Finite Group Databasebeta

← #34205865

Subgroups — #34205865

Order 20224.

630 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771864C211yesmaximal, characteristic
2#23765075C211yesmaximal, characteristic, complemented
3#23765054C211yesmaximal, characteristic, complemented
4#23763375C211yesmaximal, characteristic
5#23761529C211yesmaximal, characteristic, complemented
6#23761521C211yesmaximal, characteristic, complemented
7C79×#3011C211yesmaximal, characteristic
8#19099041—21no—
9#19098580—21no—
10#19098572—21no—
11#19098547C2211yescharacteristic
12#19098537—22no—
13#19098537—22no—
14#19098537—21no—
15#19098537C2211yescharacteristic
16#19098537C2211yescharacteristic
17#19098537C2211yescharacteristic
18#19098358—22no—
19#19098353—22no—
20C2×2−1+4×C79C2211yescharacteristic
21C2×#137×C79C411yescharacteristic, complemented
22C22×Q16×C79C411yescharacteristic, complemented
23#536×C79C2211yescharacteristic
24#452×C79—21no—
25#452×C79—21no—
26#451×C79—21no—
27#451×C79—21no—
28#419×C79C411yescharacteristic, complemented
29#416×C79C411yescharacteristic, complemented
30#408×C79C2211yescharacteristic
31C2×#11929608—21no—
32Q8×Dic79—42no—
33Q8×Dic79—42no—
34#14579120—24no—
35#14579120—21no—
36#14579120—21no—
37#14579120—21no—
38#14579088—42no—
39#14579085—42no—
40#14579080—21no—
41#14579078—21no—
42#14579076—24no—
43#14579076—21no—
44#14579076—21no—
45#14579076—21no—
46#14579076—21no—
47#14579076D412yes—
48#14579076D412yes—
492−1+4×C79—42no—
502−1+4×C79—42no—
51C2×#41×C79—21no—
52C2×#41×C79—21no—
53C2×#41×C79—21no—
54C2×#41×C79C2×C411yescharacteristic
55C2×#41×C79D411yescharacteristic
56C2×#41×C79D411yescharacteristic
57C2×#41×C79C2311yescharacteristic
58C22×Q8×C79—21no—
59C22×Q8×C79C2×C411yescharacteristic
60C22×Q8×C79C2×C411yescharacteristic
61C22×Q8×C79D411yescharacteristic
62C22×Q8×C79D411yescharacteristic
63#137×C79—42no—
64C2×Q16×C79—21no—
65C2×Q16×C79—21no—
66C2×Q16×C79—21no—
67C2×Q16×C79—42no—
68C2×SD16×C79—21no—
69C2×M16×C79C2×C411yescharacteristic
70C22×C8×C79C2×C411yescharacteristic
71C4⋊Q8×C79—21no—
72#124×C79—21no—
73#123×C79—21no—
74#123×C79—21no—
75C22⋊Q8×C79—21no—
76C22⋊Q8×C79—21no—
77#117×C79C2×C411yescharacteristic
78#103×C79—21no—
79#103×C79—21no—
80#103×C79—21no—
81#102×C79—21no—
82C22⋊C8×C79—21no—
83C22⋊C8×C79—21no—
84#11929608—41no—
85#11929608—41no—
86C2×C79⋊C8—41no—
87C4⋊Dic79—44no—
88C4⋊Dic79—41no—
89C4⋊Dic79—41no—
90C4⋊Dic79—41no—
91C22⋊Dic79—22no—
92C22⋊Dic79—22no—
93C22⋊Dic79—41no—
94C4×Dic79—44no—
95C4×Dic79—41no—
96C4×Dic79—41no—
97C4×Dic79—41no—
98#41×C79—22no—
99#41×C79—22no—
100#41×C79—41no—
101#41×C79—41no—
102#41×C79—41no—
103#41×C79—41no—
104#41×C79—41no—
105#41×C79—41no—
106#41×C79—44no—
107#41×C79—44no—
108#41×C79—44no—
109C2×Q8×C79—22no—
110C2×Q8×C79—22no—
111C2×Q8×C79—21no—
112C2×Q8×C79—21no—
113C2×Q8×C79—21no—
114C2×Q8×C79—22no—
115C2×Q8×C79—22no—
116C2×Q8×C79—21no—
117C2×Q8×C79—21no—
118C2×Q8×C79—41no—
119C2×Q8×C79—41no—
120C2×Q8×C79—41no—
121C2×Q8×C79—41no—
122C2×Q8×C79—44no—
123C2×Q8×C79C22⋊C411yescharacteristic
124C2×Q8×C79C22⋊C411yescharacteristic
125C2×Q8×C79C22⋊C411yescharacteristic
126C2×Q8×C79#4111yescharacteristic
127C2×D4×C79—21no—
128C2×D4×C79—21no—
129C2×D4×C79—21no—
130C2×D4×C79C22⋊C411yescharacteristic
131C2×D4×C79C2×D411yescharacteristic
132C2×D4×C79C2×D411yescharacteristic
133C2×D4×C79#4111yescharacteristic
134C22×C4×C79—21no—
135C22×C4×C79—21no—
136C22×C4×C79—21no—
137C22×C4×C79—21no—
138C22×C4×C79—21no—
139C22×C4×C79—21no—
140C22×C4×C79—21no—
141C22×C4×C79C22×C411yescharacteristic
142C22×C4×C79C2×D411yescharacteristic
143C22×C4×C79C2×D411yescharacteristic
144Q16×C79—42no—
145Q16×C79—42no—
146Q16×C79—42no—
147SD16×C79—42no—
148M16×C79—41no—
149C2×C8×C79—21no—
150C2×C8×C79—21no—
151C2×C8×C79—41no—
152C4⋊C4×C79—21no—
153C4⋊C4×C79—41no—
154C4⋊C4×C79—41no—
155C22⋊C4×C79—21no—
156C22⋊C4×C79—41no—
157C22⋊C4×C79—41no—
158C42×C79—21no—
159C23×C79—21no—
160C23×C79C4×D411yescharacteristic
161C23×C79#12011yescharacteristic
162C23×C79#12111yescharacteristic
163Q8×C79—24no—
164Q8×C79—21no—
165Q8×C79—21no—
166Q8×C79—21no—
167Q8×C79—21no—
168Q8×C79—41no—
169Q8×C79—42no—
170Q8×C79—42no—
171Q8×C79—41no—
172Q8×C79—41no—
173Q8×C79—81no—
174Q8×C79#9912yes—
175Q8×C79#9912yes—
176D4×C79—21no—
177D4×C79—21no—
178D4×C79—41no—
179D4×C79—41no—
180D4×C79—41no—
181D4×C79—42no—
182D4×C79—42no—
183D4×C79—42no—
184C2×C4×C79—24no—
185C2×C4×C79—21no—
186C2×C4×C79—21no—
187C2×C4×C79—21no—
188C2×C4×C79—21no—
189C2×C4×C79—21no—
190C2×C4×C79—21no—
191C2×C4×C79—21no—
192C2×C4×C79—21no—
193C2×C4×C79—21no—
194C2×C4×C79—42no—
195C2×C4×C79—42no—
196C2×C4×C79—42no—
197C2×C4×C79—42no—
198C2×C4×C79—42no—
199C2×C4×C79—41no—
200C2×C4×C79—41no—
201C2×C4×C79—81no—
202C2×C4×C79C2×C22⋊C411yescharacteristic
203C2×C4×C79C4×D411yescharacteristic
204C2×C4×C79#12111yescharacteristic
205C2×C4×C79#12311yescharacteristic
206C2×Dic79—42no—
207C2×Dic79—42no—
208C2×Dic79—81no—
209C8×C79—41no—
210C8×C79—41no—
211C79⋊C8—81no—
212C22×C79—21no—
213C22×C79—21no—
214C22×C79—21no—
215C22×C79—41no—
216C22×C79—41no—
217C22×C79—41no—
218C22×C79—41no—
219C22×C79—81no—
220C22×C79#38611yescharacteristic
221C4×C79—22no—
222C4×C79—22no—
223C4×C79—21no—
224C4×C79—21no—
225C4×C79—21no—
226C4×C79—41no—
227C4×C79—81no—
228C4×C79—81no—
229C4×C79C2×#9911yescharacteristic
230C4×C79C2×#9911yescharacteristic
231Dic79—82no—
232Dic79—82no—
233Dic79—161no—
234#13101—791nomaximal, Sylow
235C2×C79—21no—
236C2×C79—41no—
237C2×C79—41no—
238C2×C79—41no—
239C2×C79—81no—
240C2×C79#189811yescharacteristic
241#3011D7911yescharacteristic
242#2884—791no—
243#2129—791no—
244#2123—791no—
245#1796—791no—
246#1395—791no—
247#1391—791no—
248C79#1310111yescharacteristic, complemented, Sylow
249C2×2−1+4C2×D7911yescharacteristic
250C2×#137Dic7911yescharacteristic, complemented
251C22×Q16Dic7911yescharacteristic, complemented
252#536C2×D7911yescharacteristic
253#519—1581no—
254#452—21no—
255#452—21no—
256#451—21no—
257#451—21no—
258#421—1581no—
259#419—1581no—
260#419Dic7911yescharacteristic, complemented
261#416Dic7911yescharacteristic, complemented
262#413—791no—
263#410—791no—
264#410—791no—
265#410—791no—
266#410—1582no—
267#410—1582no—
268#410—1581no—
269#408C2×D7911yescharacteristic
270#354—1582no—
271#352—1582no—
2722−1+4—42no—
2732−1+4—42no—
274C2×#41—21no—
275C2×#41—21no—
276C2×#41—21no—
277C2×#41C2×Dic7911yescharacteristic
278C2×#41C22⋊D7911yescharacteristic
279C2×#41C22⋊D7911yescharacteristic
280C2×#41C22×D7911yescharacteristic
281C22×Q8—21no—
282C22×Q8C2×Dic7911yescharacteristic
283C22×Q8C2×Dic7911yescharacteristic
284C22×Q8C22⋊D7911yescharacteristic
285C22×Q8C22⋊D7911yescharacteristic
286#137—42no—
287C2×Q16—21no—
288C2×Q16—21no—
289C2×Q16—21no—
290C2×Q16—42no—
291C2×SD16—21no—
292C2×M16—1581no—
293C2×M16C2×Dic7911yescharacteristic
294C22×C8C2×Dic7911yescharacteristic
295C4⋊Q8—21no—
296#124—21no—
297#123—21no—
298#123—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C4×Q8—3162no—
302C4×Q8—3162no—
303#117—1581no—
304#117—1584no—
305#117—1581no—
306#117—1581no—
307#117C2×Dic7911yescharacteristic
308#104—3162no—
309#103—21no—
310#103—21no—
311#103—21no—
312#103—3162no—
313#102—21no—
314#101—1581no—
315#100—1581no—
316#99—792no—
317#99—792no—
318#99—1581no—
319#99—1581no—
320#99—1584no—
321#99—1581no—
322#99—1581no—
323C22⋊C8—21no—
324C22⋊C8—21no—
325#41—22no—
326#41—22no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—44no—
334#41—44no—
335#41—44no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—44no—
350C2×Q8C22⋊Dic7911yescharacteristic
351C2×Q8C22⋊Dic7911yescharacteristic
352C2×Q8C22⋊Dic7911yescharacteristic
353C2×Q8#1192962311yescharacteristic
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4C22⋊Dic7911yescharacteristic
358C2×D4D4×D7911yescharacteristic
359C2×D4D4×D7911yescharacteristic
360C2×D4#1192962311yescharacteristic
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4C22×Dic7911yescharacteristic
369C22×C4C2×C22⋊D7911yescharacteristic
370C22×C4C2×C22⋊D7911yescharacteristic
371Q16—42no—
372Q16—42no—
373Q16—42no—
374SD16—42no—
375M16—41no—
376M16—3161no—
377M16—3161no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—41no—
381C2×C8—3161no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—3164no—
386C4⋊C4—3161no—
387C4⋊C4—3161no—
388C4⋊C4—3161no—
389C22⋊C4—21no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—3161no—
393C22⋊C4—1582no—
394C22⋊C4—1582no—
395C42—21no—
396C42—3164no—
397C42—3161no—
398C42—3161no—
399C42—3161no—
400C23—21no—
401C23D4×Dic7911yescharacteristic
402C23#1457912911yescharacteristic
403C23#1457913311yescharacteristic
404Q8—24no—
405Q8—21no—
406Q8—21no—
407Q8—21no—
408Q8—21no—
409Q8—41no—
410Q8—42no—
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—81no—
415Q8#1457907612yes—
416Q8#1457907612yes—
417D4—21no—
418D4—21no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—42no—
423D4—42no—
424D4—42no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—3162no—
444C2×C4—3162no—
445C2×C4—6321no—
446C2×C4C2×C22⋊Dic7911yescharacteristic
447C2×C4D4×Dic7911yescharacteristic
448C2×C4#1457913311yescharacteristic
449C2×C4#1457914211yescharacteristic
450C8—41no—
451C8—41no—
452C8—6321no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—81no—
461C22#1909844311yescharacteristic
462C4—21no—
463C4—22no—
464C4—22no—
465C4—21no—
466C4—21no—
467C4—41no—
468C4—81no—
469C4—81no—
470C4—6322no—
471C4—6322no—
472C4—12641no—
473C4C2×#1457907611yescharacteristic
474C4C2×#1457907611yescharacteristic
475C2—21no—
476C2—41no—
477C2—41no—
478C2—41no—
479C2—81no—
480C2#2376385411yescharacteristic
subgroupG/H
centerC2#23763854
commutatorC22×C4×C79C22×C4
FrattiniC2×#41C22×D79
FittingC79×#3011C2
socleC2×C79#1898
prime radicalC2×2−1+4×C79C22
cosocleC2×#41×C79C23
hypercenter#3011D79
hypocenterC79#13101
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #3011

Complements

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

NH ≅ G/Ntype
2. #23765075C2semidirect
3. #23765054C2semidirect
5. #23761529C2semidirect
6. #23761521C2semidirect
21. C2×#137×C79C4semidirect
22. C22×Q16×C79C4semidirect
28. #419×C79C4semidirect
29. #416×C79C4semidirect
248. C79#13101semidirect
250. C2×#137Dic79semidirect
251. C22×Q16Dic79semidirect
260. #419Dic79semidirect
261. #416Dic79semidirect

Maximal subgroups

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

subgroupindex
#237718642
#237650752
#237650542
#237633752
#237615292
#237615212
C79×#30112
#1310179