Finite Group Databasebeta

← #33064777

Subgroups — #33064777

Order 18688.

932 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621059C211yesmaximal, characteristic
2#22619962C211yesmaximal, characteristic
3#22619403C211yesmaximal, characteristic
4#22609208C214yesmaximal, complemented
5#22609027C214yesmaximal, complemented
6#22608616C211yesmaximal, characteristic, complemented
7#22608360C211yesmaximal, characteristic, complemented
8#22608326C211yesmaximal, characteristic, complemented
9C2×#557×C73C211yesmaximal, characteristic, complemented
10C2×#14556622—21no—
11#17989137C2211yescharacteristic
12#17988983C2212yes—
13#17988935C2212yes—
14#17988905—21no—
15#17988905—22no—
16#17988874—22no—
17#17988872C2212yes—
18#17988809—22no—
19#17988797C2212yes—
20#17988787C2212yes—
21#17988787C2212yes—
22#17988726—21no—
23C4×#11924015—22no—
24#17988640C2211yescharacteristic
25#17988640C2211yescharacteristic
26#17988611C2211yescharacteristic
27#17988586—21no—
28#17988333C2214yes—
29Q8⋊#10596831C2214yes—
30Q8⋊#10596831C2214yes—
31#557×C73—22no—
32#557×C73—22no—
33#557×C73—22no—
34#557×C73C2214yes—
35C2×C4⋊Q8×C73—21no—
36C2×C4⋊Q8×C73C2211yescharacteristic
37C2×#125×C73—22no—
38C2×#125×C73—21no—
39C2×C4×Q8×C73—21no—
40C2×C4×Q8×C73—21no—
41C2×C4×Q8×C73C2211yescharacteristic
42C2×C4×Q8×C73C2211yescharacteristic
43C2×#11924028—21no—
44#14556622—44no—
45C2×#11924018—22no—
46C2×#11924015—22no—
47C2×#11924015—22no—
48C2×#11924012—21no—
49#14556591C2311yescharacteristic
50#14556588C2311yescharacteristic
51#14556579—21no—
52#14556565—21no—
53#14556565—21no—
54C4×D292—21no—
55C4×D292—21no—
56#14556522—22no—
57#14556522—24no—
58C4⋊#10596831—21no—
59C4⋊#10596831—21no—
60C4⋊#10596831C2312yes—
61C4⋊#10596831C2312yes—
62#14556515—22no—
63#14556514—22no—
64#14556511—24no—
65#14556511—22no—
66#14556511—22no—
67C22⋊#10596831—21no—
68C22⋊#10596831—21no—
69C4×#10596831—21no—
70C4×#10596831C2312yes—
71C22×Q8×C73—21no—
72C22×Q8×C73D411yescharacteristic
73C22×Q8×C73D411yescharacteristic
74C4⋊Q8×C73—24no—
75C4⋊Q8×C73—24no—
76C4⋊Q8×C73—21no—
77C4⋊Q8×C73—21no—
78C4⋊Q8×C73D412yes—
79C4⋊Q8×C73C2312yes—
80#125×C73—28no—
81#125×C73—24no—
82#125×C73—28no—
83#125×C73—24no—
84C4×Q8×C73—24no—
85C4×Q8×C73—24no—
86C4×Q8×C73—24no—
87C4×Q8×C73—24no—
88C4×Q8×C73—21no—
89C4×Q8×C73—21no—
90C4×Q8×C73—21no—
91C4×Q8×C73—21no—
92C4×Q8×C73D412yes—
93C4×Q8×C73D412yes—
94C4×Q8×C73C2312yes—
95C4×Q8×C73C2312yes—
96C2×C4⋊C4×C73—22no—
97C2×C4⋊C4×C73—22no—
98C2×C4⋊C4×C73—22no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73D411yescharacteristic
103C2×C4⋊C4×C73D412yes—
104C2×C4⋊C4×C73D411yescharacteristic
105C2×C42×C73—22no—
106C2×C42×C73—21no—
107C2×C42×C73—21no—
108C2×C42×C73C2311yescharacteristic
109#11924028—42no—
110#11924028—42no—
111C2×Dic146—21no—
112C2×D292—21no—
113C2×D292—21no—
114C2×D292—21no—
115C2×C22⋊D73—41no—
116C2×C4×D73—41no—
117#11924018—44no—
118#11924015—44no—
119#11924015—44no—
120#11924012—42no—
121C2×#10596831—22no—
122C2×#10596831—22no—
123C2×#10596831—22no—
124C4⋊Dic73—21no—
125C4⋊Dic73—21no—
126#11924005—41no—
127#11924005—41no—
128C2×Q8×C73—21no—
129C2×Q8×C73—21no—
130C2×Q8×C73—22no—
131C2×Q8×C73—22no—
132C2×Q8×C73—22no—
133C2×Q8×C73—42no—
134C2×Q8×C73—44no—
135C2×Q8×C73SD1614yescomplemented
136C2×Q8×C73C2×D412yes—
137C2×Q8×C73C2×D412yes—
138C22×C4×C73—21no—
139C22×C4×C73—21no—
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73—22no—
143C22×C4×C73C2×D411yescharacteristic
144C22×C4×C73C2×D411yescharacteristic
145C22×C4×C73C2×D411yescharacteristic
146C4⋊C4×C73—28no—
147C4⋊C4×C73—28no—
148C4⋊C4×C73—28no—
149C4⋊C4×C73—22no—
150C4⋊C4×C73—22no—
151C4⋊C4×C73—22no—
152C4⋊C4×C73—21no—
153C4⋊C4×C73—21no—
154C4⋊C4×C73—22no—
155C4⋊C4×C73—22no—
156C4⋊C4×C73—22no—
157C4⋊C4×C73—22no—
158C4⋊C4×C73C2×D412yes—
159C4⋊C4×C73C2×D412yes—
160C4⋊C4×C73C2×D414yes—
161C42×C73—28no—
162C42×C73—22no—
163C42×C73—22no—
164C42×C73—22no—
165C42×C73—22no—
166C42×C73C2×D411yescharacteristic
167C42×C73C2×D411yescharacteristic
168C42×C73C2×D411yescharacteristic
169C42×C73C2411yescharacteristic
170C23×C73#12011yescharacteristic
171C22×D73—41no—
172Q8×C73—22no—
173Q8×C73—24no—
174Q8×C73—22no—
175Q8×C73—22no—
176Q8×C73—42no—
177Q8×C73—44no—
178Q8×C73—44no—
179Q8×C73C2×SD1614yes—
180C2×C4×C73—22no—
181C2×C4×C73—24no—
182C2×C4×C73—22no—
183C2×C4×C73—22no—
184C2×C4×C73—22no—
185C2×C4×C73—22no—
186C2×C4×C73—24no—
187C2×C4×C73—41no—
188C2×C4×C73—42no—
189C2×C4×C73—42no—
190C2×C4×C73—42no—
191C2×C4×C73—44no—
192C2×C4×C73#12011yescharacteristic
193C2×C4×C73#12011yescharacteristic
194C2×C4×C73#12011yescharacteristic
195C2×C4×C73C2×SD1612yes—
196C2×C4×C73#13712yes—
197C2×C4×C73C22×D411yescharacteristic
198C2×C4×C73C22×D411yescharacteristic
199C2×C4×C73C22×D411yescharacteristic
200C22⋊D73—82no—
201C2×Dic73—41no—
202D292—42no—
203D292—41no—
204D292—41no—
205D292—42no—
206C4×D73—82no—
207Dic146—41no—
208Dic146—41no—
209#10596831—42no—
210#10596831—42no—
211#10596831—42no—
212C22×C73—21no—
213C22×C73—21no—
214C22×C73#44812yes—
215C22×C73C2×#12011yescharacteristic
216C2×D73—81no—
217C2×D73—81no—
218C2×D73—81no—
219C4×C73—24no—
220C4×C73—21no—
221C4×C73—21no—
222C4×C73—21no—
223C4×C73—42no—
224C4×C73—44no—
225C4×C73—42no—
226C4×C73—42no—
227C4×C73#44812yes—
228C4×C73C22×SD1611yescharacteristic
229C4×C73C2×#13711yescharacteristic
230C4×C73#57512yes—
231Dic73—81no—
232#18778—731nomaximal, Sylow
233C2×C73—22no—
234C2×C73C2×#44811yescharacteristic
235C2×C73#300511yescharacteristic
236C2×C73#302011yescharacteristic
237D73—161no—
238C2×#557D7311yescharacteristic, complemented
239#3227—731no—
240#3103—731no—
241#3037—731no—
242#1653—734no—
243#1626—734no—
244#1549—731no—
245#1487—731no—
246#1479—731no—
247C73#1877811yescharacteristic, complemented, Sylow
248C2×#137—1461no—
249#557—22no—
250#557—22no—
251#557—22no—
252#557C2×D7314yes—
253#532—731no—
254C2×C4⋊Q8—21no—
255C2×C4⋊Q8C2×D7311yescharacteristic
256C2×#125—22no—
257C2×#125—21no—
258C2×C4×Q8—21no—
259C2×C4×Q8—21no—
260C2×C4×Q8C2×D7311yescharacteristic
261C2×C4×Q8C2×D7311yescharacteristic
262#498—732no—
263#486—732no—
264#481—1461no—
265#481—1462no—
266#476—1462no—
267#475—732no—
268#464—1462no—
269#462—732no—
270#460—732no—
271#460—732no—
272#448—1461no—
273C4×SD16—1462no—
274#431—731no—
275#431—731no—
276#423—731no—
277#417—1461no—
278#332—734no—
279Q8⋊C8—734no—
280Q8⋊C8—734no—
281C2×#41—1461no—
282C22×Q8—21no—
283C22×Q8C22⋊D7311yescharacteristic
284C22×Q8C22⋊D7311yescharacteristic
285#137—2924no—
286C2×Q16—1462no—
287C2×SD16—1462no—
288C2×SD16—1462no—
289C2×M16—1461no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—731no—
295C4⋊Q8C22⋊D7312yes—
296C4⋊Q8C22×D7312yes—
297#127—731no—
298#125—28no—
299#125—24no—
300#125—28no—
301#125—24no—
302#124—1461no—
303#121—1461no—
304#121—1461no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8C22⋊D7312yes—
314C4×Q8C22⋊D7312yes—
315C4×Q8C22×D7312yes—
316C4×Q8C22×D7312yes—
317C4×D4—1461no—
318C4×D4—1461no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C22⋊D7311yescharacteristic
326C2×C4⋊C4C22⋊D7311yescharacteristic
327C2×C4⋊C4C22⋊D7312yes—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C22×D7311yescharacteristic
332#106—1462no—
333#106—1464no—
334C4⋊C8—732no—
335C4⋊C8—732no—
336C4⋊C8—1461no—
337C4⋊C8—1461no—
338#103—1462no—
339#103—1462no—
340#102—1462no—
341#102—1462no—
342#102—1464no—
343C22⋊C8—1461no—
344C22⋊C8—1461no—
345C4×C8—732no—
346C4×C8—1461no—
347#41—2922no—
348#41—2922no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—1461no—
357C2×Q8#1192401514yescomplemented
358C2×Q8C2×C22⋊D7312yes—
359C2×Q8C2×C22⋊D7312yes—
360C2×D4—2921no—
361C2×D4—1461no—
362C2×D4—1461no—
363C2×D4—1461no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—2921no—
370C22×C4C2×C22⋊D7311yescharacteristic
371C22×C4C2×C22⋊D7311yescharacteristic
372C22×C4C2×C22⋊D7311yescharacteristic
373Q16—2924no—
374SD16—2924no—
375SD16—2924no—
376M16—2922no—
377C2×C8—1462no—
378C2×C8—1462no—
379C2×C8—1462no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—1461no—
393C4⋊C4—1461no—
394C4⋊C4C2×C22⋊D7312yes—
395C4⋊C4C2×C22⋊D7312yes—
396C4⋊C4C2×C22⋊D7314yes—
397C22⋊C4—2921no—
398C22⋊C4—2921no—
399C42—28no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42C2×C22⋊D7311yescharacteristic
405C42C2×C22⋊D7311yescharacteristic
406C42C2×C22⋊D7311yescharacteristic
407C42C23×D7311yescharacteristic
408C23—2921no—
409C23#1455656011yescharacteristic
410Q8—22no—
411Q8—24no—
412Q8—22no—
413Q8—22no—
414Q8—42no—
415Q8—44no—
416Q8—44no—
417Q8—2921no—
418Q8—2921no—
419Q8C2×#1192401514yes—
420D4—2922no—
421D4—2921no—
422D4—2922no—
423D4—2921no—
424D4—5842no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—2921no—
438C2×C4—5842no—
439C2×C4#1455656011yescharacteristic
440C2×C4#1455656011yescharacteristic
441C2×C4#1455656011yescharacteristic
442C2×C4C2×#1192401512yes—
443C2×C4#1455662212yes—
444C2×C4C22×C22⋊D7311yescharacteristic
445C2×C4C22×C22⋊D7311yescharacteristic
446C2×C4C22×C22⋊D7311yescharacteristic
447C8—2922no—
448C8—2922no—
449C8—2922no—
450C22—21no—
451C22—21no—
452C22—5841no—
453C22—5841no—
454C22—5841no—
455C22#1798872612yes—
456C22C2×#1455656011yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—24no—
461C4—42no—
462C4—44no—
463C4—42no—
464C4—42no—
465C4—5841no—
466C4#1798872612yes—
467C4C22×#1192401511yescharacteristic
468C4C2×#1455662211yescharacteristic
469C4#1798938512yes—
470C2—22no—
471C2—11681no—
472C2C2×#1798872611yescharacteristic
473C2#2261917311yescharacteristic
474C2#2261929511yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#557×C73C2
socleC22×C73C2×#120
prime radical#17989137C22
hypercenterC2×#557D73
hypocenterC73#18778
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #22609208C2semidirect
5. #22609027C2semidirect
6. #22608616C2semidirect
7. #22608360C2semidirect
8. #22608326C2semidirect
9. C2×#557×C73C2semidirect
135. C2×Q8×C73SD16semidirect
238. C2×#557D73semidirect
247. C73#18778semidirect
357. C2×Q8#11924015semidirect

Maximal subgroups

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

subgroupindex
#226210592
#226199622
#226194032
#226092082
#226090272
#226086162
#226083602
#226083262
C2×#557×C732
#1877873