Finite Group Databasebeta

← #33950390

Subgroups — #33950390

Order 18688.

1810 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623334C211yesmaximal, characteristic
2#22618879C218yesmaximal, complemented
3#22618876C218yesmaximal, complemented
4#22618871C211yesmaximal, characteristic, complemented
5#22618870C212yesmaximal, complemented
6#22618866C211yesmaximal, characteristic, complemented
7#22618863C212yesmaximal, complemented
8#22618861C211yesmaximal, characteristic, complemented
9#22618493C211yesmaximal, characteristic
10#424×D73C212yesmaximal
11#22618347C212yesmaximal
12C2×#17988544C211yesmaximal, characteristic
13C73×#2961C211yesmaximal, characteristic, complemented
14C22×#11924011C2211yescharacteristic
15C2×M16×D73C2212yes—
16C22×C8×D73C2211yescharacteristic
17#17989217C414yes—
18#17989216C414yes—
19#17989214C414yes—
20#17989212C414yes—
21#17989144C2211yescharacteristic
22#17989141C2218yes—
23#17989140C2212yes—
24C2×#14556548C2211yescharacteristic
25#117×D73C2212yes—
26#17988718C2214yescomplemented
27#17988716C2218yescomplemented
28#17988715C2218yescomplemented
29#17988715C2218yescomplemented
30#17988714C2214yescomplemented
31#17988714C2218yescomplemented
32Q8×#10596831C2214yescomplemented
33C8×Dic146C2218yescomplemented
34#17988711C2214yescomplemented
35#17988615C2212yes—
36#17988614C2212yes—
37#17988613C2212yes—
38#17988612C2218yes—
39#17988612C2218yes—
40#17988611C2211yescharacteristic
41#17988610C2212yes—
42C4⋊C8×D73C2218yes—
43#17988545C2212yes—
44#17988544C2214yes—
45#17988544C2214yes—
46#17988544C2214yes—
47#17988543C2212yes—
48#17988542C2212yes—
49#17988541C2214yes—
50#17988539C2214yes—
51C4×#11924011C2214yes—
52C2×#14556501C2211yescharacteristic
53C2×C4×#10596831C2211yescharacteristic
54C2×C8×Dic73C2211yescharacteristic
55#533×C73C2211yescharacteristic
56#446×C73C2218yescomplemented
57#424×C73C2212yes—
58#423×C73C2211yescharacteristic
59#405×C73C2212yes—
60C2×#97×C73C2211yescharacteristic
61C22×C4×D73C2311yescharacteristic
62C2×#11924012C2312yes—
63C2×#11924011Q818yescomplemented
64C2×#11924011C2314yes—
65C2×#11924011C2312yes—
66C2×#11924011C2312yes—
67M16×D73—28no—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C2×C8×D73—24no—
71C2×C8×D73C2314yes—
72C2×C8×D73C2314yes—
73#14556591C2×C412yes—
74#14556590C2×C414yes—
75#14556590C2×C418yes—
76#14556589C2×C418yes—
77#14556583C2×C412yes—
78#14556582C2×C414yes—
79#14556581C2×C418yes—
80#14556581C2×C418yes—
81#14556571C2×C418yes—
82#14556570C2×C414yes—
83#14556569C2×C418yes—
84#14556569C2×C418yes—
85#14556568C2×C418yes—
86#14556567C2×C418yes—
87C4×Dic146C2318yes—
88C4×Dic146C2314yes—
89#14556550C2312yes—
90#14556549C2×C414yes—
91#14556548C2×C412yes—
92#14556548C2×C412yes—
93#14556548C2314yes—
94#14556548C2314yes—
95#14556547C2×C414yes—
96C2×#11924007C2×C412yes—
97C4⋊C4×D73C2×C414yes—
98C2×#11924005C2×C412yes—
99C22⋊C4×D73C2×C414yes—
100C2×C4×Dic73C2311yescharacteristic
101C42×D73C2314yes—
102C4⋊#10596831C2314yes—
103C4⋊#10596831C2318yes—
104#14556520C2318yes—
105#14556520C2318yes—
106#14556520C2318yes—
107#14556502C2314yes—
108#14556501C2312yes—
109#14556501C2314yes—
110#14556501C2314yes—
111#14556501C2312yes—
112C4×#10596831C2312yes—
113C4×#10596831C2312yes—
114C4×#10596831C2314yes—
115C8×Dic73C2314yes—
116C8×Dic73C2314yes—
117C8×Dic73C2314yes—
118C2×M16×C73C2312yes—
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C4⋊Q8×C73C2×C412yes—
122#125×C73C2×C412yes—
123C22⋊Q8×C73C2×C414yes—
124C4×Q8×C73C2314yes—
125#117×C73C2312yes—
126C2×C42×C73C2311yescharacteristic
127C4⋊C8×C73C2314yes—
128C4⋊C8×C73C2318yes—
129#97×C73C2314yes—
130#97×C73C2314yes—
131C4×C8×C73C2314yes—
132C23×D73C22×C411yescharacteristic
133C2×Dic146C22×C418yes—
134C2×Dic146C22×C414yes—
135C22×Dic73C22×C412yes—
136C22×Dic73C22×C411yescharacteristic
137C2×C4×D73C22×C418yes—
138C2×C4×D73C22×C412yes—
139C2×C4×D73C2×Q814yes—
140C2×C4×D73#4114yes—
141C2×C4×D73C2412yes—
142#11924012—24no—
143#11924011—28no—
144#11924011C2×Q818yes—
145#11924011C2×Q818yes—
146C2×#10596831—22no—
147C2×#10596831C2×Q814yes—
148C2×#10596831C2412yes—
149C2×#10596831C2414yes—
150C2×#10596831C2411yescharacteristic
151C2×#10596831C2411yescharacteristic
152C8×D73—28no—
153C8×D73—28no—
154C4⋊Dic73C22×C414yes—
155C4⋊Dic73C22×C418yes—
156C4⋊Dic73C22×C414yes—
157#11924007C22×C418yes—
158#11924007C22×C414yes—
159#11924007C22×C414yes—
160#11924007C22×C418yes—
161#11924007C22×C418yes—
162#11924007C22×C418yes—
163C22⋊Dic73C22×C414yes—
164#11924005C22×C418yes—
165#11924005C22×C414yes—
166#11924005C22×C414yes—
167C4×Dic73C22×C418yes—
168C4×Dic73C22×C412yes—
169C4×Dic73C2414yes—
170C4×Dic73C2412yes—
171C2×Q8×C73C22×C414yes—
172C22×C4×C73C22×C412yes—
173C22×C4×C73C2411yescharacteristic
174M16×C73—24no—
175C2×C8×C73—22no—
176C2×C8×C73C2×Q814yes—
177C2×C8×C73C2414yes—
178C2×C8×C73C2412yes—
179C2×C8×C73C2412yes—
180C4⋊C4×C73C22×C414yes—
181C4⋊C4×C73C22×C418yes—
182C4⋊C4×C73C22×C414yes—
183C22⋊C4×C73C22×C414yes—
184C42×C73C22×C411yescharacteristic
185C42×C73C22×C411yescharacteristic
186C42×C73C2414yes—
187C42×C73C2412yes—
188C23×C73C23×C411yescharacteristic
189C22×D73—42no—
190C22×D73C4×Q814yes—
191C22×D73C23×C412yes—
192Q8×C73—44no—
193C2×C4×C73—42no—
194C2×C4×C73C23×C412yes—
195C2×C4×C73C23×C412yes—
196C2×C4×C73C23×C418yes—
197C2×C4×C73C23×C411yescharacteristic
198C2×C4×C73C22×Q812yes—
199C2×C4×C73C2×#4112yes—
200C2×C4×C73C2511yescharacteristic
201C2×Dic73—42no—
202C2×Dic73C4×Q814yes—
203C2×Dic73C23×C414yes—
204C2×Dic73C23×C418yes—
205C2×Dic73C23×C412yes—
206C4×D73—48no—
207C4×D73C4×Q818yes—
208C4×D73C22×Q814yes—
209C4×D73C2×#4114yes—
210Dic146—44no—
211Dic146—48no—
212C8×C73—22no—
213C8×C73—24no—
214C8×C73C22×Q814yes—
215#10596831—22no—
216#10596831—24no—
217#10596831C22×Q812yes—
218#10596831C22×Q812yes—
219C22×C73—22no—
220C22×C73C2×C4×Q812yes—
221C22×C73C24×C411yescharacteristic
222C2×D73—24no—
223C2×D73—44no—
224C2×D73C2×C4×Q814yes—
225C4×C73—22no—
226C4×C73—28no—
227C4×C73—22no—
228C4×C73C2×C4×Q812yes—
229C4×C73C23×Q811yescharacteristic
230C4×C73C22×#4111yescharacteristic
231Dic73—28no—
232Dic73—24no—
233Dic73C2×C4×Q814yes—
234#60193—731nomaximal, Sylow
235C2×C73—22no—
236C2×C73C22×C4×Q811yescharacteristic
237C2×C73#357312yes—
238D73—24no—
239#3458—731no—
240#2962—738no—
241#2962—738no—
242#2961—732no—
243#2961—731no—
244#2961D7311yescharacteristic, complemented
245#2960—731no—
246#2960—732no—
247#2960—731no—
248#2905—731no—
249C2×#424—732no—
250#2873—732no—
251C2×#405—731no—
252C73#6019311yescharacteristic, complemented, Sylow
253C22×M16—732no—
254C22×M16—731no—
255C23×C8—731no—
256#544—734no—
257#544—734no—
258#544—734no—
259#543—734no—
260#533—738no—
261#533—732no—
262#533—731no—
263#533C2×D7311yescharacteristic
264C2×#117—732no—
265C2×#117—731no—
266#446—734no—
267#446—734no—
268#446—738no—
269#446—738no—
270#446—738no—
271#446—738no—
272#446C2×D7318yescomplemented
273C8×Q8—734no—
274C8×Q8—734no—
275C8×Q8—738no—
276#424—732no—
277#424—732no—
278#424—732no—
279#424—738no—
280#424—738no—
281#424C2×D7312yes—
282#423—732no—
283#423—731no—
284#423C2×D7311yescharacteristic
285C2×C4⋊C8—738no—
286#405—732no—
287#405—732no—
288#405—732no—
289#405—734no—
290#405—734no—
291#405—734no—
292#405—734no—
293#405C2×D7312yes—
294C4×M16—734no—
295C4×M16—734no—
296C2×#97—731no—
297C2×#97C2×D7311yescharacteristic
298C2×C4×C8—731no—
299C2×C4×C8—731no—
300C23×C4—731no—
301C2×M16—732no—
302C2×M16—738no—
303C2×M16—732no—
304C2×M16—732no—
305C2×M16—734no—
306C2×M16—1468no—
307C2×M16C22×D7312yes—
308C22×C8—731no—
309C22×C8—731no—
310C22×C8—734no—
311C22×C8—734no—
312C22×C8—1464no—
313C22×C8C22×D7311yescharacteristic
314C22×C8C22×D7311yescharacteristic
315C4⋊Q8—738no—
316C4⋊Q8—738no—
317C4⋊Q8—734no—
318C4⋊Q8—732no—
319C4⋊Q8C4×D7312yes—
320#125—738no—
321#125—738no—
322#125—734no—
323#125—732no—
324#125C4×D7312yes—
325C22⋊Q8—738no—
326C22⋊Q8—734no—
327C22⋊Q8—738no—
328C22⋊Q8—738no—
329C22⋊Q8—738no—
330C22⋊Q8—738no—
331C22⋊Q8C4×D7314yes—
332C4×Q8—734no—
333C4×Q8—738no—
334C4×Q8C22×D7314yes—
335#117—732no—
336#117—734no—
337#117—734no—
338#117—732no—
339#117—732no—
340#117—734no—
341#117—734no—
342#117C22×D7312yes—
343C2×C4⋊C4—732no—
344C2×C4⋊C4—734no—
345C2×C22⋊C4—732no—
346C2×C22⋊C4—734no—
347C2×C42—731no—
348C2×C42—734no—
349C2×C42C22×D7311yescharacteristic
350C4⋊C8—738no—
351C4⋊C8—738no—
352C4⋊C8—738no—
353C4⋊C8—738no—
354C4⋊C8—734no—
355C4⋊C8C22×D7318yes—
356C4⋊C8C22×D7314yes—
357#97—732no—
358#97—732no—
359#97—734no—
360#97—734no—
361#97—734no—
362#97C22×D7314yes—
363#97C22×D7314yes—
364C4×C8—734no—
365C4×C8—734no—
366C4×C8—734no—
367C4×C8—732no—
368C4×C8—732no—
369C4×C8—734no—
370C4×C8C22×D7314yes—
371C24—731no—
372C2×Q8—738no—
373C2×Q8—734no—
374C2×Q8C2×C4×D7314yes—
375C22×C4—732no—
376C22×C4—738no—
377C22×C4—731no—
378C22×C4—732no—
379C22×C4—734no—
380C22×C4—734no—
381C22×C4—732no—
382C22×C4C2×C4×D7312yes—
383C22×C4C23×D7311yescharacteristic
384M16—24no—
385M16—738no—
386M16—738no—
387M16—1468no—
388M16—1464no—
389C2×C8—22no—
390C2×C8—734no—
391C2×C8—731no—
392C2×C8—731no—
393C2×C8—732no—
394C2×C8—734no—
395C2×C8—1468no—
396C2×C8—1468no—
397C2×C8—1462no—
398C2×C8Q8×D7314yes—
399C2×C8C23×D7314yes—
400C2×C8C23×D7312yes—
401C2×C8C23×D7312yes—
402C4⋊C4—738no—
403C4⋊C4—734no—
404C4⋊C4—734no—
405C4⋊C4—738no—
406C4⋊C4—738no—
407C4⋊C4—738no—
408C4⋊C4—734no—
409C4⋊C4—738no—
410C4⋊C4—734no—
411C4⋊C4C2×C4×D7314yes—
412C4⋊C4C2×C4×D7314yes—
413C4⋊C4C2×C4×D7318yes—
414C22⋊C4—734no—
415C22⋊C4—738no—
416C22⋊C4—734no—
417C22⋊C4—734no—
418C22⋊C4C2×C4×D7314yes—
419C42—738no—
420C42—732no—
421C42—734no—
422C42—732no—
423C42C2×C4×D7311yescharacteristic
424C42C2×C4×D7311yescharacteristic
425C42C23×D7312yes—
426C42C23×D7314yes—
427C23—2922no—
428C23—734no—
429C23—732no—
430C23C22×C4×D7311yescharacteristic
431Q8—44no—
432Q8—2924no—
433Q8—2928no—
434C2×C4—42no—
435C2×C4—2928no—
436C2×C4—2922no—
437C2×C4—732no—
438C2×C4—734no—
439C2×C4—734no—
440C2×C4—734no—
441C2×C4—738no—
442C2×C4—738no—
443C2×C4—734no—
444C2×C4C22×C4×D7318yes—
445C2×C4C22×C4×D7312yes—
446C2×C4C22×C4×D7312yes—
447C2×C4C22×C4×D7311yescharacteristic
448C2×C4C2×Q8×D7312yes—
449C2×C4#41×D7312yes—
450C2×C4C24×D7311yescharacteristic
451C8—22no—
452C8—24no—
453C8—732no—
454C8—732no—
455C8—1462no—
456C8—1464no—
457C8C2×Q8×D7314yes—
458C22—22no—
459C22—2924no—
460C22—734no—
461C22—1464no—
462C22C4×Q8×D7312yes—
463C22C23×C4×D7311yescharacteristic
464C4—22no—
465C4—22no—
466C4—28no—
467C4—734no—
468C4—1468no—
469C4—1464no—
470C4C4×Q8×D7312yes—
471C4C22×Q8×D7311yescharacteristic
472C4C2×#41×D7311yescharacteristic
473C2—22no—
474C2—1464no—
475C2C2×C4×Q8×D7311yescharacteristic
476C2#2262464412yes—
subgroupG/H
center, FrattiniC2×C4C24×D73
commutator, socleC22×C73C24×C4
FittingC73×#2961C2
prime radicalC23×D73C22×C4
cosocleC2×C4×C73C25
hypercenter#2961D73
hypocenterC73#60193
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #22618879C2semidirect
3. #22618876C2semidirect
4. #22618871C2semidirect
5. #22618870C2semidirect
6. #22618866C2semidirect
7. #22618863C2semidirect
8. #22618861C2semidirect
13. C73×#2961C2semidirect
26. #17988718C22semidirect
27. #17988716C22semidirect
28. #17988715C22semidirect
29. #17988715C22semidirect
30. #17988714C22semidirect
31. #17988714C22semidirect
32. Q8×#10596831C22semidirect
33. C8×Dic146C22semidirect
34. #17988711C22semidirect
56. #446×C73C22semidirect
63. C2×#11924011Q8semidirect
244. #2961D73semidirect
252. C73#60193semidirect
272. #446C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226233342
#226188792
#226188762
#226188712
#226188702
#226188662
#226188632
#226188612
#226184932
#424×D732
#226183472
C2×#179885442
C73×#29612
#6019373