Finite Group Databasebeta

← #33950383

Subgroups — #33950383

Order 18688.

1882 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D73C211yesmaximal, characteristic
2#22618883C218yesmaximal, complemented
3#22618876C218yesmaximal, complemented
4#22618872C211yesmaximal, characteristic, complemented
5#22618871C211yesmaximal, characteristic, complemented
6#22618868C211yesmaximal, characteristic, complemented
7#22618864C212yesmaximal, complemented
8#22618863C212yesmaximal, complemented
9#22618492C212yesmaximal
10#423×D73C211yesmaximal, characteristic
11#22618347C212yesmaximal
12C2×#17988539C211yesmaximal, characteristic
13C73×#2961C211yesmaximal, characteristic, complemented
14C22×#11924011C2211yescharacteristic
15C22×#11924011C2211yescharacteristic
16C2×M16×D73C2212yes—
17#17989144C2211yescharacteristic
18#17989143C412yes—
19#17989142C412yes—
20#17989141C2214yes—
21#17989140C2212yes—
22#17989139C414yes—
23C4⋊Q8×D73C412yes—
24#125×D73C412yes—
25C22⋊Q8×D73C414yes—
26C4×Q8×D73C2214yes—
27#117×D73C2212yes—
28C2×C42×D73C2211yescharacteristic
29#17988718C2218yescomplemented
30#17988716C2218yescomplemented
31#17988716C2218yescomplemented
32#17988715C2218yescomplemented
33#17988715C2218yescomplemented
34C8×Dic146C2218yescomplemented
35#17988711C2218yescomplemented
36#17988615C2212yes—
37#17988614C2212yes—
38#17988613C2212yes—
39#17988612C2218yes—
40#17988611C2211yescharacteristic
41#17988610C2212yes—
42#17988609C2214yes—
43#17988609C2218yes—
44C4⋊C8×D73C2214yes—
45#17988545C2212yes—
46#17988544C2214yes—
47#17988543C2212yes—
48#17988542C2212yes—
49#17988541C2214yes—
50#17988539C2214yes—
51#17988539C2214yes—
52#17988539C2214yes—
53C4×#11924011C2214yes—
54C2×#14556502C2211yescharacteristic
55C2×C8×Dic73C2211yescharacteristic
56C2×C8×Dic73C2211yescharacteristic
57#533×C73C2211yescharacteristic
58#446×C73C2218yescomplemented
59#424×C73C2212yes—
60#423×C73C2211yescharacteristic
61#405×C73C2212yes—
62C2×#97×C73C2211yescharacteristic
63C2×Q8×D73C2×C414yes—
64C22×C4×D73C2×C412yes—
65C22×C4×D73C2311yescharacteristic
66C2×#11924012C2312yes—
67C2×#11924011—24no—
68C2×#11924011Q818yescomplemented
69C2×#11924011C2314yes—
70C2×#11924011C2312yes—
71C2×#11924011C2312yes—
72C2×#11924011C2312yes—
73C2×#11924011C2312yes—
74M16×D73—28no—
75C22×#10596831C2311yescharacteristic
76C22×#10596831C2311yescharacteristic
77C2×C8×D73C2314yes—
78#14556591C2×C412yes—
79#14556590C2×C414yes—
80#14556590C2×C414yes—
81#14556589C2×C414yes—
82#14556583C2×C412yes—
83#14556582C2×C414yes—
84#14556581C2×C418yes—
85#14556571C2×C414yes—
86#14556570C2×C414yes—
87#14556569C2×C418yes—
88#14556568C2×C418yes—
89#14556567C2×C414yes—
90Q8×Dic73C2×C414yes—
91C4×Dic146C2318yes—
92C4×Dic146C2314yes—
93#14556556C2×C414yes—
94#14556556C2×C418yes—
95#14556550C2312yes—
96#14556549C2×C414yes—
97#14556548C2314yes—
98#14556547C2×C414yes—
99C4⋊C4×D73C2×C414yes—
100C4⋊C4×D73C2×C414yes—
101C4⋊C4×D73C2×C418yes—
102C22⋊C4×D73C2×C414yes—
103C2×C4×Dic73C2×C412yes—
104C2×C4×Dic73C2311yescharacteristic
105C42×D73C2×C411yescharacteristic
106C42×D73C2×C411yescharacteristic
107C42×D73C2×C411yescharacteristic
108C42×D73C2×C411yescharacteristic
109C42×D73C2312yes—
110C42×D73C2314yes—
111C42×D73C2312yes—
112C4⋊#10596831C2314yes—
113C4⋊#10596831C2318yes—
114#14556520C2318yes—
115#14556520C2318yes—
116#14556520C2318yes—
117#14556502C2314yes—
118#14556502C2314yes—
119#14556501C2314yes—
120#14556501C2314yes—
121C4×#10596831C2314yes—
122C8×Dic73C2314yes—
123C8×Dic73C2314yes—
124C8×Dic73C2314yes—
125C8×Dic73C2314yes—
126C2×M16×C73C2312yes—
127C22×C8×C73C2311yescharacteristic
128C22×C8×C73C2311yescharacteristic
129C4⋊Q8×C73C2×C412yes—
130#125×C73C2×C412yes—
131C22⋊Q8×C73C2×C414yes—
132C4×Q8×C73C2314yes—
133#117×C73C2312yes—
134C2×C42×C73C2311yescharacteristic
135C4⋊C8×C73C2318yes—
136C4⋊C8×C73C2314yes—
137#97×C73C2314yes—
138#97×C73C2314yes—
139C4×C8×C73C2314yes—
140C23×D73C22×C411yescharacteristic
141C2×Dic146C22×C418yes—
142C2×Dic146C22×C414yes—
143Q8×D73—48no—
144C22×Dic73C22×C412yes—
145C22×Dic73C22×C411yescharacteristic
146C2×C4×D73—44no—
147C2×C4×D73C22×C412yes—
148C2×C4×D73C22×C412yes—
149C2×C4×D73C22×C412yes—
150C2×C4×D73C22×C412yes—
151C2×C4×D73C22×C418yes—
152C2×C4×D73C22×C411yescharacteristic
153C2×C4×D73C22×C411yescharacteristic
154C2×C4×D73C2×Q814yes—
155C2×C4×D73#4114yes—
156C2×C4×D73C2411yescharacteristic
157C2×C4×D73C2411yescharacteristic
158#11924012—24no—
159#11924011—28no—
160#11924011—24no—
161#11924011—24no—
162#11924011C2×Q818yes—
163#11924011C2×Q818yescomplemented
164C2×#10596831—22no—
165C2×#10596831C2×Q814yescomplemented
166C2×#10596831C2412yes—
167C2×#10596831C2414yes—
168C2×#10596831C2412yes—
169C8×D73—28no—
170C4⋊Dic73C22×C414yes—
171C4⋊Dic73C22×C414yes—
172C4⋊Dic73C22×C418yes—
173#11924007C22×C418yes—
174#11924007C22×C418yes—
175#11924007C22×C418yes—
176#11924007C22×C418yes—
177C22⋊Dic73C22×C414yes—
178#11924005C22×C418yes—
179C4×Dic73C22×C411yescharacteristic
180C4×Dic73C22×C411yescharacteristic
181C4×Dic73C22×C412yes—
182C4×Dic73C22×C412yes—
183C4×Dic73C22×C412yes—
184C4×Dic73C22×C412yes—
185C4×Dic73C22×C418yes—
186C4×Dic73C2412yes—
187C4×Dic73C2414yes—
188C2×Q8×C73C22×C414yes—
189C22×C4×C73C22×C412yes—
190C22×C4×C73C2411yescharacteristic
191M16×C73—24no—
192C2×C8×C73—22no—
193C2×C8×C73C2×Q814yescomplemented
194C2×C8×C73C2414yes—
195C2×C8×C73C2412yes—
196C2×C8×C73C2412yes—
197C4⋊C4×C73C22×C414yes—
198C4⋊C4×C73C22×C418yes—
199C4⋊C4×C73C22×C414yes—
200C22⋊C4×C73C22×C414yes—
201C42×C73C22×C411yescharacteristic
202C42×C73C22×C411yescharacteristic
203C42×C73C2414yes—
204C42×C73C2412yes—
205C23×C73C23×C411yescharacteristic
206C22×D73—42no—
207C22×D73C4×Q814yes—
208C22×D73C23×C411yescharacteristic
209C22×D73C23×C411yescharacteristic
210Q8×C73—44no—
211C2×C4×C73—42no—
212C2×C4×C73C23×C412yes—
213C2×C4×C73C23×C412yes—
214C2×C4×C73C23×C418yes—
215C2×C4×C73C23×C411yescharacteristic
216C2×C4×C73C22×Q812yes—
217C2×C4×C73C2×#4112yes—
218C2×C4×C73C2511yescharacteristic
219C2×Dic73—24no—
220C2×Dic73C4×Q814yes—
221C2×Dic73C23×C412yes—
222C2×Dic73C23×C418yes—
223C2×Dic73C23×C411yescharacteristic
224C2×Dic73C23×C411yescharacteristic
225C2×Dic73C23×C412yes—
226C4×D73—24no—
227C4×D73—24no—
228C4×D73—42no—
229C4×D73—42no—
230C4×D73—48no—
231C4×D73C4×Q814yes—
232C4×D73C4×Q814yes—
233C4×D73C22×Q812yes—
234C4×D73C22×Q812yes—
235C4×D73C2×#4112yes—
236C4×D73C2×#4112yes—
237Dic146—216no—
238Dic146—44no—
239C8×C73—22no—
240C8×C73—24no—
241C8×C73C22×Q814yes—
242#10596831—24no—
243#10596831—22no—
244#10596831C22×Q814yes—
245C22×C73—22no—
246C22×C73C2×C4×Q812yes—
247C22×C73C24×C411yescharacteristic
248C2×D73—22no—
249C2×D73—22no—
250C2×D73—44no—
251C2×D73C2×C4×Q812yes—
252C2×D73C2×C4×Q812yes—
253C4×C73—28no—
254C4×C73—22no—
255C4×C73—22no—
256C4×C73C2×C4×Q812yes—
257C4×C73C23×Q811yescharacteristic
258C4×C73C22×#4111yescharacteristic
259Dic73—28no—
260Dic73—22no—
261Dic73C2×C4×Q812yes—
262Dic73C2×C4×Q812yes—
263Dic73#56814yes—
264#60192—731nomaximal, Sylow
265C2×C73—22no—
266C2×C73C22×C4×Q811yescharacteristic
267C2×C73C2×#56812yes—
268D73—22no—
269D73—22no—
270C2×#533—731no—
271Q8×M16—738no—
272#2962—738no—
273#2961—731no—
274#2961—731no—
275#2961—731no—
276#2961D7311yescharacteristic, complemented
277#2960—732no—
278#2960—732no—
279#2905—732no—
280C2×#423—731no—
281#2873—732no—
282C2×C4×M16—731no—
283C73#6019211yescharacteristic, complemented, Sylow
284C22×M16—732no—
285C22×M16—731no—
286C22×M16—731no—
287#533—732no—
288#533—732no—
289#533—734no—
290#533—734no—
291#533—732no—
292#533—731no—
293#533C2×D7311yescharacteristic
294C2×C4⋊Q8—732no—
295C2×#125—732no—
296C2×C22⋊Q8—734no—
297C2×C4×Q8—734no—
298C2×#117—732no—
299C22×C42—731no—
300#446—738no—
301#446—738no—
302#446—738no—
303#446—738no—
304#446—738no—
305#446C2×D7318yescomplemented
306C8×Q8—738no—
307C8×Q8—738no—
308#424—732no—
309#424—732no—
310#424—732no—
311#424—738no—
312#424C2×D7312yes—
313#423—732no—
314#423—731no—
315#423—734no—
316#423—738no—
317#423C2×D7311yescharacteristic
318C2×C4⋊C8—734no—
319#405—732no—
320#405—732no—
321#405—732no—
322#405—734no—
323#405—734no—
324#405C2×D7312yes—
325C4×M16—734no—
326C4×M16—734no—
327C4×M16—734no—
328C4×M16—734no—
329C2×#97—731no—
330C2×#97C2×D7311yescharacteristic
331C2×C4×C8—731no—
332C2×C4×C8—731no—
333C22×Q8—734no—
334C23×C4—732no—
335C23×C4—731no—
336C2×M16—732no—
337C2×M16—738no—
338C2×M16—732no—
339C2×M16—732no—
340C2×M16—732no—
341C2×M16—734no—
342C2×M16—732no—
343C2×M16—1464no—
344C2×M16—1468no—
345C2×M16C22×D7312yes—
346C22×C8—731no—
347C22×C8—731no—
348C22×C8—734no—
349C22×C8C22×D7311yescharacteristic
350C22×C8C22×D7311yescharacteristic
351C4⋊Q8—734no—
352C4⋊Q8—734no—
353C4⋊Q8—734no—
354C4⋊Q8—732no—
355C4⋊Q8C4×D7312yes—
356#125—738no—
357#125—734no—
358#125—732no—
359#125C4×D7312yes—
360C22⋊Q8—734no—
361C22⋊Q8—734no—
362C22⋊Q8—738no—
363C22⋊Q8—734no—
364C22⋊Q8—738no—
365C22⋊Q8C4×D7314yes—
366C4×Q8—734no—
367C4×Q8—734no—
368C4×Q8—738no—
369C4×Q8—738no—
370C4×Q8—734no—
371C4×Q8C22×D7314yes—
372#117—732no—
373#117—734no—
374#117—734no—
375#117—734no—
376#117C22×D7312yes—
377C2×C4⋊C4—734no—
378C2×C4⋊C4—738no—
379C2×C4⋊C4—734no—
380C2×C22⋊C4—734no—
381C2×C42—732no—
382C2×C42—731no—
383C2×C42—731no—
384C2×C42—731no—
385C2×C42—731no—
386C2×C42—731no—
387C2×C42—734no—
388C2×C42—732no—
389C2×C42—732no—
390C2×C42C22×D7311yescharacteristic
391C4⋊C8—738no—
392C4⋊C8—738no—
393C4⋊C8—738no—
394C4⋊C8—734no—
395C4⋊C8—738no—
396C4⋊C8C22×D7314yes—
397C4⋊C8C22×D7318yes—
398#97—734no—
399#97—734no—
400#97—734no—
401#97—734no—
402#97C22×D7314yes—
403#97C22×D7314yes—
404C4×C8—734no—
405C4×C8—734no—
406C4×C8—734no—
407C4×C8—734no—
408C4×C8—734no—
409C4×C8C22×D7314yes—
410C24—731no—
411C2×Q8—2928no—
412C2×Q8—734no—
413C2×Q8—738no—
414C2×Q8C2×C4×D7314yes—
415C22×C4—2924no—
416C22×C4—732no—
417C22×C4—732no—
418C22×C4—738no—
419C22×C4—731no—
420C22×C4—731no—
421C22×C4—731no—
422C22×C4—734no—
423C22×C4—734no—
424C22×C4—731no—
425C22×C4—731no—
426C22×C4—732no—
427C22×C4—732no—
428C22×C4—732no—
429C22×C4C2×C4×D7312yes—
430C22×C4C23×D7311yescharacteristic
431M16—24no—
432M16—738no—
433M16—738no—
434M16—1464no—
435M16—1464no—
436M16—1468no—
437M16—1464no—
438C2×C8—22no—
439C2×C8—734no—
440C2×C8—732no—
441C2×C8—732no—
442C2×C8—734no—
443C2×C8—1468no—
444C2×C8—1462no—
445C2×C8Q8×D7314yescomplemented
446C2×C8C23×D7312yes—
447C2×C8C23×D7314yes—
448C2×C8C23×D7312yes—
449C4⋊C4—738no—
450C4⋊C4—738no—
451C4⋊C4—738no—
452C4⋊C4—738no—
453C4⋊C4—734no—
454C4⋊C4—738no—
455C4⋊C4—734no—
456C4⋊C4C2×C4×D7314yes—
457C4⋊C4C2×C4×D7318yes—
458C4⋊C4C2×C4×D7314yes—
459C22⋊C4—734no—
460C22⋊C4—738no—
461C22⋊C4C2×C4×D7314yes—
462C42—732no—
463C42—732no—
464C42—732no—
465C42—731no—
466C42—732no—
467C42—731no—
468C42—738no—
469C42—732no—
470C42—734no—
471C42C2×C4×D7311yescharacteristic
472C42C2×C4×D7311yescharacteristic
473C42C23×D7314yes—
474C42C23×D7312yes—
475C23—2922no—
476C23—734no—
477C23—731no—
478C23—731no—
479C23C22×C4×D7311yescharacteristic
480Q8—44no—
481Q8—2924no—
482Q8—14616no—
483C2×C4—42no—
484C2×C4—2922no—
485C2×C4—2922no—
486C2×C4—2928no—
487C2×C4—731no—
488C2×C4—731no—
489C2×C4—732no—
490C2×C4—734no—
491C2×C4—732no—
492C2×C4—732no—
493C2×C4—732no—
494C2×C4—732no—
495C2×C4—734no—
496C2×C4—734no—
497C2×C4—732no—
498C2×C4—738no—
499C2×C4—1464no—
500C2×C4—1464no—
501C2×C4—1464no—
502C2×C4C22×C4×D7312yes—
503C2×C4C22×C4×D7312yes—
504C2×C4C22×C4×D7318yes—
505C2×C4C22×C4×D7311yescharacteristic
506C2×C4C2×Q8×D7312yes—
507C2×C4#41×D7312yes—
508C2×C4C24×D7311yescharacteristic
509C8—22no—
510C8—24no—
511C8—734no—
512C8—1464no—
513C8—1462no—
514C8C2×Q8×D7314yes—
515C22—22no—
516C22—2924no—
517C22—732no—
518C22—732no—
519C22—1462no—
520C22—1462no—
521C22C4×Q8×D7312yes—
522C22C23×C4×D7311yescharacteristic
523C4—22no—
524C4—22no—
525C4—28no—
526C4—734no—
527C4—732no—
528C4—732no—
529C4—1462no—
530C4—1468no—
531C4C4×Q8×D7312yes—
532C4C22×Q8×D7311yescharacteristic
533C4C2×#41×D7311yescharacteristic
534C2—22no—
535C2—1462no—
536C2—1462no—
537C2C2×C4×Q8×D7311yescharacteristic
538C2#568×D7312yes—
subgroupG/H
center, FrattiniC2×C4C24×D73
commutator, socleC22×C73C24×C4
FittingC73×#2961C2
prime radicalC23×D73C22×C4
cosocleC2×C4×C73C25
hypercenter#2961D73
hypocenterC73#60192
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #22618883C2semidirect
3. #22618876C2semidirect
4. #22618872C2semidirect
5. #22618871C2semidirect
6. #22618868C2semidirect
7. #22618864C2semidirect
8. #22618863C2semidirect
13. C73×#2961C2semidirect
29. #17988718C22semidirect
30. #17988716C22semidirect
31. #17988716C22semidirect
32. #17988715C22semidirect
33. #17988715C22semidirect
34. C8×Dic146C22semidirect
35. #17988711C22semidirect
58. #446×C73C22semidirect
68. C2×#11924011Q8semidirect
163. #11924011C2×Q8semidirect
165. C2×#10596831C2×Q8semidirect
193. C2×C8×C73C2×Q8semidirect
276. #2961D73semidirect
283. C73#60192semidirect
305. #446C2×D73semidirect
445. C2×C8Q8×D73semidirect

Maximal subgroups

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

subgroupindex
#533×D732
#226188832
#226188762
#226188722
#226188712
#226188682
#226188642
#226188632
#226184922
#423×D732
#226183472
C2×#179885392
C73×#29612
#6019273