Finite Group Databasebeta

← #34272109

Subgroups — #34272109

Order 20224.

710 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773975C211yesmaximal, characteristic
2#23772872C211yesmaximal, characteristic, complemented
3C2×Q8×C79⋊C8C211yesmaximal, characteristic, complemented
4#23762846C212yesmaximal, complemented
5#23762842C212yesmaximal, complemented
6#23762792C212yesmaximal, complemented
7#23762758C212yesmaximal, complemented
8#23762158C211yesmaximal, characteristic, complemented
9#23762078C211yesmaximal, characteristic, complemented
10#23762077C211yesmaximal, characteristic, complemented
11C79×#3037C211yesmaximal, characteristic, complemented
12C2×#14579192—21no—
13#19099113C2211yescharacteristic
14#19098966C2211yescharacteristic
15#19098939C2211yescharacteristic
16#19098911C2211yescharacteristic
17#19098909C2211yescharacteristic
18#19098797—21no—
19#19098797—21no—
20#19098785—22no—
21#19098773C2212yes—
22#19098763C2212yes—
23Q8×C79⋊C8—21no—
24Q8×C79⋊C8—21no—
25Q8×C79⋊C8C2212yes—
26Q8×C79⋊C8C2212yes—
27C4×#11929609—21no—
28C4×#11929609—21no—
29#19098587C2211yescharacteristic
30C2×C4⋊(C79⋊C8)—21no—
31C2×C4⋊(C79⋊C8)C2211yescharacteristic
32#19098562—21no—
33C2×C4×C79⋊C8—21no—
34C2×C4×C79⋊C8C2211yescharacteristic
35#19098315C2212yes—
36#19098315C2212yes—
37#19098310C2212yes—
38#19098309C2212yes—
39#19098307C2212yes—
40#19098306C2212yes—
41C2×#137×C79—21no—
42#532×C79C2211yescharacteristic
43C2×C4×Q8×C79C2211yescharacteristic
44#464×C79—22no—
45#462×C79C2212yes—
46#460×C79C2212yes—
47#423×C79C2211yescharacteristic
48#417×C79—21no—
49C2×#11929624—21no—
50#14579192—44no—
51C2×#11929614—22no—
52C2×#11929611—22no—
53C2×#11929609—21no—
54C2×#11929609—21no—
55C2×#11929608—21no—
56C22×C79⋊C8—21no—
57C22×C79⋊C8D411yescharacteristic
58C22×C79⋊C8D411yescharacteristic
59#14579161C2311yescharacteristic
60#14579158C2311yescharacteristic
61#14579149—21no—
62#14579135—21no—
63#14579135—21no—
64C4×D316—21no—
65C4×D316—21no—
66#14579094—21no—
67#14579094—21no—
68C4⋊(C79⋊C8)—21no—
69C4⋊(C79⋊C8)—21no—
70C4⋊(C79⋊C8)—22no—
71C4⋊(C79⋊C8)—22no—
72C4⋊(C79⋊C8)C2312yes—
73C4⋊(C79⋊C8)C2312yes—
74#14579084—22no—
75#14579082—21no—
76#14579082—21no—
77#14579081—22no—
78#14579081—21no—
79#14579081—21no—
80C4×C79⋊C8—22no—
81C4×C79⋊C8—22no—
82C4×C79⋊C8D411yescharacteristic
83C4×C79⋊C8D411yescharacteristic
84C4×C79⋊C8C2311yescharacteristic
85C4×C79⋊C8C2311yescharacteristic
86C2×#41×C79—21no—
87C22×Q8×C79D411yescharacteristic
88#137×C79—44no—
89C2×Q16×C79—22no—
90C2×SD16×C79—22no—
91C2×M16×C79—21no—
92C4⋊Q8×C79C2311yescharacteristic
93#127×C79C2311yescharacteristic
94#124×C79—21no—
95#121×C79—21no—
96#121×C79—21no—
97C4×Q8×C79—21no—
98C4×Q8×C79—21no—
99C4×Q8×C79D412yes—
100C4×Q8×C79C2312yes—
101C4×D4×C79—21no—
102C4×D4×C79—21no—
103C2×C4⋊C4×C79—21no—
104C2×C4⋊C4×C79D411yescharacteristic
105C2×C42×C79—21no—
106C2×C42×C79C2311yescharacteristic
107C4⋊C8×C79—21no—
108C4⋊C8×C79C2312yes—
109#103×C79—22no—
110#102×C79—22no—
111#11929624—42no—
112#11929624—42no—
113C2×Dic158—21no—
114C2×D316—21no—
115C2×D316—21no—
116C2×D316—21no—
117C2×C22⋊D79—41no—
118C2×C4×D79—41no—
119#11929614—44no—
120#11929611—44no—
121#11929609—42no—
122#11929609—42no—
123#11929608—42no—
124C2×C79⋊C8—22no—
125C2×C79⋊C8—22no—
126C2×C79⋊C8—22no—
127C2×C79⋊C8—41no—
128C2×C79⋊C8—42no—
129C2×C79⋊C8C2×D411yescharacteristic
130C2×C79⋊C8C2×D411yescharacteristic
131C2×C79⋊C8C2×D411yescharacteristic
132C2×C79⋊C8C2×D411yescharacteristic
133C4⋊Dic79—21no—
134C4⋊Dic79—21no—
135(C2×D79)⋊C4—41no—
136(C2×D79)⋊C4—41no—
137#41×C79—42no—
138#41×C79—42no—
139C2×Q8×C79—21no—
140C2×Q8×C79—22no—
141C2×Q8×C79—21no—
142C2×Q8×C79D814yescomplemented
143C2×Q8×C79C2×D412yes—
144C2×D4×C79—21no—
145C2×D4×C79—21no—
146C2×D4×C79—21no—
147C2×D4×C79—41no—
148C22×C4×C79—21no—
149C22×C4×C79—21no—
150C22×C4×C79—41no—
151C22×C4×C79C2×D411yescharacteristic
152C22×C4×C79C2×D411yescharacteristic
153C22×C4×C79#4111yescharacteristic
154Q16×C79—44no—
155SD16×C79—44no—
156M16×C79—42no—
157C2×C8×C79—22no—
158C4⋊C4×C79—22no—
159C4⋊C4×C79—21no—
160C4⋊C4×C79—22no—
161C4⋊C4×C79—21no—
162C4⋊C4×C79—21no—
163C4⋊C4×C79C2×D412yes—
164C22⋊C4×C79—41no—
165C22⋊C4×C79—41no—
166C42×C79—22no—
167C42×C79—22no—
168C42×C79C2×D411yescharacteristic
169C42×C79C2×D411yescharacteristic
170C42×C79#4111yescharacteristic
171C42×C79C2411yescharacteristic
172C23×C79—41no—
173C23×C79#12111yescharacteristic
174C22×D79—41no—
175Q8×C79—22no—
176Q8×C79—24no—
177Q8×C79—41no—
178Q8×C79—41no—
179Q8×C79C2×D814yes—
180D4×C79—41no—
181D4×C79—41no—
182D4×C79—41no—
183D4×C79—41no—
184D4×C79—81no—
185D4×C79—82no—
186C2×C4×C79—22no—
187C2×C4×C79—24no—
188C2×C4×C79—21no—
189C2×C4×C79—21no—
190C2×C4×C79—22no—
191C2×C4×C79—41no—
192C2×C4×C79—42no—
193C2×C4×C79—41no—
194C2×C4×C79—82no—
195C2×C4×C79#12111yescharacteristic
196C2×C4×C79#12111yescharacteristic
197C2×C4×C79#12111yescharacteristic
198C2×C4×C79C2×D812yes—
199C2×C4×C79#13512yes—
200C2×C4×C79C22×D411yescharacteristic
201C2×C4×C79C22×D411yescharacteristic
202C2×C4×C79C2×#4111yescharacteristic
203C22⋊D79—82no—
204C2×Dic79—41no—
205D316—41no—
206D316—41no—
207D316—41no—
208D316—41no—
209D316—81no—
210C4×D79—82no—
211Dic158—41no—
212Dic158—41no—
213C8×C79—42no—
214C79⋊C8—21no—
215C79⋊C8—21no—
216C79⋊C8—21no—
217C79⋊C8—42no—
218C79⋊C8—42no—
219C79⋊C8#13712yescomplemented
220C22×C79—21no—
221C22×C79—21no—
222C22×C79—81no—
223C22×C79—81no—
224C22×C79—81no—
225C22×C79#46612yes—
226C22×C79C2×#12111yescharacteristic
227C2×D79—81no—
228C2×D79—81no—
229C2×D79—81no—
230C4×C79—21no—
231C4×C79—24no—
232C4×C79—21no—
233C4×C79—21no—
234C4×C79—42no—
235C4×C79—81no—
236C4×C79#46612yes—
237C4×C79C22×D811yescharacteristic
238C4×C79C2×#13511yescharacteristic
239C4×C79C2×#13711yescharacteristic
240C4×C79#57511yescharacteristic
241Dic79—81no—
242#19261—791nomaximal, Sylow
243C2×C79—22no—
244C2×C79—161no—
245C2×C79#303711yescharacteristic
246C2×C79C2×#46611yescharacteristic
247C2×C79#306711yescharacteristic
248D79—161no—
249#3169—791no—
250#3037—791no—
251#3037D7911yescharacteristic, complemented
252C2×C8×Q8—791no—
253#1682—792no—
254#1682—792no—
255#1674—792no—
256#1668—792no—
257#1567—791no—
258#1547—791no—
259#1547—791no—
260C79#1926111yescharacteristic, complemented, Sylow
261C2×#137—21no—
262C2×#137—1581no—
263#532—791no—
264#532C2×D7911yescharacteristic
265C2×C4×Q8C2×D7911yescharacteristic
266#500—791no—
267#493—791no—
268#486—791no—
269#486—791no—
270#466—1581no—
271#466—1581no—
272#464—22no—
273#464—1582no—
274#462—792no—
275#462C2×D7912yes—
276#460—792no—
277#460C2×D7912yes—
278C8×Q8—792no—
279C8×Q8—792no—
280C8×Q8—1581no—
281C8×Q8—1581no—
282C4×D8—1581no—
283C4×D8—1581no—
284#423—791no—
285#423C2×D7911yescharacteristic
286C2×C4⋊C8—791no—
287C2×C4⋊C8—1581no—
288#417—21no—
289#417—1581no—
290C2×C4×C8—791no—
291C2×C4×C8—1581no—
292#335—792no—
293#335—792no—
294#332—792no—
295#332—792no—
296#331—792no—
297#331—792no—
298C2×#41—21no—
299C2×#41—1581no—
300C22×Q8C22⋊D7911yescharacteristic
301#137—44no—
302#137—3164no—
303C2×Q16—22no—
304C2×Q16—1582no—
305C2×SD16—22no—
306C2×SD16—1582no—
307C2×D8—1581no—
308C2×D8—1581no—
309C2×M16—21no—
310C2×M16—1581no—
311C22×C8—791no—
312C22×C8—791no—
313C22×C8—1581no—
314C4⋊Q8—791no—
315C4⋊Q8C22×D7911yescharacteristic
316#127—791no—
317#127C22×D7911yescharacteristic
318#124—21no—
319#124—1581no—
320#121—21no—
321#121—21no—
322#121—1581no—
323#121—1581no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8C22⋊D7912yes—
327C4×Q8C22×D7912yes—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1581no—
331C4×D4—1581no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4C22⋊D7911yescharacteristic
334C2×C42—21no—
335C2×C42C22×D7911yescharacteristic
336#107—1581no—
337#107—1581no—
338C4⋊C8—21no—
339C4⋊C8—792no—
340C4⋊C8—792no—
341C4⋊C8—1582no—
342C4⋊C8—1582no—
343C4⋊C8—1581no—
344C4⋊C8—1581no—
345C4⋊C8C22×D7912yes—
346#103—22no—
347#103—1582no—
348#102—22no—
349#102—1581no—
350#102—1581no—
351#102—1581no—
352#102—1581no—
353#102—1582no—
354C4×C8—791no—
355C4×C8—791no—
356C4×C8—791no—
357C4×C8—791no—
358C4×C8—1582no—
359C4×C8—1582no—
360#41—42no—
361#41—42no—
362#41—3162no—
363#41—3162no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—1581no—
368C2×Q8#1192960914yescomplemented
369C2×Q8C2×C22⋊D7912yes—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—3161no—
375C2×D4—1581no—
376C2×D4—1581no—
377C2×D4—1581no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—41no—
381C22×C4—3161no—
382C22×C4D4×D7911yescharacteristic
383C22×C4C2×C22⋊D7911yescharacteristic
384C22×C4#1192962311yescharacteristic
385Q16—44no—
386Q16—3164no—
387SD16—44no—
388SD16—3164no—
389D8—3162no—
390D8—3162no—
391M16—42no—
392M16—3162no—
393C2×C8—22no—
394C2×C8—3162no—
395C2×C8—3161no—
396C2×C8—791no—
397C2×C8—791no—
398C2×C8—791no—
399C2×C8—791no—
400C2×C8—1582no—
401C2×C8—1582no—
402C2×C8—1582no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—1581no—
409C4⋊C4—1581no—
410C4⋊C4C2×C22⋊D7912yes—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—3161no—
414C22⋊C4—3161no—
415C42—22no—
416C42—22no—
417C42D4×D7911yescharacteristic
418C42C2×C22⋊D7911yescharacteristic
419C42#1192962311yescharacteristic
420C42C23×D7911yescharacteristic
421C23—41no—
422C23—3161no—
423C23#1457913311yescharacteristic
424Q8—24no—
425Q8—22no—
426Q8—41no—
427Q8—41no—
428Q8—3161no—
429Q8—3161no—
430Q8C2×#1192960914yes—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—81no—
436D4—82no—
437D4—3161no—
438D4—3161no—
439D4—3161no—
440D4—3161no—
441D4—6321no—
442D4—6322no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—82no—
452C2×C4—3161no—
453C2×C4—6322no—
454C2×C4#1457913311yescharacteristic
455C2×C4#1457913311yescharacteristic
456C2×C4#1457913311yescharacteristic
457C2×C4C2×#1192960912yes—
458C2×C4#1457918112yes—
459C2×C4C2×D4×D7911yescharacteristic
460C2×C4C22×C22⋊D7911yescharacteristic
461C2×C4C2×#1192962311yescharacteristic
462C8—42no—
463C8—3162no—
464C8—3162no—
465C8—792no—
466C8—1581no—
467C8—1581no—
468C8—1581no—
469C22—21no—
470C22—21no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—6321no—
475C22—6321no—
476C22—6321no—
477C22#1909879712yes—
478C22C2×#1457913311yescharacteristic
479C4—24no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—81no—
485C4—6321no—
486C4#1909879712yes—
487C4C22×#1192960911yescharacteristic
488C4C2×#1457918111yescharacteristic
489C4#137×D7911yescharacteristic
490C4#1909936211yescharacteristic
491C2—22no—
492C2—161no—
493C2—12641no—
494C2#2377286911yescharacteristic
495C2C2×#1909879711yescharacteristic
496C2#2377311711yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#3037C2
socleC22×C79C2×#121
prime radical#23773975C2
hypercenter#3037D79
hypocenterC79#19261
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
2. #23772872C2semidirect
3. C2×Q8×C79⋊C8C2semidirect
4. #23762846C2semidirect
5. #23762842C2semidirect
6. #23762792C2semidirect
7. #23762758C2semidirect
8. #23762158C2semidirect
9. #23762078C2semidirect
10. #23762077C2semidirect
11. C79×#3037C2semidirect
142. C2×Q8×C79D8semidirect
219. C79⋊C8#137semidirect
251. #3037D79semidirect
260. C79#19261semidirect
368. C2×Q8#11929609semidirect

Maximal subgroups

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

subgroupindex
#237739752
#237728722
C2×Q8×C79⋊C82
#237628462
#237628422
#237627922
#237627582
#237621582
#237620782
#237620772
C79×#30372
#1926179