Finite Group Databasebeta

← #33239409

Subgroups — #33239409

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989017C211yesmaximal, characteristic
2#22618849C211yesmaximal, characteristic, complemented
3#22611815C214yesmaximal, complemented
4#22611810C214yesmaximal, complemented
5C2×#17988416C211yesmaximal, characteristic, complemented
6#22611659C211yesmaximal, characteristic, complemented
7#22611658C211yesmaximal, characteristic, complemented
8#22611528C211yesmaximal, characteristic, complemented
9C73×#2957C211yesmaximal, characteristic, complemented
10C2×#14556613C2211yescharacteristic
11#17989353—21no—
12#17989017C412yescomplemented
13#17989017C412yescomplemented
14#17989017C2212yes—
15#17989017C2212yes—
16C2×#14556537C411yescharacteristic, complemented
17C2×#14556535C411yescharacteristic, complemented
18C2×#14556534C411yescharacteristic, complemented
19C2×#14556532C411yescharacteristic, complemented
20C2×#14556531C2211yescharacteristic
21C22×#11924003C2211yescharacteristic
22#17988710—22no—
23#17988705C2214yescomplemented
24C2×#14556527C2211yescharacteristic
25#17988605—21no—
26#17988416C2212yescomplemented
27#17988416C2212yescomplemented
28#17988394C2214yescomplemented
29#17988393C2214yescomplemented
30#17988363C2212yescomplemented
31C4×#11924003C2212yescomplemented
32C2×#135×C73C2211yescharacteristic
33#568×C73—21no—
34#444×C73—22no—
35#443×C73C2214yescomplemented
36C2×#108×C73C2211yescharacteristic
37#421×C73—21no—
38#405×C73C2211yescharacteristic
39C2×#11924028—21no—
40#14556613—21no—
41#14556613—21no—
42#14556613C2×C412yes—
43#14556613C2312yes—
44C2×Dic292C2×C411yescharacteristic
45C2×#11924017—21no—
46C2×D584C2×C411yescharacteristic
47#14556599—42no—
48#14556597—24no—
49C2×#11924012—21no—
50M16×D73—42no—
51#14556537C2×C412yes—
52#14556537C2×C412yes—
53#14556535C2×C412yes—
54#14556535C2×C412yes—
55#14556534C2×C412yes—
56#14556534C2×C412yes—
57#14556532C2×C412yes—
58#14556532C2×C412yes—
59#14556531—21no—
60#14556531—21no—
61C2×#11924003C2×C411yescharacteristic
62C2×#11924003C2×C411yescharacteristic
63C2×#11924003C2×C411yescharacteristic
64C2×#11924003C2×C411yescharacteristic
65C2×#11924003C2312yes—
66C2×#11924003C2311yescharacteristic
67C2×#11924003C2311yescharacteristic
68#14556527—21no—
69#14556527C2312yes—
70#14556519—24no—
71C2×#41×C73—21no—
72#135×C73—21no—
73#135×C73—21no—
74#135×C73C2×C412yes—
75#135×C73C2312yes—
76C2×Q16×C73C2×C411yescharacteristic
77C2×SD16×C73—21no—
78C2×D8×C73C2×C411yescharacteristic
79#131×C73—24no—
80#131×C73—42no—
81C2×M16×C73—21no—
82C2×M16×C73—42no—
83C2×M16×C73D411yescharacteristic
84C22×C8×C73C2311yescharacteristic
85#117×C73D411yescharacteristic
86#108×C73—21no—
87#108×C73C2312yes—
88#104×C73—24no—
89#97×C73D412yes—
90C4×C8×C73C2312yes—
91#11924028—22no—
92#11924028—22no—
93#11924028—41no—
94#11924028—41no—
95C2×Dic146—21no—
96C2×D292—21no—
97C2×C22⋊D73—41no—
98C2×C4×D73—41no—
99Dic292—21no—
100Dic292C22×C412yes—
101#11924017—22no—
102#11924017—22no—
103D584—21no—
104D584C22×C412yes—
105#11924012—22no—
106#11924012—41no—
107#11924011—44no—
108C2×#10596831—41no—
109C8×D73—44no—
110#11924003—22no—
111#11924003C22×C411yescharacteristic
112#11924003C22×C411yescharacteristic
113#11924003C22×C411yescharacteristic
114#11924003C22×C411yescharacteristic
115#41×C73—22no—
116#41×C73—22no—
117#41×C73—41no—
118#41×C73—41no—
119C2×Q8×C73—21no—
120C2×D4×C73—21no—
121C2×D4×C73—41no—
122C22×C4×C73—41no—
123C22×C4×C73C2×D411yescharacteristic
124Q16×C73—21no—
125Q16×C73C22×C412yes—
126SD16×C73—22no—
127SD16×C73—22no—
128D8×C73—21no—
129D8×C73C22×C412yes—
130M16×C73—22no—
131M16×C73—41no—
132M16×C73—44no—
133M16×C73D812yes—
134M16×C73D812yes—
135C2×C8×C73—41no—
136C2×C8×C73—44no—
137C2×C8×C73C22×C411yescharacteristic
138C2×C8×C73C22×C411yescharacteristic
139C2×C8×C73C2×D411yescharacteristic
140C2×C8×C73C2×D412yes—
141C2×C8×C73#4111yescharacteristic
142C2×C8×C73#4111yescharacteristic
143C2×C8×C73C2411yescharacteristic
144C4⋊C4×C73—21no—
145C22⋊C4×C73—21no—
146C42×C73C2×D412yes—
147C23×C73—41no—
148C23×C73#13511yescharacteristic
149C22×D73—41no—
150Q8×C73—22no—
151Q8×C73—41no—
152D4×C73—22no—
153D4×C73—42no—
154D4×C73—42no—
155D4×C73—41no—
156C2×C4×C73—22no—
157C2×C4×C73—41no—
158C2×C4×C73—42no—
159C2×C4×C73—42no—
160C2×C4×C73C4×D411yescharacteristic
161C2×C4×C73C4×D411yescharacteristic
162C2×C4×C73C2×D811yescharacteristic
163C2×C4×C73C2×D811yescharacteristic
164C2×C4×C73#13511yescharacteristic
165C2×C4×C73C22×D411yescharacteristic
166C22⋊D73—42no—
167C22⋊D73—42no—
168C2×Dic73—41no—
169D292—22no—
170D292—41no—
171C4×D73—42no—
172C4×D73—42no—
173Dic146—22no—
174Dic146—41no—
175C8×C73—42no—
176C8×C73C4×D411yescharacteristic
177C8×C73C4×D411yescharacteristic
178C8×C73C2×D814yes—
179C8×C73C23×C411yescharacteristic
180C8×C73C2×#4111yescharacteristic
181#10596831—42no—
182C22×C73—41no—
183C22×C73—42no—
184C22×C73—161no—
185C22×C73C4×D811yescharacteristic
186C22×C73C4×D811yescharacteristic
187C22×C73C2×#13511yescharacteristic
188C2×D73—42no—
189C2×D73—161no—
190C4×C73—42no—
191C4×C73—42no—
192C4×C73C4×D811yescharacteristic
193C4×C73C4×D811yescharacteristic
194C4×C73C2×C4×D411yescharacteristic
195C4×C73C22×D811yescharacteristic
196Dic73—42no—
197#33465—731nomaximal, Sylow
198C2×C73—21no—
199C2×C73—21no—
200C2×C73—21no—
201C2×C73—82no—
202C2×C73C2×C4×D811yescharacteristic
203D73—82no—
204C2×#508—731no—
205#2957—731no—
206#2957D7311yescharacteristic, complemented
207#2179—734no—
208#2178—734no—
209C2×#368—731no—
210#2143—731no—
211#2143—731no—
212#2108—731no—
213C73#3346511yescharacteristic, complemented, Sylow
214C2×#135—731no—
215C2×#135C2×D7311yescharacteristic
216#568—21no—
217#568—1461no—
218#508—732no—
219#508—732no—
220#508—732no—
221#508—732no—
222C2×Q32—731no—
223C2×SD32—731no—
224C2×SD32—731no—
225C2×D16—731no—
226C2×M32—731no—
227C22×C16—731no—
228#444—22no—
229#444—1462no—
230#443—734no—
231#443C2×D7314yescomplemented
232C2×#108—731no—
233C2×#108C2×D7311yescharacteristic
234#421—21no—
235#421—1461no—
236#405C2×D7311yescharacteristic
237#368—732no—
238#368—732no—
239#359—734no—
240#359—734no—
241#346—732no—
242C4×C16—732no—
243C2×#41—21no—
244C2×#41—1461no—
245#135—21no—
246#135—21no—
247#135—732no—
248#135—732no—
249#135—1461no—
250#135—1461no—
251#135C4×D7312yes—
252#135C22×D7312yes—
253C2×Q16—731no—
254C2×Q16C4×D7311yescharacteristic
255C2×SD16—21no—
256C2×SD16—1461no—
257C2×D8—731no—
258C2×D8C4×D7311yescharacteristic
259#131—24no—
260#131—42no—
261#131—2922no—
262#131—1464no—
263C2×M16—21no—
264C2×M16—42no—
265C2×M16—2922no—
266C2×M16—1461no—
267C2×M16C22⋊D7311yescharacteristic
268C22×C8C22×D7311yescharacteristic
269#117C22⋊D7311yescharacteristic
270Q32—732no—
271Q32—732no—
272SD32—732no—
273SD32—732no—
274SD32—732no—
275SD32—732no—
276D16—732no—
277D16—732no—
278M32—1461no—
279M32—1461no—
280C2×C16—731no—
281C2×C16—731no—
282C2×C16—731no—
283C2×C16—731no—
284C2×C16—731no—
285C2×C16—731no—
286C2×C16—732no—
287#108—21no—
288#108—732no—
289#108—1461no—
290#108C22×D7312yes—
291#104—24no—
292#104—1464no—
293#97C22⋊D7312yes—
294C4×C8C22×D7312yes—
295#41—22no—
296#41—22no—
297#41—41no—
298#41—41no—
299#41—2921no—
300#41—2921no—
301#41—1462no—
302#41—1462no—
303C2×Q8—21no—
304C2×Q8—1461no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—2921no—
308C2×D4—1461no—
309C22×C4—41no—
310C22×C4—2921no—
311C22×C4C2×C22⋊D7311yescharacteristic
312Q16—21no—
313Q16—732no—
314Q16—1461no—
315Q16C2×C4×D7312yes—
316SD16—22no—
317SD16—22no—
318SD16—1462no—
319SD16—1462no—
320D8—21no—
321D8—732no—
322D8—1461no—
323D8C2×C4×D7312yes—
324M16—22no—
325M16—41no—
326M16—44no—
327M16—2921no—
328M16—2924no—
329M16—1462no—
330M16#1192401312yes—
331M16#1192401312yes—
332C2×C8—41no—
333C2×C8—44no—
334C2×C8—2921no—
335C2×C8—2924no—
336C2×C8C2×C4×D7311yescharacteristic
337C2×C8C2×C4×D7311yescharacteristic
338C2×C8C2×C22⋊D7311yescharacteristic
339C2×C8C2×C22⋊D7312yes—
340C2×C8#1192402811yescharacteristic
341C2×C8#1192402811yescharacteristic
342C2×C8C23×D7311yescharacteristic
343C4⋊C4—21no—
344C22⋊C4—21no—
345C42C2×C22⋊D7312yes—
346C16—731no—
347C16—731no—
348C16—731no—
349C16—731no—
350C16—1462no—
351C23—41no—
352C23—2921no—
353C23#1455661111yescharacteristic
354Q8—22no—
355Q8—41no—
356Q8—2921no—
357Q8—1462no—
358D4—22no—
359D4—42no—
360D4—42no—
361D4—41no—
362D4—2922no—
363D4—2922no—
364D4—2921no—
365D4—1462no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—2921no—
371C2×C4—2922no—
372C2×C4—2922no—
373C2×C4C4×C22⋊D7311yescharacteristic
374C2×C4C4×C22⋊D7311yescharacteristic
375C2×C4C2×#1192401311yescharacteristic
376C2×C4C2×#1192401311yescharacteristic
377C2×C4#1455661111yescharacteristic
378C2×C4C22×C22⋊D7311yescharacteristic
379C8—42no—
380C8—2922no—
381C8C4×C22⋊D7311yescharacteristic
382C8C4×C22⋊D7311yescharacteristic
383C8C2×#1192401314yes—
384C8C22×C4×D7311yescharacteristic
385C8C2×#1192402811yescharacteristic
386C22—41no—
387C22—42no—
388C22—161no—
389C22—2922no—
390C22—11681no—
391C22C4×#1192401311yescharacteristic
392C22C4×#1192401311yescharacteristic
393C22C2×#1455661111yescharacteristic
394C4—42no—
395C4—42no—
396C4—2922no—
397C4C4×#1192401311yescharacteristic
398C4C4×#1192401311yescharacteristic
399C4C2×C4×C22⋊D7311yescharacteristic
400C4C22×#1192401311yescharacteristic
401C2—21no—
402C2—21no—
403C2—21no—
404C2—82no—
405C2—5842no—
406C2C2×C4×#1192401311yescharacteristic
subgroupG/H
centerC4C22×#11924013
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2957C2
socleC2×C73C2×C4×D8
prime radicalC2×#17989017C2
cosocleC2×C8×C73C24
hypercenter#2957D73
hypocenterC73#33465
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×M16 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
2. #22618849C2semidirect
3. #22611815C2semidirect
4. #22611810C2semidirect
5. C2×#17988416C2semidirect
6. #22611659C2semidirect
7. #22611658C2semidirect
8. #22611528C2semidirect
9. C73×#2957C2semidirect
12. #17989017C4semidirect
13. #17989017C4semidirect
16. C2×#14556537C4semidirect
17. C2×#14556535C4semidirect
18. C2×#14556534C4semidirect
19. C2×#14556532C4semidirect
23. #17988705C22semidirect
26. #17988416C22semidirect
27. #17988416C22semidirect
28. #17988394C22semidirect
29. #17988393C22semidirect
30. #17988363C22semidirect
31. C4×#11924003C22semidirect
35. #443×C73C22semidirect
206. #2957D73semidirect
213. C73#33465semidirect
231. #443C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890172
#226188492
#226118152
#226118102
C2×#179884162
#226116592
#226116582
#226115282
C73×#29572
#3346573