Finite Group Databasebeta

← #34184746

Subgroups — #34184746

Order 20224.

478 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772456C211yesmaximal, characteristic
2#23761904C211yesmaximal, characteristic
3C2×#19098300C211yesmaximal, characteristic, complemented
4#23761764C211yesmaximal, characteristic, complemented
5#23761288C211yesmaximal, characteristic, complemented
6#23761242C211yesmaximal, characteristic, complemented
7C79×#3190C211yesmaximal, characteristic
8#19098635—21no—
9#19098635—21no—
10D4×C79⋊C8—21no—
11D4×C79⋊C8—21no—
12#19098624—21no—
13#19098530—21no—
14#19098527C2211yescharacteristic
15C2×C22⋊(C79⋊C8)C2211yescharacteristic
16C2×C22⋊(C79⋊C8)C2211yescharacteristic
17C2×C79⋊#95C2211yescharacteristic
18#19098300—21no—
19#19098300—21no—
20#19098300—21no—
21#19098297—21no—
22#19098289—21no—
23#19098289—21no—
24#525×C79C2211yescharacteristic
25C2×C22⋊Q8×C79C2211yescharacteristic
26#485×C79—21no—
27#483×C79C411yescharacteristic, complemented
28#480×C79C411yescharacteristic, complemented
29#467×C79—21no—
30#465×C79—21no—
31#452×C79—21no—
32#451×C79C411yescharacteristic, complemented
33#448×C79C411yescharacteristic, complemented
34C2×C22⋊C8×C79C2211yescharacteristic
35C23×Dic79—21no—
36C2×#11929608—21no—
37C2×#11929608—21no—
38C22×C79⋊C8—21no—
39C22×C79⋊C8—21no—
40C22×C79⋊C8—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C22⋊(C79⋊C8)—21no—
44C22⋊(C79⋊C8)—21no—
45C22⋊(C79⋊C8)—21no—
46C22⋊(C79⋊C8)—21no—
47C22⋊(C79⋊C8)—21no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)D411yescharacteristic
50C22⋊(C79⋊C8)D411yescharacteristic
51C22⋊(C79⋊C8)D411yescharacteristic
52C22⋊(C79⋊C8)D411yescharacteristic
53#14579072—21no—
54C4×C79⋊C8—21no—
55C79⋊#95—21no—
56C79⋊#95—21no—
57C2×#41×C79D411yescharacteristic
58C22×Q8×C79C2×C411yescharacteristic
59C23×C4×C79C2311yescharacteristic
60C2×Q16×C79—21no—
61C2×SD16×C79—21no—
62C22×C8×C79—21no—
63#123×C79—21no—
64C22⋊Q8×C79—21no—
65C22⋊Q8×C79—21no—
66C22⋊Q8×C79—21no—
67C22⋊Q8×C79—21no—
68C22⋊Q8×C79C2×C411yescharacteristic
69C22⋊Q8×C79D411yescharacteristic
70#121×C79C2×C411yescharacteristic
71#121×C79D411yescharacteristic
72#120×C79—21no—
73C4×D4×C79—21no—
74C4×D4×C79—21no—
75#117×C79D411yescharacteristic
76C2×C4⋊C4×C79—21no—
77C2×C4⋊C4×C79C2×C411yescharacteristic
78C2×C22⋊C4×C79—21no—
79#107×C79—21no—
80#106×C79—21no—
81#103×C79—21no—
82#103×C79—21no—
83#103×C79—21no—
84#102×C79—21no—
85C22⋊C8×C79—21no—
86C22⋊C8×C79C2×C411yescharacteristic
87C22⋊C8×C79C2×C411yescharacteristic
88C22×Dic79—22no—
89C22×Dic79—41no—
90C22×Dic79—41no—
91C22×Dic79—81no—
92C22×Dic79—81no—
93#11929608—81no—
94#11929608—81no—
95C2×C79⋊C8—21no—
96C2×C79⋊C8—21no—
97C2×C79⋊C8—21no—
98C2×C79⋊C8—21no—
99C2×C79⋊C8—41no—
100C2×C79⋊C8—81no—
101C2×C79⋊C8—81no—
102C2×C79⋊C8—81no—
103C24×C79#4111yescharacteristic
104#41×C79—81no—
105C2×Q8×C79—21no—
106C2×Q8×C79—21no—
107C2×Q8×C79—81no—
108C2×Q8×C79C22⋊C411yescharacteristic
109C2×D4×C79—21no—
110C2×D4×C79—21no—
111C2×D4×C79—21no—
112C2×D4×C79C22⋊C411yescharacteristic
113C22×C4×C79—21no—
114C22×C4×C79—21no—
115C22×C4×C79—21no—
116C22×C4×C79—81no—
117C22×C4×C79C22×C411yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79C2×D411yescharacteristic
120C22×C4×C79C2×D411yescharacteristic
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79#4111yescharacteristic
123Q16×C79—81no—
124SD16×C79—81no—
125C2×C8×C79—21no—
126C2×C8×C79—21no—
127C2×C8×C79—81no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79—21no—
131C4⋊C4×C79—21no—
132C4⋊C4×C79—41no—
133C4⋊C4×C79—41no—
134C4⋊C4×C79C22⋊C411yescharacteristic
135C4⋊C4×C79C22⋊C411yescharacteristic
136C22⋊C4×C79—21no—
137C22⋊C4×C79—21no—
138C22⋊C4×C79—21no—
139C22⋊C4×C79—21no—
140C22⋊C4×C79—21no—
141C22⋊C4×C79—41no—
142C22⋊C4×C79—41no—
143C42×C79—21no—
144C23×C79—21no—
145C23×C79—21no—
146C23×C79—21no—
147C23×C79—21no—
148C23×C79—21no—
149C23×C79—41no—
150C23×C79C4×D411yescharacteristic
151C23×C79#12111yescharacteristic
152C23×C79#12311yescharacteristic
153Q8×C79—41no—
154Q8×C79—41no—
155Q8×C79—41no—
156Q8×C79—81no—
157D4×C79—41no—
158D4×C79—41no—
159D4×C79—41no—
160D4×C79—41no—
161D4×C79—41no—
162D4×C79—81no—
163C2×C4×C79—21no—
164C2×C4×C79—21no—
165C2×C4×C79—21no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—41no—
171C2×C4×C79—41no—
172C2×C4×C79—41no—
173C2×C4×C79—41no—
174C2×C4×C79—41no—
175C2×C4×C79—41no—
176C2×C4×C79—41no—
177C2×C4×C79—41no—
178C2×C4×C79—81no—
179C2×C4×C79C2×C22⋊C411yescharacteristic
180C2×C4×C79C4×D411yescharacteristic
181C2×C4×C79#12011yescharacteristic
182C2×C4×C79#12111yescharacteristic
183C2×Dic79—42no—
184C2×Dic79—81no—
185C2×Dic79—82no—
186C2×Dic79—81no—
187C2×Dic79—82no—
188C8×C79—41no—
189C8×C79—41no—
190C79⋊C8—41no—
191C79⋊C8—41no—
192C79⋊C8—41no—
193C79⋊C8—41no—
194C79⋊C8—81no—
195C22×C79—21no—
196C22×C79—21no—
197C22×C79—21no—
198C22×C79—21no—
199C22×C79—21no—
200C22×C79—21no—
201C22×C79—21no—
202C22×C79—41no—
203C22×C79—41no—
204C22×C79—41no—
205C22×C79—41no—
206C22×C79—41no—
207C22×C79—41no—
208C22×C79—41no—
209C22×C79#38611yescharacteristic
210C22×C79#44311yescharacteristic
211C22×C79#44411yescharacteristic
212C22×C79#47111yescharacteristic
213C22×C79#47311yescharacteristic
214C4×C79—21no—
215C4×C79—21no—
216C4×C79—21no—
217C4×C79—21no—
218C4×C79—41no—
219C4×C79—41no—
220C4×C79—41no—
221C4×C79—81no—
222C4×C79—81no—
223Dic79—82no—
224#9263—791nomaximal, Sylow
225C2×C79—21no—
226C2×C79—21no—
227C2×C79—21no—
228C2×C79—21no—
229C2×C79—41no—
230C2×C79—81no—
231C2×C79#187611yescharacteristic
232C2×C79#190811yescharacteristic
233C2×C79#191311yescharacteristic
234#3190D7911yescharacteristic
235#2977—791no—
236#1504—791no—
237C2×#328—791no—
238#1472—791no—
239#1298—791no—
240#1280—791no—
241C79#926311yescharacteristic, complemented, Sylow
242#525C2×D7911yescharacteristic
243C2×C22⋊Q8C2×D7911yescharacteristic
244#485—21no—
245#483Dic7911yescharacteristic, complemented
246#480Dic7911yescharacteristic, complemented
247#467—21no—
248#465—21no—
249#452—21no—
250#451Dic7911yescharacteristic, complemented
251#448Dic7911yescharacteristic, complemented
252#435—1581no—
253#435—1581no—
254C8×D4—1581no—
255C8×D4—1581no—
256#433—1581no—
257#408—1581no—
258#407—791no—
259C2×C22⋊C8—791no—
260C2×C22⋊C8—791no—
261C2×C22⋊C8C2×D7911yescharacteristic
262C2×#95—791no—
263#328—1581no—
264#328—1581no—
265#328—1581no—
266#327—1581no—
267#324—1581no—
268#324—1581no—
269C2×#41C22⋊D7911yescharacteristic
270C22×Q8C2×Dic7911yescharacteristic
271C23×C4—1581no—
272C23×C4C22×D7911yescharacteristic
273C2×Q16—21no—
274C2×SD16—21no—
275C2×M16—1581no—
276C2×M16—1581no—
277C22×C8—21no—
278C22×C8—1581no—
279C22×C8—1581no—
280C22×C8—1581no—
281#123—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286C22⋊Q8C2×Dic7911yescharacteristic
287C22⋊Q8C22⋊D7911yescharacteristic
288#121C2×Dic7911yescharacteristic
289#121C22⋊D7911yescharacteristic
290#120—21no—
291C4×D4—21no—
292C4×D4—21no—
293#117C22⋊D7911yescharacteristic
294C2×C4⋊C4—21no—
295C2×C4⋊C4C2×Dic7911yescharacteristic
296C2×C22⋊C4—21no—
297#107—21no—
298#106—21no—
299C4⋊C8—1581no—
300C4⋊C8—1581no—
301#103—21no—
302#103—21no—
303#103—21no—
304#102—21no—
305C22⋊C8—21no—
306C22⋊C8—791no—
307C22⋊C8—791no—
308C22⋊C8—791no—
309C22⋊C8—791no—
310C22⋊C8—1581no—
311C22⋊C8—1581no—
312C22⋊C8—1581no—
313C22⋊C8—1581no—
314C22⋊C8—1581no—
315C22⋊C8—1581no—
316C22⋊C8C2×Dic7911yescharacteristic
317C22⋊C8C2×Dic7911yescharacteristic
318#97—1581no—
319C4×C8—1581no—
320#95—1581no—
321#95—1581no—
322C24#1192962311yescharacteristic
323#41—81no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—81no—
327C2×Q8C22⋊Dic7911yescharacteristic
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4C22⋊Dic7911yescharacteristic
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—81no—
336C22×C4—3161no—
337C22×C4—3161no—
338C22×C4—6321no—
339C22×C4—6321no—
340C22×C4—1582no—
341C22×C4C22×Dic7911yescharacteristic
342C22×C4D4×D7911yescharacteristic
343C22×C4D4×D7911yescharacteristic
344C22×C4C2×C22⋊D7911yescharacteristic
345C22×C4C2×C22⋊D7911yescharacteristic
346C22×C4#1192962311yescharacteristic
347Q16—81no—
348SD16—81no—
349M16—6321no—
350M16—6321no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—81no—
354C2×C8—3161no—
355C2×C8—6321no—
356C2×C8—6321no—
357C2×C8—6321no—
358C2×C8—1581no—
359C2×C8—1581no—
360C2×C8—1581no—
361C2×C8—1581no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4C22⋊Dic7911yescharacteristic
369C4⋊C4C22⋊Dic7911yescharacteristic
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—21no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C42—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—41no—
384C23D4×Dic7911yescharacteristic
385C23#1457913311yescharacteristic
386C23#1457914211yescharacteristic
387Q8—41no—
388Q8—41no—
389Q8—41no—
390Q8—81no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—81no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—81no—
413C2×C4—3162no—
414C2×C4—6321no—
415C2×C4—6322no—
416C2×C4—6321no—
417C2×C4—6322no—
418C2×C4C2×C22⋊Dic7911yescharacteristic
419C2×C4D4×Dic7911yescharacteristic
420C2×C4#1457912911yescharacteristic
421C2×C4#1457913311yescharacteristic
422C8—41no—
423C8—41no—
424C8—3161no—
425C8—3161no—
426C8—3161no—
427C8—3161no—
428C8—6321no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22#1909844311yescharacteristic
444C22#1909868011yescharacteristic
445C22#1909868511yescharacteristic
446C22#1909882811yescharacteristic
447C22#1909883611yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—81no—
456C4—81no—
457C4—6322no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—81no—
464C2#2376372911yescharacteristic
465C2#2376391711yescharacteristic
466C2#2376394311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#3190C2
socleC22×C79#386
prime radical#525×C79C22
cosocleC23×C4×C79C23
hypercenter#3190D79
hypocenterC79#9263
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3190

Complements

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

NH ≅ G/Ntype
3. C2×#19098300C2semidirect
4. #23761764C2semidirect
5. #23761288C2semidirect
6. #23761242C2semidirect
27. #483×C79C4semidirect
28. #480×C79C4semidirect
32. #451×C79C4semidirect
33. #448×C79C4semidirect
241. C79#9263semidirect
245. #483Dic79semidirect
246. #480Dic79semidirect
250. #451Dic79semidirect
251. #448Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724562
#237619042
C2×#190983002
#237617642
#237612882
#237612422
C79×#31902
#926379