Finite Group Databasebeta

← #34016110

Subgroups — #34016110

Order 18688.

940 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626830C211yesmaximal, characteristic, complemented
2#22626826C211yesmaximal, characteristic, complemented
3#22625215C212yesmaximal, complemented
4#22625212C212yesmaximal, complemented
5#22625202C212yesmaximal, complemented
6#22625200C212yesmaximal, complemented
7#22625170C212yesmaximal, complemented
8#22625165C212yesmaximal, complemented
9C2×#17989207C211yesmaximal, characteristic
10#17989639—24no—
11#17989636—24no—
12#17989608C2212yes—
13#17989607C2212yes—
14#17989604C2212yes—
15#17989604C2212yes—
16#17989604C2212yes—
17#17989603C2212yes—
18#17989432C2214yes—
19#17989431C2214yes—
20#17989431C2214yes—
21#17989430C2214yes—
22#17989207—24no—
23#17989207—24no—
24C2×#14556589C411yescharacteristic
25C2×#14556586C2211yescharacteristic
26C2×#14556584C2211yescharacteristic
27C2×#14556580C2211yescharacteristic
28C2×#14556577C411yescharacteristic
29C2×#14556577C2211yescharacteristic
30C2×#14556576C411yescharacteristic
31C2×#14556574C411yescharacteristic
32C2×#14556569C411yescharacteristic
33C2×#14556569C411yescharacteristic
34C2×#14556564C411yescharacteristic
35C2×#14556556C2211yescharacteristic
36C2×C4×C22⋊D73C2211yescharacteristic
37C2×#14556549C2211yescharacteristic
38C2×C22⋊Q8×C73C411yescharacteristic
39#14556658—22no—
40#14556658—22no—
41#14556658—22no—
42#14556658—22no—
43C4⋊#10596833—22no—
44C4⋊#10596833—22no—
45C22⋊#10596833C2314yes—
46C22⋊#10596833C2314yes—
47#14556646—22no—
48#14556646—22no—
49C4×#10596833—22no—
50C4×#10596833—22no—
51C4×#10596833—22no—
52C4×#10596833—22no—
53C22×Dic146C2×C411yescharacteristic
54C22×Dic146C2×C411yescharacteristic
55C22×D292C2×C411yescharacteristic
56C22×C22⋊D73C2×C411yescharacteristic
57C22×C22⋊D73C2×C411yescharacteristic
58C22×C4×D73C2×C411yescharacteristic
59C22×C4×D73C2×C411yescharacteristic
60#14556589—24no—
61#14556586—22no—
62#14556586—22no—
63#14556584—22no—
64#14556584—22no—
65#14556580—22no—
66#14556580—22no—
67#14556577—24no—
68#14556577—22no—
69#14556577—22no—
70#14556576—24no—
71#14556574—24no—
72#14556569—24no—
73#14556569—24no—
74#14556564—24no—
75#14556556D414yes—
76#14556556D414yes—
77C4×C22⋊D73—22no—
78C4×C22⋊D73—22no—
79#14556549—22no—
80#14556549—22no—
81C2×C4⋊Dic73C2311yescharacteristic
82C2×#11924007C2×C411yescharacteristic
83C2×#11924007C2×C411yescharacteristic
84C2×#11924007C2×C411yescharacteristic
85C2×#11924007C2×C411yescharacteristic
86C2×#11924007C2×C411yescharacteristic
87C2×#11924007C2×C411yescharacteristic
88C2×C22⋊Dic73C2×C411yescharacteristic
89C2×C22⋊Dic73C2311yescharacteristic
90C2×#11924005C2×C411yescharacteristic
91C2×#11924005C2×C411yescharacteristic
92C2×#11924005C2×C411yescharacteristic
93C2×#11924005C2×C411yescharacteristic
94C2×#11924005C2×C411yescharacteristic
95C2×#11924005C2×C411yescharacteristic
96C2×#11924005C2311yescharacteristic
97C2×#11924005C2311yescharacteristic
98C2×C4×Dic73C2×C411yescharacteristic
99C2×C4×Dic73C2311yescharacteristic
100C2×C4×Dic73C2311yescharacteristic
101C2×C4×Dic73C2311yescharacteristic
102C22×Q8×C73C2×C411yescharacteristic
103C23×C4×C73C2×C411yescharacteristic
104C22⋊Q8×C73—24no—
105C2×C4⋊C4×C73C2×C411yescharacteristic
106C2×C4⋊C4×C73C2×C411yescharacteristic
107C2×C4⋊C4×C73C2×C411yescharacteristic
108C2×C22⋊C4×C73C2×C411yescharacteristic
109C2×C22⋊C4×C73C2×C411yescharacteristic
110C2×#10596833—24no—
111C2×#10596833—24no—
112C23×D73C22×C411yescharacteristic
113C23×D73C22×C411yescharacteristic
114C2×Dic146—24no—
115C2×Dic146—22no—
116C2×Dic146—81no—
117C2×Dic146C22⋊C414yes—
118C2×D292—22no—
119C2×D292—81no—
120C2×C22⋊D73—22no—
121C2×C22⋊D73—22no—
122C2×C22⋊D73—24no—
123C2×C22⋊D73—81no—
124C22×Dic73C22×C411yescharacteristic
125C22×Dic73C22×C411yescharacteristic
126C22×Dic73C22×C411yescharacteristic
127C22×Dic73C22×C411yescharacteristic
128C22×Dic73C22×C411yescharacteristic
129C22×Dic73C2411yescharacteristic
130C2×C4×D73—22no—
131C2×C4×D73—24no—
132C2×C4×D73—22no—
133C2×C4×D73—81no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136#11924007—22no—
137#11924007—22no—
138#11924007—22no—
139#11924007—22no—
140#11924007C22⋊C414yes—
141#11924007C22⋊C414yes—
142C22⋊Dic73—22no—
143C22⋊Dic73—21no—
144C22⋊Dic73—21no—
145#11924005—22no—
146#11924005—22no—
147#11924005—22no—
148#11924005—22no—
149#11924005—22no—
150#11924005—22no—
151#11924005—21no—
152#11924005—21no—
153#11924005—21no—
154#11924005—21no—
155C4×Dic73—22no—
156C4×Dic73C2×D412yes—
157C4×Dic73C2×D412yes—
158C4×Dic73C2×D412yes—
159C4×Dic73C2×D412yes—
160C4×Dic73C2×D412yes—
161C4×Dic73C2×D412yes—
162C24×C73C22×C411yescharacteristic
163C2×Q8×C73—22no—
164C2×Q8×C73—81no—
165C22×C4×C73—22no—
166C22×C4×C73—81no—
167C22×C4×C73C22×C411yescharacteristic
168C22×C4×C73C22×C411yescharacteristic
169C22×C4×C73C22×C411yescharacteristic
170C22×C4×C73C22×C411yescharacteristic
171C22×C4×C73C22×C411yescharacteristic
172C22×C4×C73C22×C411yescharacteristic
173C4⋊C4×C73—22no—
174C4⋊C4×C73—22no—
175C4⋊C4×C73C22⋊C414yes—
176C22⋊C4×C73—22no—
177C22⋊C4×C73—22no—
178C23×C73—21no—
179C23×C73—22no—
180C23×C73—41no—
181C23×C73—41no—
182C23×C73C23×C411yescharacteristic
183C22×D73—21no—
184C22×D73—21no—
185C22×D73—22no—
186C22×D73—22no—
187C22×D73—41no—
188C22×D73—41no—
189C22×D73—41no—
190C22×D73—41no—
191Q8×C73—82no—
192C2×C4×C73—22no—
193C2×C4×C73—21no—
194C2×C4×C73—21no—
195C2×C4×C73—21no—
196C2×C4×C73—22no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—41no—
200C2×C4×C73—41no—
201C2×C4×C73—82no—
202C2×C4×C73C2×C22⋊C412yes—
203C2×C4×C73C2×C22⋊C412yes—
204C2×C4×C73C2×C22⋊C412yes—
205C22⋊D73—44no—
206C22⋊D73—82no—
207C2×Dic73—22no—
208C2×Dic73—21no—
209C2×Dic73—21no—
210C2×Dic73—41no—
211C2×Dic73—41no—
212C2×Dic73—41no—
213C2×Dic73—41no—
214C2×Dic73C2×C22⋊C412yes—
215C2×Dic73C2×C22⋊C412yes—
216C2×Dic73C2×C22⋊C412yes—
217C2×Dic73#13114yes—
218C2×Dic73C22×D411yescharacteristic
219C2×Dic73C22×D411yescharacteristic
220D292—82no—
221C4×D73—44no—
222C4×D73—82no—
223Dic146—28no—
224Dic146—82no—
225#10596833—44no—
226#10596833—44no—
227C22×C73—21no—
228C22×C73—21no—
229C22×C73—41no—
230C22×C73—41no—
231C22×C73—41no—
232C22×C73—82no—
233C22×C73C22×C22⋊C411yescharacteristic
234C22×C73C2×#13112yes—
235C2×D73—41no—
236C2×D73—41no—
237C2×D73—41no—
238C2×D73—41no—
239C2×D73—41no—
240C2×D73—41no—
241C2×D73—82no—
242C2×D73—82no—
243C4×C73—22no—
244C4×C73—22no—
245C4×C73—22no—
246C4×C73—41no—
247C4×C73—41no—
248C4×C73—41no—
249Dic73—22no—
250Dic73—21no—
251Dic73—21no—
252Dic73—22no—
253Dic73—41no—
254Dic73—41no—
255Dic73#40814yes—
256#17390—731nomaximal, Sylow
257C2×C73—22no—
258C2×C73—81no—
259C2×C73C2×#40811yescharacteristic
260C2×C73#288411yescharacteristic
261C2×C73#288811yescharacteristic
262D73—81no—
263D73—81no—
264C2×#542—731no—
265#2983—731no—
266#2982—731no—
267#1473—732no—
268#1473—732no—
269#1469—732no—
270#1469—732no—
271#1460—732no—
272#1458—732no—
273C73#1739011yescharacteristic, complemented, Sylow
274#542—1464no—
275#542—1464no—
276C2×C4⋊Q8—731no—
277C2×#126—731no—
278C2×#126—731no—
279C2×#124—731no—
280C2×#124—731no—
281C2×#124—731no—
282C2×#123—731no—
283C2×#123—731no—
284C2×C22⋊Q8—731no—
285C2×C22⋊Q8—731no—
286C2×C22⋊Q8#6440811yescharacteristic
287C2×#121—731no—
288C2×C4×Q8—731no—
289C2×C4×D4—731no—
290C2×#117—731no—
291#446—1464no—
292C8×Q8—1464no—
293#432—732no—
294#432—732no—
295#431—732no—
296#431—732no—
297#431—732no—
298#431—732no—
299#324—734no—
300#323—734no—
301#323—734no—
302#323—734no—
303C22×Q8—731no—
304C22×Q8—731no—
305C22×Q8C2×#6440811yescharacteristic
306C22×D4—731no—
307C22×D4—731no—
308C22×D4—731no—
309C23×C4—731no—
310C23×C4—731no—
311C23×C4C2×#6440811yescharacteristic
312C4⋊Q8—1464no—
313#126—1462no—
314#126—1462no—
315#126—1462no—
316#126—1462no—
317#124—1462no—
318#124—1462no—
319#124—1462no—
320#124—1462no—
321#124—1464no—
322#123—1464no—
323#123—1464no—
324C22⋊Q8—24no—
325C22⋊Q8—1464no—
326C22⋊Q8—1464no—
327#121—1464no—
328C4×Q8—734no—
329C4×Q8—734no—
330C4×D4—1462no—
331C4×D4—1462no—
332#117—1462no—
333#117—1462no—
334C2×C4⋊C4—731no—
335C2×C4⋊C4—731no—
336C2×C4⋊C4—731no—
337C2×C4⋊C4—731no—
338C2×C4⋊C4—731no—
339C2×C4⋊C4—731no—
340C2×C4⋊C4—731no—
341C2×C4⋊C4C2×#6440811yescharacteristic
342C2×C4⋊C4C2×#6440811yescharacteristic
343C2×C4⋊C4C2×#6440811yescharacteristic
344C2×C22⋊C4—731no—
345C2×C22⋊C4—731no—
346C2×C22⋊C4—731no—
347C2×C22⋊C4—731no—
348C2×C22⋊C4—731no—
349C2×C22⋊C4—731no—
350C2×C22⋊C4—731no—
351C2×C22⋊C4—731no—
352C2×C22⋊C4—731no—
353C2×C22⋊C4—731no—
354C2×C22⋊C4C2×#6440811yescharacteristic
355C2×C22⋊C4C2×#6440811yescharacteristic
356C2×C42—731no—
357C2×C42—731no—
358C2×C42—731no—
359C2×C42—731no—
360C4⋊C8—1462no—
361C4⋊C8—1462no—
362C4⋊C8—1462no—
363C4⋊C8—1462no—
364C4⋊C8—1462no—
365C4⋊C8—1462no—
366C22⋊C8—734no—
367C22⋊C8—734no—
368#97—1462no—
369#97—1462no—
370C4×C8—1462no—
371C4×C8—1462no—
372C4×C8—1462no—
373C4×C8—1462no—
374C24—731no—
375C24—731no—
376C24C22×#6440811yescharacteristic
377C2×Q8—22no—
378C2×Q8—81no—
379C2×Q8—5841no—
380C2×Q8—734no—
381C2×Q8—1462no—
382C2×Q8—1464no—
383C2×D4—5841no—
384C2×D4—5841no—
385C2×D4—1462no—
386C2×D4—1464no—
387C2×D4—1462no—
388C2×D4—1462no—
389C22×C4—22no—
390C22×C4—81no—
391C22×C4—5841no—
392C22×C4—731no—
393C22×C4—731no—
394C22×C4—731no—
395C22×C4—731no—
396C22×C4—731no—
397C22×C4—731no—
398C22×C4—1462no—
399C22×C4—1464no—
400C22×C4—1462no—
401C22×C4C22×#6440811yescharacteristic
402C22×C4C22×#6440811yescharacteristic
403C22×C4C22×#6440811yescharacteristic
404C22×C4C22×#6440811yescharacteristic
405C22×C4C22×#6440811yescharacteristic
406C22×C4C22×#6440811yescharacteristic
407C2×C8—1464no—
408C2×C8—1464no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—734no—
412C4⋊C4—734no—
413C4⋊C4—1462no—
414C4⋊C4—1462no—
415C4⋊C4—1462no—
416C4⋊C4—1461no—
417C4⋊C4—1461no—
418C4⋊C4—1462no—
419C4⋊C4C22⋊#6440814yes—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—1461no—
423C22⋊C4—1461no—
424C22⋊C4—1461no—
425C22⋊C4—1461no—
426C22⋊C4—1462no—
427C22⋊C4—1462no—
428C22⋊C4—1462no—
429C22⋊C4—1461no—
430C22⋊C4—1461no—
431C22⋊C4—1462no—
432C22⋊C4—1462no—
433C22⋊C4—1462no—
434C22⋊C4—1462no—
435C42—732no—
436C42—732no—
437C42—732no—
438C42—732no—
439C42—732no—
440C42—732no—
441C42—1462no—
442C23—21no—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—2921no—
447C23—2921no—
448C23—2921no—
449C23—2921no—
450C23—1461no—
451C23—1461no—
452C23—1462no—
453C23—1462no—
454C23C23×#6440811yescharacteristic
455Q8—82no—
456Q8—5842no—
457Q8—1468no—
458D4—2924no—
459D4—5842no—
460D4—5842no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—2924no—
472C2×C4—2921no—
473C2×C4—2921no—
474C2×C4—2921no—
475C2×C4—2921no—
476C2×C4—5842no—
477C2×C4—732no—
478C2×C4—734no—
479C2×C4—731no—
480C2×C4—731no—
481C2×C4—732no—
482C2×C4—732no—
483C2×C4—1461no—
484C2×C4—1461no—
485C2×C4—1462no—
486C2×C4C2×C22⋊#6440812yes—
487C2×C4C2×C22⋊#6440812yes—
488C2×C4C2×C22⋊#6440812yes—
489C8—2924no—
490C8—2924no—
491C22—21no—
492C22—21no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—82no—
497C22—2921no—
498C22—2921no—
499C22—2921no—
500C22—2921no—
501C22—2921no—
502C22—2921no—
503C22—5842no—
504C22—5842no—
505C22C22×C22⋊#6440811yescharacteristic
506C22D73⋊#13112yes—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—2921no—
514C4—2921no—
515C4—734no—
516C4—1462no—
517C4—1461no—
518C4—1461no—
519C4—1462no—
520C2—22no—
521C2—81no—
522C2—5841no—
523C2—5841no—
524C2D73⋊#40811yescharacteristic
525C2#2262653211yescharacteristic
526C2#2262654111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×C22⋊Q8×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC2×#14556564C4
cosocleC22×Dic73C24
hypercenterC2×C22⋊Q8#64408
hypocenterC73#17390
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22626830C2semidirect
2. #22626826C2semidirect
3. #22625215C2semidirect
4. #22625212C2semidirect
5. #22625202C2semidirect
6. #22625200C2semidirect
7. #22625170C2semidirect
8. #22625165C2semidirect
273. C73#17390semidirect

Maximal subgroups

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

subgroupindex
#226268302
#226268262
#226252152
#226252122
#226252022
#226252002
#226251702
#226251652
C2×#179892072
#1739073