Finite Group Databasebeta

← #34205830

Subgroups — #34205830

Order 20224.

624 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771883C211yesmaximal, characteristic
2#23765095C211yesmaximal, characteristic, complemented
3#23765054C211yesmaximal, characteristic, complemented
4#23763375C211yesmaximal, characteristic
5#23761529C211yesmaximal, characteristic, complemented
6#23761523C211yesmaximal, characteristic, complemented
7C79×#3011C211yesmaximal, characteristic
8#19099330—21no—
9#19098580—21no—
10#19098572—21no—
11#19098547—22no—
12#19098547C2211yescharacteristic
13#19098547C2211yescharacteristic
14#19098547C2211yescharacteristic
15C2×#14579080—22no—
16C2×#14579080—21no—
17#19098537C2211yescharacteristic
18#19098362—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
31#14579168—42no—
32#14579168—42no—
33C2×#11929608—24no—
34C2×#11929608—21no—
35C2×#11929608—21no—
36C2×#11929608—21no—
37#14579120—21no—
38#14579088—42no—
39#14579085—42no—
40#14579080—24no—
41#14579080—21no—
42#14579080—21no—
43#14579080D412yes—
44#14579078—21no—
45#14579078—21no—
46#14579078D412yes—
47#14579076—21no—
48#14579076—21no—
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—22no—
85#11929608—22no—
86#11929608—44no—
87#11929608—41no—
88#11929608—41no—
89#11929608—41no—
90#11929608—41no—
91C2×C79⋊C8—44no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94C2×C79⋊C8—41no—
95C4⋊Dic79—41no—
96C22⋊Dic79—41no—
97C4×Dic79—41no—
98#41×C79—22no—
99#41×C79—22no—
100#41×C79—22no—
101#41×C79—22no—
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—21no—
110C2×Q8×C79—21no—
111C2×Q8×C79—21no—
112C2×Q8×C79—22no—
113C2×Q8×C79—22no—
114C2×Q8×C79—21no—
115C2×Q8×C79—21no—
116C2×Q8×C79—41no—
117C2×Q8×C79—41no—
118C2×Q8×C79—41no—
119C2×Q8×C79—41no—
120C2×Q8×C79—44no—
121C2×Q8×C79—41no—
122C2×Q8×C79—41no—
123C2×Q8×C79C22⋊C411yescharacteristic
124C2×Q8×C79C22⋊C411yescharacteristic
125C2×Q8×C79C22⋊C411yescharacteristic
126C2×Q8×C79C2×D411yescharacteristic
127C2×D4×C79—21no—
128C2×D4×C79—21no—
129C2×D4×C79—21no—
130C2×D4×C79C22⋊C411yescharacteristic
131C2×D4×C79C2×D411yescharacteristic
132C2×D4×C79#4111yescharacteristic
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#12111yescharacteristic
162C23×C79#12311yescharacteristic
163Q8×C79—21no—
164Q8×C79—21no—
165Q8×C79—21no—
166Q8×C79—21no—
167Q8×C79—21no—
168Q8×C79—41no—
169Q8×C79—42no—
170Q8×C79—42no—
171Q8×C79—42no—
172Q8×C79—41no—
173Q8×C79—41no—
174Q8×C79—81no—
175Q8×C79#9912yes—
176D4×C79—24no—
177D4×C79—21no—
178D4×C79—41no—
179D4×C79—41no—
180D4×C79—41no—
181D4×C79—42no—
182D4×C79—42no—
183D4×C79#9912yes—
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#12011yescharacteristic
205C2×C4×C79#12111yescharacteristic
206C2×Dic79—81no—
207C8×C79—41no—
208C8×C79—41no—
209C79⋊C8—42no—
210C79⋊C8—42no—
211C79⋊C8—81no—
212C22×C79—21no—
213C22×C79—21no—
214C22×C79—22no—
215C22×C79—21no—
216C22×C79—41no—
217C22×C79—41no—
218C22×C79—41no—
219C22×C79—81no—
220C22×C79#38611yescharacteristic
221C4×C79—21no—
222C4×C79—22no—
223C4×C79—21no—
224C4×C79—21no—
225C4×C79—41no—
226C4×C79—41no—
227C4×C79—81no—
228C4×C79—81no—
229C4×C79C2×#9911yescharacteristic
230C4×C79C2×#9911yescharacteristic
231Dic79—161no—
232#13096—791nomaximal, Sylow
233C2×C79—21no—
234C2×C79—41no—
235C2×C79—41no—
236C2×C79—41no—
237C2×C79—42no—
238C2×C79#189811yescharacteristic
239#3011D7911yescharacteristic
240#2889—791no—
241#2135—791no—
242#2123—791no—
243#1796—791no—
244#1395—791no—
245#1392—791no—
246C79#1309611yescharacteristic, complemented, Sylow
247C2×2−1+4C2×D7911yescharacteristic
248C2×#137Dic7911yescharacteristic, complemented
249C22×Q16Dic7911yescharacteristic, complemented
250#568—1581no—
251#536C2×D7911yescharacteristic
252#452—21no—
253#452—21no—
254#451—21no—
255#451—21no—
256#421—1581no—
257#419—1581no—
258#419Dic7911yescharacteristic, complemented
259#416Dic7911yescharacteristic, complemented
260#413—791no—
261#413—791no—
262#413—791no—
263#413—1582no—
264C2×#101—1582no—
265C2×#101—1581no—
266#410—791no—
267#408C2×D7911yescharacteristic
268#356—1582no—
269#352—1582no—
2702−1+4—42no—
2712−1+4—42no—
272C2×#41—21no—
273C2×#41—21no—
274C2×#41—21no—
275C2×#41C2×Dic7911yescharacteristic
276C2×#41C22⋊D7911yescharacteristic
277C2×#41C22⋊D7911yescharacteristic
278C2×#41C22×D7911yescharacteristic
279C22×Q8—21no—
280C22×Q8C2×Dic7911yescharacteristic
281C22×Q8C2×Dic7911yescharacteristic
282C22×Q8C22⋊D7911yescharacteristic
283C22×Q8C22⋊D7911yescharacteristic
284#137—42no—
285C2×Q16—21no—
286C2×Q16—21no—
287C2×Q16—21no—
288C2×Q16—42no—
289C2×SD16—21no—
290#131—3162no—
291#131—3162no—
292C2×M16—1581no—
293C2×M16—1584no—
294C2×M16—1581no—
295C2×M16—1581no—
296C2×M16C2×Dic7911yescharacteristic
297C22×C8C2×Dic7911yescharacteristic
298C4⋊Q8—21no—
299#124—21no—
300#123—21no—
301#123—21no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#117—1581no—
305#117C2×Dic7911yescharacteristic
306#104—3162no—
307#103—21no—
308#103—21no—
309#103—21no—
310#103—3162no—
311#102—21no—
312#101—792no—
313#101—1581no—
314#101—1584no—
315#101—1581no—
316#100—792no—
317#100—1581no—
318#100—1581no—
319#99—1581no—
320#99—1581no—
321C22⋊C8—21no—
322C22⋊C8—21no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—44no—
332#41—44no—
333#41—44no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—44no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8C22⋊Dic7911yescharacteristic
349C2×Q8C22⋊Dic7911yescharacteristic
350C2×Q8C22⋊Dic7911yescharacteristic
351C2×Q8D4×D7911yescharacteristic
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4C22⋊Dic7911yescharacteristic
356C2×D4D4×D7911yescharacteristic
357C2×D4#1192962311yescharacteristic
358C2×D4#1192962311yescharacteristic
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4C22×Dic7911yescharacteristic
367C22×C4C2×C22⋊D7911yescharacteristic
368C22×C4C2×C22⋊D7911yescharacteristic
369Q16—42no—
370Q16—42no—
371Q16—42no—
372SD16—42no—
373M16—41no—
374M16—3164no—
375M16—3161no—
376M16—3161no—
377M16—3161no—
378M16—3161no—
379M16—1582no—
380M16—1582no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—41no—
384C2×C8—3164no—
385C2×C8—3161no—
386C2×C8—3161no—
387C2×C8—3161no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—3161no—
392C22⋊C4—21no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—3161no—
396C42—21no—
397C42—3161no—
398C23—21no—
399C23D4×Dic7911yescharacteristic
400C23#1457913311yescharacteristic
401C23#1457914211yescharacteristic
402Q8—21no—
403Q8—21no—
404Q8—21no—
405Q8—21no—
406Q8—21no—
407Q8—41no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—41no—
412Q8—41no—
413Q8—81no—
414Q8#1457907612yes—
415D4—24no—
416D4—21no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—42no—
421D4—42no—
422D4#1457907612yes—
423C2×C4—24no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4—6321no—
442C2×C4C2×C22⋊Dic7911yescharacteristic
443C2×C4D4×Dic7911yescharacteristic
444C2×C4#1457912911yescharacteristic
445C2×C4#1457913311yescharacteristic
446C8—41no—
447C8—41no—
448C8—3162no—
449C8—3162no—
450C8—6321no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—81no—
459C22#1909844311yescharacteristic
460C4—21no—
461C4—21no—
462C4—22no—
463C4—21no—
464C4—41no—
465C4—41no—
466C4—81no—
467C4—81no—
468C4—12641no—
469C4C2×#1457907611yescharacteristic
470C4C2×#1457907611yescharacteristic
471C2—21no—
472C2—41no—
473C2—41no—
474C2—41no—
475C2—42no—
476C2#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#13096
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. #23765095C2semidirect
3. #23765054C2semidirect
5. #23761529C2semidirect
6. #23761523C2semidirect
21. C2×#137×C79C4semidirect
22. C22×Q16×C79C4semidirect
28. #419×C79C4semidirect
29. #416×C79C4semidirect
246. C79#13096semidirect
248. C2×#137Dic79semidirect
249. C22×Q16Dic79semidirect
258. #419Dic79semidirect
259. #416Dic79semidirect

Maximal subgroups

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

subgroupindex
#237718832
#237650952
#237650542
#237633752
#237615292
#237615232
C79×#30112
#1309679