Finite Group Databasebeta

← #34292800

Subgroups — #34292800

Order 20224.

1174 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
1C2×#125×D79C211yesmaximal, characteristic
2C2×#19098585C211yesmaximal, characteristic
3C4×#14579119C211yesmaximal, characteristic
4#23764358C211yesmaximal, characteristic, complemented
5#23764357C211yesmaximal, characteristic, complemented
6#23764356C212yesmaximal, complemented
7#23764355C211yesmaximal, characteristic, complemented
8#23764354C212yesmaximal, complemented
9#23763118C212yesmaximal
10#23763118C212yesmaximal
11C79×#1989C211yesmaximal, characteristic, complemented
12C22×#11929607C2212yes—
13C2×#14579153C2211yescharacteristic
14C2×#14579152C2211yescharacteristic
15C2×#14579152C2211yescharacteristic
16C2×#14579151—21no—
17C2×#14579151—21no—
18#125×D79—24no—
19#125×D79C418yescomplemented
20C2×#14579119C2212yes—
21C2×C4⋊C4×D79—21no—
22C2×C4⋊C4×D79—21no—
23C2×C4⋊C4×D79C2211yescharacteristic
24C2×C4⋊C4×D79C2211yescharacteristic
25C2×C42×D79C2211yescharacteristic
26#19098585—24no—
27C2×C4⋊(C79⋊C8)C2211yescharacteristic
28C2×Dic79⋊C8C2212yes—
29C4⋊C4×Dic79C2212yes—
30C4×C4⋊Dic79C2211yescharacteristic
31C4×(C2×D79)⋊C4—21no—
32#19098410—21no—
33#19098408—21no—
34C42×Dic79—21no—
35#19098326C2212yes—
36#19098326C2212yes—
37#19098325C2212yes—
38#19098325C2212yes—
39#19098325C2212yes—
40#19098325C2212yes—
41C2×#125×C79C2211yescharacteristic
42C2×C4⋊C8×C79C2211yescharacteristic
43C4×C4⋊C4×C79C2211yescharacteristic
44#340×C79C2212yes—
45#340×C79C2212yes—
46C22×C4×D79—21no—
47C22×C4×D79—21no—
48C22×C4×D79D411yescharacteristic
49C22×C4×D79D411yescharacteristic
50C22×C4×D79C2311yescharacteristic
51C2×#11929607—24no—
52C2×#11929607—28no—
53C22×C79⋊C8C2312yes—
54#14579153—22no—
55#14579153C2×C414yes—
56#14579152—22no—
57#14579152—22no—
58#14579152C2×C414yes—
59#14579152C2×C414yes—
60#14579151—28no—
61#14579151—28no—
62#14579119—24no—
63#14579119Q818yes—
64C2×C4⋊Dic79—21no—
65C2×C4⋊Dic79—21no—
66C2×C4⋊Dic79C2311yescharacteristic
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2312yes—
69C2×Dic79⋊C4—21no—
70C2×Dic79⋊C4—21no—
71C2×Dic79⋊C4—21no—
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C4⋊C4×D79—28no—
76C4⋊C4×D79—28no—
77C4⋊C4×D79—22no—
78C4⋊C4×D79—22no—
79C4⋊C4×D79C2×C414yes—
80C4⋊C4×D79C2×C414yes—
81C2×(C2×D79)⋊C4—22no—
82C2×C4×Dic79—22no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—22no—
85C2×C4×Dic79D411yescharacteristic
86C2×C4×Dic79D411yescharacteristic
87C2×C4×Dic79C2311yescharacteristic
88C42×D79C2×C414yes—
89C42×D79D414yes—
90C4⋊(C79⋊C8)—22no—
91Dic79⋊C8—24no—
92C79⋊#95—21no—
93C79⋊#95—22no—
94C22×C8×C79C2312yes—
95#125×C79—22no—
96#125×C79C2×C414yes—
97C2×C4⋊C4×C79—21no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79C2311yescharacteristic
100C2×C4⋊C4×C79C2311yescharacteristic
101C2×C4⋊C4×C79C2312yes—
102C2×C42×C79—21no—
103C2×C42×C79C2311yescharacteristic
104C4⋊C8×C79—22no—
105#95×C79—21no—
106C23×D79C2×D411yescharacteristic
107C22×Dic79—22no—
108C22×Dic79—21no—
109C22×Dic79—21no—
110C22×Dic79C2×D411yescharacteristic
111C22×Dic79C2×D411yescharacteristic
112C22×Dic79C2×D411yescharacteristic
113C2×C4×D79—24no—
114C2×C4×D79—24no—
115C2×C4×D79—41no—
116C2×C4×D79—41no—
117C2×C4×D79—42no—
118C2×C4×D79—42no—
119C2×C4×D79—44no—
120C2×C4×D79—42no—
121C2×C4×D79C22⋊C414yes—
122C2×C4×D79C22⋊C414yes—
123C2×C4×D79C22×C412yes—
124C2×C4×D79C2×D412yes—
125C2×C4×D79C2×Q814yes—
126C2×C4×D79#4114yes—
127#11929607—48no—
128C2×C79⋊C8—22no—
129C2×C79⋊C8—24no—
130C4⋊Dic79—24no—
131C4⋊Dic79—24no—
132C4⋊Dic79—21no—
133C4⋊Dic79—21no—
134C4⋊Dic79—22no—
135C4⋊Dic79C22×C412yes—
136C4⋊Dic79C22×C412yes—
137C4⋊Dic79C2×Q814yes—
138Dic79⋊C4—24no—
139Dic79⋊C4—24no—
140Dic79⋊C4—24no—
141Dic79⋊C4—24no—
142Dic79⋊C4—24no—
143Dic79⋊C4—24no—
144(C2×D79)⋊C4—28no—
145C4×Dic79—28no—
146C4×Dic79—42no—
147C4×Dic79—44no—
148C4×Dic79C22⋊C414yes—
149C4×Dic79C22⋊C414yes—
150C4×Dic79C22×C412yes—
151C4×Dic79C2×D412yes—
152C22×C4×C79—22no—
153C22×C4×C79—21no—
154C22×C4×C79—21no—
155C22×C4×C79C2×D411yescharacteristic
156C22×C4×C79C2×D411yescharacteristic
157C22×C4×C79C2411yescharacteristic
158C2×C8×C79—22no—
159C2×C8×C79—24no—
160C4⋊C4×C79—24no—
161C4⋊C4×C79—24no—
162C4⋊C4×C79—21no—
163C4⋊C4×C79—21no—
164C4⋊C4×C79—22no—
165C4⋊C4×C79C22×C412yes—
166C4⋊C4×C79C22×C412yes—
167C4⋊C4×C79C2×Q814yes—
168C42×C79—42no—
169C42×C79C22×C412yes—
170C42×C79C2×D412yes—
171C23×C79C22×D411yescharacteristic
172C22×D79—21no—
173C22×D79—22no—
174C22×D79—21no—
175C22×D79C2×C22⋊C412yes—
176C22×D79C22⋊Q814yes—
177C2×C4×C79—21no—
178C2×C4×C79—21no—
179C2×C4×C79—22no—
180C2×C4×C79—22no—
181C2×C4×C79—22no—
182C2×C4×C79—24no—
183C2×C4×C79—42no—
184C2×C4×C79—41no—
185C2×C4×C79—42no—
186C2×C4×C79—41no—
187C2×C4×C79—42no—
188C2×C4×C79C2×C22⋊C412yes—
189C2×C4×C79C2×C22⋊C412yes—
190C2×C4×C79C23×C411yescharacteristic
191C2×C4×C79C22×D411yescharacteristic
192C2×C4×C79C22×Q812yes—
193C2×C4×C79C2×#4112yes—
194C2×Dic79—22no—
195C2×Dic79—21no—
196C2×Dic79—21no—
197C2×Dic79—22no—
198C2×Dic79—22no—
199C2×Dic79—24no—
200C2×Dic79—42no—
201C2×Dic79—42no—
202C2×Dic79—41no—
203C2×Dic79—42no—
204C2×Dic79—41no—
205C2×Dic79C2×C22⋊C412yes—
206C2×Dic79C2×C22⋊C412yes—
207C2×Dic79C2×C22⋊C412yes—
208C2×Dic79C22⋊Q814yes—
209C4×D79—44no—
210C4×D79—44no—
211C4×D79—48no—
212C4×D79—48no—
213C4×D79C4×Q818yes—
214C4×D79#12414yes—
215C4×D79C4⋊Q814yes—
216C8×C79—42no—
217C79⋊C8—42no—
218C22×C79C22×C22⋊C411yescharacteristic
219C22×C79C2×C22⋊Q812yes—
220C22×C79#57511yescharacteristic
221C22×C79#57512yes—
222C22×C79#57511yescharacteristic
223C2×D79—24no—
224C2×D79—24no—
225C2×D79—22no—
226C2×D79—22no—
227C2×D79#39114yes—
228C4×C79—22no—
229C4×C79—22no—
230C4×C79—44no—
231C4×C79—42no—
232C4×C79—42no—
233C4×C79C2×C4×Q812yes—
234C4×C79C2×#12411yescharacteristic
235C4×C79C2×C4⋊Q811yescharacteristic
236Dic79—22no—
237Dic79—22no—
238Dic79—44no—
239Dic79—42no—
240Dic79—42no—
241Dic79#39114yes—
242#21314—791nomaximal, Sylow
243C2×C79C2×#39111yescharacteristic
244C2×C79#289611yescharacteristic
245C2×C79#289611yescharacteristic
246C2×C79#308012yes—
247C2×C79#308012yes—
248D79—24no—
249C22×#125—791no—
250C2×#423—791no—
251C4×#117—791no—
252#1989—791no—
253#1989—791no—
254#1989—792no—
255#1989—792no—
256#1989—791no—
257#1989D7911yescharacteristic, complemented
258#1736—792no—
259#1736—792no—
260C79#2131411yescharacteristic, complemented, Sylow
261C22×M16—792no—
262C2×#125—798no—
263C2×#125—791no—
264C2×#125—791no—
265C2×#125—791no—
266C2×#125—1581no—
267C2×#125—1581no—
268C2×#125—1584no—
269C2×#125C2×D7911yescharacteristic
270C2×#117—792no—
271C22×C4⋊C4—791no—
272C22×C4⋊C4—791no—
273C22×C4⋊C4—1581no—
274C22×C4⋊C4—1581no—
275C22×C42—791no—
276#423—1584no—
277C2×C4⋊C8—792no—
278C2×C4⋊C8—791no—
279C2×C4⋊C8C2×D7911yescharacteristic
280C4×C4⋊C4—792no—
281C4×C4⋊C4—791no—
282C4×C4⋊C4C2×D7911yescharacteristic
283C4×C22⋊C4—1581no—
284C4×C22⋊C4—1581no—
285#376—1581no—
286C43—1581no—
287#340—792no—
288#340—792no—
289#340—792no—
290#340—792no—
291#340—792no—
292#340—792no—
293#340C2×D7912yes—
294#340C2×D7912yes—
295C23×C4—791no—
296C23×C4—791no—
297C23×C4—791no—
298C23×C4—1581no—
299C23×C4—1581no—
300C2×M16—1584no—
301C2×M16—1588no—
302C22×C8—792no—
303C22×C8C22×D7912yes—
304#125—22no—
305#125—794no—
306#125—794no—
307#125—794no—
308#125—1588no—
309#125—1588no—
310#125—1582no—
311#125—1582no—
312#125—1582no—
313#125C4×D7914yes—
314#117—798no—
315#117—1584no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—791no—
319C2×C4⋊C4—791no—
320C2×C4⋊C4—792no—
321C2×C4⋊C4—794no—
322C2×C4⋊C4—794no—
323C2×C4⋊C4—1582no—
324C2×C4⋊C4—1582no—
325C2×C4⋊C4—1588no—
326C2×C4⋊C4—1588no—
327C2×C4⋊C4—1581no—
328C2×C4⋊C4—1581no—
329C2×C4⋊C4—1581no—
330C2×C4⋊C4—1581no—
331C2×C4⋊C4—1581no—
332C2×C4⋊C4—1581no—
333C2×C4⋊C4—1581no—
334C2×C4⋊C4—1581no—
335C2×C4⋊C4C22×D7911yescharacteristic
336C2×C4⋊C4C22×D7911yescharacteristic
337C2×C4⋊C4C22×D7912yes—
338C2×C22⋊C4—1582no—
339C2×C42—21no—
340C2×C42—791no—
341C2×C42—791no—
342C2×C42—791no—
343C2×C42—794no—
344C2×C42—794no—
345C2×C42—1582no—
346C2×C42—1581no—
347C2×C42—1582no—
348C2×C42C22×D7911yescharacteristic
349C4⋊C8—22no—
350C4⋊C8—1584no—
351C4⋊C8—1582no—
352#95—21no—
353#95—1582no—
354#95—1581no—
355C24—791no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—3161no—
360C22×C4—3162no—
361C22×C4—3161no—
362C22×C4—3162no—
363C22×C4—3164no—
364C22×C4—3162no—
365C22×C4—794no—
366C22×C4—794no—
367C22×C4—791no—
368C22×C4—794no—
369C22×C4—794no—
370C22×C4—792no—
371C22×C4—792no—
372C22×C4—791no—
373C22×C4—791no—
374C22×C4—1584no—
375C22×C4—1584no—
376C22×C4—1582no—
377C22×C4—1581no—
378C22×C4—1581no—
379C22×C4D4×D7911yescharacteristic
380C22×C4D4×D7911yescharacteristic
381C22×C4C23×D7911yescharacteristic
382M16—3168no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—1584no—
386C2×C8—1582no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—794no—
393C4⋊C4—792no—
394C4⋊C4—792no—
395C4⋊C4—1584no—
396C4⋊C4—1584no—
397C4⋊C4—1584no—
398C4⋊C4—1584no—
399C4⋊C4—1584no—
400C4⋊C4—1584no—
401C4⋊C4—1582no—
402C4⋊C4—1584no—
403C4⋊C4—1584no—
404C4⋊C4—1581no—
405C4⋊C4—1581no—
406C4⋊C4C2×C4×D7912yes—
407C4⋊C4C2×C4×D7912yes—
408C4⋊C4Q8×D7914yes—
409C22⋊C4—1588no—
410C42—42no—
411C42—3162no—
412C42—3164no—
413C42—794no—
414C42—794no—
415C42—792no—
416C42—792no—
417C42—1588no—
418C42C2×C4×D7912yes—
419C42D4×D7912yes—
420C23—792no—
421C23—794no—
422C23—1581no—
423C23—1582no—
424C23—1581no—
425C23C2×D4×D7911yescharacteristic
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—3164no—
438C2×C4—3164no—
439C2×C4—3168no—
440C2×C4—3168no—
441C2×C4—3161no—
442C2×C4—3162no—
443C2×C4—3162no—
444C2×C4—3162no—
445C2×C4—3161no—
446C2×C4—792no—
447C2×C4—794no—
448C2×C4—798no—
449C2×C4—794no—
450C2×C4—794no—
451C2×C4—792no—
452C2×C4—792no—
453C2×C4—1582no—
454C2×C4—1582no—
455C2×C4—1584no—
456C2×C4—1581no—
457C2×C4—1582no—
458C2×C4—1581no—
459C2×C4C22⋊C4×D7912yes—
460C2×C4C22⋊C4×D7912yes—
461C2×C4C22×C4×D7911yescharacteristic
462C2×C4C2×D4×D7911yescharacteristic
463C2×C4C2×Q8×D7912yes—
464C2×C4#41×D7912yes—
465C8—42no—
466C8—3162no—
467C22—794no—
468C22—1582no—
469C22—1582no—
470C22—1584no—
471C22—1584no—
472C22C2×C22⋊C4×D7911yescharacteristic
473C22C22⋊Q8×D7912yes—
474C22#1909935911yescharacteristic
475C22#1909936012yes—
476C22#1909936211yescharacteristic
477C4—22no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—44no—
482C4—3162no—
483C4—3162no—
484C4—3164no—
485C4—794no—
486C4—1582no—
487C4—1582no—
488C4C4×Q8×D7912yes—
489C4#124×D7911yescharacteristic
490C4C4⋊Q8×D7911yescharacteristic
491C2—1584no—
492C2#391×D7911yescharacteristic
493C2#2377191511yescharacteristic
494C2#2377191611yescharacteristic
495C2#2377321712yes—
496C2#2377321912yes—
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1989C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
cosocleC22×C4×C79C24
hypercenter#1989D79
hypocenterC79#21314
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. #23764358C2semidirect
5. #23764357C2semidirect
6. #23764356C2semidirect
7. #23764355C2semidirect
8. #23764354C2semidirect
11. C79×#1989C2semidirect
19. #125×D79C4semidirect
257. #1989D79semidirect
260. C79#21314semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D792
C2×#190985852
C4×#145791192
#237643582
#237643572
#237643562
#237643552
#237643542
#237631182
C79×#19892
#2131479