Finite Group Databasebeta

← #34191713

Subgroups — #34191713

Order 20224.

556 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23768349C211yesmaximal, characteristic
2#23764360C211yesmaximal, characteristic, complemented
3#23763397C211yesmaximal, characteristic
4C2×#19098310C211yesmaximal, characteristic, complemented
5C2×#19098290C211yesmaximal, characteristic
6#23761241C211yesmaximal, characteristic, complemented
7C79×#1948C211yesmaximal, characteristic, complemented
8C2×#14579161C2211yescharacteristic
9C2×#14579141—21no—
10C2×#14579141—21no—
11C2×C4×D316C2211yescharacteristic
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×#14579081—21no—
14C2×C4×C79⋊C8C2211yescharacteristic
15#19098500—22no—
16#19098465—21no—
17#19098433—21no—
18#19098325—22no—
19#19098325—21no—
20#19098310—22no—
21#19098290—22no—
22C2×#106×C79—21no—
23C2×C4⋊C8×C79C2211yescharacteristic
24#384×C79C2211yescharacteristic
25C4×C4⋊C4×C79C2211yescharacteristic
26#340×C79—21no—
27C22×Dic158—21no—
28C22×D316D411yescharacteristic
29C22×C4×D79—21no—
30C22×C79⋊C8—21no—
31C22×C79⋊C8—21no—
32#14579161—21no—
33#14579161—21no—
34#14579161—21no—
35#14579161—21no—
36#14579141—42no—
37#14579141—42no—
38#14579141—42no—
39#14579141—42no—
40C4×D316—21no—
41C4×D316—21no—
42C4×D316—21no—
43C4×D316—21no—
44C2×C4⋊Dic79—22no—
45C2×C4⋊Dic79—22no—
46C2×C4⋊Dic79—21no—
47C2×C4⋊Dic79D411yescharacteristic
48C2×C4⋊Dic79D412yes—
49C2×(C2×D79)⋊C4—22no—
50C2×(C2×D79)⋊C4—22no—
51C2×(C2×D79)⋊C4—21no—
52C4⋊(C79⋊C8)—21no—
53C4⋊(C79⋊C8)—21no—
54#14579081—42no—
55C4×C79⋊C8—21no—
56C4×C79⋊C8—21no—
57C22×C8×C79—21no—
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79D411yescharacteristic
62C2×C4⋊C4×C79D411yescharacteristic
63C2×C42×C79—21no—
64C2×C42×C79C2311yescharacteristic
65#106×C79—42no—
66C4⋊C8×C79—21no—
67C4⋊C8×C79D412yes—
68#95×C79—21no—
69C23×D79—21no—
70C2×Dic158—41no—
71C2×Dic158—41no—
72C2×Dic158—41no—
73C2×Dic158—41no—
74C2×Dic158—41no—
75C2×Dic158—41no—
76C2×D316—21no—
77C2×D316—21no—
78C2×D316—21no—
79C2×D316—21no—
80C2×D316—21no—
81C2×D316SD1612yes—
82C22×Dic79—22no—
83C22×Dic79—21no—
84C2×C4×D79—41no—
85C2×C4×D79—41no—
86C2×C4×D79—41no—
87C2×C4×D79—41no—
88C2×C4×D79—41no—
89C2×C4×D79—41no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94C2×C79⋊C8—41no—
95C2×C79⋊C8—41no—
96C4⋊Dic79—21no—
97C4⋊Dic79—22no—
98C4⋊Dic79—21no—
99C4⋊Dic79—22no—
100C4⋊Dic79—44no—
101C4⋊Dic79—44no—
102C4⋊Dic79—41no—
103C4⋊Dic79—41no—
104(C2×D79)⋊C4—44no—
105(C2×D79)⋊C4—44no—
106(C2×D79)⋊C4—41no—
107(C2×D79)⋊C4—41no—
108C22×C4×C79—21no—
109C22×C4×C79—21no—
110C22×C4×C79—41no—
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C2×C8×C79—22no—
115C2×C8×C79—41no—
116C2×C8×C79—41no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—42no—
120C4⋊C4×C79—41no—
121C4⋊C4×C79—41no—
122C4⋊C4×C79—41no—
123C4⋊C4×C79—41no—
124C4⋊C4×C79D812yes—
125C4⋊C4×C79SD1612yes—
126C42×C79—41no—
127C42×C79—41no—
128C42×C79C2×D411yescharacteristic
129C42×C79#4111yescharacteristic
130C42×C79#4111yescharacteristic
131C42×C79#4111yescharacteristic
132C23×C79#12011yescharacteristic
133C22×D79—22no—
134C22×D79—41no—
135C22×D79—41no—
136C22×D79—41no—
137C22×D79—41no—
138C22×D79—41no—
139C22×D79—41no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—22no—
143C2×C4×C79—22no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79—81no—
149C2×C4×C79—81no—
150C2×C4×C79—81no—
151C2×C4×C79#12111yescharacteristic
152C2×C4×C79#12111yescharacteristic
153C2×C4×C79#12111yescharacteristic
154C2×C4×C79#12311yescharacteristic
155C2×C4×C79#12311yescharacteristic
156C2×C4×C79#12411yescharacteristic
157C2×C4×C79C2×D811yescharacteristic
158C2×C4×C79C2×SD1611yescharacteristic
159C2×C4×C79C2×SD1611yescharacteristic
160C2×C4×C79#13511yescharacteristic
161C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C79#13711yescharacteristic
164C2×C4×C79#13711yescharacteristic
165C2×Dic79—41no—
166C2×Dic79—42no—
167C2×Dic79—42no—
168C2×Dic79—41no—
169C2×Dic79—42no—
170C2×Dic79—41no—
171D316—22no—
172D316—22no—
173D316—41no—
174D316—41no—
175C4×D79—81no—
176C4×D79—81no—
177C4×D79—81no—
178C4×D79—81no—
179Dic158—81no—
180Dic158—81no—
181Dic158—81no—
182Dic158—81no—
183C8×C79—42no—
184C79⋊C8—42no—
185C79⋊C8—81no—
186C22×C79#39411yescharacteristic
187C22×C79#44711yescharacteristic
188C22×C79#44811yescharacteristic
189C22×C79#45011yescharacteristic
190C22×C79#45211yescharacteristic
191C22×C79#45411yescharacteristic
192C22×C79#45611yescharacteristic
193C2×D79—42no—
194C2×D79—42no—
195C2×D79—42no—
196C2×D79—81no—
197C2×D79—81no—
198C2×D79—81no—
199C2×D79—81no—
200C4×C79—21no—
201C4×C79—21no—
202C4×C79—42no—
203C4×C79—42no—
204C4×C79—161no—
205C4×C79#45911yescharacteristic
206C4×C79#46011yescharacteristic
207C4×C79#46511yescharacteristic
208C4×C79#47011yescharacteristic
209C4×C79#48311yescharacteristic
210C4×C79#48511yescharacteristic
211C4×C79#48611yescharacteristic
212C4×C79#49011yescharacteristic
213Dic79—82no—
214Dic79—81no—
215#10420—791nomaximal, Sylow
216C2×C79#162711yescharacteristic
217C2×C79#163511yescharacteristic
218C2×C79#164711yescharacteristic
219C2×C79#165111yescharacteristic
220C2×C79#203511yescharacteristic
221C2×C79#203811yescharacteristic
222C2×C79#204111yescharacteristic
223D79—82no—
224#2542—791no—
225#1990—791no—
226#1948D7911yescharacteristic, complemented
227#1802—791no—
228C2×#332—791no—
229C2×D4⋊C8—791no—
230#1279—791no—
231C79#1042011yescharacteristic, complemented, Sylow
232C2×C4⋊Q8—791no—
233C2×C22⋊Q8—1581no—
234C2×C22⋊Q8—1581no—
235C2×C4×D4—791no—
236C2×#106—21no—
237C2×C4⋊C8—791no—
238C2×C4⋊C8C2×D7911yescharacteristic
239C2×#102—1581no—
240C2×C4×C8—791no—
241#399—1582no—
242#390—1581no—
243#384—1581no—
244#384C2×D7911yescharacteristic
245C4×C4⋊C4C2×D7911yescharacteristic
246#340—21no—
247#340—1581no—
248#340—1582no—
249#332—1582no—
250D4⋊C8—1582no—
251C22×Q8—1581no—
252C22×D4—791no—
253C23×C4—1581no—
254C22×C8—21no—
255C22×C8—1581no—
256C22×C8—1581no—
257C4⋊Q8—1581no—
258C4⋊Q8—1581no—
259C4⋊Q8—1581no—
260C4⋊Q8—1581no—
261C22⋊Q8—3162no—
262C22⋊Q8—3162no—
263C22⋊Q8—3162no—
264C22⋊Q8—3162no—
265C4×D4—1581no—
266C4×D4—1581no—
267C4×D4—1581no—
268C4×D4—1581no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—791no—
273C2×C4⋊C4—792no—
274C2×C4⋊C4—1582no—
275C2×C4⋊C4—1582no—
276C2×C4⋊C4—1581no—
277C2×C4⋊C4C22⋊D7911yescharacteristic
278C2×C4⋊C4C22⋊D7911yescharacteristic
279C2×C22⋊C4—1582no—
280C2×C22⋊C4—1582no—
281C2×C22⋊C4—1581no—
282C2×C42—21no—
283C2×C42C22×D7911yescharacteristic
284#106—42no—
285C4⋊C8—21no—
286C4⋊C8—1581no—
287C4⋊C8—1581no—
288C4⋊C8D31612yes—
289#102—3162no—
290C4×C8—1581no—
291C4×C8—1581no—
292#95—21no—
293C24—1581no—
294C2×Q8—3161no—
295C2×Q8—3161no—
296C2×Q8—3161no—
297C2×Q8—3161no—
298C2×Q8—3161no—
299C2×Q8—3161no—
300C2×D4—792no—
301C2×D4—1581no—
302C2×D4—1581no—
303C2×D4—1581no—
304C2×D4—1581no—
305C2×D4—1581no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—3161no—
310C22×C4—3161no—
311C22×C4—3161no—
312C22×C4—3161no—
313C22×C4—3161no—
314C22×C4—3161no—
315C22×C4—1582no—
316C22×C4—1581no—
317C22×C4D4×D7911yescharacteristic
318C22×C4D4×D7911yescharacteristic
319C22×C4C2×C22⋊D7911yescharacteristic
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—3161no—
324C2×C8—3161no—
325C2×C8—3161no—
326C2×C8—3161no—
327C2×C8—3161no—
328C2×C8—1582no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—3164no—
337C4⋊C4—3164no—
338C4⋊C4—3161no—
339C4⋊C4—3161no—
340C4⋊C4—1581no—
341C4⋊C4—1582no—
342C4⋊C4—1581no—
343C4⋊C4—1582no—
344C4⋊C4#1192960912yes—
345C4⋊C4#1192961112yes—
346C22⋊C4—3164no—
347C22⋊C4—3164no—
348C22⋊C4—3161no—
349C22⋊C4—3161no—
350C42—41no—
351C42—41no—
352C42C2×D31611yescharacteristic
353C42#1192962411yescharacteristic
354C42#1192962511yescharacteristic
355C42#1192962511yescharacteristic
356C23—3161no—
357C23—3161no—
358C23—3161no—
359C23—3161no—
360C23—3161no—
361C23—3161no—
362C23—1582no—
363C23#1457912911yescharacteristic
364Q8—6321no—
365Q8—6321no—
366Q8—6321no—
367Q8—6321no—
368D4—3161no—
369D4—3161no—
370D4—1582no—
371D4—1582no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—3162no—
384C2×C4—3161no—
385C2×C4—3162no—
386C2×C4—3161no—
387C2×C4—3161no—
388C2×C4—3162no—
389C2×C4—6321no—
390C2×C4—6321no—
391C2×C4—6321no—
392C2×C4—6321no—
393C2×C4#1457913511yescharacteristic
394C2×C4#1457913611yescharacteristic
395C2×C4#1457913611yescharacteristic
396C2×C4#1457914611yescharacteristic
397C2×C4#1457914611yescharacteristic
398C2×C4#1457915011yescharacteristic
399C2×C4C2×#1192960911yescharacteristic
400C2×C4SD16×D7911yescharacteristic
401C2×C4C2×#1192961111yescharacteristic
402C2×C4#1457918211yescharacteristic
403C2×C4#1457918611yescharacteristic
404C2×C4#1457918611yescharacteristic
405C2×C4#1457919011yescharacteristic
406C2×C4#1457919011yescharacteristic
407C8—42no—
408C8—3162no—
409C8—6321no—
410C22—3162no—
411C22—3162no—
412C22—3162no—
413C22—6321no—
414C22—6321no—
415C22—6321no—
416C22—6321no—
417C22#1909848011yescharacteristic
418C22#1909869611yescharacteristic
419C22#1909870011yescharacteristic
420C22#1909871311yescharacteristic
421C22#1909872211yescharacteristic
422C22#1909873411yescharacteristic
423C22#1909874411yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—42no—
428C4—161no—
429C4—6321no—
430C4—6322no—
431C4#1909875811yescharacteristic
432C4#1909876311yescharacteristic
433C4#1909879311yescharacteristic
434C4#1909882211yescharacteristic
435C4#1909889511yescharacteristic
436C4#1909890711yescharacteristic
437C4#1909891111yescharacteristic
438C4#1909892911yescharacteristic
439C2—6322no—
440C2#2376250111yescharacteristic
441C2#2376254911yescharacteristic
442C2#2376263311yescharacteristic
443C2#2376266211yescharacteristic
444C2#2376459811yescharacteristic
445C2#2376461911yescharacteristic
446C2#2376463411yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1948C2
socleC23×C79#120
prime radicalC22×D316D4
hypercenter#1948D79
hypocenterC79#10420
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. #23764360C2semidirect
4. C2×#19098310C2semidirect
6. #23761241C2semidirect
7. C79×#1948C2semidirect
226. #1948D79semidirect
231. C79#10420semidirect

Maximal subgroups

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

subgroupindex
#237683492
#237643602
#237633972
C2×#190983102
C2×#190982902
#237612412
C79×#19482
#1042079