Finite Group Databasebeta

← #34442953

Subgroups — #34442953

Order 20224.

556 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#23773933C211yesmaximal, characteristic, complemented
2#23772247C211yesmaximal, characteristic
3#23765559C211yesmaximal, characteristic, complemented
4#23765558C211yesmaximal, characteristic, complemented
5#23765557C211yesmaximal, characteristic, complemented
6#23765556C211yesmaximal, characteristic, complemented
7#23765546C211yesmaximal, characteristic, complemented
8#23765535C211yesmaximal, characteristic, complemented
9#23765534C211yesmaximal, characteristic, complemented
10#23765187C211yesmaximal, characteristic, complemented
11#23765183C211yesmaximal, characteristic, complemented
12#23765157C211yesmaximal, characteristic, complemented
13#23765114C211yesmaximal, characteristic
14#357×D79C211yesmaximal, characteristic
15C79×#2226C211yesmaximal, characteristic, complemented
16C2×D8×D79C2211yescharacteristic
17#19099190—21no—
18#19099035—21no—
19C2×#14579100C2211yescharacteristic
20C2×C16×D79C2211yescharacteristic
21#19098971—21no—
22#19098969—21no—
23#19098967C2211yescharacteristic, complemented
24#19098965C2211yescharacteristic, complemented
25#19098964C2211yescharacteristic, complemented
26#19098673—21no—
27#19098672—21no—
28C4×D632C2211yescharacteristic
29D8×Dic79C2211yescharacteristic
30#19098645C2211yescharacteristic
31#19098604—21no—
32#19098600C2211yescharacteristic
33#19098600C2211yescharacteristic
34#107×D79C2211yescharacteristic
35#19098566—21no—
36#19098517C2211yescharacteristic
37#19098391C2211yescharacteristic, complemented
38#19098390C2211yescharacteristic, complemented
39#19098389C2211yescharacteristic, complemented
40#19098388C2211yescharacteristic, complemented
41#19098388C2211yescharacteristic, complemented
42#19098388C2211yescharacteristic, complemented
43C4⋊(C79⋊C16)C2211yescharacteristic, complemented
44Dic79⋊C16C2211yescharacteristic, complemented
45Dic79⋊C16C2211yescharacteristic, complemented
46#19098365C2211yescharacteristic
47#19098365C2211yescharacteristic
48#19098364C2211yescharacteristic
49#19098364C2211yescharacteristic
50#19098363C2211yescharacteristic
51#19098363C2211yescharacteristic
52#500×C79C2211yescharacteristic, complemented
53C4×D8×C79C2211yescharacteristic
54#367×C79C2211yescharacteristic, complemented
55#367×C79C2211yescharacteristic, complemented
56C4⋊C16×C79C2211yescharacteristic, complemented
57#357×C79C2211yescharacteristic
58#357×C79C2211yescharacteristic
59C2×D4×D79—21no—
60C2×D632C2311yescharacteristic
61C2×#11929609—21no—
62D8×D79—22no—
63D8×D79Q814yescomplemented
64C2×#11929607D411yescharacteristic
65C2×C8×D79C2311yescharacteristic
66#14579160—21no—
67#14579159—41no—
68#14579153—21no—
69#14579152—21no—
70#14579151—41no—
71#14579139—41no—
72#14579137—41no—
73C4×D316—21no—
74D4×Dic79—21no—
75#14579123—21no—
76C4×C22⋊D79—41no—
77(C22⋊D79)⋊C4—41no—
78#14579119—21no—
79#14579119—21no—
80#14579118D411yescharacteristic
81C4⋊C4×D79—21no—
82C2×(C2×D79)⋊C4—41no—
83C22⋊C4×D79—41no—
84#14579100—22no—
85C2×C79⋊C16C2311yescharacteristic
86C2×C79⋊C16C2311yescharacteristic
87C16×D79—22no—
88#14579095C2311yescharacteristic
89#14579095C2311yescharacteristic
90#14579095C2311yescharacteristic
91#14579094—21no—
92#14579094—21no—
93#14579093—21no—
94#14579092—21no—
95#14579092—21no—
96#14579092—21no—
97#14579083—21no—
98#14579082—21no—
99#14579081—21no—
100#14579072D411yescharacteristic
101#14579071D411yescharacteristic
102C8×Dic79C2311yescharacteristic
103C2×D8×C79C2311yescharacteristic
104C4⋊Q8×C79—21no—
105C4×D4×C79—21no—
106C2×C16×C79C2311yescharacteristic
107C2×C16×C79C2311yescharacteristic
108#107×C79—21no—
109#107×C79C2311yescharacteristic
110#107×C79C2311yescharacteristic
111#107×C79C2311yescharacteristic
112#102×C79—21no—
113C4×C8×C79C2311yescharacteristic
114C23×D79—41no—
115C2×Dic158—41no—
116C2×D316—21no—
117C2×C22⋊D79—41no—
118D4×D79—24no—
119D4×D79—42no—
120C22×Dic79—41no—
121C2×C4×D79—41no—
122C2×C4×D79—41no—
123C2×C4×D79—41no—
124C2×C4×D79C2×D411yescharacteristic
125D632—21no—
126D632C2×Q812yes—
127#11929609—24no—
128#11929607—22no—
129C2×C79⋊C8C2×D411yescharacteristic
130C2×C79⋊C8C2×D411yescharacteristic
131C8×D79C2×Q812yes—
132C8×D79#4112yes—
133C4⋊Dic79—21no—
134C4⋊Dic79—21no—
135C4⋊Dic79—21no—
136C4⋊Dic79—41no—
137Dic79⋊C4—41no—
138Dic79⋊C4—41no—
139Dic79⋊C4—41no—
140Dic79⋊C4—41no—
141Dic79⋊C4—41no—
142Dic79⋊C4D811yescharacteristic
143Dic79⋊C4D811yescharacteristic
144C22⋊Dic79—41no—
145(C2×D79)⋊C4—41no—
146(C2×D79)⋊C4—41no—
147(C2×D79)⋊C4—41no—
148(C2×D79)⋊C4—41no—
149(C2×D79)⋊C4D811yescharacteristic
150(C2×D79)⋊C4D811yescharacteristic
151C4×Dic79—41no—
152C4×Dic79—41no—
153C4×Dic79C2×D411yescharacteristic
154C79⋊C16—21no—
155C79⋊C16—21no—
156C2×Q8×C79—41no—
157C2×D4×C79—21no—
158C22×C4×C79—41no—
159D8×C79—21no—
160D8×C79C2×Q812yes—
161C2×C8×C79C2×D411yescharacteristic
162C2×C8×C79C2411yescharacteristic
163C4⋊C4×C79—21no—
164C4⋊C4×C79—21no—
165C4⋊C4×C79—21no—
166C4⋊C4×C79—41no—
167C22⋊C4×C79—41no—
168C42×C79C2×D411yescharacteristic
169C16×C79—21no—
170C16×C79—21no—
171C23×C79—41no—
172C22×D79—41no—
173C22×D79—44no—
174C22×D79—161no—
175C22×D79—161no—
176C22×D79C2×D811yescharacteristic
177Q8×C79—161no—
178D4×C79—22no—
179D4×C79—41no—
180C2×C4×C79—41no—
181C2×C4×C79—41no—
182C2×C4×C79—41no—
183C2×C4×C79—161no—
184C2×C4×C79C2×D811yescharacteristic
185C2×C4×C79C2×D811yescharacteristic
186C2×C4×C79C22×D411yescharacteristic
187C22⋊D79—44no—
188C2×Dic79—41no—
189C2×Dic79—41no—
190C2×Dic79—41no—
191C2×Dic79—161no—
192C2×Dic79C2×D811yescharacteristic
193C2×Dic79C2×D811yescharacteristic
194C2×Dic79C2×D811yescharacteristic
195D316—22no—
196D316—41no—
197C4×D79—161no—
198C4×D79—161no—
199C4×D79—161no—
200C4×D79C22⋊Q812yes—
201C4×D79#13712yes—
202Dic158—161no—
203C8×C79—21no—
204C8×C79C22×Q811yescharacteristic
205C8×C79C2×#4111yescharacteristic
206C79⋊C8—21no—
207C79⋊C8C22⋊Q812yes—
208C22×C79—42no—
209C22×C79—81no—
210C22×C79—81no—
211C22×C79C22×D811yescharacteristic
212C2×D79—21no—
213C2×D79—21no—
214C2×D79—42no—
215C2×D79—81no—
216C2×D79—81no—
217C2×D79—164no—
218C2×D79#47412yes—
219C4×C79—21no—
220C4×C79—21no—
221C4×C79—81no—
222C4×C79—81no—
223C4×C79—81no—
224C4×C79C2×C22⋊Q811yescharacteristic
225C4×C79C2×#13711yescharacteristic
226Dic79—21no—
227Dic79—21no—
228Dic79—81no—
229Dic79—81no—
230Dic79—81no—
231Dic79#47412yes—
232#33631—791nomaximal, Sylow
233C2×C79—82no—
234C2×C79C2×#47411yescharacteristic
235C2×C79#341511yescharacteristic
236C2×C79#341611yescharacteristic
237D79—22no—
238D79—82no—
239#3164—791no—
240#2948—791no—
241#2228—791no—
242#2228—791no—
243#2228—791no—
244#2228—791no—
245#2226—791no—
246#2226D7911yescharacteristic, complemented
247#2224—791no—
248#2224—791no—
249#2159—791no—
250#2158—791no—
251#2151—791no—
252#2140—791no—
253C2×#357—791no—
254C79#3363111yescharacteristic, complemented, Sylow
255C22×D8—791no—
256#544—1581no—
257#518—1581no—
258C2×M32—791no—
259C22×C16—791no—
260#501—1581no—
261#501—1581no—
262#500—791no—
263#500C2×D7911yescharacteristic, complemented
264#499—791no—
265#499—791no—
266#442—1581no—
267#442—1581no—
268C4×D8—791no—
269C4×D8—791no—
270C4×D8—791no—
271C4×D8C2×D7911yescharacteristic
272#428—1581no—
273#427—791no—
274#427—791no—
275C2×#107—791no—
276#418—1581no—
277#405—791no—
278#367—791no—
279#367—791no—
280#367C2×D7911yescharacteristic, complemented
281#367C2×D7911yescharacteristic, complemented
282#366—791no—
283#366—791no—
284#366—791no—
285#366—791no—
286C4⋊C16—791no—
287C4⋊C16—791no—
288C4⋊C16—791no—
289C4⋊C16C2×D7911yescharacteristic, complemented
290#357—791no—
291#357—791no—
292#357—791no—
293#357—791no—
294#357—791no—
295#357—791no—
296#357C2×D7911yescharacteristic
297#357C2×D7911yescharacteristic
298C22×D4—1581no—
299C2×D8—794no—
300C2×D8—791no—
301C2×D8—1582no—
302C2×D8—1581no—
303C2×D8C22×D7911yescharacteristic
304C2×M16—791no—
305C22×C8—791no—
306C4⋊Q8—21no—
307C4⋊Q8—3161no—
308C4⋊Q8—1581no—
309#125—3161no—
310#125—1581no—
311#125—1581no—
312C22⋊Q8—3161no—
313C22⋊Q8—3161no—
314C4×D4—21no—
315C4×D4—3161no—
316C4×D4—3161no—
317C4×D4—1581no—
318C4×D4—1581no—
319C4×D4—1581no—
320#117—791no—
321#117—1581no—
322#117—1581no—
323C2×C4⋊C4—1581no—
324C2×C22⋊C4—3161no—
325C2×C22⋊C4—3161no—
326M32—1582no—
327C2×C16—791no—
328C2×C16—791no—
329C2×C16—1582no—
330C2×C16C22×D7911yescharacteristic
331C2×C16C22×D7911yescharacteristic
332#107—21no—
333#107—791no—
334#107—791no—
335#107—791no—
336#107—1581no—
337#107—1581no—
338#107C22×D7911yescharacteristic
339#107C22×D7911yescharacteristic
340#107C22×D7911yescharacteristic
341#106—1581no—
342#106—1581no—
343#106—1581no—
344#106—1581no—
345#102—21no—
346#102—1581no—
347#102—1581no—
348#102—1581no—
349#97—791no—
350#97—791no—
351C4×C8—791no—
352C4×C8C22×D7911yescharacteristic
353C24—3161no—
354C2×Q8—41no—
355C2×Q8—3161no—
356C2×D4—21no—
357C2×D4—3162no—
358C2×D4—3161no—
359C2×D4—1584no—
360C2×D4—1581no—
361C22×C4—41no—
362C22×C4—3161no—
363C22×C4—3161no—
364C22×C4—3161no—
365C22×C4—3161no—
366C22×C4—791no—
367D8—21no—
368D8—792no—
369D8—1584no—
370D8—1581no—
371D8Q8×D7912yes—
372M16—1582no—
373C2×C8—791no—
374C2×C8—792no—
375C2×C8—792no—
376C2×C8—791no—
377C2×C8D4×D7911yescharacteristic
378C2×C8C23×D7911yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—3161no—
384C4⋊C4—3161no—
385C4⋊C4—3161no—
386C4⋊C4—3161no—
387C4⋊C4—3161no—
388C4⋊C4—3161no—
389C4⋊C4—791no—
390C4⋊C4—791no—
391C4⋊C4—1581no—
392C4⋊C4—1581no—
393C4⋊C4—1581no—
394C22⋊C4—41no—
395C22⋊C4—3161no—
396C22⋊C4—3161no—
397C22⋊C4—3161no—
398C22⋊C4—3161no—
399C22⋊C4—3161no—
400C22⋊C4—791no—
401C22⋊C4—791no—
402C42—3161no—
403C42—3161no—
404C42—791no—
405C42D4×D7911yescharacteristic
406C16—21no—
407C16—21no—
408C16—1581no—
409C16—1581no—
410C23—41no—
411C23—3164no—
412C23—3161no—
413C23—791no—
414C23—12641no—
415C23—12641no—
416Q8—161no—
417Q8—12641no—
418D4—22no—
419D4—41no—
420D4—3164no—
421D4—3161no—
422D4—1582no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—3161no—
428C2×C4—3161no—
429C2×C4—3161no—
430C2×C4—791no—
431C2×C4—792no—
432C2×C4—792no—
433C2×C4—791no—
434C2×C4—791no—
435C2×C4—12641no—
436C2×C4—12641no—
437C2×C4—12641no—
438C2×C4—12641no—
439C2×C4D8×D7911yescharacteristic
440C2×C4D8×D7911yescharacteristic
441C2×C4C2×D4×D7911yescharacteristic
442C8—21no—
443C8—792no—
444C8—1581no—
445C8C2×Q8×D7911yescharacteristic
446C8#41×D7911yescharacteristic
447C22—42no—
448C22—81no—
449C22—81no—
450C22—3162no—
451C22—792no—
452C22—6321no—
453C22—6321no—
454C22—1581no—
455C22—1581no—
456C22—12644no—
457C22C2×D8×D7911yescharacteristic
458C4—21no—
459C4—21no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—792no—
464C4—6321no—
465C4—6321no—
466C4—6321no—
467C4—1581no—
468C4—1581no—
469C4C22⋊Q8×D7911yescharacteristic
470C4#137×D7911yescharacteristic
471C2—82no—
472C2—6322no—
473C2—1582no—
474C2#474×D7911yescharacteristic
475C2#2377654011yescharacteristic
476C2#2377654211yescharacteristic
subgroupG/H
centerC22C2×D8×D79
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#2226C2
socleC22×C79C22×D8
prime radicalC2×D8×D79C22
hypercenter#2226D79
hypocenterC79#33631
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2226

Complements

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

NH ≅ G/Ntype
1. #23773933C2semidirect
3. #23765559C2semidirect
4. #23765558C2semidirect
5. #23765557C2semidirect
6. #23765556C2semidirect
7. #23765546C2semidirect
8. #23765535C2semidirect
9. #23765534C2semidirect
10. #23765187C2semidirect
11. #23765183C2semidirect
12. #23765157C2semidirect
15. C79×#2226C2semidirect
23. #19098967C22semidirect
24. #19098965C22semidirect
25. #19098964C22semidirect
37. #19098391C22semidirect
38. #19098390C22semidirect
39. #19098389C22semidirect
40. #19098388C22semidirect
41. #19098388C22semidirect
42. #19098388C22semidirect
43. C4⋊(C79⋊C16)C22semidirect
44. Dic79⋊C16C22semidirect
45. Dic79⋊C16C22semidirect
52. #500×C79C22semidirect
54. #367×C79C22semidirect
55. #367×C79C22semidirect
56. C4⋊C16×C79C22semidirect
63. D8×D79Q8semidirect
246. #2226D79semidirect
254. C79#33631semidirect
263. #500C2×D79semidirect
280. #367C2×D79semidirect
281. #367C2×D79semidirect
289. C4⋊C16C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237739332
#237722472
#237655592
#237655582
#237655572
#237655562
#237655462
#237655352
#237655342
#237651872
#237651832
#237651572
#237651142
#357×D792
C79×#22262
#3363179