Finite Group Databasebeta

← C2×#398×D79 #34454479

Subgroups — C2×#398×D79 #34454479

Order 20224.

4158 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D79C211yesmaximal, characteristic
2C22×C4⋊C4×D79C211yesmaximal, characteristic
3C22×C4⋊C4×D79C212yesmaximal
4C2×#19098497C212yesmaximal, complemented
5C2×#19098496C212yesmaximal, complemented
6C2×#19098495C211yesmaximal, characteristic, complemented
7C2×#19098494C211yesmaximal, characteristic, complemented
8C2×#19098493C211yesmaximal, characteristic, complemented
9#398×D79C2116yesmaximal, direct product factor
10C2×#95×D79C211yesmaximal, characteristic
11C2×#95×D79C212yesmaximal
12C2×#398×C79C211yesmaximal, characteristic, complemented
13C23×C4×D79C2212yes—
14C23×C4×D79C2211yescharacteristic
15C23×C4×D79C2211yescharacteristic
16C23×C4×D79C2211yescharacteristic
17C23×C4×D79C2212yes—
18C22×C4⋊Dic79C2211yescharacteristic
19C22×C4⋊Dic79C2212yes—
20C22×C4⋊Dic79C2211yescharacteristic
21C22×Dic79⋊C4C2212yes—
22C22×Dic79⋊C4C2211yescharacteristic
23C22×Dic79⋊C4C2212yes—
24C22×Dic79⋊C4C2212yes—
25C22×Dic79⋊C4C2211yescharacteristic
26C2×C4⋊C4×D79—216no—
27C2×C4⋊C4×D79—28no—
28C2×C4⋊C4×D79—28no—
29C2×C4⋊C4×D79C2218yes—
30C2×C4⋊C4×D79C2218yes—
31C2×C4⋊C4×D79C22116yes—
32#19098497C22116yescomplemented
33#19098496C22116yescomplemented
34#19098495C2218yescomplemented
35#19098494C2218yescomplemented
36#19098493C2218yescomplemented
37C2×C79⋊#95C2211yescharacteristic
38C2×C79⋊#95C2212yes—
39C2×C79⋊#95C2212yes—
40C2×C79⋊#95C2212yes—
41C2×C79⋊#95C2212yes—
42#95×D79C2218yes—
43#95×D79C22116yes—
44C22×C4⋊C4×C79C2211yescharacteristic
45C22×C4⋊C4×C79C2211yescharacteristic
46C22×C4⋊C4×C79C2212yes—
47#398×C79C2218yescomplemented
48C2×#95×C79C2211yescharacteristic
49C2×#95×C79C2212yes—
50C24×D79C2311yescharacteristic
51C23×Dic79C2311yescharacteristic
52C23×Dic79C2312yes—
53C23×Dic79C2311yescharacteristic
54C23×Dic79C2311yescharacteristic
55C23×Dic79C2312yes—
56C22×C4×D79—24no—
57C22×C4×D79—28no—
58C22×C4×D79—24no—
59C22×C4×D79—28no—
60C22×C4×D79—28no—
61C22×C4×D79—28no—
62C22×C4×D79—28no—
63C22×C4×D79—24no—
64C22×C4×D79—28no—
65C22×C4×D79—28no—
66C22×C4×D79D418yes—
67C22×C4×D79D418yes—
68C22×C4×D79Q8116yes—
69C22×C4×D79C2314yes—
70C22×C4×D79C2318yes—
71C22×C4×D79C2314yes—
72C22×C4×D79C2314yes—
73C22×C4×D79C2318yes—
74C2×C4⋊Dic79—24no—
75C2×C4⋊Dic79—24no—
76C2×C4⋊Dic79—28no—
77C2×C4⋊Dic79C2314yes—
78C2×C4⋊Dic79C2318yes—
79C2×C4⋊Dic79C2314yes—
80C2×Dic79⋊C4—28no—
81C2×Dic79⋊C4—24no—
82C2×Dic79⋊C4—24no—
83C2×Dic79⋊C4—28no—
84C2×Dic79⋊C4—28no—
85C2×Dic79⋊C4C2318yes—
86C2×Dic79⋊C4C2314yes—
87C2×Dic79⋊C4C2318yes—
88C2×Dic79⋊C4C2318yes—
89C2×Dic79⋊C4C2314yes—
90C4⋊C4×D79—232no—
91C4⋊C4×D79—232no—
92C4⋊C4×D79—264no—
93C79⋊#95C2314yes—
94C79⋊#95C2318yes—
95C79⋊#95C2318yes—
96C79⋊#95C2318yes—
97C79⋊#95C2318yes—
98C23×C4×C79C2311yescharacteristic
99C23×C4×C79C2312yes—
100C23×C4×C79C2311yescharacteristic
101C23×C4×C79C2311yescharacteristic
102C23×C4×C79C2312yes—
103C2×C4⋊C4×C79—28no—
104C2×C4⋊C4×C79—24no—
105C2×C4⋊C4×C79—24no—
106C2×C4⋊C4×C79C2314yes—
107C2×C4⋊C4×C79C2314yes—
108C2×C4⋊C4×C79C2318yes—
109#95×C79C2314yes—
110#95×C79C2318yes—
111C23×D79C2×D414yes—
112C23×D79C2×D414yes—
113C23×D79C2×Q818yes—
114C23×D79#4114yes—
115C23×D79#4118yes—
116C23×D79C2412yes—
117C22×Dic79—24no—
118C22×Dic79—22no—
119C22×Dic79—24no—
120C22×Dic79—24no—
121C22×Dic79—22no—
122C22×Dic79—22no—
123C22×Dic79—24no—
124C22×Dic79—24no—
125C22×Dic79—24no—
126C22×Dic79—24no—
127C22×Dic79C2×D414yes—
128C22×Dic79C2×D414yes—
129C22×Dic79C2×Q818yes—
130C22×Dic79C2412yes—
131C22×Dic79C2414yes—
132C22×Dic79C2412yes—
133C22×Dic79C2412yes—
134C22×Dic79C2414yes—
135C2×C4×D79—232no—
136C2×C4×D79—264no—
137C2×C4×D79—232no—
138C2×C4×D79—216no—
139C2×C4×D79—28no—
140C2×C4×D79—216no—
141C2×C4×D79—216no—
142C2×C4×D79—28no—
143C2×C4×D79—28no—
144C2×C4×D79—216no—
145C2×C4×D79—216no—
146C2×C4×D79—216no—
147C2×C4×D79—216no—
148C2×C4×D79—432no—
149C2×C4×D79—416no—
150C2×C4×D79C2×D4116yes—
151C2×C4×D79C2×D4116yes—
152C2×C4×D79C2×Q8132yes—
153C4⋊Dic79—28no—
154C4⋊Dic79—216no—
155C4⋊Dic79—28no—
156Dic79⋊C4—28no—
157Dic79⋊C4—216no—
158Dic79⋊C4—216no—
159Dic79⋊C4—216no—
160Dic79⋊C4—28no—
161C24×C79C2411yescharacteristic
162C22×C4×C79—22no—
163C22×C4×C79—24no—
164C22×C4×C79—22no—
165C22×C4×C79—24no—
166C22×C4×C79—24no—
167C22×C4×C79—24no—
168C22×C4×C79—24no—
169C22×C4×C79—22no—
170C22×C4×C79—24no—
171C22×C4×C79—24no—
172C22×C4×C79C2×D414yes—
173C22×C4×C79C2×D414yes—
174C22×C4×C79C2×Q818yes—
175C22×C4×C79C2412yes—
176C22×C4×C79C2414yes—
177C22×C4×C79C2412yes—
178C22×C4×C79C2412yes—
179C22×C4×C79C2414yes—
180C4⋊C4×C79—216no—
181C4⋊C4×C79—28no—
182C4⋊C4×C79—28no—
183C23×C79C22×D412yes—
184C23×C79C22×D412yes—
185C23×C79C22×Q814yes—
186C23×C79C2×#4112yes—
187C23×C79C2×#4114yes—
188C23×C79C2511yescharacteristic
189C22×D79#121116yes—
190C22×D79C22⋊Q8132yes—
191C22×D79#123116yes—
192C22×D79#125132yes—
193C22×D79C4⋊Q8116yes—
194C22×D79C22×D414yes—
195C22×D79C22×D414yes—
196C22×D79C22×Q818yes—
197C22×D79C2×#4114yes—
198C22×D79C2×#4118yes—
199C2×C4×C79—28no—
200C2×C4×C79—216no—
201C2×C4×C79—28no—
202C2×C4×C79—22no—
203C2×C4×C79—24no—
204C2×C4×C79—22no—
205C2×C4×C79—24no—
206C2×C4×C79—24no—
207C2×C4×C79—24no—
208C2×C4×C79—24no—
209C2×C4×C79—22no—
210C2×C4×C79—24no—
211C2×C4×C79—24no—
212C2×C4×C79—48no—
213C2×C4×C79—44no—
214C2×C4×C79C22×D414yes—
215C2×C4×C79C22×D414yes—
216C2×C4×C79C22×Q818yes—
217C2×Dic79—28no—
218C2×Dic79—216no—
219C2×Dic79—28no—
220C2×Dic79—22no—
221C2×Dic79—24no—
222C2×Dic79—22no—
223C2×Dic79—24no—
224C2×Dic79—24no—
225C2×Dic79—24no—
226C2×Dic79—24no—
227C2×Dic79—22no—
228C2×Dic79—24no—
229C2×Dic79—24no—
230C2×Dic79—48no—
231C2×Dic79—44no—
232C2×Dic79C22×D414yes—
233C2×Dic79C22×D414yes—
234C2×Dic79C22×Q818yes—
235C4×D79—232no—
236C4×D79—264no—
237C4×D79—232no—
238C4×D79—432no—
239C4×D79—416no—
240C22×C79C2×#12114yes—
241C22×C79C2×C22⋊Q818yes—
242C22×C79C2×#12314yes—
243C22×C79C2×#12518yes—
244C22×C79C2×C4⋊Q814yes—
245C22×C79C23×D411yescharacteristic
246C22×C79C23×D411yescharacteristic
247C22×C79C23×Q812yes—
248C22×C79C22×#4111yescharacteristic
249C22×C79C22×#4112yes—
250C2×D79#398164yesdirect product factor
251C2×D79C2×#12118yes—
252C2×D79C2×C22⋊Q8116yes—
253C2×D79C2×#12318yes—
254C2×D79C2×#125116yes—
255C2×D79C2×C4⋊Q818yes—
256C4×C79—24no—
257C4×C79—28no—
258C4×C79—24no—
259C4×C79—44no—
260C4×C79—42no—
261Dic79—24no—
262Dic79—28no—
263Dic79—24no—
264Dic79—44no—
265Dic79—42no—
266C22×#398—791nomaximal, Sylow
267C2×C79C2×#39818yescomplemented
268C2×C79C22×#12111yescharacteristic
269C2×C79C22×C22⋊Q812yes—
270C2×C79C22×#12311yescharacteristic
271C2×C79C22×#12512yes—
272C2×C79C22×C4⋊Q811yescharacteristic
273D79C2×#398116yesdirect product factor
274C23×C4⋊C4—791no—
275C23×C4⋊C4—792no—
276C23×C4⋊C4—791no—
277C2×#398—7916no—
278C2×#398—791no—
279C2×#398—792no—
280C2×#398—791no—
281C2×#398—791no—
282C2×#398—792no—
283C2×#398D7911yescharacteristic, direct product factor
284C22×#95—791no—
285C22×#95—792no—
286C79C22×#39811yescharacteristic, complemented, Sylow
287C24×C4—791no—
288C24×C4—791no—
289C24×C4—792no—
290C24×C4—792no—
291C24×C4—791no—
292C22×C4⋊C4—791no—
293C22×C4⋊C4—792no—
294C22×C4⋊C4—792no—
295C22×C4⋊C4—791no—
296C22×C4⋊C4—792no—
297C22×C4⋊C4—792no—
298C22×C4⋊C4—791no—
299C22×C4⋊C4—791no—
300C22×C4⋊C4—798no—
301C22×C4⋊C4—7916no—
302C22×C4⋊C4—798no—
303C22×C4⋊C4—1588no—
304C22×C4⋊C4—15816no—
305C22×C4⋊C4—1588no—
306C22×C4⋊C4C2×D7911yescharacteristic
307C22×C4⋊C4C2×D7912yes—
308C22×C4⋊C4C2×D7911yescharacteristic
309#398—798no—
310#398—798no—
311#398—7916no—
312#398—798no—
313#398—7916no—
314#398C2×D7918yesdirect product factor
315C2×#95—792no—
316C2×#95—791no—
317C2×#95—792no—
318C2×#95—792no—
319C2×#95—792no—
320C2×#95—798no—
321C2×#95—7916no—
322C2×#95C2×D7912yes—
323C2×#95C2×D7911yescharacteristic
324C25—791no—
325C23×C4—791no—
326C23×C4—791no—
327C23×C4—792no—
328C23×C4—798no—
329C23×C4—798no—
330C23×C4—7916no—
331C23×C4—794no—
332C23×C4—794no—
333C23×C4—798no—
334C23×C4—792no—
335C23×C4—791no—
336C23×C4—798no—
337C23×C4—794no—
338C23×C4—1588no—
339C23×C4—1588no—
340C23×C4—1584no—
341C23×C4—1588no—
342C23×C4—1588no—
343C23×C4—1584no—
344C23×C4—1588no—
345C23×C4—1584no—
346C23×C4—1588no—
347C23×C4—1588no—
348C23×C4C22×D7912yes—
349C23×C4C22×D7911yescharacteristic
350C23×C4C22×D7911yescharacteristic
351C23×C4C22×D7911yescharacteristic
352C23×C4C22×D7912yes—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—28no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—798no—
357C2×C4⋊C4—794no—
358C2×C4⋊C4—798no—
359C2×C4⋊C4—798no—
360C2×C4⋊C4—794no—
361C2×C4⋊C4—794no—
362C2×C4⋊C4—798no—
363C2×C4⋊C4—794no—
364C2×C4⋊C4—15832no—
365C2×C4⋊C4—15864no—
366C2×C4⋊C4—15832no—
367C2×C4⋊C4—1588no—
368C2×C4⋊C4—1588no—
369C2×C4⋊C4—1584no—
370C2×C4⋊C4—1584no—
371C2×C4⋊C4—1588no—
372C2×C4⋊C4—1588no—
373C2×C4⋊C4—1584no—
374C2×C4⋊C4—1584no—
375C2×C4⋊C4C22×D7914yes—
376C2×C4⋊C4C22×D7914yes—
377C2×C4⋊C4C22×D7918yes—
378#95—794no—
379#95—798no—
380#95—798no—
381#95—798no—
382#95—798no—
383#95C22×D7918yes—
384#95C22×D7914yes—
385C24—798no—
386C24—794no—
387C24—798no—
388C24—792no—
389C24—794no—
390C24—794no—
391C24C23×D7911yescharacteristic
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—31632no—
403C22×C4—31616no—
404C22×C4—798no—
405C22×C4—794no—
406C22×C4—792no—
407C22×C4—792no—
408C22×C4—794no—
409C22×C4—794no—
410C22×C4—7932no—
411C22×C4—7916no—
412C22×C4—7916no—
413C22×C4—792no—
414C22×C4—794no—
415C22×C4—15816no—
416C22×C4—15816no—
417C22×C4—1588no—
418C22×C4—15816no—
419C22×C4—15816no—
420C22×C4—1588no—
421C22×C4—15816no—
422C22×C4—15832no—
423C22×C4—15864no—
424C22×C4—15832no—
425C22×C4—1588no—
426C22×C4—15816no—
427C22×C4—15816no—
428C22×C4—1584no—
429C22×C4—1584no—
430C22×C4—1584no—
431C22×C4—1584no—
432C22×C4—1582no—
433C22×C4—1582no—
434C22×C4—1584no—
435C22×C4—1584no—
436C22×C4—1582no—
437C22×C4—1584no—
438C22×C4D4×D7914yes—
439C22×C4D4×D7914yes—
440C22×C4Q8×D7918yes—
441C22×C4C23×D7912yes—
442C22×C4C23×D7914yes—
443C22×C4C23×D7914yes—
444C22×C4C23×D7912yes—
445C22×C4C23×D7912yes—
446C4⋊C4—28no—
447C4⋊C4—216no—
448C4⋊C4—28no—
449C4⋊C4—15816no—
450C4⋊C4—15816no—
451C4⋊C4—1588no—
452C4⋊C4—1588no—
453C4⋊C4—15816no—
454C4⋊C4—15816no—
455C4⋊C4—1588no—
456C4⋊C4—1588no—
457C23—7916no—
458C23—7932no—
459C23—7916no—
460C23—7932no—
461C23—7916no—
462C23—794no—
463C23—794no—
464C23—798no—
465C23—794no—
466C23—798no—
467C23C2×D4×D7912yes—
468C23C2×D4×D7912yes—
469C23C2×Q8×D7914yes—
470C23#41×D7912yes—
471C23#41×D7914yes—
472C23C24×D7911yescharacteristic
473C2×C4—28no—
474C2×C4—216no—
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—31632no—
489C2×C4—31616no—
490C2×C4—3168no—
491C2×C4—3164no—
492C2×C4—794no—
493C2×C4—794no—
494C2×C4—798no—
495C2×C4—15832no—
496C2×C4—15864no—
497C2×C4—15832no—
498C2×C4—1584no—
499C2×C4—1584no—
500C2×C4—1582no—
501C2×C4—1584no—
502C2×C4—1584no—
503C2×C4—1588no—
504C2×C4—15816no—
505C2×C4—1588no—
506C2×C4—1582no—
507C2×C4—1584no—
508C2×C4—1582no—
509C2×C4—1584no—
510C2×C4—1584no—
511C2×C4C2×D4×D7914yes—
512C2×C4C2×D4×D7914yes—
513C2×C4C2×Q8×D7918yes—
514C22—7964no—
515C22—798no—
516C22—7916no—
517C22—798no—
518C22—7916no—
519C22—798no—
520C22#121×D7914yes—
521C22C22⋊Q8×D7918yes—
522C22#123×D7914yes—
523C22#125×D7918yes—
524C22C4⋊Q8×D7914yes—
525C22C22×D4×D7911yescharacteristic
526C22C22×D4×D7911yescharacteristic
527C22C22×Q8×D7912yes—
528C22C2×#41×D7911yescharacteristic
529C22C2×#41×D7912yes—
530C4—24no—
531C4—28no—
532C4—24no—
533C4—44no—
534C4—42no—
535C4—3164no—
536C4—3162no—
537C4—1584no—
538C4—1588no—
539C4—1584no—
540C2—7916no—
541C2#398×D7918yesdirect product factor
542C2C2×#121×D7911yescharacteristic
543C2C2×C22⋊Q8×D7912yes—
544C2C2×#123×D7911yescharacteristic
545C2C2×#125×D7912yes—
546C2C2×C4⋊Q8×D7911yescharacteristic
subgroupG/H
centerC24C23×D79
commutator, cosocleC23×C79C25
FrattiniC23C24×D79
FittingC2×#398×C79C2
socleC24×C79C24
prime radicalC24×D79C23
hypercenterC2×#398D79
hypocenterC79C22×#398
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
4. C2×#19098497C2semidirect
5. C2×#19098496C2semidirect
6. C2×#19098495C2semidirect
7. C2×#19098494C2semidirect
8. C2×#19098493C2semidirect
9. #398×D79C2direct (N × H)
12. C2×#398×C79C2semidirect
32. #19098497C22semidirect
33. #19098496C22semidirect
34. #19098495C22semidirect
35. #19098494C22semidirect
36. #19098493C22semidirect
47. #398×C79C22semidirect
250. C2×D79#398direct (N × H)
267. C2×C79C2×#398semidirect
273. D79C2×#398direct (N × H)
283. C2×#398D79direct (N × H)
286. C79C22×#398semidirect
314. #398C2×D79direct (N × H)
541. C2#398×D79direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D792
C2×#190984972
C2×#190984962
C2×#190984952
C2×#190984942
C2×#190984932
#398×D792
C2×#95×D792
C2×#398×C792
C22×#39879