Finite Group Databasebeta

← #33240377

Subgroups — #33240377

Order 18688.

696 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623065C212yesmaximal, complemented
2#22620359C211yesmaximal, characteristic, complemented
3#22612012C212yesmaximal, complemented
4#22612007C212yesmaximal, complemented
5#22612000C214yesmaximal, complemented
6C2×#17988408C211yesmaximal, characteristic, complemented
7#22611678C212yesmaximal, complemented
8C2×#443×C73C211yesmaximal, characteristic, complemented
9C2×#14556613—21no—
10C2×#14556613C2212yes—
11#17989137—21no—
12#17989028—22no—
13#17989027C2214yes—
14#17989023—22no—
15#17989022C2214yes—
16#17989017—24no—
17C2×#14556531C2212yes—
18#17988971—21no—
19#17988971C2212yescomplemented
20#17988967C2211yescharacteristic, complemented
21#17988964C2211yescharacteristic, complemented
22#17988960—21no—
23#17988408C2212yescomplemented
24#17988408C2212yescomplemented
25#17988402C2214yescomplemented
26#17988399C2214yescomplemented
27C2×#135×C73C2211yescharacteristic
28C2×#131×C73—21no—
29#443×C73—21no—
30#443×C73—21no—
31#443×C73C2212yes—
32#443×C73C2212yes—
33C2×#108×C73C2211yescharacteristic
34C2×#104×C73—21no—
35C2×C4×C8×C73C2211yescharacteristic
36C2×#11924028—22no—
37#14556613—22no—
38#14556613—22no—
39#14556613—22no—
40#14556613—22no—
41#14556613—41no—
42#14556613—41no—
43#14556613—41no—
44#14556613—41no—
45C2×Dic292—21no—
46C2×Dic292C2312yes—
47C2×#11924017—22no—
48C2×#11924017—22no—
49C2×D584—21no—
50C2×D584C2312yes—
51#14556591—21no—
52#14556588—21no—
53#14556579—22no—
54#14556565—42no—
55C4×D292—42no—
56#14556537—24no—
57#14556535—24no—
58#14556534—24no—
59#14556532—24no—
60#14556531—22no—
61#14556531C2314yes—
62C2×#11924003—22no—
63C2×#41×C73—21no—
64#135×C73—21no—
65#135×C73—21no—
66#135×C73D412yescomplemented
67#135×C73C2312yes—
68C2×Q16×C73D411yescharacteristic, complemented
69C2×SD16×C73—21no—
70C2×D8×C73D411yescharacteristic, complemented
71#131×C73—22no—
72#131×C73—22no—
73#131×C73—41no—
74#131×C73—41no—
75C2×M16×C73—21no—
76C2×M16×C73—41no—
77C22×C8×C73—41no—
78C22×C8×C73D411yescharacteristic
79C22×C8×C73C2311yescharacteristic
80C2×C42×C73D411yescharacteristic
81#108×C73—21no—
82#108×C73C2312yes—
83#104×C73—22no—
84#104×C73—22no—
85C4×C8×C73D411yescharacteristic
86C4×C8×C73D411yescharacteristic
87C4×C8×C73C2311yescharacteristic
88C4×C8×C73C2311yescharacteristic
89#11924028—42no—
90#11924028—42no—
91#11924028—42no—
92#11924028—42no—
93C2×Dic146—22no—
94C2×D292—22no—
95C2×D292—42no—
96C2×C22⋊D73—42no—
97C2×C4×D73—42no—
98Dic292—22no—
99Dic292—22no—
100Dic292—41no—
101Dic292—41no—
102#11924017—42no—
103#11924017—42no—
104#11924017—42no—
105#11924017—42no—
106D584—22no—
107D584—22no—
108D584—41no—
109D584—41no—
110C4⋊Dic73—42no—
111#11924005—44no—
112#11924003—24no—
113#41×C73—22no—
114#41×C73—22no—
115#41×C73—41no—
116#41×C73—41no—
117C2×Q8×C73—21no—
118C2×D4×C73—21no—
119C2×D4×C73—41no—
120C22×C4×C73—21no—
121C22×C4×C73—41no—
122C22×C4×C73C2×D411yescharacteristic
123Q16×C73—21no—
124Q16×C73C2×D412yes—
125SD16×C73—22no—
126SD16×C73—22no—
127D8×C73—21no—
128D8×C73C2×D412yes—
129M16×C73—22no—
130M16×C73—41no—
131M16×C73—42no—
132M16×C73—42no—
133C2×C8×C73—41no—
134C2×C8×C73—42no—
135C2×C8×C73—42no—
136C2×C8×C73D811yescharacteristic
137C2×C8×C73D811yescharacteristic
138C2×C8×C73D811yescharacteristic
139C2×C8×C73D811yescharacteristic
140C2×C8×C73C2×D411yescharacteristic
141C2×C8×C73C2×D411yescharacteristic
142C2×C8×C73C2×D411yescharacteristic
143C2×C8×C73C2×D411yescharacteristic
144C2×C8×C73C2×D411yescharacteristic
145C2×C8×C73#4111yescharacteristic
146C2×C8×C73#4111yescharacteristic
147C2×C8×C73C2411yescharacteristic
148C42×C73—21no—
149C42×C73C2×D411yescharacteristic
150C42×C73C2×D411yescharacteristic
151C23×C73—41no—
152C23×C73AGL(1,ℤ/8)11yescharacteristic
153C22×D73—42no—
154Q8×C73—22no—
155Q8×C73—41no—
156D4×C73—22no—
157D4×C73—42no—
158D4×C73—42no—
159D4×C73—41no—
160C2×C4×C73—21no—
161C2×C4×C73—21no—
162C2×C4×C73—42no—
163C2×C4×C73—41no—
164C2×C4×C73—42no—
165C2×C4×C73—42no—
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73C2×D811yescharacteristic
169C2×C4×C73C2×D811yescharacteristic
170C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C73C22×D411yescharacteristic
172C22⋊D73—82no—
173C22⋊D73—82no—
174C2×Dic73—42no—
175D292—42no—
176D292—42no—
177D292—84no—
178C4×D73—82no—
179C4×D73—82no—
180Dic146—42no—
181Dic146—42no—
182C8×C73—42no—
183C8×C73#12111yescharacteristic
184C8×C73#12111yescharacteristic
185C8×C73C2×D811yescharacteristic
186C8×C73C2×D811yescharacteristic
187C8×C73C2×D811yescharacteristic
188C8×C73C2×D811yescharacteristic
189C8×C73C22×D411yescharacteristic
190C8×C73C2×#4111yescharacteristic
191C22×C73—41no—
192C22×C73—42no—
193C22×C73—82no—
194C22×C73#45911yescharacteristic
195C22×C73#45911yescharacteristic
196C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
197C2×D73—84no—
198C2×D73—82no—
199C4×C73—22no—
200C4×C73—41no—
201C4×C73—42no—
202C4×C73#45911yescharacteristic
203C4×C73#45911yescharacteristic
204C4×C73C2×#12111yescharacteristic
205C4×C73C22×D811yescharacteristic
206Dic73—82no—
207#33594—731nomaximal, Sylow
208C2×C73—21no—
209C2×C73—21no—
210C2×C73—21no—
211C2×C73—82no—
212C2×C73C2×#45911yescharacteristic
213D73—162no—
214#3414—732no—
215#3147—731no—
216C2×#443D7311yescharacteristic, complemented
217#2214—732no—
218#2213—732no—
219#2212—734no—
220C2×#364—731no—
221#2148—732no—
222C73#3359411yescharacteristic, complemented, Sylow
223C2×#135—732no—
224C2×#135—1461no—
225C2×#135C2×D7311yescharacteristic
226C2×#131—21no—
227#532—1461no—
228#510—734no—
229#510—1462no—
230#509—734no—
231#509—1462no—
232#508—1464no—
233C2×M32—732no—
234#495—732no—
235#495—1461no—
236#494—731no—
237#493—731no—
238#492—1461no—
239#443—21no—
240#443—21no—
241#443C2×D7312yes—
242#443C2×D7312yes—
243C2×#108C2×D7311yescharacteristic
244C2×#104—21no—
245C2×C4×C8C2×D7311yescharacteristic
246#364—732no—
247#364—732no—
248#362—734no—
249#361—734no—
250C2×#41—21no—
251C2×#41—1462no—
252#135—21no—
253#135—21no—
254#135—2921no—
255#135—2921no—
256#135—2921no—
257#135—2921no—
258#135—1462no—
259#135—1462no—
260#135—1462no—
261#135—1462no—
262#135D29212yescomplemented
263#135C22×D7312yes—
264C2×Q16—732no—
265C2×Q16—1461no—
266C2×Q16D29211yescharacteristic, complemented
267C2×SD16—21no—
268C2×SD16—1462no—
269C2×SD16—1462no—
270C2×D8—732no—
271C2×D8—1461no—
272C2×D8D29211yescharacteristic, complemented
273#131—22no—
274#131—22no—
275#131—41no—
276#131—41no—
277C2×M16—21no—
278C2×M16—41no—
279C22×C8—41no—
280C22×C8C22⋊D7311yescharacteristic
281C22×C8C22×D7311yescharacteristic
282C4⋊Q8—1461no—
283#127—1461no—
284#124—1462no—
285#121—2922no—
286C4×D4—2922no—
287C2×C42C22⋊D7311yescharacteristic
288Q32—1464no—
289SD32—1464no—
290SD32—1464no—
291D16—1464no—
292M32—734no—
293M32—1462no—
294C2×C16—1462no—
295#108—21no—
296#108C22×D7312yes—
297#104—22no—
298#104—22no—
299C4×C8C22⋊D7311yescharacteristic
300C4×C8C22⋊D7311yescharacteristic
301C4×C8C22×D7311yescharacteristic
302C4×C8C22×D7311yescharacteristic
303#41—22no—
304#41—22no—
305#41—41no—
306#41—41no—
307#41—2922no—
308#41—2922no—
309#41—2922no—
310#41—2922no—
311C2×Q8—21no—
312C2×Q8—1462no—
313C2×D4—21no—
314C2×D4—41no—
315C2×D4—2922no—
316C2×D4—2922no—
317C2×D4—1462no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—2922no—
321C22×C4C2×C22⋊D7311yescharacteristic
322Q16—21no—
323Q16—2921no—
324Q16—2921no—
325Q16—1462no—
326Q16—1462no—
327Q16C2×D29212yes—
328SD16—22no—
329SD16—22no—
330SD16—2922no—
331SD16—2922no—
332SD16—2922no—
333SD16—2922no—
334D8—21no—
335D8—2921no—
336D8—2921no—
337D8—1462no—
338D8—1462no—
339D8C2×D29212yes—
340M16—22no—
341M16—41no—
342M16—42no—
343M16—42no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8#1192401311yescharacteristic
348C2×C8#1192401311yescharacteristic
349C2×C8#1192401311yescharacteristic
350C2×C8#1192401311yescharacteristic
351C2×C8C2×C22⋊D7311yescharacteristic
352C2×C8C2×C22⋊D7311yescharacteristic
353C2×C8C2×C22⋊D7311yescharacteristic
354C2×C8C2×D29211yescharacteristic
355C2×C8C2×D29211yescharacteristic
356C2×C8#1192402811yescharacteristic
357C2×C8#1192402811yescharacteristic
358C2×C8C23×D7311yescharacteristic
359C4⋊C4—2922no—
360C22⋊C4—2924no—
361C42—21no—
362C42C2×C22⋊D7311yescharacteristic
363C42C2×C22⋊D7311yescharacteristic
364C16—1464no—
365C23—41no—
366C23—2922no—
367C23#1455661611yescharacteristic
368Q8—22no—
369Q8—41no—
370Q8—2922no—
371Q8—2922no—
372D4—22no—
373D4—42no—
374D4—42no—
375D4—41no—
376D4—2922no—
377D4—2922no—
378D4—5844no—
379D4—5842no—
380D4—5842no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—2922no—
388C2×C4—5842no—
389C2×C4—5842no—
390C2×C4#1455656511yescharacteristic
391C2×C4#1455656511yescharacteristic
392C2×C4C2×#1192401311yescharacteristic
393C2×C4C2×#1192401311yescharacteristic
394C2×C4#1455661611yescharacteristic
395C2×C4C22×C22⋊D7311yescharacteristic
396C8—42no—
397C8#1455656511yescharacteristic
398C8#1455656511yescharacteristic
399C8C2×#1192401311yescharacteristic
400C8C2×#1192401311yescharacteristic
401C8C2×#1192401311yescharacteristic
402C8C2×#1192401311yescharacteristic
403C8C22×D29211yescharacteristic
404C8C2×#1192402811yescharacteristic
405C22—41no—
406C22—42no—
407C22—82no—
408C22—5844no—
409C22—5842no—
410C22#1798878211yescharacteristic
411C22#1798878211yescharacteristic
412C22C2×#1455661611yescharacteristic
413C4—22no—
414C4—41no—
415C4—42no—
416C4—5842no—
417C4#1798878211yescharacteristic
418C4#1798878211yescharacteristic
419C4C2×#1455656511yescharacteristic
420C4C22×#1192401311yescharacteristic
421C2—21no—
422C2—21no—
423C2—21no—
424C2—82no—
425C2—11682no—
426C2C2×#1798878211yescharacteristic
subgroupG/H
centerC4#17988782
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC2×#443×C73C2
socleC2×C73C2×#459
hypercenterC2×#443D73
hypocenterC73#33594
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
1. #22623065C2semidirect
2. #22620359C2semidirect
3. #22612012C2semidirect
4. #22612007C2semidirect
5. #22612000C2semidirect
6. C2×#17988408C2semidirect
7. #22611678C2semidirect
8. C2×#443×C73C2semidirect
19. #17988971C22semidirect
20. #17988967C22semidirect
21. #17988964C22semidirect
23. #17988408C22semidirect
24. #17988408C22semidirect
25. #17988402C22semidirect
26. #17988399C22semidirect
66. #135×C73D4semidirect
68. C2×Q16×C73D4semidirect
70. C2×D8×C73D4semidirect
216. C2×#443D73semidirect
222. C73#33594semidirect
262. #135D292semidirect
266. C2×Q16D292semidirect
272. C2×D8D292semidirect

Maximal subgroups

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

subgroupindex
#226230652
#226203592
#226120122
#226120072
#226120002
C2×#179884082
#226116782
C2×#443×C732
#3359473