Finite Group Databasebeta

← #33240448

Subgroups — #33240448

Order 18688.

690 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989017C211yesmaximal, characteristic, complemented
2C22×#14556531C211yesmaximal, characteristic, complemented
3#22619512C211yesmaximal, characteristic, complemented
4#22612046C212yesmaximal, complemented
5#22612042C212yesmaximal, complemented
6#22612037C212yesmaximal, complemented
7#22612032C212yesmaximal, complemented
8#22611713C211yesmaximal, characteristic, complemented
9#22611638C211yesmaximal, characteristic, complemented
10#22611636C211yesmaximal, characteristic, complemented
11C73×#3050C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13#17989098—21no—
14#17989017—24no—
15C2×#14556537C2211yescharacteristic, complemented
16C2×#14556535C2211yescharacteristic, complemented
17C2×#14556534C2211yescharacteristic, complemented
18C2×#14556532C2211yescharacteristic, complemented
19C2×#14556531—24no—
20C2×#14556531C2212yes—
21C2×#14556531C2212yes—
22C22×#11924003C2211yescharacteristic
23C22×#11924003C2211yescharacteristic
24#17988828C2212yes—
25#17988823C2212yes—
26#17988816—22no—
27#17988625C2211yescharacteristic
28#17988584—21no—
29#17988414C2212yescomplemented
30#17988412C2212yescomplemented
31#17988391C2212yescomplemented
32#17988390C2212yescomplemented
33#17988388C2212yescomplemented
34#17988387C2212yescomplemented
35C2×#135×C73C2211yescharacteristic
36C23×C8×C73C2211yescharacteristic
37#525×C73—21no—
38#467×C73C2212yes—
39#466×C73C2212yes—
40#465×C73—22no—
41#427×C73C2211yescharacteristic
42#416×C73—21no—
43C2×#11924028—21no—
44#14556613—22no—
45#14556613—22no—
46C2×Dic292C2311yescharacteristic
47C2×#11924017—21no—
48C2×D584C2311yescharacteristic
49#14556574—41no—
50#14556570—22no—
51#14556565—22no—
52#14556560—41no—
53C4×C22⋊D73—42no—
54#14556550—21no—
55#14556537—22no—
56#14556535—22no—
57#14556534—22no—
58#14556532—22no—
59#14556531—24no—
60#14556531—24no—
61C2×#11924003—22no—
62C2×#11924003D414yescomplemented
63C2×#11924003C2312yes—
64C2×#11924003C2311yescharacteristic
65C2×#11924003C2311yescharacteristic
66#14556525C2312yes—
67#14556523—21no—
68#14556516—22no—
69#14556513—22no—
70C2×#41×C73—21no—
71C23×C4×C73D411yescharacteristic
72#135×C73—22no—
73#135×C73—22no—
74C2×Q16×C73C2311yescharacteristic
75C2×SD16×C73—21no—
76C2×D8×C73C2311yescharacteristic
77C22×C8×C73—22no—
78C22×C8×C73—22no—
79C22×C8×C73D411yescharacteristic
80C22×C8×C73D412yes—
81C22×C8×C73C2312yes—
82C22×C8×C73C2311yescharacteristic
83#123×C73—41no—
84C22⋊Q8×C73—22no—
85#121×C73—22no—
86#120×C73—41no—
87C4×D4×C73—42no—
88#117×C73—21no—
89#107×C73C2312yes—
90#106×C73—21no—
91#103×C73—22no—
92#102×C73—22no—
93#11924028—42no—
94#11924028—42no—
95C2×Dic146—21no—
96C2×D292—21no—
97C2×C22⋊D73—42no—
98C2×C22⋊D73—41no—
99C2×C4×D73—41no—
100Dic292—21no—
101Dic292—21no—
102#11924017—42no—
103D584—21no—
104D584—21no—
105C4⋊Dic73—22no—
106#11924007—42no—
107C22⋊Dic73—41no—
108C22⋊Dic73—42no—
109#11924005—42no—
110C4×Dic73—41no—
111#11924003—22no—
112#11924003C2×D412yes—
113#11924003C2×D412yes—
114C24×C73D811yescharacteristic
115#41×C73—42no—
116#41×C73—42no—
117C2×Q8×C73—21no—
118C2×D4×C73—21no—
119C2×D4×C73—42no—
120C2×D4×C73—41no—
121C22×C4×C73—22no—
122C22×C4×C73—22no—
123C22×C4×C73—41no—
124C22×C4×C73D812yes—
125C22×C4×C73D811yescharacteristic
126C22×C4×C73C2×D412yes—
127C22×C4×C73C2×D411yescharacteristic
128Q16×C73—21no—
129Q16×C73—21no—
130SD16×C73—42no—
131D8×C73—21no—
132D8×C73—21no—
133C2×C8×C73—22no—
134C2×C8×C73—24no—
135C2×C8×C73—22no—
136C2×C8×C73—22no—
137C2×C8×C73C2×D412yes—
138C2×C8×C73C2×D412yes—
139C2×C8×C73C2×D411yescharacteristic
140C2×C8×C73#4112yes—
141C2×C8×C73C2411yescharacteristic
142C4⋊C4×C73—22no—
143C4⋊C4×C73—42no—
144C22⋊C4×C73—41no—
145C22⋊C4×C73—42no—
146C22⋊C4×C73—42no—
147C42×C73—41no—
148C23×C73—22no—
149C23×C73—42no—
150C23×C73—41no—
151C23×C73C2×D812yes—
152C23×C73C2×D811yescharacteristic
153C22×D73—41no—
154Q8×C73—41no—
155Q8×C73—41no—
156D4×C73—41no—
157D4×C73—41no—
158D4×C73—84no—
159D4×C73—82no—
160C2×C4×C73—22no—
161C2×C4×C73—22no—
162C2×C4×C73—22no—
163C2×C4×C73—24no—
164C2×C4×C73—42no—
165C2×C4×C73—41no—
166C2×C4×C73—82no—
167C2×C4×C73#12112yes—
168C2×C4×C73C2×D812yes—
169C2×C4×C73C2×D811yescharacteristic
170C2×C4×C73#13512yes—
171C2×C4×C73C22×D411yescharacteristic
172C22⋊D73—84no—
173C22⋊D73—82no—
174C2×Dic73—42no—
175C2×Dic73—41no—
176D292—41no—
177D292—41no—
178C4×D73—82no—
179Dic146—41no—
180Dic146—41no—
181C8×C73—22no—
182C8×C73#12112yes—
183C8×C73C22×D411yescharacteristic
184C8×C73C2×#4111yescharacteristic
185C22×C73—22no—
186C22×C73—22no—
187C22×C73—22no—
188C22×C73—22no—
189C22×C73—44no—
190C22×C73—82no—
191C22×C73—81no—
192C22×C73#46612yes—
193C22×C73C22×D811yescharacteristic
194C2×D73—82no—
195C2×D73—81no—
196C4×C73—22no—
197C4×C73—82no—
198C4×C73—81no—
199C4×C73#46612yes—
200C4×C73C2×#12111yescharacteristic
201C4×C73C2×#13511yescharacteristic
202Dic73—82no—
203Dic73—81no—
204#33604—731nomaximal, Sylow
205C2×C73—22no—
206C2×C73—42no—
207C2×C73—161no—
208C2×C73C2×#46611yescharacteristic
209C2×C73#341412yes—
210D73—161no—
211C2×#508—731no—
212C22×M32—731no—
213#3050—731no—
214#3050D7311yescharacteristic, complemented
215#2220—732no—
216#2220—732no—
217#2219—732no—
218#2218—732no—
219#2158—731no—
220#2138—731no—
221#2138—731no—
222C73#3360411yescharacteristic, complemented, Sylow
223C2×#135—731no—
224C2×#135C2×D7311yescharacteristic
225C23×C8C2×D7311yescharacteristic
226#525—21no—
227#525—1461no—
228#508—1464no—
229C2×Q32—731no—
230C2×SD32—731no—
231C2×SD32—731no—
232C2×D16—731no—
233C2×M32—732no—
234C2×M32—732no—
235C2×M32—1464no—
236C22×C16—731no—
237C22×C16—731no—
238#467—732no—
239#467C2×D7312yes—
240#466—732no—
241#466C2×D7312yes—
242#465—22no—
243#465—1462no—
244#427—731no—
245#427C2×D7311yescharacteristic
246#416—21no—
247#416—1461no—
248#367—732no—
249#366—732no—
250#358—732no—
251#358—732no—
252#357—732no—
253#357—732no—
254C2×#41—21no—
255C2×#41—1461no—
256C23×C4C22⋊D7311yescharacteristic
257#135—22no—
258#135—22no—
259#135—1462no—
260#135—1462no—
261C2×Q16—731no—
262C2×Q16C22×D7311yescharacteristic
263C2×SD16—21no—
264C2×SD16—1461no—
265C2×D8—731no—
266C2×D8C22×D7311yescharacteristic
267C22×C8—22no—
268C22×C8—22no—
269C22×C8C22⋊D7311yescharacteristic
270C22×C8C22⋊D7312yes—
271C22×C8C22×D7312yes—
272C22×C8C22×D7311yescharacteristic
273#123—41no—
274#123—2921no—
275C22⋊Q8—22no—
276C22⋊Q8—1462no—
277#121—22no—
278#121—1462no—
279#120—41no—
280#120—2921no—
281C4×D4—42no—
282C4×D4—2922no—
283#117—21no—
284#117—1461no—
285Q32—1462no—
286SD32—1462no—
287SD32—1462no—
288D16—1462no—
289M32—1464no—
290M32—1464no—
291C2×C16—734no—
292C2×C16—731no—
293C2×C16—731no—
294C2×C16—732no—
295C2×C16—1462no—
296#107—732no—
297#107C22×D7312yes—
298#106—21no—
299#106—1461no—
300#103—22no—
301#103—1462no—
302#102—22no—
303#102—1462no—
304C24#1192401311yescharacteristic
305#41—42no—
306#41—42no—
307#41—2922no—
308#41—2922no—
309C2×Q8—21no—
310C2×Q8—1461no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4—41no—
314C2×D4—2922no—
315C2×D4—2921no—
316C2×D4—1461no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—2921no—
321C22×C4#1192401311yescharacteristic
322C22×C4#1192401312yes—
323C22×C4C2×C22⋊D7312yes—
324C22×C4C2×C22⋊D7311yescharacteristic
325Q16—21no—
326Q16—21no—
327Q16—1461no—
328Q16—1461no—
329SD16—42no—
330SD16—2922no—
331D8—21no—
332D8—21no—
333D8—1461no—
334D8—1461no—
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8D4×D7312yes—
340C2×C8C2×C22⋊D7312yes—
341C2×C8C2×C22⋊D7311yescharacteristic
342C2×C8#1192402712yes—
343C2×C8C23×D7311yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—42no—
346C4⋊C4—2922no—
347C4⋊C4—1462no—
348C22⋊C4—41no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—2921no—
352C22⋊C4—2922no—
353C22⋊C4—2922no—
354C42—41no—
355C42—2921no—
356C16—732no—
357C16—732no—
358C16—1462no—
359C23—22no—
360C23—42no—
361C23—41no—
362C23—2921no—
363C23C2×#1192401312yes—
364C23C2×#1192401311yescharacteristic
365Q8—41no—
366Q8—41no—
367Q8—2921no—
368Q8—2921no—
369D4—41no—
370D4—41no—
371D4—84no—
372D4—82no—
373D4—2921no—
374D4—2921no—
375D4—5844no—
376D4—5842no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—82no—
384C2×C4—2922no—
385C2×C4—2921no—
386C2×C4—5842no—
387C2×C4#1455656312yes—
388C2×C4C2×#1192401312yes—
389C2×C4C2×#1192401311yescharacteristic
390C2×C4#1455661112yes—
391C2×C4C22×C22⋊D7311yescharacteristic
392C8—22no—
393C8#1455656312yes—
394C8C2×D4×D7311yescharacteristic
395C8C2×#1192402711yescharacteristic
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—44no—
401C22—82no—
402C22—81no—
403C22—5842no—
404C22—5841no—
405C22#1798882112yes—
406C22C22×#1192401311yescharacteristic
407C4—22no—
408C4—82no—
409C4—81no—
410C4—5842no—
411C4—5841no—
412C4#1798882112yes—
413C4C2×#1455656311yescharacteristic
414C4C2×#1455661111yescharacteristic
415C2—22no—
416C2—42no—
417C2—161no—
418C2—11681no—
419C2C2×#1798882111yescharacteristic
420C2#2262306512yes—
subgroupG/H
centerC2×C4C2×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3050C2
socleC22×C73C22×D8
hypercenter#3050D73
hypocenterC73#33604
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#17989017C2semidirect
2. C22×#14556531C2semidirect
3. #22619512C2semidirect
4. #22612046C2semidirect
5. #22612042C2semidirect
6. #22612037C2semidirect
7. #22612032C2semidirect
8. #22611713C2semidirect
9. #22611638C2semidirect
10. #22611636C2semidirect
11. C73×#3050C2semidirect
15. C2×#14556537C22semidirect
16. C2×#14556535C22semidirect
17. C2×#14556534C22semidirect
18. C2×#14556532C22semidirect
29. #17988414C22semidirect
30. #17988412C22semidirect
31. #17988391C22semidirect
32. #17988390C22semidirect
33. #17988388C22semidirect
34. #17988387C22semidirect
62. C2×#11924003D4semidirect
214. #3050D73semidirect
222. C73#33604semidirect

Maximal subgroups

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

subgroupindex
C2×#179890172
C22×#145565312
#226195122
#226120462
#226120422
#226120372
#226120322
#226117132
#226116382
#226116362
C73×#30502
#3360473