Finite Group Databasebeta

← #34204000

Subgroups — #34204000

Order 20224.

882 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098379C211yesmaximal, characteristic, complemented
2C2×#19098370C211yesmaximal, characteristic, complemented
3#23765112C211yesmaximal, characteristic, complemented
4C2×#19098367C211yesmaximal, characteristic, complemented
5C2×#19098347C211yesmaximal, characteristic, complemented
6#23763464C211yesmaximal, characteristic, complemented
7C2×#578×C79C211yesmaximal, characteristic
8C2×#14579101C2211yescharacteristic
9C22×C79⋊C16C2211yescharacteristic
10C2×#14579097C2211yescharacteristic
11C2×#14579095C2211yescharacteristic
12#19098564—21no—
13#19098379—22no—
14#19098370—22no—
15#19098367—22no—
16#19098367—21no—
17#19098364—21no—
18#19098347—22no—
19#19098331—21no—
20C2×2−1+4×C79—21no—
21#578×C79—22no—
22#578×C79—22no—
23#578×C79—22no—
24#578×C79C414yescomplemented
25C2×#137×C79—22no—
26C2×#137×C79—21no—
27C2×#135×C79—21no—
28C2×#135×C79C2211yescharacteristic
29C22×Q16×C79—21no—
30C22×Q16×C79C2211yescharacteristic
31C2×#131×C79C2211yescharacteristic
32C2×#11929608—21no—
33C2×C4⋊Dic79—21no—
34#14579101—21no—
35#14579101—21no—
36C2×C79⋊C16—21no—
37C2×C79⋊C16—21no—
38C2×C79⋊C16—21no—
39C2×C79⋊C16—21no—
40#14579097—21no—
41#14579097—21no—
42#14579095—21no—
43#14579095—21no—
44#14579085—41no—
45#14579082—41no—
462−1+4×C79—24no—
472−1+4×C79—24no—
482−1+4×C79—42no—
492−1+4×C79—42no—
50C2×#41×C79—22no—
51C2×#41×C79—21no—
52C2×#41×C79—42no—
53C2×#41×C79—41no—
54C2×#41×C79D411yescharacteristic
55C22×Q8×C79—22no—
56C22×Q8×C79—21no—
57C22×Q8×C79—41no—
58#137×C79—28no—
59#137×C79—24no—
60#137×C79—28no—
61#137×C79—24no—
62#135×C79—24no—
63#135×C79—24no—
64#135×C79—21no—
65#135×C79—21no—
66#135×C79C2×C412yes—
67#135×C79D412yes—
68C2×Q16×C79—22no—
69C2×Q16×C79—24no—
70C2×Q16×C79—24no—
71C2×Q16×C79—21no—
72C2×Q16×C79—21no—
73C2×Q16×C79—21no—
74C2×Q16×C79C2×C412yes—
75C2×Q16×C79D412yes—
76C2×Q16×C79D411yescharacteristic
77C2×Q16×C79D412yes—
78C2×SD16×C79—22no—
79C2×SD16×C79—21no—
80C2×D8×C79D411yescharacteristic
81#131×C79—21no—
82#131×C79—21no—
83#131×C79C2×C412yes—
84#131×C79D412yes—
85C2×M16×C79—21no—
86C2×M16×C79D411yescharacteristic
87C22×C8×C79—21no—
88C22×C8×C79C2311yescharacteristic
89C22×Dic79—41no—
90#11929608—41no—
91#11929608—41no—
92C2×C79⋊C8—41no—
93C4⋊Dic79—41no—
94C4⋊Dic79—41no—
95C79⋊C16—41no—
96C79⋊C16—41no—
97#41×C79—21no—
98#41×C79—21no—
99#41×C79—28no—
100#41×C79—22no—
101#41×C79—22no—
102#41×C79—44no—
103#41×C79—48no—
104#41×C79—48no—
105#41×C79—44no—
106#41×C79—44no—
107#41×C79—41no—
108#41×C79—41no—
109#41×C79C22⋊C412yes—
110#41×C79SD1612yes—
111C2×Q8×C79—28no—
112C2×Q8×C79—22no—
113C2×Q8×C79—22no—
114C2×Q8×C79—21no—
115C2×Q8×C79—22no—
116C2×Q8×C79—42no—
117C2×Q8×C79—44no—
118C2×Q8×C79—44no—
119C2×Q8×C79—44no—
120C2×Q8×C79—41no—
121C2×Q8×C79—41no—
122C2×Q8×C79—41no—
123C2×Q8×C79D811yescharacteristic
124C2×D4×C79—21no—
125C2×D4×C79—21no—
126C2×D4×C79—42no—
127C2×D4×C79—41no—
128C2×D4×C79D811yescharacteristic
129C22×C4×C79—21no—
130C22×C4×C79—42no—
131C22×C4×C79—42no—
132C22×C4×C79—41no—
133C22×C4×C79—41no—
134C22×C4×C79C2×D411yescharacteristic
135Q16×C79—28no—
136Q16×C79—22no—
137Q16×C79—21no—
138Q16×C79—22no—
139Q16×C79—22no—
140Q16×C79C22⋊C412yes—
141Q16×C79C22⋊C414yes—
142SD16×C79—28no—
143SD16×C79—22no—
144SD16×C79—22no—
145D8×C79—21no—
146D8×C79C22⋊C412yes—
147M16×C79—22no—
148M16×C79—22no—
149M16×C79—21no—
150M16×C79C22⋊C412yes—
151C2×C8×C79—22no—
152C2×C8×C79—22no—
153C2×C8×C79—21no—
154C2×C8×C79C22×C411yescharacteristic
155C2×C8×C79C2×D411yescharacteristic
156C2×C8×C79C2×D411yescharacteristic
157C2×C8×C79C2×D411yescharacteristic
158C2×C8×C79C2×D411yescharacteristic
159C2×C8×C79C2×D411yescharacteristic
160C23×C79—21no—
161C23×C79—41no—
162C23×C79AGL(1,ℤ/8)11yescharacteristic
163Q8×C79—21no—
164Q8×C79—24no—
165Q8×C79—22no—
166Q8×C79—48no—
167Q8×C79—42no—
168Q8×C79—42no—
169Q8×C79—41no—
170Q8×C79—42no—
171Q8×C79#10212yes—
172D4×C79—22no—
173D4×C79—22no—
174D4×C79—22no—
175D4×C79—21no—
176D4×C79—48no—
177D4×C79—42no—
178D4×C79—42no—
179D4×C79—41no—
180D4×C79#10212yes—
181C2×C4×C79—21no—
182C2×C4×C79—22no—
183C2×C4×C79—22no—
184C2×C4×C79—48no—
185C2×C4×C79—48no—
186C2×C4×C79—42no—
187C2×C4×C79—42no—
188C2×C4×C79—41no—
189C2×C4×C79—42no—
190C2×C4×C79—42no—
191C2×C4×C79—42no—
192C2×C4×C79C2×C22⋊C411yescharacteristic
193C2×C4×C79#12011yescharacteristic
194C2×C4×C79#12011yescharacteristic
195C2×C4×C79C2×D811yescharacteristic
196C2×C4×C79C2×SD1611yescharacteristic
197C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
198C2×Dic79—81no—
199C2×Dic79—161no—
200C8×C79—22no—
201C8×C79C2×C22⋊C411yescharacteristic
202C8×C79C2×C22⋊C411yescharacteristic
203C8×C79#12011yescharacteristic
204C8×C79#12011yescharacteristic
205C79⋊C8—81no—
206C22×C79—21no—
207C22×C79—21no—
208C22×C79—22no—
209C22×C79—41no—
210C22×C79—41no—
211C22×C79—42no—
212C22×C79—81no—
213C22×C79—81no—
214C22×C79#41811yescharacteristic
215C22×C79#44711yescharacteristic
216C22×C79#45011yescharacteristic
217C4×C79—22no—
218C4×C79—44no—
219C4×C79—42no—
220C4×C79#37911yescharacteristic
221C4×C79C2×#10211yescharacteristic
222C4×C79#44711yescharacteristic
223C4×C79#45011yescharacteristic
224Dic79—161no—
225#12749—791nomaximal, Sylow
226C2×C79—21no—
227C2×C79—21no—
228C2×C79—42no—
229C2×C79—82no—
230C2×C79#178611yescharacteristic
231C2×C79#219311yescharacteristic
232C2×C79#219511yescharacteristic
233C2×#578D7911yescharacteristic
234C2×#362—791no—
235C2×#359—791no—
236#2139—791no—
237C2×#358—791no—
238C2×#350—791no—
239#1816—791no—
240C79#1274911yescharacteristic, complemented, Sylow
241C2×2−1+4—21no—
242#578—22no—
243#578—22no—
244#578—22no—
245#578Dic7914yescomplemented
246C2×#137—22no—
247C2×#137—21no—
248C2×#135—21no—
249C2×#135C2×D7911yescharacteristic
250C22×Q16—21no—
251C22×Q16C2×D7911yescharacteristic
252C2×#131C2×D7911yescharacteristic
253C2×M32—791no—
254C22×C16—791no—
255C2×#108—791no—
256C2×#107—791no—
257#417—1581no—
258#362—1582no—
259#359—1582no—
260#358—1582no—
261#358—1581no—
262#357—1581no—
263#350—1582no—
264#343—1581no—
2652−1+4—24no—
2662−1+4—24no—
2672−1+4—42no—
2682−1+4—42no—
269C2×#41—22no—
270C2×#41—21no—
271C2×#41—42no—
272C2×#41—41no—
273C2×#41C22⋊D7911yescharacteristic
274C22×Q8—22no—
275C22×Q8—21no—
276C22×Q8—41no—
277#137—24no—
278#137—28no—
279#137—28no—
280#137—24no—
281#135—24no—
282#135—24no—
283#135—21no—
284#135—21no—
285#135C2×Dic7912yes—
286#135C22⋊D7912yes—
287C2×Q16—22no—
288C2×Q16—24no—
289C2×Q16—24no—
290C2×Q16—21no—
291C2×Q16—21no—
292C2×Q16—21no—
293C2×Q16C2×Dic7912yes—
294C2×Q16C22⋊D7911yescharacteristic
295C2×Q16C22⋊D7912yes—
296C2×Q16C22⋊D7912yes—
297C2×SD16—22no—
298C2×SD16—21no—
299C2×D8C22⋊D7911yescharacteristic
300#131—21no—
301#131—21no—
302#131C2×Dic7912yes—
303#131C22⋊D7912yes—
304C2×M16—21no—
305C2×M16—1581no—
306C2×M16C22⋊D7911yescharacteristic
307C22×C8—21no—
308C22×C8C22×D7911yescharacteristic
309C2×C4⋊C4—1581no—
310M32—1581no—
311M32—1581no—
312C2×C16—1581no—
313C2×C16—1581no—
314C2×C16—1581no—
315C2×C16—1581no—
316#108—1581no—
317#108—1581no—
318#107—1581no—
319#107—1581no—
320#103—3161no—
321#102—3161no—
322#41—21no—
323#41—21no—
324#41—28no—
325#41—22no—
326#41—22no—
327#41—44no—
328#41—48no—
329#41—48no—
330#41—44no—
331#41—44no—
332#41—41no—
333#41—41no—
334#41C22⋊Dic7912yes—
335#41#1192961212yes—
336C2×Q8—28no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—42no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8#1192960911yescharacteristic
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—42no—
352C2×D4—41no—
353C2×D4#1192960911yescharacteristic
354C22×C4—21no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—3161no—
360C22×C4C2×C22⋊D7911yescharacteristic
361Q16—28no—
362Q16—22no—
363Q16—21no—
364Q16—22no—
365Q16—22no—
366Q16C22⋊Dic7912yes—
367Q16C22⋊Dic7914yes—
368SD16—28no—
369SD16—22no—
370SD16—22no—
371D8—21no—
372D8C22⋊Dic7912yes—
373M16—21no—
374M16—22no—
375M16—22no—
376M16—3161no—
377M16—3161no—
378M16C22⋊Dic7912yes—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—3161no—
383C2×C8C22×Dic7911yescharacteristic
384C2×C8C2×C22⋊D7911yescharacteristic
385C2×C8C2×C22⋊D7911yescharacteristic
386C2×C8C2×C22⋊D7911yescharacteristic
387C2×C8C2×C22⋊D7911yescharacteristic
388C2×C8C2×C22⋊D7911yescharacteristic
389C4⋊C4—3161no—
390C4⋊C4—3161no—
391C16—3161no—
392C16—3161no—
393C23—21no—
394C23—41no—
395C23#1457918711yescharacteristic
396Q8—21no—
397Q8—24no—
398Q8—22no—
399Q8—48no—
400Q8—42no—
401Q8—41no—
402Q8—42no—
403Q8—42no—
404Q8#1457908212yes—
405D4—22no—
406D4—22no—
407D4—22no—
408D4—21no—
409D4—48no—
410D4—42no—
411D4—42no—
412D4—41no—
413D4#1457908212yes—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—6321no—
426C2×C4—12641no—
427C2×C4C2×C22⋊Dic7911yescharacteristic
428C2×C4#1457913011yescharacteristic
429C2×C4#1457913011yescharacteristic
430C2×C4C2×#1192960911yescharacteristic
431C2×C4C2×#1192961211yescharacteristic
432C2×C4#1457918711yescharacteristic
433C8—22no—
434C8—6321no—
435C8C2×C22⋊Dic7911yescharacteristic
436C8C2×C22⋊Dic7911yescharacteristic
437C8#1457913011yescharacteristic
438C8#1457913011yescharacteristic
439C22—21no—
440C22—21no—
441C22—22no—
442C22—41no—
443C22—41no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22#1909856811yescharacteristic
448C22#1909869711yescharacteristic
449C22#1909871411yescharacteristic
450C4—22no—
451C4—44no—
452C4—42no—
453C4—12641no—
454C4#1909841911yescharacteristic
455C4C2×#1457908211yescharacteristic
456C4#1909869711yescharacteristic
457C4#1909871411yescharacteristic
458C2—21no—
459C2—21no—
460C2—42no—
461C2—82no—
462C2#2376333111yescharacteristic
463C2#2376536411yescharacteristic
464C2#2376537811yescharacteristic
subgroupG/H
centerC22#19098697
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
Fitting, prime radicalC2×#578×C79C2
socleC22×C79#447
cosocleC22×C8×C79C23
hypercenterC2×#578D79
hypocenterC79#12749
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. C2×#19098379C2semidirect
2. C2×#19098370C2semidirect
3. #23765112C2semidirect
4. C2×#19098367C2semidirect
5. C2×#19098347C2semidirect
6. #23763464C2semidirect
24. #578×C79C4semidirect
240. C79#12749semidirect
245. #578Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190983792
C2×#190983702
#237651122
C2×#190983672
C2×#190983472
#237634642
C2×#578×C792
#1274979