Finite Group Databasebeta

← #33950520

Subgroups — #33950520

Order 18688.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623320C211yesmaximal, characteristic
2#22618910C218yesmaximal, complemented
3#22618901C218yesmaximal, complemented
4#22618871C211yesmaximal, characteristic, complemented
5#22618861C211yesmaximal, characteristic, complemented
6#22618643C211yesmaximal, characteristic, complemented
7#22618632C212yesmaximal, complemented
8#22618628C212yesmaximal, complemented
9#22618578C212yesmaximal, complemented
10#22618568C211yesmaximal, characteristic, complemented
11#131×Dic73C212yesmaximal, complemented
12C2×#17988544C211yesmaximal, characteristic, complemented
13C73×#2932C211yesmaximal, characteristic, complemented
14C2×#14556598C2212yes—
15C22×#11924011C2211yescharacteristic
16C22×C8×D73C2211yescharacteristic
17#17989206C414yes—
18#17989198C412yes—
19#17989191C412yes—
20#17989182C414yes—
21#17989174C414yes—
22#17989144C2211yescharacteristic
23#17989117C2218yes—
24#17989094C2212yes—
25#41×Dic73C2212yes—
26C2×#14556548C2211yescharacteristic
27#17988715C2218yescomplemented
28#17988714C2214yescomplemented
29#17988711C2214yescomplemented
30#17988665C2214yescomplemented
31#17988663C2218yes—
32#17988661C2218yes—
33#17988658C2218yes—
34D4×#10596831C2214yescomplemented
35#17988650C2218yes—
36#17988647C2218yes—
37#17988645C2218yescomplemented
38#17988641C2218yescomplemented
39#17988615C2212yes—
40#17988613C2212yes—
41#17988611C2211yescharacteristic
42#17988554C2212yes—
43#17988550C2212yes—
44#17988545C2212yes—
45#17988545C2214yescomplemented
46#17988544C2214yes—
47#17988544C2214yes—
48#17988542C2212yes—
49#17988542C2214yescomplemented
50M16×Dic73C2212yes—
51M16×Dic73C2214yescomplemented
52C2×#14556501C2211yescharacteristic
53C2×C4×#10596831C2211yescharacteristic
54C2×C8×Dic73C2211yescharacteristic
55C2×C8×Dic73C2214yescomplemented
56C2×#131×C73C2212yes—
57#532×C73C2211yescharacteristic
58#435×C73C2218yescomplemented
59#423×C73C2211yescharacteristic
60#408×C73C2212yes—
61C2×#97×C73C2211yescharacteristic
62C22×C4×D73C2311yescharacteristic
63#14556598—28no—
64C2×#11924012C2312yes—
65C2×#11924012C2314yes—
66C2×#11924011—24no—
67C2×#11924011C2314yes—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2314yes—
70C22×#10596831C2311yescharacteristic
71C2×C8×D73—24no—
72C2×C8×D73C2312yes—
73C2×C8×D73C2312yes—
74#14556591C2×C412yes—
75#14556586C2×C418yes—
76#14556584C2×C418yes—
77#14556583C2×C412yes—
78#14556581C2×C418yes—
79#14556577C2×C418yes—
80#14556575C2×C418yes—
81#14556572C2×C414yes—
82#14556572C2×C418yes—
83#14556570C2×C414yes—
84#14556568C2×C418yes—
85#14556567C2×C414yes—
86#14556563C2×C418yes—
87#14556562C2×C414yes—
88#14556559C2×C414yes—
89Q8×Dic73C2×C414yes—
90C4×Dic146C2314yes—
91D4×Dic73C2×C418yes—
92D4×Dic73C2×C414yes—
93C4×C22⋊D73C2318yes—
94#14556550C2312yes—
95#14556550C2314yes—
96#14556548C2×C412yes—
97#14556548C2×C412yes—
98#14556548C2314yes—
99C2×#11924007C2×C412yes—
100C2×#11924005C2×C412yes—
101C2×C4×Dic73C2314yes—
102C2×C4×Dic73C2311yescharacteristic
103C4⋊#10596831C2314yes—
104#14556520C2318yes—
105#14556520C2318yes—
106C22⋊#10596831C2318yes—
107#14556503C2318yes—
108#14556501C2314yes—
109#14556501C2314yes—
110#14556501C2314yes—
111#14556501C2314yes—
112C4×#10596831C2312yes—
113C4×#10596831C2312yes—
114C8×Dic73C2314yes—
115C8×Dic73C2314yes—
116C8×Dic73C2314yes—
117C8×Dic73C2312yes—
118C8×Dic73C2312yes—
119C2×#41×C73C2312yes—
120#131×C73—28no—
121C2×M16×C73C2312yes—
122C2×M16×C73C2314yes—
123C22×C8×C73C2311yescharacteristic
124C22×C8×C73C2314yes—
125C22×C8×C73C2311yescharacteristic
126C4⋊Q8×C73C2×C411yescharacteristic
127#127×C73C2×C411yescharacteristic
128#124×C73C2×C412yes—
129#121×C73C2×C414yes—
130C4×D4×C73C2314yes—
131C2×C42×C73C2311yescharacteristic
132C4⋊C8×C73C2314yes—
133C22⋊C8×C73C2318yes—
134#97×C73C2314yes—
135C23×D73C22×C411yescharacteristic
136C2×Dic146C22×C414yes—
137C2×C22⋊D73C22×C418yes—
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C412yes—
140C22×Dic73C22×C418yes—
141C2×C4×D73—22no—
142C2×C4×D73—22no—
143C2×C4×D73C22×C412yes—
144C2×C4×D73C2412yes—
145#11924012—24no—
146#11924012—28no—
147#11924011—28no—
148C2×#10596831—22no—
149C2×#10596831—28no—
150C2×#10596831#4114yes—
151C2×#10596831C2414yes—
152C2×#10596831C2412yes—
153C2×#10596831C2411yescharacteristic
154C2×#10596831C2411yescharacteristic
155C8×D73—24no—
156C8×D73—24no—
157C4⋊Dic73C22×C414yes—
158C4⋊Dic73C22×C418yes—
159C4⋊Dic73C22×C414yes—
160#11924007C22×C414yes—
161#11924007C22×C414yes—
162#11924007C22×C418yes—
163#11924007C22×C418yes—
164C22⋊Dic73C22×C414yes—
165C22⋊Dic73C22×C418yes—
166C22⋊Dic73C22×C418yes—
167#11924005C22×C414yes—
168#11924005C22×C414yes—
169#11924005C22×C418yes—
170C4×Dic73C22×C412yes—
171C4×Dic73C22×C418yes—
172C4×Dic73C2412yes—
173C4×Dic73C2414yes—
174#41×C73—24no—
175#41×C73—24no—
176C2×Q8×C73C22×C412yes—
177C2×D4×C73C22×C414yes—
178C2×D4×C73C22×C412yes—
179C2×D4×C73C22×C414yes—
180C22×C4×C73C22×C412yes—
181C22×C4×C73C2414yes—
182C22×C4×C73C2411yescharacteristic
183M16×C73—24no—
184M16×C73—28no—
185C2×C8×C73—22no—
186C2×C8×C73—28no—
187C2×C8×C73#4114yes—
188C2×C8×C73C2412yes—
189C2×C8×C73C2414yes—
190C2×C8×C73C2412yes—
191C4⋊C4×C73C22×C414yes—
192C22⋊C4×C73C22×C418yes—
193C42×C73C22×C411yescharacteristic
194C42×C73C22×C411yescharacteristic
195C42×C73C2412yes—
196C23×C73C23×C414yes—
197C23×C73C23×C411yescharacteristic
198C22×D73—22no—
199C22×D73—42no—
200C22×D73C23×C412yes—
201Q8×C73—24no—
202D4×C73—28no—
203D4×C73—24no—
204D4×C73—44no—
205C2×C4×C73—24no—
206C2×C4×C73—24no—
207C2×C4×C73—42no—
208C2×C4×C73C23×C412yes—
209C2×C4×C73C23×C412yes—
210C2×C4×C73C23×C414yes—
211C2×C4×C73C23×C411yescharacteristic
212C2×C4×C73C2×#4112yes—
213C2×C4×C73C2×#4112yes—
214C2×C4×C73C2511yescharacteristic
215C22⋊D73—48no—
216C2×Dic73—22no—
217C2×Dic73—48no—
218C2×Dic73—42no—
219C2×Dic73C23×C414yes—
220C2×Dic73C23×C418yes—
221C2×Dic73C23×C412yes—
222C4×D73—24no—
223C4×D73—22no—
224C4×D73—22no—
225Dic146—44no—
226C8×C73—24no—
227C8×C73—22no—
228C8×C73C2×#4114yes—
229#10596831—22no—
230#10596831—24no—
231#10596831C2×#4112yes—
232#10596831C2×#4112yes—
233C22×C73—22no—
234C22×C73—24no—
235C22×C73—28no—
236C22×C73C4×#4112yes—
237C22×C73C24×C411yescharacteristic
238C2×D73—22no—
239C2×D73—24no—
240C2×D73—44no—
241C4×C73—24no—
242C4×C73—22no—
243C4×C73—22no—
244C4×C73C4×#4112yes—
245C4×C73C22×#4111yescharacteristic
246C4×C73C22×#4111yescharacteristic
247Dic73—22no—
248Dic73—28no—
249Dic73—24no—
250#60206—731nomaximal, Sylow
251C2×C73—22no—
252C2×C73—44no—
253C2×C73C2×C4×#4111yescharacteristic
254C2×C73#357312yes—
255D73—42no—
256#3455—731no—
257#2967—738no—
258#2966—738no—
259#2961—731no—
260#2960—731no—
261#2932D7311yescharacteristic, complemented
262#2931—731no—
263#2930—732no—
264#2929—732no—
265#2923—732no—
266#2921—731no—
267C4×#131—732no—
268C2×#405—731no—
269C73#6020611yescharacteristic, complemented, Sylow
270C2×#131—732no—
271C2×#131C2×D7312yes—
272C22×M16—731no—
273C23×C8—731no—
274#542—734no—
275#541—732no—
276#540—732no—
277#539—734no—
278#538—734no—
279#533—731no—
280#532C2×D7311yescharacteristic
281#529—738no—
282#525—732no—
283C4×#41—732no—
284C2×#117—731no—
285#446—734no—
286#446—738no—
287C8×Q8—734no—
288#436—738no—
289#436—734no—
290#435—738no—
291#435—738no—
292#435C2×D7318yescomplemented
293C8×D4—734no—
294C8×D4—738no—
295#433—738no—
296#433—738no—
297#432—738no—
298#424—732no—
299#424—732no—
300#423—731no—
301#423C2×D7311yescharacteristic
302#408—732no—
303#408C2×D7312yes—
304#407—732no—
305#405—732no—
306#405—734no—
307#405—732no—
308#405—734no—
309#405—734no—
310#405—734no—
311C4×M16—732no—
312C4×M16—734no—
313C2×#97—731no—
314C2×#97C2×D7311yescharacteristic
315C2×C4×C8—731no—
316C2×C4×C8—734no—
317C2×C4×C8—731no—
318C2×#41C22×D7312yes—
319C23×C4—731no—
320#131—28no—
321#131—1468no—
322C2×M16—732no—
323C2×M16—734no—
324C2×M16—734no—
325C2×M16—1464no—
326C2×M16C22×D7312yes—
327C2×M16C22×D7314yes—
328C22×C8—731no—
329C22×C8—731no—
330C22×C8—734no—
331C22×C8—732no—
332C22×C8—732no—
333C22×C8—1464no—
334C22×C8C22×D7311yescharacteristic
335C22×C8C22×D7314yes—
336C22×C8C22×D7311yescharacteristic
337C4⋊Q8—732no—
338C4⋊Q8C4×D7311yescharacteristic
339#127C4×D7311yescharacteristic
340#126—738no—
341#126—738no—
342#125—738no—
343#125—732no—
344#124—738no—
345#124C4×D7312yes—
346#123—734no—
347#123—738no—
348#123—738no—
349C22⋊Q8—734no—
350C22⋊Q8—734no—
351C22⋊Q8—738no—
352#121—734no—
353#121—738no—
354#121C4×D7314yes—
355#120—734no—
356C4×Q8—734no—
357C4×Q8—734no—
358C4×D4—738no—
359C4×D4—734no—
360C4×D4—738no—
361C4×D4C22×D7314yes—
362#117—732no—
363#117—734no—
364#117—732no—
365#117—732no—
366#117—734no—
367C2×C4⋊C4—732no—
368C2×C22⋊C4—732no—
369C2×C42—731no—
370C2×C42—734no—
371C2×C42C22×D7311yescharacteristic
372C4⋊C8—738no—
373C4⋊C8—738no—
374C4⋊C8—734no—
375C4⋊C8C22×D7314yes—
376C22⋊C8—738no—
377C22⋊C8—738no—
378C22⋊C8C22×D7318yes—
379#97—734no—
380#97—734no—
381#97—734no—
382#97—734no—
383#97C22×D7314yes—
384C4×C8—732no—
385C4×C8—732no—
386C4×C8—734no—
387C4×C8—734no—
388C4×C8—734no—
389C4×C8—732no—
390C4×C8—732no—
391C24—731no—
392#41—24no—
393#41—24no—
394C2×Q8—734no—
395C2×Q8C2×C4×D7312yes—
396C2×D4—738no—
397C2×D4C2×C4×D7314yes—
398C2×D4C2×C4×D7312yes—
399C2×D4C2×C4×D7314yes—
400C22×C4—732no—
401C22×C4—738no—
402C22×C4—731no—
403C22×C4—732no—
404C22×C4—732no—
405C22×C4—1462no—
406C22×C4—1462no—
407C22×C4C2×C4×D7312yes—
408C22×C4C23×D7314yes—
409C22×C4C23×D7311yescharacteristic
410M16—24no—
411M16—28no—
412M16—1468no—
413M16—1464no—
414M16—1468no—
415C2×C8—22no—
416C2×C8—28no—
417C2×C8—734no—
418C2×C8—731no—
419C2×C8—731no—
420C2×C8—734no—
421C2×C8—732no—
422C2×C8—1464no—
423C2×C8—1464no—
424C2×C8—1468no—
425C2×C8—1462no—
426C2×C8#1192402714yes—
427C2×C8C23×D7314yes—
428C2×C8C23×D7312yes—
429C2×C8C23×D7312yes—
430C4⋊C4—734no—
431C4⋊C4—734no—
432C4⋊C4—738no—
433C4⋊C4—738no—
434C4⋊C4—734no—
435C4⋊C4—738no—
436C4⋊C4—734no—
437C4⋊C4C2×C4×D7314yes—
438C22⋊C4—734no—
439C22⋊C4—738no—
440C22⋊C4—738no—
441C22⋊C4—734no—
442C22⋊C4—734no—
443C22⋊C4—738no—
444C22⋊C4C2×C4×D7318yes—
445C42—732no—
446C42—738no—
447C42—732no—
448C42—734no—
449C42C2×C4×D7311yescharacteristic
450C42C2×C4×D7311yescharacteristic
451C42C23×D7312yes—
452C23—2922no—
453C23—732no—
454C23—1462no—
455C23C22×C4×D7314yes—
456C23C22×C4×D7311yescharacteristic
457Q8—24no—
458Q8—2924no—
459D4—28no—
460D4—24no—
461D4—44no—
462D4—2928no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—2922no—
467C2×C4—2928no—
468C2×C4—734no—
469C2×C4—738no—
470C2×C4—732no—
471C2×C4—1462no—
472C2×C4—1462no—
473C2×C4—1462no—
474C2×C4—1464no—
475C2×C4C22×C4×D7314yes—
476C2×C4C22×C4×D7312yes—
477C2×C4C22×C4×D7312yes—
478C2×C4C22×C4×D7311yescharacteristic
479C2×C4#41×D7312yes—
480C2×C4C2×#1192402712yes—
481C2×C4C24×D7311yescharacteristic
482C8—22no—
483C8—24no—
484C8—732no—
485C8—732no—
486C8—1464no—
487C8—1462no—
488C8C2×#1192402714yes—
489C22—22no—
490C22—28no—
491C22—24no—
492C22—2924no—
493C22—1464no—
494C22—1462no—
495C22C4×#1192402712yes—
496C22C23×C4×D7311yescharacteristic
497C4—22no—
498C4—22no—
499C4—24no—
500C4—1464no—
501C4—1468no—
502C4—1462no—
503C4C4×#1192402712yes—
504C4C2×#41×D7311yescharacteristic
505C4C22×#1192402711yescharacteristic
506C2—22no—
507C2—44no—
508C2—2922no—
509C2C2×C4×#1192402711yescharacteristic
510C2#2262464412yes—
subgroupG/H
center, FrattiniC2×C4C24×D73
commutator, socleC22×C73C24×C4
FittingC73×#2932C2
prime radical#22623320C2
cosocleC2×C4×C73C25
hypercenter#2932D73
hypocenterC73#60206
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #22618910C2semidirect
3. #22618901C2semidirect
4. #22618871C2semidirect
5. #22618861C2semidirect
6. #22618643C2semidirect
7. #22618632C2semidirect
8. #22618628C2semidirect
9. #22618578C2semidirect
10. #22618568C2semidirect
11. #131×Dic73C2semidirect
12. C2×#17988544C2semidirect
13. C73×#2932C2semidirect
27. #17988715C22semidirect
28. #17988714C22semidirect
29. #17988711C22semidirect
30. #17988665C22semidirect
34. D4×#10596831C22semidirect
37. #17988645C22semidirect
38. #17988641C22semidirect
45. #17988545C22semidirect
49. #17988542C22semidirect
51. M16×Dic73C22semidirect
55. C2×C8×Dic73C22semidirect
58. #435×C73C22semidirect
261. #2932D73semidirect
269. C73#60206semidirect
292. #435C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226233202
#226189102
#226189012
#226188712
#226188612
#226186432
#226186322
#226186282
#226185782
#226185682
#131×Dic732
C2×#179885442
C73×#29322
#6020673