Finite Group Databasebeta

← #33241541

Subgroups — #33241541

Order 18688.

558 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620512C211yesmaximal, characteristic, complemented
2#22620505C211yesmaximal, characteristic
3#22612212C211yesmaximal, characteristic, complemented
4#22612185C211yesmaximal, characteristic, complemented
5#22612157C211yesmaximal, characteristic, complemented
6#22612153C211yesmaximal, characteristic, complemented
7#22612029C211yesmaximal, characteristic, complemented
8#22612027C211yesmaximal, characteristic, complemented
9#22612025C211yesmaximal, characteristic, complemented
10#22611786C211yesmaximal, characteristic, complemented
11SD32×Dic73C211yesmaximal, characteristic, complemented
12#22611782C211yesmaximal, characteristic, complemented
13#22611780C211yesmaximal, characteristic, complemented
14C2×C16×Dic73C211yesmaximal, characteristic, complemented
15C73×#2217C211yesmaximal, characteristic, complemented
16#17989139—21no—
17#17989132—21no—
18C2×#14556536C2211yescharacteristic
19C2×#14556535C2211yescharacteristic
20C2×#14556534C2211yescharacteristic
21C22×#11924003C2211yescharacteristic
22#17988991C2211yescharacteristic
23#17988991C2211yescharacteristic
24#17988965C2211yescharacteristic
25#17988962C2211yescharacteristic
26#17988936—21no—
27#17988932—21no—
28#17988828C2211yescharacteristic
29#17988826—21no—
30#17988823C2211yescharacteristic
31#17988821—21no—
32Q16×Dic73C2211yescharacteristic
33#17988681C2211yescharacteristic
34#17988679—21no—
35D8×Dic73C2211yescharacteristic
36#17988669C2211yescharacteristic
37#17988667—21no—
38C2×C8×Dic73C2211yescharacteristic
39#17988415C2211yescharacteristic
40#17988414C2211yescharacteristic
41#17988414C2211yescharacteristic
42#17988392C2211yescharacteristic
43#17988391C2211yescharacteristic
44#17988390C2211yescharacteristic
45#17988389C2211yescharacteristic
46#17988388C2211yescharacteristic
47#17988387C2211yescharacteristic
48C16×Dic73C2211yescharacteristic
49C16×Dic73C2211yescharacteristic
50C16×Dic73C2211yescharacteristic
51C16×Dic73C2211yescharacteristic
52C2×SD32×C73C2211yescharacteristic
53C22×C16×C73C2211yescharacteristic
54#467×C73C2211yescharacteristic
55#466×C73C2211yescharacteristic
56#367×C73C2211yescharacteristic
57#358×C73C2211yescharacteristic
58#357×C73C2211yescharacteristic
59C2×#11924027—41no—
60C2×Dic292C2311yescharacteristic
61C2×#11924018—21no—
62C2×D584C2311yescharacteristic
63C2×#11924013—21no—
64C22×#10596831D411yescharacteristic
65#14556590—21no—
66#14556590—21no—
67#14556589—21no—
68#14556587—21no—
69#14556581—41no—
70#14556577—41no—
71#14556570—21no—
72#14556568—41no—
73#14556565—21no—
74#14556563—41no—
75Q8×Dic73—21no—
76#14556556—21no—
77#14556556—41no—
78D4×Dic73—21no—
79#14556553—21no—
80#14556551—41no—
81#14556547—41no—
82C2×C4×Dic73D411yescharacteristic
83#14556536—22no—
84#14556535—22no—
85#14556534—22no—
86C2×#11924003—22no—
87C2×#11924003C2311yescharacteristic
88C2×#11924003C2311yescharacteristic
89#14556525C2311yescharacteristic
90#14556525C2311yescharacteristic
91#14556524—21no—
92#14556524—21no—
93#14556516—21no—
94#14556515—21no—
95#14556514—21no—
96#14556513—21no—
97#14556512—21no—
98#14556511—21no—
99C8×Dic73D411yescharacteristic
100C8×Dic73D411yescharacteristic
101C8×Dic73C2311yescharacteristic
102C8×Dic73C2311yescharacteristic
103C2×Q16×C73C2311yescharacteristic
104C2×D8×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22⋊Q8×C73—21no—
107#121×C73—21no—
108SD32×C73—22no—
109C2×C16×C73—22no—
110C2×C16×C73C2311yescharacteristic
111C2×C16×C73C2311yescharacteristic
112#107×C73C2311yescharacteristic
113#107×C73C2311yescharacteristic
114#103×C73—21no—
115#102×C73—21no—
116#11924027—84no—
117C2×Dic146—21no—
118C2×Dic146—41no—
119C2×Dic146—41no—
120C2×D292—21no—
121C2×C22⋊D73—41no—
122C2×C22⋊D73—41no—
123C2×C22⋊D73—41no—
124C22×Dic73—41no—
125C22×Dic73D811yescharacteristic
126C22×Dic73D811yescharacteristic
127C2×C4×D73—41no—
128Dic292—21no—
129Dic292—21no—
130#11924018—42no—
131D584—21no—
132D584—21no—
133#11924013—42no—
134C2×#10596831—22no—
135C2×#10596831C2×D411yescharacteristic
136C2×#10596831C2×D411yescharacteristic
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—41no—
140#11924007—41no—
141#11924007—41no—
142#11924007—41no—
143#11924007—41no—
144#11924007—41no—
145#11924007—41no—
146C22⋊Dic73—41no—
147C22⋊Dic73—41no—
148#11924005—41no—
149#11924005—41no—
150C4×Dic73—41no—
151C4×Dic73—41no—
152C4×Dic73D811yescharacteristic
153C4×Dic73D811yescharacteristic
154C4×Dic73C2×D411yescharacteristic
155C4×Dic73C2×D411yescharacteristic
156#11924003—21no—
157#11924003#4112yescomplemented
158C2×Q8×C73—21no—
159C2×D4×C73—21no—
160C2×D4×C73—41no—
161C22×C4×C73C2×D411yescharacteristic
162Q16×C73—21no—
163Q16×C73—21no—
164D8×C73—21no—
165D8×C73—21no—
166C2×C8×C73—21no—
167C2×C8×C73—21no—
168C2×C8×C73C2×D411yescharacteristic
169C2×C8×C73C2411yescharacteristic
170C4⋊C4×C73—21no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—41no—
173C22⋊C4×C73—41no—
174C22⋊C4×C73—41no—
175C16×C73—21no—
176C16×C73#4112yescomplemented
177C23×C73—41no—
178C23×C73C2×D811yescharacteristic
179C22×D73—41no—
180Q8×C73—41no—
181Q8×C73—41no—
182D4×C73—41no—
183D4×C73—41no—
184D4×C73—82no—
185C2×C4×C73—21no—
186C2×C4×C73—21no—
187C2×C4×C73—41no—
188C2×C4×C73—41no—
189C2×C4×C73—41no—
190C2×C4×C73C2×D811yescharacteristic
191C2×C4×C73C22×D411yescharacteristic
192C22⋊D73—82no—
193C22⋊D73—82no—
194C22⋊D73—82no—
195C2×Dic73—22no—
196C2×Dic73—41no—
197C2×Dic73—41no—
198C2×Dic73—41no—
199C2×Dic73—82no—
200C2×Dic73SD3214yescomplemented
201C2×Dic73C2×D811yescharacteristic
202C2×Dic73C2×D811yescharacteristic
203C2×Dic73C2×D811yescharacteristic
204C2×Dic73C2×D811yescharacteristic
205D292—41no—
206D292—41no—
207C4×D73—82no—
208Dic146—41no—
209Dic146—41no—
210Dic146—82no—
211Dic146—82no—
212C8×C73—21no—
213C8×C73C2×#4111yescharacteristic
214C8×C73C2×#4111yescharacteristic
215#10596831—21no—
216#10596831#13512yes—
217C22×C73—21no—
218C22×C73—21no—
219C22×C73—81no—
220C22×C73—81no—
221C22×C73—81no—
222C22×C73C2×SD3212yes—
223C22×C73C22×D811yescharacteristic
224C2×D73—81no—
225C2×D73—81no—
226C2×D73—81no—
227C4×C73—21no—
228C4×C73—81no—
229C4×C73—81no—
230C4×C73—81no—
231C4×C73#53211yescharacteristic
232C4×C73C2×#13511yescharacteristic
233Dic73—21no—
234Dic73—81no—
235Dic73—81no—
236Dic73—81no—
237Dic73C2×SD3212yes—
238Dic73C2×SD3212yes—
239Dic73#50812yes—
240#33736—731nomaximal, Sylow
241C2×C73—22no—
242C2×C73—161no—
243C2×C73#316911yescharacteristic
244C2×C73C22×SD3211yescharacteristic
245C2×C73C2×#50811yescharacteristic
246D73—161no—
247#3170—731no—
248#3169—731no—
249#2254—731no—
250#2248—731no—
251#2241—731no—
252#2240—731no—
253#2217—731no—
254#2217—731no—
255#2217—731no—
256#2217D7311yescharacteristic, complemented
257C4×SD32—731no—
258C4×SD32—731no—
259C4×SD32—731no—
260C4×SD32—731no—
261C2×C4×C16—731no—
262C73#3373611yescharacteristic, complemented, Sylow
263#533—1461no—
264#532—1461no—
265C2×SD32—731no—
266C2×SD32—731no—
267C2×SD32—731no—
268C2×SD32C2×D7311yescharacteristic
269C22×C16—731no—
270C22×C16C2×D7311yescharacteristic
271#500—731no—
272#500—731no—
273#494—731no—
274#493—731no—
275#487—1461no—
276#486—1461no—
277#467—731no—
278#467—1461no—
279#467C2×D7311yescharacteristic
280#466—731no—
281#466—1461no—
282#466C2×D7311yescharacteristic
283C4×Q16—731no—
284C4×Q16—731no—
285C4×Q16—1461no—
286C4×D8—731no—
287C4×D8—731no—
288C4×D8—1461no—
289C2×C4×C8—731no—
290#367—731no—
291#367—731no—
292#367—731no—
293#367C2×D7311yescharacteristic
294#358—731no—
295#358—731no—
296#358—731no—
297#358C2×D7311yescharacteristic
298#357—731no—
299#357—731no—
300#357—731no—
301#357C2×D7311yescharacteristic
302C4×C16—731no—
303C4×C16—731no—
304C4×C16—731no—
305C4×C16—731no—
306C2×#41—2921no—
307C2×Q16—731no—
308C2×Q16—1461no—
309C2×Q16C22×D7311yescharacteristic
310C2×D8—731no—
311C2×D8—1461no—
312C2×D8C22×D7311yescharacteristic
313C22×C8—731no—
314C22×C8C22×D7311yescharacteristic
315C4⋊Q8—1461no—
316C4⋊Q8—1461no—
317C4⋊Q8—1461no—
318#127—1461no—
319#125—2921no—
320#124—2921no—
321C22⋊Q8—21no—
322C22⋊Q8—2921no—
323C22⋊Q8—1461no—
324#121—21no—
325#121—2921no—
326#121—1461no—
327C4×Q8—2921no—
328C4×Q8—1461no—
329C4×Q8—1461no—
330C4×D4—2921no—
331C4×D4—1461no—
332C4×D4—1461no—
333#117—2921no—
334C2×C42—731no—
335SD32—22no—
336SD32—1462no—
337SD32—1462no—
338SD32—1462no—
339C2×C16—22no—
340C2×C16—731no—
341C2×C16—731no—
342C2×C16—1462no—
343C2×C16C22×D7311yescharacteristic
344C2×C16C22×D7311yescharacteristic
345#107—731no—
346#107—731no—
347#107—1461no—
348#107—1461no—
349#107C22×D7311yescharacteristic
350#107C22×D7311yescharacteristic
351#103—21no—
352#103—1461no—
353#103—1461no—
354#103—1461no—
355#102—21no—
356#102—1461no—
357#102—1461no—
358#102—1461no—
359C4×C8—731no—
360C4×C8—731no—
361C4×C8—731no—
362C4×C8—731no—
363#41—5844no—
364C2×Q8—21no—
365C2×Q8—2921no—
366C2×Q8—2921no—
367C2×Q8—1461no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—2921no—
371C2×D4—2921no—
372C2×D4—2921no—
373C2×D4—1461no—
374C22×C4—2921no—
375C22×C4—2921no—
376C22×C4—731no—
377C22×C4—731no—
378C22×C4D4×D7311yescharacteristic
379Q16—21no—
380Q16—21no—
381Q16—2922no—
382Q16—1461no—
383Q16—1461no—
384D8—21no—
385D8—21no—
386D8—2922no—
387D8—1461no—
388D8—1461no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—731no—
392C2×C8—731no—
393C2×C8—1462no—
394C2×C8D4×D7311yescharacteristic
395C2×C8C23×D7311yescharacteristic
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—2921no—
400C4⋊C4—2921no—
401C4⋊C4—2921no—
402C4⋊C4—2921no—
403C4⋊C4—2921no—
404C4⋊C4—2921no—
405C4⋊C4—2921no—
406C4⋊C4—1461no—
407C4⋊C4—1461no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—2921no—
411C22⋊C4—2921no—
412C22⋊C4—2921no—
413C22⋊C4—2921no—
414C42—2921no—
415C42—2921no—
416C42—731no—
417C42—731no—
418C42—731no—
419C42—731no—
420C16—21no—
421C16—732no—
422C16—1461no—
423C16#1192402712yescomplemented
424C23—41no—
425C23—2921no—
426C23D8×D7311yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—2921no—
430Q8—2921no—
431Q8—5842no—
432Q8—5842no—
433D4—41no—
434D4—41no—
435D4—82no—
436D4—2921no—
437D4—2921no—
438D4—5842no—
439D4—5842no—
440D4—5842no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—2921no—
447C2×C4—2921no—
448C2×C4—2921no—
449C2×C4—5842no—
450C2×C4—5842no—
451C2×C4—734no—
452C2×C4—731no—
453C2×C4—731no—
454C2×C4—731no—
455C2×C4—731no—
456C2×C4—1462no—
457C2×C4D8×D7311yescharacteristic
458C2×C4C2×D4×D7311yescharacteristic
459C8—21no—
460C8—732no—
461C8—1461no—
462C8#41×D7311yescharacteristic
463C8C2×#1192402711yescharacteristic
464C22—21no—
465C22—21no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—5841no—
470C22—5841no—
471C22—5841no—
472C22SD32×D7312yes—
473C22C2×D8×D7311yescharacteristic
474C4—21no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—5841no—
479C4—5841no—
480C4—5841no—
481C4—732no—
482C4—732no—
483C4—732no—
484C4—1461no—
485C4#1798913211yescharacteristic
486C4#135×D7311yescharacteristic
487C2—22no—
488C2—161no—
489C2—11681no—
490C2#2262050511yescharacteristic
491C2C2×SD32×D7311yescharacteristic
492C2#508×D7311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2217C2
socleC22×C73C22×D8
prime radical#22620505C2
hypercenter#2217D73
hypocenterC73#33736
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. #22620512C2semidirect
3. #22612212C2semidirect
4. #22612185C2semidirect
5. #22612157C2semidirect
6. #22612153C2semidirect
7. #22612029C2semidirect
8. #22612027C2semidirect
9. #22612025C2semidirect
10. #22611786C2semidirect
11. SD32×Dic73C2semidirect
12. #22611782C2semidirect
13. #22611780C2semidirect
14. C2×C16×Dic73C2semidirect
15. C73×#2217C2semidirect
157. #11924003#41semidirect
176. C16×C73#41semidirect
200. C2×Dic73SD32semidirect
256. #2217D73semidirect
262. C73#33736semidirect
423. C16#11924027semidirect

Maximal subgroups

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

subgroupindex
#226205122
#226205052
#226122122
#226121852
#226121572
#226121532
#226120292
#226120272
#226120252
#226117862
SD32×Dic732
#226117822
#226117802
C2×C16×Dic732
C73×#22172
#3373673