Finite Group Databasebeta

← #33240524

Subgroups — #33240524

Order 18688.

694 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989022C211yesmaximal, characteristic, complemented
2C2×#17989004C211yesmaximal, characteristic, complemented
3#22619579C211yesmaximal, characteristic, complemented
4#22612046C212yesmaximal, complemented
5#22612032C212yesmaximal, complemented
6#22612029C212yesmaximal, complemented
7#22612015C212yesmaximal, complemented
8#22611717C211yesmaximal, characteristic, complemented
9#22611638C211yesmaximal, characteristic, complemented
10C2×#17988387C211yesmaximal, characteristic, complemented
11C73×#3061C211yesmaximal, characteristic, complemented
12C22×D584C2211yescharacteristic
13#17989151—21no—
14#17989022—24no—
15C2×#14556534C2212yes—
16C2×#14556532C2212yes—
17#17989004—24no—
18C2×#14556531C2211yescharacteristic
19C2×#14556531C2212yescomplemented
20C22×#11924003C2211yescharacteristic
21C22×#11924003C2212yescomplemented
22#17988840—22no—
23#17988823C2212yes—
24#17988823C2212yes—
25#17988625C2211yescharacteristic
26#17988593—21no—
27#17988414C2212yescomplemented
28#17988412C2212yescomplemented
29#17988391C2211yescharacteristic, complemented
30#17988391C2211yescharacteristic, complemented
31#17988388C2211yescharacteristic, complemented
32#17988388C2211yescharacteristic, complemented
33#17988387C2212yes—
34#17988387C2212yes—
35C2×#135×C73C2211yescharacteristic
36C2×#131×C73C2211yescharacteristic
37#537×C73—21no—
38#468×C73—22no—
39#467×C73C2212yescomplemented
40#466×C73C2212yescomplemented
41C2×#107×C73C2211yescharacteristic
42#417×C73—21no—
43C22×D292—21no—
44C2×D584—22no—
45C2×D584—21no—
46C2×D584—22no—
47C2×D584C2312yes—
48#14556587—41no—
49#14556580—41no—
50#14556565—22no—
51#14556565—22no—
52#14556559—42no—
53#14556550—21no—
54#14556534—24no—
55#14556532—24no—
56#14556531—24no—
57#14556531D414yescomplemented
58C2×#11924003—24no—
59C2×#11924003—22no—
60C2×#11924003C2311yescharacteristic
61C2×#11924003C2311yescharacteristic
62C2×#11924003C2312yes—
63#14556525C2311yescharacteristic
64#14556525C2311yescharacteristic
65#14556523—21no—
66#14556513—22no—
67#14556513—22no—
68C2×#41×C73—21no—
69C2×#41×C73D411yescharacteristic
70#135×C73—22no—
71#135×C73—22no—
72C2×Q16×C73C2311yescharacteristic
73C2×SD16×C73—21no—
74C2×D8×C73C2311yescharacteristic
75#131×C73—22no—
76#131×C73—22no—
77C2×M16×C73D411yescharacteristic
78C2×M16×C73D412yes—
79C22×C8×C73C2312yes—
80C22×C8×C73C2311yescharacteristic
81C22⋊Q8×C73—22no—
82C22⋊Q8×C73—41no—
83#121×C73—22no—
84#121×C73—41no—
85#121×C73—42no—
86C2×C4⋊C4×C73—21no—
87#107×C73—21no—
88#107×C73C2312yes—
89#103×C73—22no—
90#102×C73—22no—
91C23×D73—41no—
92C2×D292—22no—
93C2×D292—42no—
94C2×D292—42no—
95C2×D292—41no—
96C2×C22⋊D73—42no—
97C2×C22⋊D73—42no—
98D584—22no—
99D584—22no—
100D584—42no—
101C4⋊Dic73—21no—
102C4⋊Dic73—21no—
103C22⋊Dic73—41no—
104#11924005—42no—
105#11924005—42no—
106C4×Dic73—41no—
107#11924003—21no—
108#11924003—21no—
109#11924003C2×D414yes—
110#41×C73—22no—
111#41×C73—22no—
112#41×C73—42no—
113#41×C73—42no—
114C2×Q8×C73—21no—
115C2×Q8×C73D811yescharacteristic
116C2×D4×C73—21no—
117C2×D4×C73—41no—
118C2×D4×C73—42no—
119C2×D4×C73D812yes—
120C2×D4×C73D811yescharacteristic
121C22×C4×C73—41no—
122C22×C4×C73—41no—
123C22×C4×C73C2×D412yes—
124C22×C4×C73C2×D411yescharacteristic
125Q16×C73—21no—
126Q16×C73—21no—
127SD16×C73—42no—
128D8×C73—21no—
129D8×C73—21no—
130M16×C73—22no—
131M16×C73—24no—
132C2×C8×C73—22no—
133C2×C8×C73—22no—
134C2×C8×C73C2×D412yes—
135C2×C8×C73C2×D412yes—
136C2×C8×C73C2×D411yescharacteristic
137C2×C8×C73#4112yes—
138C2×C8×C73C2411yescharacteristic
139C4⋊C4×C73—22no—
140C4⋊C4×C73—41no—
141C4⋊C4×C73—42no—
142C22⋊C4×C73—42no—
143C22⋊C4×C73—42no—
144C23×C73—41no—
145C23×C73C2×D812yes—
146C23×C73C2×D811yescharacteristic
147C22×D73—42no—
148C22×D73—82no—
149C22×D73—161no—
150C22×D73—161no—
151Q8×C73—22no—
152Q8×C73—41no—
153Q8×C73—41no—
154D4×C73—24no—
155D4×C73—22no—
156D4×C73—41no—
157D4×C73—41no—
158D4×C73—82no—
159D4×C73—162no—
160C2×C4×C73—22no—
161C2×C4×C73—22no—
162C2×C4×C73—42no—
163C2×C4×C73—41no—
164C2×C4×C73—82no—
165C2×C4×C73—161no—
166C2×C4×C73#12112yes—
167C2×C4×C73C2×D812yes—
168C2×C4×C73C2×D811yescharacteristic
169C2×C4×C73#13512yes—
170C2×C4×C73C22×D411yescharacteristic
171C22⋊D73—162no—
172C22⋊D73—162no—
173C2×Dic73—41no—
174C2×Dic73—41no—
175D292—42no—
176D292—42no—
177D292—82no—
178C8×C73—22no—
179C8×C73#12112yes—
180C8×C73C22×D411yescharacteristic
181C8×C73C2×#4111yescharacteristic
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—22no—
185C22×C73—22no—
186C22×C73—22no—
187C22×C73—81no—
188C22×C73—81no—
189C22×C73—81no—
190C22×C73#46612yes—
191C22×C73C22×D811yescharacteristic
192C2×D73—82no—
193C2×D73—82no—
194C2×D73—82no—
195C2×D73—164no—
196C4×C73—22no—
197C4×C73—82no—
198C4×C73—81no—
199C4×C73#46612yes—
200C4×C73C2×#12111yescharacteristic
201C4×C73C2×#13511yescharacteristic
202Dic73—81no—
203Dic73—81no—
204#33612—731nomaximal, Sylow
205C2×C73—22no—
206C2×C73—42no—
207C2×C73—161no—
208C2×C73C2×#46611yescharacteristic
209C2×C73#341511yescharacteristic
210C2×C73#341611yescharacteristic
211D73—162no—
212C2×#509—731no—
213C2×#504—731no—
214#3061D7311yescharacteristic, complemented
215#3058—731no—
216#2220—732no—
217#2218—732no—
218#2217—732no—
219#2215—732no—
220#2159—731no—
221#2138—731no—
222C2×#357—731no—
223C73#3361211yescharacteristic, complemented, Sylow
224C2×#135C2×D7311yescharacteristic
225C22×D8—731no—
226C2×#131C2×D7311yescharacteristic
227#537—21no—
228#535—1461no—
229#509—1464no—
230C2×SD32—732no—
231C2×D16—732no—
232#504—1464no—
233C2×M32—731no—
234C2×M32—732no—
235C22×C16—731no—
236C22×C16—732no—
237#469—1462no—
238#468—22no—
239#467C2×D7312yescomplemented
240#466—732no—
241#466—732no—
242#466C2×D7312yescomplemented
243#427—731no—
244C2×#107C2×D7311yescharacteristic
245#418—1461no—
246#417—21no—
247#367—732no—
248#366—732no—
249#358—731no—
250#358—731no—
251#357—731no—
252#357—731no—
253#357—732no—
254#357—732no—
255C2×#41—21no—
256C2×#41C22⋊D7311yescharacteristic
257C22×D4—1461no—
258#135—22no—
259#135—22no—
260C2×Q16C22×D7311yescharacteristic
261C2×SD16—21no—
262C2×D8—732no—
263C2×D8—1462no—
264C2×D8—1461no—
265C2×D8—1462no—
266C2×D8C22×D7311yescharacteristic
267#131—22no—
268#131—22no—
269C2×M16C22⋊D7311yescharacteristic
270C2×M16C22⋊D7312yes—
271C22×C8C22×D7312yes—
272C22×C8C22×D7311yescharacteristic
273#127—2921no—
274#124—2921no—
275C22⋊Q8—22no—
276C22⋊Q8—41no—
277#121—22no—
278#121—41no—
279#121—42no—
280#121—1462no—
281#121—1462no—
282#120—2922no—
283#117—1461no—
284C2×C4⋊C4—21no—
285SD32—1464no—
286D16—1464no—
287M32—734no—
288M32—1464no—
289C2×C16—732no—
290C2×C16—731no—
291C2×C16—731no—
292C2×C16—1464no—
293C2×C16—1462no—
294#107—21no—
295#107—731no—
296#107—731no—
297#107C22×D7312yes—
298#106—1461no—
299#103—22no—
300#102—22no—
301#102—1462no—
302#102—1462no—
303C24—2921no—
304#41—22no—
305#41—22no—
306#41—42no—
307#41—42no—
308C2×Q8—21no—
309C2×Q8#1192401311yescharacteristic
310C2×D4—21no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—2922no—
314C2×D4—2922no—
315C2×D4—2922no—
316C2×D4—2921no—
317C2×D4—2922no—
318C2×D4—1462no—
319C2×D4#1192401312yes—
320C2×D4#1192401311yescharacteristic
321C22×C4—41no—
322C22×C4—41no—
323C22×C4C2×C22⋊D7312yes—
324C22×C4C2×C22⋊D7311yescharacteristic
325Q16—21no—
326Q16—21no—
327SD16—42no—
328D8—21no—
329D8—21no—
330D8—2922no—
331D8—1462no—
332D8—1462no—
333M16—22no—
334M16—24no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8D4×D7312yes—
338C2×C8C2×C22⋊D7312yes—
339C2×C8C2×C22⋊D7311yescharacteristic
340C2×C8#1192402712yes—
341C2×C8C23×D7311yescharacteristic
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—1461no—
346C4⋊C4—1461no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—2921no—
350C22⋊C4—2922no—
351C22⋊C4—2922no—
352C42—2921no—
353C16—734no—
354C16—1461no—
355C16—1461no—
356C23—41no—
357C23—2922no—
358C23—5842no—
359C23—11681no—
360C23—11681no—
361C23C2×#1192401312yes—
362C23C2×#1192401311yescharacteristic
363Q8—22no—
364Q8—41no—
365Q8—41no—
366D4—24no—
367D4—22no—
368D4—41no—
369D4—41no—
370D4—82no—
371D4—162no—
372D4—2922no—
373D4—2922no—
374D4—5842no—
375D4—11682no—
376D4—11682no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—82no—
382C2×C4—161no—
383C2×C4—2921no—
384C2×C4—2921no—
385C2×C4#1455656312yes—
386C2×C4C2×#1192401312yes—
387C2×C4C2×#1192401311yescharacteristic
388C2×C4#1455661112yes—
389C2×C4C22×C22⋊D7311yescharacteristic
390C8—22no—
391C8#1455656312yes—
392C8C2×D4×D7311yescharacteristic
393C8C2×#1192402711yescharacteristic
394C22—21no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—5842no—
403C22—5842no—
404C22—5842no—
405C22—11684no—
406C22#1798882112yes—
407C22C22×#1192401311yescharacteristic
408C4—22no—
409C4—82no—
410C4—81no—
411C4—5841no—
412C4—5841no—
413C4#1798882112yes—
414C4C2×#1455656311yescharacteristic
415C4C2×#1455661111yescharacteristic
416C2—22no—
417C2—42no—
418C2—161no—
419C2—11682no—
420C2C2×#1798882111yescharacteristic
421C2#2262307011yescharacteristic
422C2#2262307511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3061C2
socleC22×C73C22×D8
hypercenter#3061D73
hypocenterC73#33612
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

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

NH ≅ G/Ntype
1. C2×#17989022C2semidirect
2. C2×#17989004C2semidirect
3. #22619579C2semidirect
4. #22612046C2semidirect
5. #22612032C2semidirect
6. #22612029C2semidirect
7. #22612015C2semidirect
8. #22611717C2semidirect
9. #22611638C2semidirect
10. C2×#17988387C2semidirect
11. C73×#3061C2semidirect
19. C2×#14556531C22semidirect
21. C22×#11924003C22semidirect
27. #17988414C22semidirect
28. #17988412C22semidirect
29. #17988391C22semidirect
30. #17988391C22semidirect
31. #17988388C22semidirect
32. #17988388C22semidirect
39. #467×C73C22semidirect
40. #466×C73C22semidirect
57. #14556531D4semidirect
214. #3061D73semidirect
223. C73#33612semidirect
239. #467C2×D73semidirect
242. #466C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890222
C2×#179890042
#226195792
#226120462
#226120322
#226120292
#226120152
#226117172
#226116382
C2×#179883872
C73×#30612
#3361273