Finite Group Databasebeta

← #33084828

Subgroups — #33084828

Order 18688.

866 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988641C211yesmaximal, characteristic
2C2×#17988610C211yesmaximal, characteristic
3#22612325C211yesmaximal, characteristic
4#22609807C211yesmaximal, characteristic, complemented
5#22609805C211yesmaximal, characteristic, complemented
6#22609804C211yesmaximal, characteristic, complemented
7#22609802C211yesmaximal, characteristic, complemented
8#22609791C211yesmaximal, characteristic, complemented
9#22609789C211yesmaximal, characteristic, complemented
10#22609787C211yesmaximal, characteristic, complemented
11#22609787C211yesmaximal, characteristic, complemented
12#22609717C211yesmaximal, characteristic
13#22609715C211yesmaximal, characteristic
14#22609712C211yesmaximal, characteristic
15C73×#1753C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C22×C4×Dic73C2211yescharacteristic
18#17988641C418yescomplemented
19#17988610C418yescomplemented
20C2×#14556520C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×C22⋊#10596831C2211yescharacteristic
27C2×#14556502C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30C2×#14556501C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32C2×C4×#10596831C2211yescharacteristic
33C2×C8×Dic73C2211yescharacteristic
34C2×C8×Dic73C2211yescharacteristic
35#17988454C412yes—
36#17988451C412yes—
37#17988448C412yes—
38#17988445C412yes—
39C4×#11924007C2211yescharacteristic
40C4×#11924007C2211yescharacteristic
41C4×C22⋊Dic73C2211yescharacteristic
42#17988433C2211yescharacteristic
43#17988433C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988341C2211yescharacteristic
47#17988341C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50C22×M16×C73C2211yescharacteristic
51C2×C22⋊C8×C73C2211yescharacteristic
52C2×#97×C73C2211yescharacteristic
53C2×C4×C8×C73C2211yescharacteristic
54C4×C22⋊C4×C73C2211yescharacteristic
55#336×C73C2211yescharacteristic
56#336×C73C2211yescharacteristic
57C23×Dic73C2×C411yescharacteristic
58C23×Dic73C2×C411yescharacteristic
59C2×#11924012—41no—
60C2×#11924012—41no—
61C2×#11924012C2×C414yes—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C2×#11924007C2×C412yes—
67C2×#11924007C2×C412yes—
68C2×C22⋊Dic73C2×C412yes—
69C2×C4×Dic73C2×C411yescharacteristic
70C2×C4×Dic73C2×C411yescharacteristic
71C2×C4×Dic73C2×C414yes—
72C2×C4×Dic73C2×C414yes—
73C2×C4×Dic73C2311yescharacteristic
74C2×C4×Dic73C2311yescharacteristic
75C2×C4×Dic73C2311yescharacteristic
76C2×C4×Dic73C2311yescharacteristic
77#14556520C2×C414yes—
78#14556520C2×C414yes—
79#14556520C2×C414yes—
80#14556520C2×C414yes—
81#14556520C2×C414yes—
82#14556520C2×C414yes—
83C22⋊#10596831C2×C414yes—
84#14556502—41no—
85#14556501—41no—
86#14556501—41no—
87#14556501C2×C414yes—
88#14556501C2×C414yes—
89C4×#10596831—41no—
90C8×Dic73—41no—
91C8×Dic73—41no—
92#14556498C2×C412yes—
93#14556498C2×C412yes—
94#14556498C2×C412yes—
95#14556498C2×C412yes—
96#14556498C2×C412yes—
97#14556498C2×C412yes—
98#14556498C2×C412yes—
99C23×C4×C73C2311yescharacteristic
100C2×M16×C73—41no—
101C2×M16×C73—41no—
102C2×M16×C73C2×C414yes—
103C22×C8×C73C2311yescharacteristic
104C22×C8×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C2×C22⋊C4×C73C2×C412yes—
108C2×C42×C73C2311yescharacteristic
109C2×C42×C73C2311yescharacteristic
110C22⋊C8×C73C2×C414yes—
111#97×C73—41no—
112C4×C8×C73—41no—
113#95×C73C2×C412yes—
114C22×Dic73—42no—
115C22×Dic73—42no—
116C22×Dic73C4214yes—
117C22×Dic73C4214yes—
118C22×Dic73C22×C412yes—
119C22×Dic73C22×C412yes—
120C22×Dic73C22×C411yescharacteristic
121C22×Dic73C22×C411yescharacteristic
122C22×Dic73C22×C411yescharacteristic
123C22×Dic73C22×C411yescharacteristic
124#11924012—44no—
125C2×#10596831—21no—
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831C22×C412yes—
134C2×#10596831C22×C412yes—
135C2×#10596831C22×C412yes—
136C2×#10596831C22×C412yes—
137#11924007—42no—
138#11924007—42no—
139C22⋊Dic73—42no—
140C4×Dic73—21no—
141C4×Dic73—21no—
142C4×Dic73—21no—
143C4×Dic73—21no—
144C4×Dic73C4214yes—
145C4×Dic73C4214yes—
146C4×Dic73C22×C412yes—
147C4×Dic73C22×C412yes—
148C4×Dic73C22×C412yes—
149C4×Dic73C22×C412yes—
150C24×C73C22×C411yescharacteristic
151C22×C4×C73—21no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73C22×C412yes—
156C22×C4×C73C22×C412yes—
157C22×C4×C73C22×C411yescharacteristic
158C22×C4×C73C22×C412yes—
159C22×C4×C73C22×C412yes—
160C22×C4×C73C2411yescharacteristic
161M16×C73—44no—
162C2×C8×C73—21no—
163C2×C8×C73—21no—
164C2×C8×C73—21no—
165C2×C8×C73—21no—
166C2×C8×C73—21no—
167C2×C8×C73—21no—
168C2×C8×C73—21no—
169C2×C8×C73—21no—
170C2×C8×C73C22×C412yes—
171C2×C8×C73C22×C412yes—
172C2×C8×C73C22×C412yes—
173C2×C8×C73C22×C412yes—
174C22⋊C4×C73—42no—
175C42×C73—21no—
176C42×C73—21no—
177C42×C73—21no—
178C42×C73—21no—
179C23×C73—21no—
180C23×C73—21no—
181C23×C73—22no—
182C23×C73—22no—
183C23×C73C2×C4212yes—
184C23×C73C23×C411yescharacteristic
185C2×C4×C73—21no—
186C2×C4×C73—21no—
187C2×C4×C73—22no—
188C2×C4×C73—22no—
189C2×C4×C73—22no—
190C2×C4×C73—22no—
191C2×C4×C73—22no—
192C2×C4×C73—22no—
193C2×C4×C73—22no—
194C2×C4×C73—22no—
195C2×C4×C73—22no—
196C2×C4×C73—22no—
197C2×C4×C73C2×C4212yes—
198C2×C4×C73C23×C411yescharacteristic
199C2×C4×C73C23×C411yescharacteristic
200C2×C4×C73H(5,𝔽2)11yescharacteristic
201C2×C4×C73H(5,𝔽2)11yescharacteristic
202C2×C4×C732−1+411yescharacteristic
203C2×C4×C732−1+411yescharacteristic
204C2×Dic73—22no—
205C2×Dic73—22no—
206C2×Dic73—22no—
207C2×Dic73—22no—
208C2×Dic73—22no—
209C2×Dic73—22no—
210C2×Dic73—22no—
211C2×Dic73—22no—
212C2×Dic73—22no—
213C2×Dic73—22no—
214C2×Dic73—44no—
215C2×Dic73—44no—
216C2×Dic73C2×C4212yes—
217C2×Dic73C2×C4212yes—
218C2×Dic73C2×C4212yes—
219C2×Dic73C2×C4212yes—
220C8×C73—22no—
221C8×C73—22no—
222C8×C73—22no—
223C8×C73—22no—
224#10596831—22no—
225#10596831—22no—
226#10596831—22no—
227#10596831—22no—
228C22×C73—22no—
229C22×C73—22no—
230C22×C73—22no—
231C22×C73—42no—
232C22×C73—42no—
233C22×C73C22×C4211yescharacteristic
234C22×C73#52011yescharacteristic
235C22×C73#52011yescharacteristic
236C22×C73#56812yes—
237C22×C73#56812yes—
238C4×C73—22no—
239C4×C73—42no—
240C4×C73—42no—
241C4×C73#51811yescharacteristic
242C4×C73#51911yescharacteristic
243C4×C73#52011yescharacteristic
244C4×C73#52011yescharacteristic
245Dic73—22no—
246Dic73—22no—
247Dic73—22no—
248Dic73—22no—
249Dic73—42no—
250Dic73—42no—
251#20551—731nomaximal, Sylow
252C2×C73—42no—
253C2×C73#227411yescharacteristic
254C2×C73#287311yescharacteristic
255C2×C73#287311yescharacteristic
256C2×C73#299512yes—
257C2×C73#299512yes—
258C2×#432—731no—
259C2×#423—731no—
260#2283—731no—
261#1771—731no—
262#1771—731no—
263#1771—731no—
264#1771—731no—
265#1768—731no—
266#1768—731no—
267#1768—731no—
268#1768—731no—
269#1753—731no—
270#1753—731no—
271#1753—731no—
272#1753D7311yescharacteristic
273C73#2055111yescharacteristic, complemented, Sylow
274C22×M16—731no—
275C22×M16C2×D7311yescharacteristic
276C22×C42—731no—
277#432—738no—
278#423—738no—
279C2×C4⋊C8—731no—
280C2×C4⋊C8—731no—
281C2×C4⋊C8—731no—
282C2×C4⋊C8—731no—
283C2×C4⋊C8—731no—
284C2×C4⋊C8—731no—
285C2×C22⋊C8—731no—
286C2×C22⋊C8C2×D7311yescharacteristic
287C2×#97—731no—
288C2×#97—731no—
289C2×#97—731no—
290C2×#97—731no—
291C2×#97—731no—
292C2×#97C2×D7311yescharacteristic
293C2×C4×C8—731no—
294C2×C4×C8—731no—
295C2×C4×C8—731no—
296C2×C4×C8C2×D7311yescharacteristic
297#383—732no—
298#382—732no—
299#381—732no—
300#380—732no—
301C4×C4⋊C4—731no—
302C4×C4⋊C4—731no—
303C4×C22⋊C4—731no—
304C4×C22⋊C4C2×D7311yescharacteristic
305#376—731no—
306#376—731no—
307#336—731no—
308#336—731no—
309#336—731no—
310#336—731no—
311#336—731no—
312#336—731no—
313#336C2×D7311yescharacteristic
314#336C2×D7311yescharacteristic
315C23×C4—731no—
316C23×C4—731no—
317C23×C4C22×D7311yescharacteristic
318C2×M16—41no—
319C2×M16—41no—
320C2×M16—2921no—
321C2×M16—2921no—
322C2×M16—734no—
323C2×M16C4×D7314yes—
324C22×C8—731no—
325C22×C8—731no—
326C22×C8—731no—
327C22×C8—731no—
328C22×C8C22×D7311yescharacteristic
329C22×C8C22×D7311yescharacteristic
330C22×C8C22×D7311yescharacteristic
331C22×C8C22×D7311yescharacteristic
332C2×C4⋊C4—732no—
333C2×C4⋊C4—732no—
334C2×C22⋊C4—732no—
335C2×C22⋊C4C4×D7312yes—
336C2×C42—731no—
337C2×C42—731no—
338C2×C42—734no—
339C2×C42—734no—
340C2×C42—731no—
341C2×C42—731no—
342C2×C42—731no—
343C2×C42—731no—
344C2×C42C22×D7311yescharacteristic
345C2×C42C22×D7311yescharacteristic
346C4⋊C8—734no—
347C4⋊C8—734no—
348C4⋊C8—734no—
349C4⋊C8—734no—
350C4⋊C8—734no—
351C4⋊C8—734no—
352C22⋊C8—734no—
353C22⋊C8C4×D7314yes—
354#97—41no—
355#97—2921no—
356#97—2921no—
357#97—2921no—
358#97—734no—
359#97—734no—
360C4×C8—41no—
361C4×C8—2921no—
362C4×C8—2921no—
363C4×C8—2921no—
364#95—732no—
365#95—732no—
366#95—732no—
367#95—732no—
368#95—732no—
369#95—732no—
370#95—732no—
371#95C4×D7312yes—
372C24C2×C4×D7311yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—2922no—
378C22×C4—2922no—
379C22×C4—734no—
380C22×C4—734no—
381C22×C4—731no—
382C22×C4—731no—
383C22×C4—731no—
384C22×C4—731no—
385C22×C4—732no—
386C22×C4—732no—
387C22×C4C2×C4×D7312yes—
388C22×C4C2×C4×D7312yes—
389C22×C4C2×C4×D7311yescharacteristic
390C22×C4C2×C4×D7312yes—
391C22×C4C2×C4×D7312yes—
392C22×C4C23×D7311yescharacteristic
393M16—44no—
394M16—2924no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—732no—
404C2×C8—732no—
405C2×C8—732no—
406C2×C8—732no—
407C2×C8—1461no—
408C2×C8—1461no—
409C2×C8—1461no—
410C2×C8—1461no—
411C2×C8—1461no—
412C2×C8—1461no—
413C2×C8—1461no—
414C2×C8—1461no—
415C2×C8C2×C4×D7312yes—
416C2×C8C2×C4×D7312yes—
417C2×C8C2×C4×D7312yes—
418C2×C8C2×C4×D7312yes—
419C4⋊C4—2922no—
420C4⋊C4—2922no—
421C22⋊C4—42no—
422C22⋊C4—2922no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—734no—
428C42—734no—
429C42—732no—
430C42—732no—
431C42—732no—
432C42—732no—
433C42—1461no—
434C42—1461no—
435C42—1461no—
436C42—1461no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23C42×D7312yes—
442C23C22×C4×D7311yescharacteristic
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—2924no—
456C2×C4—2924no—
457C2×C4—732no—
458C2×C4—732no—
459C2×C4—732no—
460C2×C4—732no—
461C2×C4—1462no—
462C2×C4—1462no—
463C2×C4—1462no—
464C2×C4—1462no—
465C2×C4—1462no—
466C2×C4—1462no—
467C2×C4—1462no—
468C2×C4—1462no—
469C2×C4—1462no—
470C2×C4—1462no—
471C2×C4C42×D7312yes—
472C2×C4C22×C4×D7311yescharacteristic
473C2×C4C22×C4×D7311yescharacteristic
474C2×C4#1455663611yescharacteristic
475C2×C4#1455663611yescharacteristic
476C2×C4#1455663811yescharacteristic
477C2×C4#1455663811yescharacteristic
478C8—22no—
479C8—22no—
480C8—22no—
481C8—22no—
482C8—1462no—
483C8—1462no—
484C8—1462no—
485C8—1462no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—42no—
490C22—42no—
491C22C2×C42×D7311yescharacteristic
492C22#1798906611yescharacteristic
493C22#1798906611yescharacteristic
494C22#1798935212yes—
495C22#1798935312yes—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—2922no—
500C4—2922no—
501C4—1462no—
502C4—1462no—
503C4—1462no—
504C4—1462no—
505C4#1798906011yescharacteristic
506C4#1798906411yescharacteristic
507C4#1798906611yescharacteristic
508C4#1798906611yescharacteristic
509C2—42no—
510C2#2261228611yescharacteristic
511C2#2261834711yescharacteristic
512C2#2261834811yescharacteristic
513C2#2261909512yes—
514C2#2261910212yes—
subgroupG/H
center, FrattiniC22×C4C23×D73
commutatorC22×C73C22×C42
FittingC73×#1753C2
socleC23×C73C23×C4
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenter#1753D73
hypocenterC73#20551
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #22609807C2semidirect
5. #22609805C2semidirect
6. #22609804C2semidirect
7. #22609802C2semidirect
8. #22609791C2semidirect
9. #22609789C2semidirect
10. #22609787C2semidirect
11. #22609787C2semidirect
18. #17988641C4semidirect
19. #17988610C4semidirect
273. C73#20551semidirect

Maximal subgroups

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

subgroupindex
C2×#179886412
C2×#179886102
#226123252
#226098072
#226098052
#226098042
#226098022
#226097912
#226097892
#226097872
#226097172
#226097152
#226097122
C73×#17532
#2055173