Finite Group Databasebeta

← #34258192

Subgroups — #34258192

Order 20224.

610 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773437C211yesmaximal, characteristic
2#23772329C211yesmaximal, characteristic
3C8×Q8×D79C211yesmaximal, characteristic
4#23762274C211yesmaximal, characteristic, complemented
5#23762272C211yesmaximal, characteristic, complemented
6#23762270C211yesmaximal, characteristic, complemented
7#23762268C211yesmaximal, characteristic, complemented
8#23762220C211yesmaximal, characteristic, complemented
9#23762217C211yesmaximal, characteristic, complemented
10#23762217C211yesmaximal, characteristic, complemented
11#23762169C211yesmaximal, characteristic
12#23761827C211yesmaximal, characteristic
13#23761801C211yesmaximal, characteristic
14#23761691C211yesmaximal, characteristic
15C79×#1579C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17C4×Q8×D79C2211yescharacteristic
18#19098900C411yescharacteristic
19#19098897C411yescharacteristic
20#19098857C411yescharacteristic
21#19098856C411yescharacteristic
22#19098772C411yescharacteristic
23#19098771C411yescharacteristic
24#19098720C411yescharacteristic
25#19098717C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098692—21no—
28#19098690C2211yescharacteristic
29Q8×C79⋊C8C2211yescharacteristic
30C8×Dic158—21no—
31C8×Dic158—21no—
32C8×Dic158C2211yescharacteristic
33C8×Dic158C2211yescharacteristic
34#19098687—21no—
35#19098687C2211yescharacteristic
36C4×Dic316C2211yescharacteristic
37C4×#11929614C2211yescharacteristic
38C4×#11929614C2211yescharacteristic
39#19098614C2211yescharacteristic
40#19098614C2211yescharacteristic
41#19098588—21no—
42#19098585C2211yescharacteristic
43C4⋊C8×D79—21no—
44C4⋊C8×D79C2211yescharacteristic
45#19098517—21no—
46C4×C8×D79—21no—
47C4×C8×D79C2211yescharacteristic
48#19098314C2211yescharacteristic
49#19098313C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54Q8⋊(C79⋊C8)C2211yescharacteristic
55#19098293C2211yescharacteristic
56#19098293C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098284C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×Q8×D79C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929614C2×C411yescharacteristic
71C2×#11929607—21no—
72C2×C8×D79—21no—
73C2×C8×D79D411yescharacteristic
74C2×C8×D79D411yescharacteristic
75#14579160C2×C411yescharacteristic
76#14579159C2×C411yescharacteristic
77#14579151—21no—
78#14579141C2×C411yescharacteristic
79#14579139—21no—
80#14579137C2×C411yescharacteristic
81Q8×Dic79C2×C411yescharacteristic
82C4×Dic158—21no—
83C4×Dic158—21no—
84C4×Dic158C2311yescharacteristic
85C4×Dic158C2311yescharacteristic
86#14579126—21no—
87#14579126C2×C411yescharacteristic
88#14579118—21no—
89C4⋊C4×D79—21no—
90C4⋊C4×D79C2×C411yescharacteristic
91C42×D79—21no—
92C42×D79C2311yescharacteristic
93#14579095C2×C411yescharacteristic
94#14579094C2×C411yescharacteristic
95#14579094C2×C411yescharacteristic
96C4⋊(C79⋊C8)—21no—
97C4⋊(C79⋊C8)—21no—
98C4⋊(C79⋊C8)C2311yescharacteristic
99C4⋊(C79⋊C8)C2311yescharacteristic
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8C2×C411yescharacteristic
108Dic79⋊C8C2×C411yescharacteristic
109#14579086C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579085C2×C411yescharacteristic
112#14579085C2×C411yescharacteristic
113#14579084C2×C411yescharacteristic
114#14579084C2×C411yescharacteristic
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8C2×C411yescharacteristic
117(C2×D79)⋊C8C2×C411yescharacteristic
118#14579072—21no—
119#14579071—21no—
120C4×C79⋊C8—21no—
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C8×Dic79—21no—
125C8×Dic79—21no—
126C8×Dic79—21no—
127C8×Dic79D411yescharacteristic
128C8×Dic79D411yescharacteristic
129C2×Q16×C79C2×C411yescharacteristic
130C4×Q8×C79C2311yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132#107×C79C2×C411yescharacteristic
133C4⋊C8×C79—21no—
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#103×C79C2×C411yescharacteristic
139C4×C8×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158—21no—
146C2×Dic158C22×C411yescharacteristic
147C2×Dic158C22×C411yescharacteristic
148Q8×D79—22no—
149Q8×D79—22no—
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79C22×C411yescharacteristic
153C2×C4×D79C2×D411yescharacteristic
154C2×C4×D79#4111yescharacteristic
155Dic316—22no—
156#11929614—22no—
157#11929614—22no—
158#11929607—42no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C8×D79—22no—
167C8×D79—22no—
168C8×D79—42no—
169C4⋊Dic79—21no—
170C4⋊Dic79—21no—
171C4⋊Dic79C22×C411yescharacteristic
172C4⋊Dic79C22×C411yescharacteristic
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183C4×Dic79—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79C22×C411yescharacteristic
188C4×Dic79C2×D411yescharacteristic
189C4×Dic79#4111yescharacteristic
190C2×Q8×C79C22×C411yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192Q16×C79—22no—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79C22×C411yescharacteristic
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C2×D411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C4⋊C4×C79—21no—
201C4⋊C4×C79—21no—
202C4⋊C4×C79C22×C411yescharacteristic
203C4⋊C4×C79C22×C411yescharacteristic
204C42×C79—21no—
205C42×C79—21no—
206C42×C79C2411yescharacteristic
207C22×D79C4×D411yescharacteristic
208Q8×C79—21no—
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79#13112yes—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79C23×C411yescharacteristic
217C2×C4×C79C22×D411yescharacteristic
218C2×C4×C79C2×#4111yescharacteristic
219C2×Dic79—21no—
220C2×Dic79—21no—
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79C4×D411yescharacteristic
224C2×Dic79C4×D411yescharacteristic
225C2×Dic79C4×D411yescharacteristic
226C4×D79—24no—
227C4×D79—21no—
228C4×D79—21no—
229C4×D79—41no—
230C4×D79—42no—
231C4×D79#13712yes—
232C4×D79#13712yes—
233Dic158—24no—
234Dic158—21no—
235Dic158—21no—
236Dic158—21no—
237Dic158—42no—
238Dic158#13112yes—
239C8×C79—21no—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—41no—
244C8×C79—41no—
245C8×C79—41no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—41no—
251C79⋊C8—41no—
252C79⋊C8—41no—
253C22×C79C2×C4×D411yescharacteristic
254C2×D79—21no—
255C2×D79—21no—
256C2×D79#44112yes—
257C4×C79—22no—
258C4×C79—21no—
259C4×C79—41no—
260C4×C79—41no—
261C4×C79—41no—
262C4×C79C2×#13111yescharacteristic
263C4×C79C2×#13111yescharacteristic
264C4×C79C2×#13711yescharacteristic
265C4×C79C2×#13711yescharacteristic
266Dic79—21no—
267Dic79—22no—
268Dic79—21no—
269Dic79—41no—
270Dic79—41no—
271Dic79—41no—
272Dic79#44112yes—
273#17906—791nomaximal, Sylow
274C2×C79#292911yescharacteristic
275C2×C79C2×#44111yescharacteristic
276C2×C79#295511yescharacteristic
277D79—22no—
278#3104—791no—
279#2960—791no—
280C2×C8×Q8—791no—
281#1587—791no—
282#1587—791no—
283#1587—791no—
284#1587—791no—
285#1579—791no—
286#1579—791no—
287#1579—791no—
288#1579D7911yescharacteristic, complemented
289#1570—791no—
290#1487—791no—
291#1481—791no—
292#1452—791no—
293C79#1790611yescharacteristic, complemented, Sylow
294#533—791no—
295C2×C4×Q8—791no—
296#484—791no—
297#484—791no—
298#477—791no—
299#477—791no—
300#462—791no—
301#462—791no—
302#451—791no—
303#451—791no—
304#446—791no—
305#446—791no—
306#446—1581no—
307C8×Q8—791no—
308C8×Q8—791no—
309C8×Q8—791no—
310C8×Q8—791no—
311C8×Q8—1581no—
312C8×Q8—1581no—
313C8×Q8—1581no—
314C8×Q8C2×D7911yescharacteristic
315C8×Q8C2×D7911yescharacteristic
316C4×Q16—791no—
317C4×Q16—791no—
318C4×Q16—791no—
319C4×Q16C2×D7911yescharacteristic
320#431—791no—
321#431—791no—
322#424—1581no—
323#423—791no—
324C2×C4⋊C8—791no—
325C2×C4⋊C8—1581no—
326#405—1581no—
327C2×C4×C8—791no—
328C2×C4×C8—1581no—
329#334—791no—
330#334—791no—
331#334—791no—
332#334C2×D7911yescharacteristic
333Q8⋊C8—791no—
334Q8⋊C8—791no—
335Q8⋊C8—791no—
336Q8⋊C8—791no—
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8C2×D7911yescharacteristic
340Q8⋊C8C2×D7911yescharacteristic
341#322—791no—
342#322—791no—
343#322—791no—
344#322C2×D7911yescharacteristic
345C22×Q8—791no—
346C2×Q16—791no—
347C2×Q16—791no—
348C2×Q16—791no—
349C2×Q16C4×D7911yescharacteristic
350C2×M16—1581no—
351C22×C8—791no—
352C22×C8—791no—
353C22×C8—1581no—
354C4⋊Q8—791no—
355C4⋊Q8—791no—
356#125—1581no—
357C22⋊Q8—791no—
358C22⋊Q8—791no—
359C22⋊Q8—1581no—
360C4×Q8—791no—
361C4×Q8—791no—
362C4×Q8—791no—
363C4×Q8—791no—
364C4×Q8—1581no—
365C4×Q8—1581no—
366C4×Q8—1581no—
367C4×Q8C22×D7911yescharacteristic
368C4×Q8C22×D7911yescharacteristic
369#117—1581no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—1581no—
372C2×C42—791no—
373C2×C42—1581no—
374#107—791no—
375#107—791no—
376#107—791no—
377#107C4×D7911yescharacteristic
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—791no—
381C4⋊C8—791no—
382C4⋊C8—791no—
383C4⋊C8—791no—
384C4⋊C8—1581no—
385C4⋊C8—1581no—
386C4⋊C8—1581no—
387C4⋊C8—1581no—
388C4⋊C8—1581no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8C22×D7911yescharacteristic
394C4⋊C8C22×D7911yescharacteristic
395#103—791no—
396#103—791no—
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103C4×D7911yescharacteristic
402#103C4×D7911yescharacteristic
403C22⋊C8—791no—
404C22⋊C8—791no—
405C22⋊C8—1581no—
406#97—1581no—
407#97—1581no—
408C4×C8—21no—
409C4×C8—21no—
410C4×C8—791no—
411C4×C8—791no—
412C4×C8—791no—
413C4×C8—791no—
414C4×C8—791no—
415C4×C8—1581no—
416C4×C8—1581no—
417C4×C8—1581no—
418C4×C8—1581no—
419C4×C8C22×D7911yescharacteristic
420C4×C8C22×D7911yescharacteristic
421C4×C8C22×D7911yescharacteristic
422C2×Q8—791no—
423C2×Q8—791no—
424C2×Q8—1581no—
425C2×Q8—1581no—
426C2×Q8—1582no—
427C2×Q8—1582no—
428C2×Q8C2×C4×D7911yescharacteristic
429C2×Q8C2×C4×D7911yescharacteristic
430C22×C4—791no—
431C22×C4—791no—
432C22×C4—791no—
433C22×C4—1581no—
434C22×C4—1581no—
435Q16—22no—
436Q16—1582no—
437Q16—1582no—
438Q16—1582no—
439M16—3162no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—3162no—
444C2×C8—791no—
445C2×C8—791no—
446C2×C8—791no—
447C2×C8—791no—
448C2×C8—1582no—
449C2×C8—1582no—
450C2×C8—1581no—
451C2×C8—1581no—
452C2×C8—1581no—
453C2×C8C2×C4×D7911yescharacteristic
454C2×C8C2×C4×D7911yescharacteristic
455C2×C8D4×D7911yescharacteristic
456C2×C8D4×D7911yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—791no—
460C4⋊C4—791no—
461C4⋊C4—1581no—
462C4⋊C4—1581no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4C2×C4×D7911yescharacteristic
472C4⋊C4C2×C4×D7911yescharacteristic
473C22⋊C4—1581no—
474C22⋊C4—1581no—
475C42—21no—
476C42—21no—
477C42—791no—
478C42—791no—
479C42—791no—
480C42—1581no—
481C42—1581no—
482C42—1581no—
483C42—1581no—
484C42C23×D7911yescharacteristic
485C23—791no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—3162no—
490Q8—792no—
491Q8—1584no—
492Q8—1581no—
493Q8—1581no—
494Q8—1581no—
495Q8#1457916712yes—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—3161no—
501C2×C4—3162no—
502C2×C4—791no—
503C2×C4—792no—
504C2×C4—792no—
505C2×C4—791no—
506C2×C4—791no—
507C2×C4—1584no—
508C2×C4—1581no—
509C2×C4—1581no—
510C2×C4—1581no—
511C2×C4—1581no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4C22×C4×D7911yescharacteristic
515C2×C4C2×D4×D7911yescharacteristic
516C2×C4#41×D7911yescharacteristic
517C8—21no—
518C8—21no—
519C8—21no—
520C8—21no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—3161no—
525C8—3161no—
526C8—3161no—
527C8—1581no—
528C8—1581no—
529C8—1581no—
530C8—1581no—
531C22—792no—
532C22—1581no—
533C22—1581no—
534C22C4×D4×D7911yescharacteristic
535C4—21no—
536C4—22no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—3161no—
541C4—3161no—
542C4—3161no—
543C4—792no—
544C4—1581no—
545C4—1581no—
546C4—1582no—
547C4#131×D7911yescharacteristic
548C4C2×#1457916711yescharacteristic
549C4#137×D7911yescharacteristic
550C4#137×D7911yescharacteristic
551C2—1582no—
552C2#2377208911yescharacteristic
553C2#441×D7911yescharacteristic
554C2#2377230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1579C2
socleC22×C79C2×C4×D4
prime radicalC22×D79C4×D4
cosocleC42×C79C24
hypercenter#1579D79
hypocenterC79#17906
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1579

Complements

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

NH ≅ G/Ntype
4. #23762274C2semidirect
5. #23762272C2semidirect
6. #23762270C2semidirect
7. #23762268C2semidirect
8. #23762220C2semidirect
9. #23762217C2semidirect
10. #23762217C2semidirect
15. C79×#1579C2semidirect
288. #1579D79semidirect
293. C79#17906semidirect

Maximal subgroups

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

subgroupindex
#237734372
#237723292
C8×Q8×D792
#237622742
#237622722
#237622702
#237622682
#237622202
#237622172
#237621692
#237618272
#237618012
#237616912
C79×#15792
#1790679