Finite Group Databasebeta

← #34034839

Subgroups — #34034839

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989709C211yesmaximal, characteristic, complemented
2#22627182C211yesmaximal, characteristic, complemented
3#22627152C212yesmaximal, complemented
4#22627151C212yesmaximal, complemented
5#22627150C212yesmaximal, complemented
6#22627149C212yesmaximal, complemented
7#22627127C211yesmaximal, characteristic, complemented
8#22627115C211yesmaximal, characteristic, complemented
9#22627114C211yesmaximal, characteristic, complemented
10#22627094C211yesmaximal, characteristic, complemented
11#22626704C211yesmaximal, characteristic
12#17989709—21no—
13#17989709—21no—
14#17989709C2212yescomplemented
15#17989709C2211yescharacteristic, complemented
16#17989709C2212yescomplemented
17#17989709C2211yescharacteristic, complemented
18#17989708—21no—
19#17989708—21no—
20#17989708C2211yescharacteristic, complemented
21#17989708C2211yescharacteristic, complemented
22C2×#14556688—21no—
23C2×#14556688C2211yescharacteristic
24#17989706—21no—
25#17989706C2211yescharacteristic
26#17989706C2211yescharacteristic
27C2×#14556686—21no—
28C2×#14556686—21no—
29C2×#14556686C2211yescharacteristic
30#17989692C2212yescomplemented
31#17989691C2212yescomplemented
32#17989688C2212yescomplemented
33#17989688C2212yescomplemented
34#17989688C2212yescomplemented
35#17989687C2212yescomplemented
36#17989682C2212yescomplemented
37#64408⋊C16C2212yescomplemented
38C2×#14556679C2211yescharacteristic
39D73⋊#131C2211yescharacteristic
40#17989633C412yes—
41#17989632C412yes—
42#17989631C2212yes—
43#17989630C2212yes—
44#17989602C411yescharacteristic
45#17989586C411yescharacteristic
46#17989584C411yescharacteristic
47#17989553C2211yescharacteristic
48#17989386C411yescharacteristic
49#14556688—21no—
50#14556688—21no—
51#14556688—22no—
52#14556688—22no—
53#14556688—21no—
54#14556688—21no—
55#14556688C2312yes—
56#14556688C2311yescharacteristic
57#14556688C2311yescharacteristic
58#14556687—21no—
59#14556687—21no—
60#14556687—21no—
61#14556687—21no—
62#14556686—21no—
63#14556686—21no—
64#14556686—22no—
65#14556686—22no—
66#14556686—21no—
67#14556686D411yescharacteristic
68#14556686D411yescharacteristic
69#14556686C2312yes—
70#14556686C2311yescharacteristic
71#14556686C2311yescharacteristic
72C2×#11924040—21no—
73C2×#11924040—21no—
74C2×#11924040—21no—
75C2×#11924040D411yescharacteristic
76C2×#11924040D411yescharacteristic
77#14556679—21no—
78#14556679—21no—
79#14556679C2×C412yes—
80#14556679C2×C411yescharacteristic
81#14556679C2311yescharacteristic
82#14556679C2312yes—
83#14556678—21no—
84#14556678—21no—
85#14556678C2×C411yescharacteristic
86#14556678C2311yescharacteristic
87C2×#11924038—21no—
88C2×#11924038C2×C411yescharacteristic
89D73⋊M16—21no—
90D73⋊M16C2×C411yescharacteristic
91C2×#11924036—21no—
92C2×#11924036—21no—
93C2×#11924036C2311yescharacteristic
94#14556670C2×C411yescharacteristic
95#14556662C2×C412yes—
96#14556661C2×C412yes—
97#14556656C2×C412yes—
98#14556656C2×C412yes—
99#14556656C2×C412yes—
100#14556655C2×C412yes—
101#14556645C2×C412yes—
102C8×#64408C2312yes—
103C2×#11924029C2×C411yescharacteristic
104#41×D73C2×C411yescharacteristic
105#14556623C812yes—
106#14556622C811yescharacteristic
107#14556619C811yescharacteristic
108#14556618C812yes—
109#14556616C811yescharacteristic
110#14556614C2×C412yes—
111#14556612C2×C412yes—
112M16×D73C2×C411yescharacteristic
113#137×C73C811yescharacteristic
114#11924040—21no—
115#11924040—21no—
116#11924040—22no—
117#11924040—22no—
118#11924040C2×D411yescharacteristic
119#11924040C2×D411yescharacteristic
120#11924040C2×D411yescharacteristic
121#11924040C2×D411yescharacteristic
122#11924038—21no—
123#11924038—21no—
124#11924038—22no—
125#11924038—21no—
126#11924038—22no—
127#11924038C22×C411yescharacteristic
128#11924038C22×C411yescharacteristic
129#11924038C22×C412yes—
130#11924037—21no—
131#11924037—21no—
132#11924037—21no—
133#11924036—21no—
134#11924036—21no—
135#11924036—22no—
136#11924036—21no—
137#11924036—22no—
138#11924036C22×C411yescharacteristic
139#11924036C2×D411yescharacteristic
140#11924036#4111yescharacteristic
141#11924036C2411yescharacteristic
142C2×#10596833—21no—
143C2×#10596833—21no—
144C2×#10596833—21no—
145C2×#10596833C2×D411yescharacteristic
146C2×#10596833#4111yescharacteristic
147C4⋊#64408—21no—
148C22⋊#64408—21no—
149C4×#64408C22×C412yes—
150#11924029—21no—
151#11924029—21no—
152#11924029C2×C812yes—
153#11924029C22×C412yes—
154#11924029C22×C411yescharacteristic
155#11924028—21no—
156#11924028C2×C811yescharacteristic
157#11924027—21no—
158#11924027C22×C411yescharacteristic
159Q8×D73C2×C811yescharacteristic
160C2×D292—21no—
161C2×D292C2×C811yescharacteristic
162D4×D73—21no—
163D4×D73C2×C811yescharacteristic
164C2×C4×D73—21no—
165C2×C4×D73—21no—
166C2×C4×D73C22×C411yescharacteristic
167#11924018C2×C812yes—
168#11924017C2×C812yes—
169#11924015C2×C812yes—
170#11924015C2×C812yes—
171D584C2×C812yes—
172#11924013C2×C812yes—
173#11924012C2×C811yescharacteristic
174#11924011C2×C812yes—
175C8×D73C22×C412yes—
176#41×C73C2×C811yescharacteristic
177C2×Q8×C73C2×C811yescharacteristic
178Q16×C73C2×C812yes—
179SD16×C73C2×C812yes—
180M16×C73C2×C811yescharacteristic
181C22×D73—21no—
182C22×D73—21no—
183C22×D73#13111yescharacteristic
184C2×#64408—22no—
185Q8×C73—21no—
186Q8×C73C22×C812yes—
187Q8×C73C22×C811yescharacteristic
188D4×C73—21no—
189D4×C73C22×C811yescharacteristic
190C2×C4×C73—21no—
191C2×C4×C73—21no—
192C2×C4×C73C22×C811yescharacteristic
193C22⋊D73—21no—
194C22⋊D73—21no—
195C22⋊D73—21no—
196C2×Dic73—21no—
197C2×Dic73C4×D411yescharacteristic
198D292—21no—
199D292—22no—
200D292—22no—
201D292—21no—
202D292C22×C812yes—
203D292C22×C811yescharacteristic
204C4×D73—21no—
205C4×D73—21no—
206C4×D73—21no—
207C4×D73—22no—
208C4×D73—22no—
209C4×D73C4×D411yescharacteristic
210C4×D73C22×C811yescharacteristic
211C4×D73#13111yescharacteristic
212C4×D73C23×C411yescharacteristic
213Dic146—21no—
214Dic146C22×C811yescharacteristic
215#10596833—21no—
216#10596833—21no—
217#10596833—22no—
218#10596833C4×D411yescharacteristic
219#10596833C4×D411yescharacteristic
220#10596833C22×D411yescharacteristic
221#10596833C2×#4111yescharacteristic
222C8×C73C22×C812yes—
223#10596831C22×C812yes—
224C22×C73—21no—
225C22×C73C8×D411yescharacteristic
226C2×D73—22no—
227C2×D73—21no—
228C2×D73—41no—
229C2×D73—42no—
230C2×D73—42no—
231C2×D73C8×D411yescharacteristic
232C2×D73C2×#13111yescharacteristic
233#64408—42no—
234C4×C73—21no—
235C4×C73—22no—
236C4×C73C8×D411yescharacteristic
237C4×C73C23×C811yescharacteristic
238Dic73—21no—
239Dic73C8×D411yescharacteristic
240Dic73C2×C4×D411yescharacteristic
241#33441—731nomaximal, Sylow
242C2×C73—21no—
243C2×C73—41no—
244C2×C73C2×C8×D411yescharacteristic
245D73—21no—
246D73—21no—
247D73—42no—
248D73—41no—
249C2×#504—731no—
250C2×#504—731no—
251#2955—731no—
252#2171—732no—
253#2171—732no—
254#2170—732no—
255#2170—732no—
256#2153—731no—
257#2117—731no—
258#2117—731no—
259#2108—731no—
260C73#3344111yescharacteristic, complemented, Sylow
261#575—731no—
262C2×#131—731no—
263C2×#131—731no—
264#504—731no—
265#504—731no—
266#504—732no—
267#504—731no—
268#504—732no—
269#504—731no—
270#504—1461no—
271#504—1461no—
272#504—1461no—
273#504—1461no—
274C2×M32—731no—
275C2×M32—731no—
276C2×M32—731no—
277C2×M32—1461no—
278C2×M32—1461no—
279C22×C16—731no—
280C22×C16—1461no—
281C22×C16—1461no—
282#444—732no—
283#444—732no—
284#443—732no—
285#443—732no—
286#430—731no—
287#421—731no—
288#421—731no—
289#405—731no—
290#364—732no—
291#364—732no—
292#350—732no—
293#350—732no—
294#350—732no—
295#350—732no—
296#346—732no—
297C4×C16—732no—
298C2×#41—731no—
299C2×#41—731no—
300#137—731no—
301#137—732no—
302#137#1059684211yescharacteristic
303AGL(1,ℤ/8)—731no—
304AGL(1,ℤ/8)—732no—
305AGL(1,ℤ/8)—731no—
306#135—732no—
307#135—732no—
308#131—731no—
309#131—731no—
310#131—732no—
311#131—731no—
312#131—731no—
313#131—732no—
314#131—1461no—
315#131—1461no—
316#131—1461no—
317#131—1461no—
318C2×M16—731no—
319C2×M16—731no—
320C2×M16—731no—
321C2×M16—1461no—
322C2×M16—1461no—
323C22×C8—731no—
324C22×C8—1461no—
325C22×C8—1461no—
326#117—731no—
327M32—732no—
328M32—731no—
329M32—731no—
330M32—1461no—
331M32—1461no—
332M32—1461no—
333M32—1462no—
334M32—1461no—
335M32—1462no—
336M32—1461no—
337M32—1461no—
338M32—1461no—
339M32—1461no—
340C2×C16—731no—
341C2×C16—731no—
342C2×C16—731no—
343C2×C16—731no—
344C2×C16—731no—
345C2×C16—731no—
346C2×C16—732no—
347C2×C16—1461no—
348C2×C16—1461no—
349C2×C16—1461no—
350C2×C16—1462no—
351C2×C16—1461no—
352C2×C16—1462no—
353C2×C16—1461no—
354C2×C16—1461no—
355#108—732no—
356#108—732no—
357#104—732no—
358#104—732no—
359#104—732no—
360#104—732no—
361#97—732no—
362C4×C8—732no—
363#41—732no—
364#41—731no—
365#41—732no—
366#41—731no—
367#41—731no—
368#41—1461no—
369#41—1461no—
370#41—1461no—
371#41—1461no—
372#41C2×#1059684211yescharacteristic
373C2×Q8—731no—
374C2×Q8C2×#1059684211yescharacteristic
375C2×D4—731no—
376C2×D4—731no—
377C2×D4—1461no—
378C2×D4—1461no—
379C22×C4—731no—
380C22×C4—1461no—
381C22×C4—1461no—
382Q16—732no—
383Q16C2×#1059684212yes—
384SD16—732no—
385SD16—732no—
386SD16—732no—
387SD16C2×#1059684212yes—
388D8—732no—
389D8—732no—
390M16—731no—
391M16—732no—
392M16—731no—
393M16—731no—
394M16—732no—
395M16—1461no—
396M16—1461no—
397M16—1461no—
398M16—1462no—
399M16—1461no—
400M16—1462no—
401M16—1461no—
402M16—1461no—
403M16C2×#1059684211yescharacteristic
404C2×C8—731no—
405C2×C8—731no—
406C2×C8—731no—
407C2×C8—731no—
408C2×C8—731no—
409C2×C8—731no—
410C2×C8—732no—
411C2×C8—1461no—
412C2×C8—1461no—
413C2×C8—1461no—
414C2×C8—1462no—
415C2×C8—1461no—
416C2×C8—1462no—
417C2×C8—1461no—
418C2×C8—1461no—
419C4⋊C4—1461no—
420C22⋊C4—1461no—
421C42—732no—
422C16—731no—
423C16—731no—
424C16—731no—
425C16—731no—
426C16—1462no—
427C16—1461no—
428C16—1461no—
429C16—1462no—
430C23—731no—
431C23—1461no—
432C23—1461no—
433Q8—21no—
434Q8—731no—
435Q8—1461no—
436Q8C22×#1059684211yescharacteristic
437Q8C22×#1059684212yes—
438D4—21no—
439D4—732no—
440D4—731no—
441D4—1461no—
442D4—1462no—
443D4—1462no—
444D4—1461no—
445D4—1461no—
446D4—1461no—
447D4—1461no—
448D4C22×#1059684211yescharacteristic
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—731no—
452C2×C4—731no—
453C2×C4—731no—
454C2×C4—731no—
455C2×C4—731no—
456C2×C4—1461no—
457C2×C4—1462no—
458C2×C4—1462no—
459C2×C4—1461no—
460C2×C4—1462no—
461C2×C4—1461no—
462C2×C4—1461no—
463C2×C4C22×#1059684211yescharacteristic
464C8—731no—
465C8—731no—
466C8—731no—
467C8—731no—
468C8—732no—
469C8—1461no—
470C8—1461no—
471C8—1462no—
472C8C22×#1059684212yes—
473C22—21no—
474C22—2922no—
475C22—2921no—
476C22—2922no—
477C22—731no—
478C22—731no—
479C22—1462no—
480C22—1461no—
481C22D4×#1059684211yescharacteristic
482C4—21no—
483C4—22no—
484C4—2922no—
485C4—731no—
486C4—731no—
487C4—1461no—
488C4D4×#1059684211yescharacteristic
489C4C23×#1059684211yescharacteristic
490C2—21no—
491C2—41no—
492C2—2922no—
493C2—2921no—
494C2—1461no—
495C2—1461no—
496C2C2×D4×#1059684211yescharacteristic
subgroupG/H
centerC2C2×D4×#10596842
commutatorC4×C73C23×C8
FrattiniC4C23×#10596842
Fitting#137×C73C8
socleC2×C73C2×C8×D4
prime radical#17989386C4
cosocle#11924036C24
hypercenter#137#10596842
hypocenterC73#33441
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#17989709C2semidirect
2. #22627182C2semidirect
3. #22627152C2semidirect
4. #22627151C2semidirect
5. #22627150C2semidirect
6. #22627149C2semidirect
7. #22627127C2semidirect
8. #22627115C2semidirect
9. #22627114C2semidirect
10. #22627094C2semidirect
14. #17989709C22semidirect
15. #17989709C22semidirect
16. #17989709C22semidirect
17. #17989709C22semidirect
20. #17989708C22semidirect
21. #17989708C22semidirect
30. #17989692C22semidirect
31. #17989691C22semidirect
32. #17989688C22semidirect
33. #17989688C22semidirect
34. #17989688C22semidirect
35. #17989687C22semidirect
36. #17989682C22semidirect
37. #64408⋊C16C22semidirect
260. C73#33441semidirect

Maximal subgroups

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

subgroupindex
C2×#179897092
#226271822
#226271522
#226271512
#226271502
#226271492
#226271272
#226271152
#226271142
#226270942
#226267042
#3344173