192D4×C79—22no—
193D4×C79—22no—
194D4×C79—21no—
195D4×C79—41no—
196D4×C79—41no—
197D4×C79—81no—
198D4×C79—81no—
199D4×C79C2×D812yes—
200C2×C4×C79—21no—
201C2×C4×C79—22no—
202C2×C4×C79—22no—
203C2×C4×C79—41no—
204C2×C4×C79—81no—
205C2×C4×C79—81no—
206C2×C4×C79#12111yescharacteristic
207C2×C4×C79#12111yescharacteristic
208C2×C4×C79C2×D811yescharacteristic
209C2×C4×C79C2×D811yescharacteristic
210C2×C4×C79#13511yescharacteristic
211C2×C4×C79C22×D411yescharacteristic
212C22⋊D79—81no—
213C22⋊D79—81no—
214C2×Dic79—41no—
215D316—41no—
216D316—41no—
217C4×D79—81no—
218C4×D79—81no—
219Dic158—41no—
220Dic158—41no—
221C8×C79—22no—
222C8×C79—42no—
223C8×C79#12111yescharacteristic
224C8×C79#12111yescharacteristic
225C8×C79C22×D411yescharacteristic
226C8×C79C2×#4111yescharacteristic
227C79⋊C8—42no—
228C22×C79—22no—
229C22×C79—41no—
230C22×C79—42no—
231C22×C79—81no—
232C22×C79—161no—
233C22×C79#46611yescharacteristic
234C22×C79#46611yescharacteristic
235C22×C79C2×#13511yescharacteristic
236C2×D79—81no—
237C2×D79—161no—
238C4×C79—22no—
239C4×C79—81no—
240C4×C79#46611yescharacteristic
241C4×C79#46611yescharacteristic
242C4×C79C2×#12111yescharacteristic
243C4×C79C22×D811yescharacteristic
244Dic79—81no—
245#33617—791nomaximal, Sylow
246C2×C79—21no—
247C2×C79—21no—
248C2×C79—21no—
249C2×C79—42no—
250C2×C79—161no—
251C2×C79C2×#46611yescharacteristic
252D79—161no—
253C2×#508—791no—
254C2×#504—791no—
255#3067—791no—
256#3067D7911yescharacteristic, complemented
257#2223—792no—
258#2223—792no—
259#2222—792no—
260#2221—792no—
261C2×#368—791no—
262#2148—791no—
263#2148—791no—
264C79#3361711yescharacteristic, complemented, Sylow
265#575—21no—
266#575—1581no—
267C2×#135—791no—
268C2×#135C2×D7911yescharacteristic
269C2×#131C2×D7911yescharacteristic
270#508—1581no—
271#508—1581no—
272#508—1581no—
273#508—1581no—
274C2×Q32—791no—
275C2×SD32—791no—
276C2×SD32—791no—
277C2×D16—791no—
278#504—792no—
279#504—792no—
280#504—1581no—
281#504—1581no—
282C2×M32—791no—
283C2×M32—1581no—
284C22×C16—791no—
285C22×C16—1581no—
286#473—792no—
287#473C2×D7912yes—
288#472—792no—
289#472C2×D7912yes—
290#471—22no—
291#471—1582no—
292C2×#108—791no—
293C2×#108C2×D7911yescharacteristic
294#413—21no—
295#413—1581no—
296#368—792no—
297#368—792no—
298#362—792no—
299#362—792no—
300#361—792no—
301#361—792no—
302C2×#41—21no—
303C2×#41—1581no—
304C2×#41C22⋊D7911yescharacteristic
305#137—22no—
306#137—41no—
307#137—3161no—
308#137—1582no—
309AGL(1,ℤ/8)—22no—
310AGL(1,ℤ/8)—41no—
311AGL(1,ℤ/8)—3161no—
312AGL(1,ℤ/8)—1582no—
313#135—21no—
314#135—21no—
315#135—21no—
316#135—21no—
317#135—42no—
318#135—3162no—
319#135—1581no—
320#135—1581no—
321#135—1581no—
322#135—1581no—
323C2×Q16—791no—
324C2×Q16C22×D7911yescharacteristic
325C2×SD16—21no—
326C2×SD16—1581no—
327C2×D8—791no—
328C2×D8C22×D7911yescharacteristic
329#131—21no—
330#131—21no—
331#131C22⋊D7912yes—
332#131C22×D7912yes—
333C2×M16—21no—
334C2×M16—21no—
335C2×M16—1581no—
336C2×M16C22⋊D7911yescharacteristic
337C22×C8—21no—
338C22×C8C22×D7911yescharacteristic
339Q32—1581no—
340Q32—1581no—
341SD32—1581no—
342SD32—1581no—
343SD32—1581no—
344SD32—1581no—
345D16—1581no—
346D16—1581no—
347M32—792no—
348M32—1582no—
349M32—1582no—
350M32—1581no—
351C2×C16—791no—
352C2×C16—791no—
353C2×C16—791no—
354C2×C16—791no—
355C2×C16—791no—
356C2×C16—791no—
357C2×C16—1582no—
358C2×C16—1582no—
359C2×C16—1581no—
360#108—21no—
361#108—792no—
362#108—1581no—
363#108C22×D7912yes—
364#101—22no—
365#101—1582no—
366#100—22no—
367#100—1582no—
368#41—21no—
369#41—21no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—3161no—
375#41—3161no—
376#41—3161no—
377#41—3161no—
378#41#1192960912yes—
379#41C2×C22⋊D7912yes—
380C2×Q8—21no—
381C2×Q8—1581no—
382C2×Q8#1192960911yescharacteristic
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—41no—
386C2×D4—3161no—
387C2×D4—1581no—
388C2×D4#1192960911yescharacteristic
389C22×C4—21no—
390C22×C4—41no—
391C22×C4—3161no—
392C22×C4C2×C22⋊D7911yescharacteristic
393Q16—21no—
394Q16—21no—
395Q16—42no—
396Q16—3162no—
397Q16—1581no—
398Q16—1581no—
399SD16—42no—
400SD16—42no—
401SD16—41no—
402SD16—41no—
403SD16—3162no—
404SD16—3162no—
405SD16—3161no—
406SD16—3161no—
407D8—21no—
408D8—21no—
409D8—42no—
410D8—3162no—
411D8—1581no—
412D8—1581no—
413M16—21no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—41no—
418M16—3161no—
419M16—1582no—
420M16C2×C22⋊D7912yes—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—41no—
425C2×C8—3161no—
426C2×C8D4×D7911yescharacteristic
427C2×C8D4×D7911yescharacteristic
428C2×C8C2×C22⋊D7911yescharacteristic
429C2×C8#1192962311yescharacteristic
430C2×C8#1192962311yescharacteristic
431C2×C8C23×D7911yescharacteristic
432C16—791no—
433C16—791no—
434C16—791no—
435C16—791no—
436C16—1582no—
437C23—21no—
438C23—41no—
439C23—3161no—
440C23#1457918111yescharacteristic
441Q8—21no—
442Q8—41no—
443Q8—41no—
444Q8—3161no—
445Q8—3161no—
446Q8C2×#1192960912yes—
447D4—22no—
448D4—22no—
449D4—21no—
450D4—41no—
451D4—41no—
452D4—81no—
453D4—81no—
454D4—3161no—
455D4—3161no—
456D4—6321no—
457D4—6321no—
458D4C2×#1192960912yes—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—3161no—
466C2×C4—6321no—
467C2×C4—6321no—
468C2×C4#1457913311yescharacteristic
469C2×C4#1457913311yescharacteristic
470C2×C4C2×#1192960911yescharacteristic
471C2×C4C2×#1192960911yescharacteristic
472C2×C4#1457918111yescharacteristic
473C2×C4C22×C22⋊D7911yescharacteristic
474C8—22no—
475C8—42no—
476C8—3162no—
477C8#1457913311yescharacteristic
478C8#1457913311yescharacteristic
479C8C2×D4×D7911yescharacteristic
480C8C2×#1192962311yescharacteristic
481C22—22no—
482C22—41no—
483C22—42no—
484C22—81no—
485C22—161no—
486C22—6321no—
487C22—12641no—
488C22#1909879711yescharacteristic
489C22#1909879711yescharacteristic
490C22C2×#1457918111yescharacteristic
491C4—22no—
492C4—81no—
493C4—6321no—
494C4#1909879711yescharacteristic
495C4#1909879711yescharacteristic
496C4C2×#1457913311yescharacteristic
497C4C22×#1192960911yescharacteristic
498C2—21no—
499C2—21no—
500C2—21no—
501C2—42no—
502C2—161no—
503C2—12641no—
504C2C2×#1909879711yescharacteristic
subgroupG/H
centerC4#19098797
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3067C2
socleC2×C79C2×#466
hypercenter#3067D79
hypocenterC79#33617
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. C2×#19098993C2semidirect
2. C2×#19098980C2semidirect
3. #23773120C2semidirect
4. #23765532C2semidirect
5. #23765528C2semidirect
6. #23765523C2semidirect
7. #23765518C2semidirect
8. C2×#19098392C2semidirect
9. #23765149C2semidirect
10. #23765147C2semidirect
11. C79×#3067C2semidirect
18. C2×#14579107C22semidirect
19. C2×#14579105C22semidirect
20. C2×#14579104C22semidirect
21. C2×#14579102C22semidirect
35. #19098392C22semidirect
36. #19098392C22semidirect
37. #19098379C22semidirect
38. #19098378C22semidirect
39. #19098376C22semidirect
40. #19098375C22semidirect
256. #3067D79semidirect
264. C79#33617semidirect

Maximal subgroups

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

subgroupindex
C2×#190989932
C2×#190989802
#237731202
#237655322
#237655282
#237655232
#237655182
C2×#190983922
#237651492
#237651472
C79×#30672
#3361779