Finite Group Databasebeta

← #34441677

Subgroups — #34441677

Order 20224.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776533C211yesmaximal, characteristic
2#443×D79C212yesmaximal, complemented
3#23765286C211yesmaximal, characteristic, complemented
4#23765285C211yesmaximal, characteristic, complemented
5#23765283C212yesmaximal, complemented
6#23765282C211yesmaximal, characteristic, complemented
7#23765279C212yesmaximal, complemented
8#23765193C211yesmaximal, characteristic, complemented
9#23765126C212yesmaximal, complemented
10C4×#14579100C211yesmaximal, characteristic, complemented
11C79×#2178C211yesmaximal, characteristic, complemented
12#135×D79C2212yes—
13#131×D79—22no—
14#19099005C412yescomplemented
15#19099002C412yescomplemented
16#19099000C412yescomplemented
17#19098997C412yescomplemented
18#19098995C2211yescharacteristic
19#19098993C2212yes—
20C2×#14579100C2211yescharacteristic
21C2×#14579100C2211yescharacteristic
22#19098681C2212yescomplemented
23#19098680C2212yescomplemented
24#19098679C2212yescomplemented
25#19098678—22no—
26#19098677—22no—
27#108×D79C2212yes—
28#104×D79—22no—
29C4×C8×D79C2211yescharacteristic
30#19098393C2211yescharacteristic, complemented
31#19098392C2212yescomplemented
32#19098371C2212yescomplemented
33#19098370C2212yescomplemented
34#19098369C2212yescomplemented
35#19098338C2211yescharacteristic, complemented
36#19098338C2211yescharacteristic, complemented
37C4×C79⋊C16C2211yescharacteristic, complemented
38#508×C79C2211yescharacteristic
39#443×C79C2212yescomplemented
40#368×C79C2211yescharacteristic, complemented
41#359×C79C2212yescomplemented
42C4×C16×C79C2211yescharacteristic, complemented
43#41×D79—22no—
44#14579184C2×C412yes—
45#14579183C2312yes—
46#14579182—22no—
47#14579181—22no—
48#14579180C2×C412yes—
49Q16×D79C2×C412yes—
50SD16×D79—22no—
51D8×D79C2×C412yes—
52#14579169—22no—
53#14579169—42no—
54#14579168—22no—
55#14579167—22no—
56#14579167—42no—
57M16×D79—22no—
58M16×D79—42no—
59C2×C8×D79—42no—
60C2×C8×D79D411yescharacteristic
61C2×C8×D79C2311yescharacteristic
62C42×D79D411yescharacteristic
63Dic632C2×C411yescharacteristic, complemented
64#14579107C2×C412yescomplemented
65#14579106C2×C412yescomplemented
66#14579105C2×C412yescomplemented
67#14579104C2×C412yescomplemented
68D1264C2×C411yescharacteristic, complemented
69#14579102C2×C412yescomplemented
70#14579100—22no—
71#14579100C2×C412yes—
72#14579100C2×C412yes—
73C2×C79⋊C16C2311yescharacteristic
74C2×C79⋊C16C2311yescharacteristic
75#14579097C2312yes—
76#14579096—22no—
77#14579089—22no—
78#14579088—22no—
79#14579087—22no—
80C4×C79⋊C8D411yescharacteristic
81C8×Dic79D411yescharacteristic
82C8×Dic79C2311yescharacteristic
83#135×C79C2312yes—
84#131×C79—22no—
85Q32×C79C2×C411yescharacteristic, complemented
86SD32×C79C2×C412yescomplemented
87D16×C79C2×C411yescharacteristic, complemented
88C2×C16×C79C2311yescharacteristic
89C2×C16×C79C2311yescharacteristic
90#108×C79C2312yes—
91#104×C79—22no—
92C4×C8×C79C2311yescharacteristic
93#11929625—22no—
94#11929624—22no—
95#11929624—42no—
96#11929623—22no—
97#11929623—42no—
98Q8×D79—22no—
99D4×D79—22no—
100D4×D79—42no—
101C2×C4×D79—21no—
102C2×C4×D79—42no—
103C2×C4×D79C2×D411yescharacteristic
104Dic316C22×C412yes—
105#11929614—22no—
106#11929613—22no—
107#11929612—22no—
108#11929611—22no—
109D632C22×C412yes—
110#11929609—22no—
111#11929608—22no—
112#11929608—42no—
113#11929607—42no—
114#11929607—42no—
115#11929607—42no—
116C2×C79⋊C8—42no—
117C2×C79⋊C8C2×D411yescharacteristic
118C2×C79⋊C8C2×D411yescharacteristic
119C8×D79—42no—
120C8×D79—42no—
121C8×D79—42no—
122C8×D79D812yes—
123C8×D79D812yes—
124C8×D79C22×C412yes—
125C8×D79#4112yes—
126C4×Dic79—21no—
127C4×Dic79C2×D411yescharacteristic
128C79⋊C16—21no—
129C79⋊C16C22×C412yes—
130#41×C79—22no—
131Q16×C79C22×C412yes—
132SD16×C79—22no—
133D8×C79C22×C412yes—
134M16×C79—22no—
135M16×C79—42no—
136C2×C8×C79—42no—
137C2×C8×C79C2×D411yescharacteristic
138C2×C8×C79C2411yescharacteristic
139C42×C79C2×D411yescharacteristic
140C16×C79—21no—
141C16×C79C22×C411yescharacteristic
142C16×C79C22×C411yescharacteristic
143C22×D79—42no—
144C22×D79#13511yescharacteristic
145Q8×C79—22no—
146D4×C79—22no—
147D4×C79—42no—
148C2×C4×C79—21no—
149C2×C4×C79—42no—
150C2×C4×C79C22×D411yescharacteristic
151C22⋊D79—42no—
152C22⋊D79—42no—
153C22⋊D79—42no—
154C2×Dic79—21no—
155C2×Dic79—42no—
156C2×Dic79#13511yescharacteristic
157D316—22no—
158D316—42no—
159C4×D79—42no—
160C4×D79—42no—
161C4×D79—42no—
162C4×D79—42no—
163C4×D79C4×D412yes—
164C4×D79C2×D812yes—
165Dic158—22no—
166Dic158—42no—
167C8×C79—42no—
168C8×C79C2×D811yescharacteristic
169C8×C79C2×D811yescharacteristic
170C8×C79C23×C411yescharacteristic
171C8×C79C2×#4111yescharacteristic
172C79⋊C8—42no—
173C79⋊C8C4×D412yes—
174C79⋊C8C2×D812yes—
175C22×C79—42no—
176C22×C79C2×#13511yescharacteristic
177C2×D79—41no—
178C2×D79—42no—
179C2×D79—84no—
180C2×D79C4×D812yes—
181C4×C79—22no—
182C4×C79—42no—
183C4×C79C2×C4×D411yescharacteristic
184C4×C79C22×D811yescharacteristic
185Dic79—41no—
186Dic79—42no—
187Dic79C4×D812yes—
188#33463—791nomaximal, Sylow
189C2×C79—21no—
190C2×C79—82no—
191C2×C79C2×C4×D811yescharacteristic
192D79—22no—
193D79—82no—
194#3414—791no—
195C2×#443—792no—
196#2179—791no—
197#2179—791no—
198#2179—792no—
199#2178—792no—
200#2178—791no—
201#2178D7911yescharacteristic, complemented
202#2161—791no—
203#2143—792no—
204C4×M32—791no—
205C79#3346311yescharacteristic, complemented, Sylow
206C2×#135—792no—
207C2×#131—1582no—
208#510—792no—
209#510—792no—
210#509—792no—
211#509—792no—
212#508—792no—
213#508—791no—
214#508C2×D7911yescharacteristic
215C2×M32—791no—
216C2×M32—791no—
217#443—792no—
218#443—792no—
219#443—792no—
220#443—1582no—
221#443—1582no—
222#443C2×D7912yescomplemented
223C2×#108—792no—
224C2×#104—1582no—
225C2×C4×C8—791no—
226#368—791no—
227#368—792no—
228#368C2×D7911yescharacteristic, complemented
229#359—792no—
230#359—792no—
231#359—792no—
232#359C2×D7912yescomplemented
233#346—791no—
234#346—791no—
235C4×C16—791no—
236C4×C16C2×D7911yescharacteristic, complemented
237C2×#41—1582no—
238#135—792no—
239#135—792no—
240#135—792no—
241#135—1582no—
242#135—1582no—
243#135C22×D7912yes—
244C2×Q16—792no—
245C2×SD16—1582no—
246C2×D8—792no—
247#131—22no—
248#131—3162no—
249#131—3162no—
250#131—1582no—
251#131—1582no—
252#131—1582no—
253C2×M16—3162no—
254C2×M16—1582no—
255C22×C8—3162no—
256C22×C8—791no—
257C22×C8—791no—
258C2×C42—791no—
259Q32—792no—
260Q32—791no—
261Q32C4×D7911yescharacteristic, complemented
262SD32—792no—
263SD32—792no—
264SD32—792no—
265SD32C4×D7912yescomplemented
266D16—792no—
267D16—791no—
268D16C4×D7911yescharacteristic, complemented
269M32—792no—
270M32—792no—
271M32—1582no—
272C2×C16—791no—
273C2×C16—791no—
274C2×C16C22×D7911yescharacteristic
275C2×C16C22×D7911yescharacteristic
276#108—792no—
277#108—1582no—
278#108C22×D7912yes—
279#104—22no—
280#104—1582no—
281#104—1582no—
282#104—1582no—
283C4×C8—791no—
284C4×C8—791no—
285C4×C8—791no—
286C4×C8C22×D7911yescharacteristic
287#41—22no—
288#41—3162no—
289#41—3162no—
290#41—1582no—
291#41—1582no—
292#41—1582no—
293C2×Q8—1582no—
294C2×D4—3162no—
295C2×D4—1582no—
296C22×C4—3162no—
297C22×C4—791no—
298C22×C4—1581no—
299Q16—792no—
300Q16—1582no—
301Q16C2×C4×D7912yes—
302SD16—22no—
303SD16—1582no—
304SD16—1582no—
305SD16—1582no—
306D8—792no—
307D8—1582no—
308D8C2×C4×D7912yes—
309M16—22no—
310M16—42no—
311M16—3162no—
312M16—3162no—
313M16—3162no—
314M16—3162no—
315M16—1582no—
316C2×C8—42no—
317C2×C8—3162no—
318C2×C8—3162no—
319C2×C8—3162no—
320C2×C8—3162no—
321C2×C8—791no—
322C2×C8—791no—
323C2×C8—792no—
324C2×C8—792no—
325C2×C8—792no—
326C2×C8—792no—
327C2×C8D4×D7911yescharacteristic
328C2×C8C23×D7911yescharacteristic
329C42—791no—
330C42—1581no—
331C42D4×D7911yescharacteristic
332C16—21no—
333C16—792no—
334C16—1581no—
335C16C2×C4×D7911yescharacteristic
336C16C2×C4×D7911yescharacteristic
337C23—3162no—
338C23—791no—
339Q8—22no—
340Q8—3162no—
341Q8—1582no—
342D4—22no—
343D4—42no—
344D4—3162no—
345D4—3162no—
346D4—3162no—
347D4—3162no—
348D4—1582no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—3162no—
352C2×C4—3162no—
353C2×C4—3162no—
354C2×C4—3162no—
355C2×C4—3162no—
356C2×C4—791no—
357C2×C4—792no—
358C2×C4—792no—
359C2×C4—1581no—
360C2×C4C2×D4×D7911yescharacteristic
361C8—42no—
362C8—3162no—
363C8—792no—
364C8—792no—
365C8D8×D7911yescharacteristic
366C8D8×D7911yescharacteristic
367C8C22×C4×D7911yescharacteristic
368C8#41×D7911yescharacteristic
369C22—42no—
370C22—3161no—
371C22—3162no—
372C22—792no—
373C22—6324no—
374C22#135×D7911yescharacteristic
375C4—22no—
376C4—42no—
377C4—3161no—
378C4—3162no—
379C4—792no—
380C4C4×D4×D7911yescharacteristic
381C4C2×D8×D7911yescharacteristic
382C2—21no—
383C2—82no—
384C2—6322no—
385C2—1582no—
386C2C4×D8×D7911yescharacteristic
subgroupG/H
centerC8D8×D79
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2178C2
socleC2×C79C2×C4×D8
prime radical#23776533C2
cosocleC2×C8×C79C24
hypercenter#2178D79
hypocenterC79#33463
LCSG ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C8 ◁ C2×C8 ◁ C4×C8 ◁ #2178

Complements

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

NH ≅ G/Ntype
2. #443×D79C2semidirect
3. #23765286C2semidirect
4. #23765285C2semidirect
5. #23765283C2semidirect
6. #23765282C2semidirect
7. #23765279C2semidirect
8. #23765193C2semidirect
9. #23765126C2semidirect
10. C4×#14579100C2semidirect
11. C79×#2178C2semidirect
14. #19099005C4semidirect
15. #19099002C4semidirect
16. #19099000C4semidirect
17. #19098997C4semidirect
22. #19098681C22semidirect
23. #19098680C22semidirect
24. #19098679C22semidirect
30. #19098393C22semidirect
31. #19098392C22semidirect
32. #19098371C22semidirect
33. #19098370C22semidirect
34. #19098369C22semidirect
35. #19098338C22semidirect
36. #19098338C22semidirect
37. C4×C79⋊C16C22semidirect
39. #443×C79C22semidirect
40. #368×C79C22semidirect
41. #359×C79C22semidirect
42. C4×C16×C79C22semidirect
63. Dic632C2×C4semidirect
64. #14579107C2×C4semidirect
65. #14579106C2×C4semidirect
66. #14579105C2×C4semidirect
67. #14579104C2×C4semidirect
68. D1264C2×C4semidirect
69. #14579102C2×C4semidirect
85. Q32×C79C2×C4semidirect
86. SD32×C79C2×C4semidirect
87. D16×C79C2×C4semidirect
201. #2178D79semidirect
205. C79#33463semidirect
222. #443C2×D79semidirect
228. #368C2×D79semidirect
232. #359C2×D79semidirect
236. C4×C16C2×D79semidirect
261. Q32C4×D79semidirect
265. SD32C4×D79semidirect
268. D16C4×D79semidirect

Maximal subgroups

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

subgroupindex
#237765332
#443×D792
#237652862
#237652852
#237652832
#237652822
#237652792
#237651932
#237651262
C4×#145791002
C79×#21782
#3346379