Finite Group Databasebeta

← #34184534

Subgroups — #34184534

Order 20224.

724 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773400C211yesmaximal, characteristic
2#23764228C211yesmaximal, characteristic, complemented
3#23761894C211yesmaximal, characteristic, complemented
4#23761825C211yesmaximal, characteristic, complemented
5#23761303C211yesmaximal, characteristic, complemented
6#23761233C211yesmaximal, characteristic, complemented
7C42×D4×C79C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098876—21no—
10#19098843C412yescomplemented
11#19098758C412yescomplemented
12#19098697—21no—
13C4×#11929609—22no—
14#19098616C2211yescharacteristic
15#19098616C2211yescharacteristic
16#19098587C2211yescharacteristic
17C2×#14579089—21no—
18C2×#14579089—21no—
19#19098296—22no—
20D4⋊(C79⋊C8)—22no—
21C2×C4×D4×C79—21no—
22C2×C4×D4×C79C2211yescharacteristic
23C22×C42×C79—21no—
24C4×C4⋊C4×C79—21no—
25C4×C4⋊C4×C79C2211yescharacteristic
26C4×C22⋊C4×C79—21no—
27C4×C22⋊C4×C79—22no—
28C43×C79C2211yescharacteristic
29C2×#11929624—21no—
30C2×#11929609—22no—
31C2×#11929608—21no—
32#14579161C2×C411yescharacteristic, complemented
33#14579158C2×C411yescharacteristic, complemented
34#14579149—21no—
35#14579135—21no—
36#14579135—21no—
37C4×D316—21no—
38C4×D316—21no—
39#14579094—22no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)C2×C412yescomplemented
44#14579089—42no—
45#14579089—42no—
46#14579082—22no—
47#14579081—22no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C22×D4×C79D411yescharacteristic
53C23×C4×C79—22no—
54C23×C4×C79—21no—
55C4×D4×C79—28no—
56C4×D4×C79—21no—
57C4×D4×C79—21no—
58C4×D4×C79C2×C412yes—
59C4×D4×C79D412yes—
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79D411yescharacteristic
62C2×C22⋊C4×C79—22no—
63C2×C22⋊C4×C79—21no—
64C2×C42×C79—21no—
65C2×C42×C79—22no—
66C2×C42×C79—21no—
67C2×C42×C79—21no—
68C2×C42×C79—42no—
69C2×C42×C79—42no—
70C2×C42×C79D411yescharacteristic
71C2×C42×C79D411yescharacteristic
72C2×C42×C79C2311yescharacteristic
73#95×C79—22no—
74#95×C79D412yes—
75#11929624—41no—
76#11929624—41no—
77#11929624—41no—
78#11929624—41no—
79C2×Dic158—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×D316—21no—
83C2×C22⋊D79—41no—
84C2×C4×D79—41no—
85#11929609—44no—
86#11929608—41no—
87#11929608—41no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C4⋊Dic79—21no—
92C4⋊Dic79—21no—
93(C2×D79)⋊C4—41no—
94(C2×D79)⋊C4—41no—
95C24×C79—21no—
96C2×D4×C79—24no—
97C2×D4×C79—21no—
98C2×D4×C79C22⋊C412yes—
99C22×C4×C79—21no—
100C22×C4×C79—28no—
101C22×C4×C79—21no—
102C22×C4×C79—21no—
103C22×C4×C79—22no—
104C22×C4×C79—22no—
105C22×C4×C79—22no—
106C22×C4×C79—44no—
107C22×C4×C79—44no—
108C22×C4×C79—41no—
109C22×C4×C79—41no—
110C22×C4×C79—41no—
111C22×C4×C79—41no—
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115C4⋊C4×C79—24no—
116C4⋊C4×C79—21no—
117C4⋊C4×C79C22⋊C412yes—
118C22⋊C4×C79—28no—
119C22⋊C4×C79—22no—
120C22⋊C4×C79—22no—
121C42×C79—24no—
122C42×C79—21no—
123C42×C79—21no—
124C42×C79—21no—
125C42×C79—42no—
126C42×C79—44no—
127C42×C79—41no—
128C42×C79—41no—
129C42×C79D812yescomplemented
130C42×C79C22×C411yescharacteristic
131C42×C79C2×D411yescharacteristic
132C42×C79#4111yescharacteristic
133C42×C79#4111yescharacteristic
134C23×C79—24no—
135C23×C79—22no—
136C23×C79—42no—
137C23×C79—41no—
138C23×C79—41no—
139C23×C79#12011yescharacteristic
140C22×D79—41no—
141D4×C79—24no—
142D4×C79—22no—
143D4×C79#10414yescomplemented
144C2×C4×C79—22no—
145C2×C4×C79—22no—
146C2×C4×C79—22no—
147C2×C4×C79—22no—
148C2×C4×C79—22no—
149C2×C4×C79—24no—
150C2×C4×C79—22no—
151C2×C4×C79—41no—
152C2×C4×C79—416no—
153C2×C4×C79—41no—
154C2×C4×C79—42no—
155C2×C4×C79—42no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—42no—
160C2×C4×C79—42no—
161C2×C4×C79—42no—
162C2×C4×C79—42no—
163C2×C4×C79C2×C22⋊C411yescharacteristic
164C2×C4×C79C4×D411yescharacteristic
165C2×C4×C79C4×D411yescharacteristic
166C2×C4×C79#12111yescharacteristic
167C2×C4×C79#12111yescharacteristic
168C2×C4×C79#12311yescharacteristic
169C2×C4×C79C2×D811yescharacteristic
170C2×C4×C79#13511yescharacteristic
171C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
173C22⋊D79—81no—
174C22⋊D79—81no—
175C2×Dic79—41no—
176D316—42no—
177D316—41no—
178D316—41no—
179D316—81no—
180C4×D79—81no—
181C4×D79—81no—
182Dic158—41no—
183Dic158—41no—
184C79⋊C8—42no—
185C79⋊C8—81no—
186C79⋊C8—81no—
187C22×C79—21no—
188C22×C79—21no—
189C22×C79—24no—
190C22×C79—44no—
191C22×C79—44no—
192C22×C79—42no—
193C22×C79—42no—
194C22×C79#38611yescharacteristic
195C22×C79#44711yescharacteristic
196C22×C79#44911yescharacteristic
197C2×D79—81no—
198C2×D79—81no—
199C2×D79—81no—
200C4×C79—24no—
201C4×C79—21no—
202C4×C79—21no—
203C4×C79—21no—
204C4×C79—42no—
205C4×C79—44no—
206C4×C79—42no—
207C4×C79C2×#10412yes—
208C4×C79C4×D811yescharacteristic
209C4×C79#44211yescharacteristic
210C4×C79#45911yescharacteristic
211C4×C79#46311yescharacteristic
212Dic79—81no—
213#9233—791nomaximal, Sylow
214C2×C79—21no—
215C2×C79—21no—
216C2×C79—44no—
217C2×C79#187911yescharacteristic
218C2×C79#189611yescharacteristic
219C2×C79#190611yescharacteristic
220D79—161no—
221#3100—791no—
222C42×D4D7911yescharacteristic, complemented
223#1962—791no—
224#1502—791no—
225#1486—791no—
226#1303—791no—
227#1276—791no—
228C79#923311yescharacteristic, complemented, Sylow
229#532—791no—
230C2×C4×D4—21no—
231C2×C4×D4C2×D7911yescharacteristic
232C22×C42—21no—
233#480—1581no—
234#474—792no—
235#459—792no—
236#447—1581no—
237C4×D8—1582no—
238#431—791no—
239#431—791no—
240#423—791no—
241C2×#104—1581no—
242C2×#104—1581no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4C2×D7911yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4—22no—
247C43C2×D7911yescharacteristic
248#327—1582no—
249D4⋊C8—1582no—
250C2×#41—1581no—
251C22×D4C22⋊D7911yescharacteristic
252C23×C4—22no—
253C23×C4—21no—
254C2×D8—1582no—
255C2×M16—1581no—
256C4⋊Q8—791no—
257#127—791no—
258#124—1581no—
259#121—1581no—
260#121—1581no—
261C4×D4—28no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—1581no—
265C4×D4—1581no—
266C4×D4C4×D7912yes—
267C4×D4D31612yes—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C22⋊D7911yescharacteristic
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C22⋊D7911yescharacteristic
279C2×C42C22⋊D7911yescharacteristic
280C2×C42C22×D7911yescharacteristic
281#107—1582no—
282C4⋊C8—792no—
283C4⋊C8—1581no—
284C4⋊C8—1581no—
285C4⋊C8—1581no—
286#104—3162no—
287#104—3162no—
288#102—1582no—
289#102—1582no—
290C22⋊C8—1581no—
291C22⋊C8—1581no—
292C4×C8—1581no—
293C4×C8—1581no—
294#95—22no—
295#95C22⋊D7912yes—
296C24—21no—
297#41—3161no—
298#41—3161no—
299#41—3161no—
300#41—3161no—
301C2×Q8—1581no—
302C2×D4—24no—
303C2×D4—21no—
304C2×D4—3161no—
305C2×D4—1581no—
306C2×D4—1581no—
307C2×D4—1581no—
308C2×D4(C2×D79)⋊C412yes—
309C22×C4—21no—
310C22×C4—28no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—3161no—
323C22×C4C2×C22⋊D7911yescharacteristic
324C22×C4C2×C22⋊D7911yescharacteristic
325C22×C4C2×C22⋊D7911yescharacteristic
326D8—3164no—
327M16—3161no—
328M16—3161no—
329C2×C8—3161no—
330C2×C8—3161no—
331C2×C8—1582no—
332C4⋊C4—24no—
333C4⋊C4—21no—
334C4⋊C4—1581no—
335C4⋊C4—1581no—
336C4⋊C4(C2×D79)⋊C412yes—
337C22⋊C4—28no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—3161no—
341C22⋊C4—3161no—
342C42—24no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—44no—
347C42—42no—
348C42—41no—
349C42—41no—
350C42#1192960912yescomplemented
351C42C2×C4×D7911yescharacteristic
352C42C2×D31611yescharacteristic
353C42#1192962411yescharacteristic
354C42#1192962411yescharacteristic
355C23—24no—
356C23—22no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—3161no—
361C23#1457913011yescharacteristic
362Q8—3161no—
363Q8—3161no—
364D4—24no—
365D4—22no—
366D4—3161no—
367D4—3161no—
368D4—3162no—
369D4—6321no—
370D4—6321no—
371D4—6321no—
372D4#1457908914yescomplemented
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—416no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—3161no—
393C2×C4—6321no—
394C2×C4—6321no—
395C2×C4C2×(C2×D79)⋊C411yescharacteristic
396C2×C4C4×C22⋊D7911yescharacteristic
397C2×C4C4×C22⋊D7911yescharacteristic
398C2×C4#1457913511yescharacteristic
399C2×C4#1457913511yescharacteristic
400C2×C4#1457914411yescharacteristic
401C2×C4C2×#1192960911yescharacteristic
402C2×C4#1457918111yescharacteristic
403C2×C4#1457918611yescharacteristic
404C2×C4#1457918711yescharacteristic
405C8—3162no—
406C8—6321no—
407C8—6321no—
408C22—21no—
409C22—21no—
410C22—24no—
411C22—44no—
412C22—44no—
413C22—42no—
414C22—42no—
415C22—6321no—
416C22—6321no—
417C22—6321no—
418C22#1909844211yescharacteristic
419C22#1909869711yescharacteristic
420C22#1909870811yescharacteristic
421C4—24no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—44no—
426C4—42no—
427C4—42no—
428C4—6321no—
429C4C2×#1457908912yes—
430C4C4×#1192960911yescharacteristic
431C4#1909867311yescharacteristic
432C4#1909875811yescharacteristic
433C4#1909877811yescharacteristic
434C2—21no—
435C2—21no—
436C2—44no—
437C2—12641no—
438C2#2376374811yescharacteristic
439C2#2376384311yescharacteristic
440C2#2376390111yescharacteristic
subgroupG/H
centerC2×C4#14579135
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC42×D4×C79C2
socleC22×C79#386
prime radical#23773400C2
cosocleC2×C42×C79C23
hypercenterC42×D4D79
hypocenterC79#9233
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #23764228C2semidirect
3. #23761894C2semidirect
4. #23761825C2semidirect
5. #23761303C2semidirect
6. #23761233C2semidirect
7. C42×D4×C79C2semidirect
10. #19098843C4semidirect
11. #19098758C4semidirect
32. #14579161C2×C4semidirect
33. #14579158C2×C4semidirect
43. C4⋊(C79⋊C8)C2×C4semidirect
129. C42×C79D8semidirect
143. D4×C79#104semidirect
222. C42×D4D79semidirect
228. C79#9233semidirect
350. C42#11929609semidirect
372. D4#14579089semidirect

Maximal subgroups

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

subgroupindex
#237734002
#237642282
#237618942
#237618252
#237613032
#237612332
C42×D4×C792
#923379