Finite Group Databasebeta

← #33239334

Subgroups — #33239334

Order 18688.

660 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×#17989026C211yesmaximal, characteristic
2#22618801C211yesmaximal, characteristic
3#22618794C211yesmaximal, characteristic, complemented
4#22611803C211yesmaximal, characteristic, complemented
5#22611799C211yesmaximal, characteristic, complemented
6#22611792C211yesmaximal, characteristic, complemented
7#22611789C211yesmaximal, characteristic, complemented
8C4×#14556537C211yesmaximal, characteristic, complemented
9SD32×Dic73C211yesmaximal, characteristic, complemented
10C4×#14556535C211yesmaximal, characteristic, complemented
11#22611716C211yesmaximal, characteristic, complemented
12#22611635C211yesmaximal, characteristic
13#358×D73C211yesmaximal, characteristic, complemented
14#22611529C211yesmaximal, characteristic, complemented
15C73×#2175C211yesmaximal, characteristic, complemented
16C2×#14556610C2211yescharacteristic
17C2×Q16×D73C2211yescharacteristic
18#17989066—21no—
19#17989063—21no—
20#17989026C418yescomplemented
21C2×Dic584C2211yescharacteristic
22C2×#14556537C2211yescharacteristic
23C2×#14556535C2211yescharacteristic
24C2×#14556530C2211yescharacteristic
25C2×C16×D73C2211yescharacteristic
26#17988692—21no—
27#17988691—21no—
28#17988687—21no—
29#17988684—21no—
30C4×Dic292C2211yescharacteristic
31C4×Dic292C2211yescharacteristic
32Q16×Dic73C2211yescharacteristic
33#17988681C2211yescharacteristic
34#17988681C2211yescharacteristic
35D8×Dic73C2211yescharacteristic
36#17988624C2211yescharacteristic
37#107×D73C2211yescharacteristic
38#17988594—21no—
39#17988585—21no—
40#17988541C2211yescharacteristic
41#17988415C2211yescharacteristic
42#17988414C2211yescharacteristic
43#17988412C2211yescharacteristic
44#17988392C2211yescharacteristic
45#17988392C2211yescharacteristic
46#17988391C2211yescharacteristic
47#17988390C2211yescharacteristic
48#17988390C2211yescharacteristic
49#17988388C2211yescharacteristic
50#17988362C2211yescharacteristic
51C4×#11924003C2211yescharacteristic
52C16×Dic73C2211yescharacteristic
53C2×SD32×C73C2211yescharacteristic
54C4×Q16×C73C2211yescharacteristic
55C4×D8×C73C2211yescharacteristic
56#366×C73C2211yescharacteristic
57#358×C73C2211yescharacteristic
58#357×C73C2211yescharacteristic
59#346×C73C2211yescharacteristic
60C2×#11924027—21no—
61C2×Q8×D73—21no—
62#14556610—22no—
63#14556610C2×C414yes—
64C2×Dic292C2311yescharacteristic
65C2×Dic292C2311yescharacteristic
66C2×#11924018—21no—
67Q16×D73—22no—
68Q16×D73C2×C414yescomplemented
69C2×#11924016—21no—
70C2×#11924011D411yescharacteristic
71C2×C8×D73C2311yescharacteristic
72Q8×Dic73—21no—
73C4×Dic146—21no—
74C4×Dic146—21no—
75C4×Dic146—41no—
76#14556556—21no—
77#14556556—21no—
78#14556556—41no—
79D4×Dic73—21no—
80C4×C22⋊D73—41no—
81#14556551—41no—
82#14556549—21no—
83#14556549—41no—
84#14556548—41no—
85#14556548D411yescharacteristic
86#14556547—41no—
87C2×#11924007—41no—
88C4⋊C4×D73—21no—
89Dic584C2×C414yescomplemented
90#14556537C2×C414yescomplemented
91#14556535C2×C414yescomplemented
92#14556530C2×C414yescomplemented
93C2×#11924003C2311yescharacteristic
94C2×#11924003C2311yescharacteristic
95C16×D73—22no—
96#14556525C2311yescharacteristic
97#14556525C2311yescharacteristic
98#14556524—21no—
99#14556522—21no—
100#14556516—21no—
101#14556516—21no—
102#14556515—21no—
103#14556514—21no—
104#14556514—21no—
105#14556512—21no—
106#14556502D411yescharacteristic
107#14556501D411yescharacteristic
108C8×Dic73C2311yescharacteristic
109C2×Q16×C73C2311yescharacteristic
110C2×D8×C73C2311yescharacteristic
111C4×Q8×C73—21no—
112C4×D4×C73—21no—
113SD32×C73C2×C414yescomplemented
114C2×C16×C73C2311yescharacteristic
115C2×C16×C73C2311yescharacteristic
116#107×C73C2311yescharacteristic
117#107×C73C2311yescharacteristic
118#103×C73—21no—
119#102×C73—21no—
120C4×C8×C73C2311yescharacteristic
121#11924027—24no—
122#11924027—42no—
123C2×Dic146—21no—
124C2×Dic146—21no—
125C2×Dic146—41no—
126Q8×D73—24no—
127Q8×D73—42no—
128C2×C22⋊D73—41no—
129C22×Dic73—41no—
130C22×Dic73—41no—
131C2×C4×D73—41no—
132C2×C4×D73—41no—
133C2×C4×D73C2×D411yescharacteristic
134Dic292—21no—
135Dic292—21no—
136Dic292C22×C412yes—
137Dic292C22×C412yes—
138#11924018—24no—
139#11924016—24no—
140#11924011—22no—
141C2×#10596831C2×D411yescharacteristic
142C2×#10596831C2×D411yescharacteristic
143C8×D73C22×C412yes—
144C8×D73#4112yes—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—41no—
148#11924007—41no—
149#11924007—41no—
150#11924007—41no—
151#11924007—41no—
152#11924007—41no—
153#11924007D811yescharacteristic
154#11924007D811yescharacteristic
155C22⋊Dic73—41no—
156#11924005—41no—
157#11924005D811yescharacteristic
158#11924005D811yescharacteristic
159C4×Dic73—41no—
160C4×Dic73—41no—
161C4×Dic73—41no—
162C4×Dic73—41no—
163C4×Dic73—41no—
164C4×Dic73—41no—
165C4×Dic73C2×D411yescharacteristic
166#11924003—21no—
167#11924003C22×C412yes—
168C2×Q8×C73—21no—
169C2×D4×C73—21no—
170C22×C4×C73—41no—
171Q16×C73—21no—
172Q16×C73C22×C412yes—
173D8×C73—21no—
174D8×C73C22×C412yes—
175C2×C8×C73C2×D411yescharacteristic
176C2×C8×C73C2411yescharacteristic
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—41no—
180C22⋊C4×C73—41no—
181C42×C73—41no—
182C42×C73C2×D411yescharacteristic
183C16×C73—21no—
184C16×C73C22×C412yes—
185C23×C73—41no—
186C22×D73C2×D811yescharacteristic
187Q8×C73—22no—
188Q8×C73—41no—
189D4×C73—22no—
190D4×C73—41no—
191C2×C4×C73—41no—
192C2×C4×C73—41no—
193C2×C4×C73—41no—
194C2×C4×C73—161no—
195C2×C4×C73C2×D811yescharacteristic
196C2×C4×C73C2×D811yescharacteristic
197C2×C4×C73C22×D411yescharacteristic
198C22⋊D73—44no—
199C2×Dic73—41no—
200C2×Dic73—41no—
201C2×Dic73—44no—
202C2×Dic73—41no—
203C2×Dic73—41no—
204C2×Dic73—161no—
205C2×Dic73C2×D811yescharacteristic
206C2×Dic73C2×D811yescharacteristic
207C2×Dic73C2×D811yescharacteristic
208C4×D73—44no—
209C4×D73—161no—
210C4×D73C4×D412yes—
211C4×D73#13512yes—
212Dic146—22no—
213Dic146—22no—
214Dic146—41no—
215Dic146—41no—
216Dic146—44no—
217C8×C73—21no—
218C8×C73C23×C411yescharacteristic
219C8×C73C2×#4111yescharacteristic
220#10596831—21no—
221#10596831C4×D412yes—
222C22×C73—42no—
223C22×C73—81no—
224C22×C73—81no—
225C22×C73C22×D811yescharacteristic
226C2×D73—21no—
227C2×D73—21no—
228C2×D73C4×D812yes—
229C4×C73—21no—
230C4×C73—21no—
231C4×C73—42no—
232C4×C73—81no—
233C4×C73—81no—
234C4×C73C2×C4×D411yescharacteristic
235C4×C73C2×#13511yescharacteristic
236Dic73—21no—
237Dic73—21no—
238Dic73—42no—
239Dic73—42no—
240Dic73—81no—
241Dic73—81no—
242Dic73C4×D812yes—
243#33455—731nomaximal, Sylow
244C2×C73—82no—
245C2×C73C2×C4×D811yescharacteristic
246C2×C73#341611yescharacteristic
247C2×C73#341711yescharacteristic
248D73—22no—
249C2×#510—731no—
250#2951—731no—
251#2950—731no—
252#2176—731no—
253#2176—731no—
254#2175—731no—
255#2175D7311yescharacteristic, complemented
256C4×Q32—731no—
257C4×Q32—731no—
258C4×SD32—731no—
259C4×SD32—731no—
260#2159—731no—
261#2138—731no—
262C2×#358—731no—
263#2108—731no—
264C73#3345511yescharacteristic, complemented, Sylow
265C2×#135—731no—
266C22×Q16—731no—
267#520—1461no—
268#519—1461no—
269#510—738no—
270C2×Q32—731no—
271C2×Q32—731no—
272C2×SD32—731no—
273C2×SD32C2×D7311yescharacteristic
274C2×M32—731no—
275C22×C16—731no—
276#441—1461no—
277#441—1461no—
278#440—1461no—
279#440—1461no—
280C4×Q16—731no—
281C4×Q16—731no—
282C4×Q16—731no—
283C4×Q16—731no—
284C4×Q16—731no—
285C4×Q16C2×D7311yescharacteristic
286C4×D8—731no—
287C4×D8C2×D7311yescharacteristic
288#427—731no—
289C2×#107—731no—
290#419—1461no—
291#417—1461no—
292#405—731no—
293#367—731no—
294#367—731no—
295#366—731no—
296#366C2×D7311yescharacteristic
297#358—731no—
298#358—731no—
299#358—731no—
300#358—731no—
301#358—731no—
302#358C2×D7311yescharacteristic
303#357—731no—
304#357C2×D7311yescharacteristic
305#346—731no—
306#346C2×D7311yescharacteristic
307C4×C16—731no—
308C4×C16—731no—
309C2×#41—1461no—
310C22×Q8—1461no—
311#135—734no—
312#135—1462no—
313C2×Q16—734no—
314C2×Q16—731no—
315C2×Q16—731no—
316C2×Q16—1462no—
317C2×Q16—1461no—
318C2×Q16C22×D7311yescharacteristic
319C2×SD16—1461no—
320C2×D8C22×D7311yescharacteristic
321C2×M16—731no—
322C22×C8—731no—
323C4×Q8—21no—
324C4×Q8—2921no—
325C4×Q8—2921no—
326C4×Q8—1461no—
327C4×Q8—1461no—
328C4×Q8—1461no—
329C4×Q8—1461no—
330C4×Q8—1461no—
331C4×D4—21no—
332C4×D4—2921no—
333C4×D4—2921no—
334C4×D4—1461no—
335#117—2921no—
336#117—2921no—
337#117—2921no—
338#117—731no—
339#117—1461no—
340C2×C4⋊C4—2921no—
341C2×C4⋊C4—1461no—
342Q32—734no—
343Q32—734no—
344SD32—734no—
345SD32C4×D7314yescomplemented
346M32—734no—
347C2×C16—731no—
348C2×C16—731no—
349C2×C16—1462no—
350C2×C16C22×D7311yescharacteristic
351C2×C16C22×D7311yescharacteristic
352#107—731no—
353#107—731no—
354#107—1461no—
355#107C22×D7311yescharacteristic
356#107C22×D7311yescharacteristic
357#106—1461no—
358#103—21no—
359#103—1461no—
360#103—1461no—
361#103—1461no—
362#103—1461no—
363#103—1461no—
364#102—21no—
365#102—1461no—
366#97—731no—
367#97—731no—
368C4×C8—731no—
369C4×C8C22×D7311yescharacteristic
370#41—2922no—
371#41—1464no—
372C2×Q8—21no—
373C2×Q8—2921no—
374C2×Q8—2922no—
375C2×Q8—1464no—
376C2×Q8—1461no—
377C2×Q8—1461no—
378C2×D4—21no—
379C2×D4—2921no—
380C22×C4—41no—
381C22×C4—2921no—
382C22×C4—2921no—
383C22×C4—2921no—
384C22×C4—2921no—
385C22×C4—731no—
386Q16—21no—
387Q16—732no—
388Q16—732no—
389Q16—1464no—
390Q16—1461no—
391Q16—1461no—
392Q16C2×C4×D7312yes—
393SD16—1464no—
394D8—21no—
395D8C2×C4×D7312yes—
396M16—1462no—
397C2×C8—731no—
398C2×C8—732no—
399C2×C8—732no—
400C2×C8—731no—
401C2×C8D4×D7311yescharacteristic
402C2×C8C23×D7311yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—2921no—
407C4⋊C4—2921no—
408C4⋊C4—2921no—
409C4⋊C4—2921no—
410C4⋊C4—2921no—
411C4⋊C4—2921no—
412C4⋊C4—731no—
413C4⋊C4—731no—
414C4⋊C4—1461no—
415C4⋊C4—1461no—
416C22⋊C4—41no—
417C22⋊C4—2921no—
418C22⋊C4—2921no—
419C22⋊C4—731no—
420C22⋊C4—731no—
421C42—41no—
422C42—2921no—
423C42—2921no—
424C42—2921no—
425C42—2921no—
426C42—2921no—
427C42—2921no—
428C42—731no—
429C42D4×D7311yescharacteristic
430C16—21no—
431C16—732no—
432C16—1461no—
433C16C2×C4×D7312yes—
434C23—41no—
435C23—731no—
436Q8—22no—
437Q8—41no—
438Q8—2924no—
439Q8—2921no—
440Q8—2921no—
441Q8—1462no—
442Q8—1462no—
443D4—22no—
444D4—41no—
445D4—2924no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—161no—
450C2×C4—2924no—
451C2×C4—2921no—
452C2×C4—2921no—
453C2×C4—2924no—
454C2×C4—2921no—
455C2×C4—2921no—
456C2×C4—731no—
457C2×C4—732no—
458C2×C4—732no—
459C2×C4—731no—
460C2×C4—731no—
461C2×C4—11681no—
462C2×C4—11681no—
463C2×C4D8×D7311yescharacteristic
464C2×C4D8×D7311yescharacteristic
465C2×C4C2×D4×D7311yescharacteristic
466C8—21no—
467C8—732no—
468C8—1461no—
469C8C22×C4×D7311yescharacteristic
470C8#41×D7311yescharacteristic
471C22—42no—
472C22—81no—
473C22—81no—
474C22—732no—
475C22—1461no—
476C22—1461no—
477C22C2×D8×D7311yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—2922no—
484C4—2922no—
485C4—5841no—
486C4—5841no—
487C4—732no—
488C4—1461no—
489C4—1461no—
490C4C4×D4×D7311yescharacteristic
491C4#135×D7311yescharacteristic
492C2—82no—
493C2—1462no—
494C2C4×D8×D7311yescharacteristic
495C2#2262307711yescharacteristic
496C2#2262308311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2175C2
socleC22×C73C22×D8
prime radicalC2×#14556610C22
cosocleC2×C8×C73C24
hypercenter#2175D73
hypocenterC73#33455
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
3. #22618794C2semidirect
4. #22611803C2semidirect
5. #22611799C2semidirect
6. #22611792C2semidirect
7. #22611789C2semidirect
8. C4×#14556537C2semidirect
9. SD32×Dic73C2semidirect
10. C4×#14556535C2semidirect
11. #22611716C2semidirect
13. #358×D73C2semidirect
14. #22611529C2semidirect
15. C73×#2175C2semidirect
20. #17989026C4semidirect
68. Q16×D73C2×C4semidirect
89. Dic584C2×C4semidirect
90. #14556537C2×C4semidirect
91. #14556535C2×C4semidirect
92. #14556530C2×C4semidirect
113. SD32×C73C2×C4semidirect
255. #2175D73semidirect
264. C73#33455semidirect
345. SD32C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890262
#226188012
#226187942
#226118032
#226117992
#226117922
#226117892
C4×#145565372
SD32×Dic732
C4×#145565352
#226117162
#226116352
#358×D732
#226115292
C73×#21752
#3345573