Finite Group Databasebeta

← #33241131

Subgroups — #33241131

Order 18688.

532 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618800C211yesmaximal, characteristic, complemented
2#22612136C211yesmaximal, characteristic, complemented
3#22612119C211yesmaximal, characteristic
4#22612080C211yesmaximal, characteristic, complemented
5#22612070C211yesmaximal, characteristic, complemented
6#22611990C211yesmaximal, characteristic, complemented
7#22611990C211yesmaximal, characteristic, complemented
8#22611864C211yesmaximal, characteristic, complemented
9#22611847C211yesmaximal, characteristic
10#22611800C211yesmaximal, characteristic, complemented
11#22611795C211yesmaximal, characteristic, complemented
12C4×#14556537C211yesmaximal, characteristic, complemented
13C4×#14556535C211yesmaximal, characteristic, complemented
14#22611738C211yesmaximal, characteristic, complemented
15C73×#3175C211yesmaximal, characteristic
16C22×Dic292C2211yescharacteristic
17#17989064—21no—
18C2×#14556537C2211yescharacteristic
19C2×#14556537C2211yescharacteristic
20C2×#14556535C2211yescharacteristic
21C2×#14556535C2211yescharacteristic
22#17988695—21no—
23#17988695—21no—
24C4×Dic292C2211yescharacteristic
25C4×Dic292C2211yescharacteristic
26C4×Dic292C2211yescharacteristic
27C4×Dic292C2211yescharacteristic
28#17988625C2211yescharacteristic
29#17988597—21no—
30#17988414C2211yescharacteristic
31#17988414C2211yescharacteristic
32#17988412C2211yescharacteristic
33#17988412C2211yescharacteristic
34C4⋊#11924003C2211yescharacteristic
35C4⋊#11924003C2211yescharacteristic
36#17988391C2211yescharacteristic
37#17988391C2211yescharacteristic
38#17988390C2211yescharacteristic
39#17988390C2211yescharacteristic
40#17988390C2211yescharacteristic
41#17988390C2211yescharacteristic
42#17988388C2211yescharacteristic
43#17988388C2211yescharacteristic
44C22⋊#11924003C2211yescharacteristic
45C22⋊#11924003C2211yescharacteristic
46#17988363C2211yescharacteristic
47C4×#11924003C2211yescharacteristic
48#542×C73—21no—
49#500×C73C2211yescharacteristic
50#495×C73C2211yescharacteristic
51#489×C73—21no—
52#468×C73—21no—
53#467×C73C2211yescharacteristic
54#466×C73C2211yescharacteristic
55#440×C73—21no—
56C4×Q16×C73C2211yescharacteristic
57C4×D8×C73C2211yescharacteristic
58#405×C73C2211yescharacteristic
59C22×Dic146—21no—
60C2×Dic292—22no—
61C2×Dic292—22no—
62C2×Dic292—21no—
63C2×Dic292C2311yescharacteristic
64C2×Dic292C2311yescharacteristic
65C4×Dic146—21no—
66C4×Dic146—21no—
67C4×Dic146—21no—
68C4×Dic146—21no—
69#14556556—41no—
70#14556556—41no—
71#14556550—21no—
72#14556547—41no—
73#14556547—41no—
74#14556537—22no—
75#14556537—22no—
76#14556535—22no—
77#14556535—22no—
78C2×#11924003C2311yescharacteristic
79C2×#11924003C2311yescharacteristic
80C2×#11924003C2311yescharacteristic
81C2×#11924003C2311yescharacteristic
82#14556525C2311yescharacteristic
83#14556525C2311yescharacteristic
84#14556523—21no—
85#14556516—21no—
86#14556516—21no—
87#14556516—21no—
88#14556516—21no—
89C2×Q16×C73C2311yescharacteristic
90C2×SD16×C73—21no—
91C2×D8×C73C2311yescharacteristic
92C2×M16×C73D411yescharacteristic
93C22×C8×C73C2311yescharacteristic
94C4⋊Q8×C73—21no—
95#126×C73—41no—
96#124×C73—21no—
97#124×C73—41no—
98#123×C73—41no—
99C22⋊Q8×C73—21no—
100C22⋊Q8×C73—41no—
101#121×C73—21no—
102C4×Q8×C73—21no—
103C4×D4×C73—21no—
104#117×C73D411yescharacteristic
105#107×C73—21no—
106#107×C73C2311yescharacteristic
107#107×C73C2311yescharacteristic
108#103×C73—21no—
109#103×C73—21no—
110#102×C73—21no—
111#102×C73—21no—
112#97×C73D411yescharacteristic
113#97×C73D411yescharacteristic
114C4×C8×C73C2311yescharacteristic
115C4×C8×C73C2311yescharacteristic
116C2×Dic146—24no—
117C2×Dic146—21no—
118C2×Dic146—21no—
119C2×Dic146—42no—
120C2×Dic146—41no—
121C22×Dic73—41no—
122Dic292—24no—
123Dic292—21no—
124Dic292—21no—
125Dic292#4112yes—
126Dic292#4112yes—
127C4⋊Dic73—21no—
128C4⋊Dic73—21no—
129#11924007—41no—
130#11924007—41no—
131#11924007—41no—
132#11924007—41no—
133C22⋊Dic73—41no—
134C4×Dic73—41no—
135C4×Dic73—41no—
136C4×Dic73—41no—
137C4×Dic73—41no—
138C4×Dic73—41no—
139#11924003—21no—
140#11924003—21no—
141#11924003—21no—
142#11924003—21no—
143C2×Q8×C73—21no—
144C2×Q8×C73—41no—
145C2×D4×C73—21no—
146C2×D4×C73—41no—
147C22×C4×C73—41no—
148C22×C4×C73C2×D411yescharacteristic
149Q16×C73—21no—
150Q16×C73—21no—
151SD16×C73—42no—
152D8×C73—21no—
153D8×C73—21no—
154M16×C73—41no—
155C2×C8×C73—21no—
156C2×C8×C73—21no—
157C2×C8×C73C2×D411yescharacteristic
158C2×C8×C73C2×D411yescharacteristic
159C2×C8×C73C2×D411yescharacteristic
160C2×C8×C73C2411yescharacteristic
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—41no—
164C4⋊C4×C73—41no—
165C4⋊C4×C73—41no—
166C4⋊C4×C73D811yescharacteristic
167C4⋊C4×C73D811yescharacteristic
168C22⋊C4×C73—41no—
169C22⋊C4×C73—41no—
170C22⋊C4×C73—41no—
171C22⋊C4×C73—41no—
172C22⋊C4×C73—41no—
173C22⋊C4×C73D811yescharacteristic
174C22⋊C4×C73D811yescharacteristic
175C42×C73—41no—
176C42×C73C2×D411yescharacteristic
177C42×C73C2×D411yescharacteristic
178C23×C73—41no—
179C23×C73C2×D811yescharacteristic
180Q8×C73—41no—
181Q8×C73—41no—
182Q8×C73—161no—
183D4×C73—41no—
184D4×C73—41no—
185D4×C73—161no—
186C2×C4×C73—21no—
187C2×C4×C73—21no—
188C2×C4×C73—41no—
189C2×C4×C73—41no—
190C2×C4×C73—41no—
191C2×C4×C73—161no—
192C2×C4×C73C2×D811yescharacteristic
193C2×C4×C73C2×D811yescharacteristic
194C2×C4×C73C2×D811yescharacteristic
195C2×C4×C73C2×D811yescharacteristic
196C2×C4×C73C2×D811yescharacteristic
197C2×C4×C73C22×D411yescharacteristic
198C2×Dic73—44no—
199C2×Dic73—41no—
200C2×Dic73—41no—
201C2×Dic73—41no—
202C2×Dic73—41no—
203Dic146—22no—
204Dic146—22no—
205Dic146—44no—
206Dic146—41no—
207Dic146—41no—
208C8×C73—21no—
209C8×C73—21no—
210C8×C73—21no—
211C8×C73C2×#4111yescharacteristic
212C8×C73C2×#4111yescharacteristic
213C22×C73—21no—
214C22×C73—21no—
215C22×C73—21no—
216C22×C73—81no—
217C22×C73—81no—
218C22×C73—81no—
219C22×C73C22×D811yescharacteristic
220C4×C73—21no—
221C4×C73—21no—
222C4×C73—21no—
223C4×C73—21no—
224C4×C73—21no—
225C4×C73—81no—
226C4×C73—81no—
227C4×C73—81no—
228C4×C73#52511yescharacteristic
229C4×C73#57511yescharacteristic
230Dic73—42no—
231Dic73—42no—
232Dic73—81no—
233Dic73—81no—
234#33681—731nomaximal, Sylow
235C2×C73—41no—
236C2×C73—161no—
237C2×C73#310011yescharacteristic
238C2×C73#341611yescharacteristic
239C2×C73#341711yescharacteristic
240#3175D7311yescharacteristic
241#2950—731no—
242#2237—731no—
243#2234—731no—
244#2227—731no—
245#2225—731no—
246#2211—731no—
247#2211—731no—
248#2188—731no—
249#2185—731no—
250#2176—731no—
251#2175—731no—
252C4×Q32—731no—
253C4×SD32—731no—
254#2164—731no—
255C73#3368111yescharacteristic, complemented, Sylow
256C22×Q16—731no—
257#542—21no—
258#519—1461no—
259C2×Q32—731no—
260C2×Q32—731no—
261C2×SD32—731no—
262C2×SD32—731no—
263#500C2×D7311yescharacteristic
264#495C2×D7311yescharacteristic
265#489—21no—
266#468—21no—
267#467C2×D7311yescharacteristic
268#466C2×D7311yescharacteristic
269#441—1461no—
270#441—1461no—
271#440—21no—
272C4×Q16—731no—
273C4×Q16—731no—
274C4×Q16—731no—
275C4×Q16—731no—
276C4×Q16C2×D7311yescharacteristic
277C4×D8C2×D7311yescharacteristic
278#427—731no—
279#419—1461no—
280#405C2×D7311yescharacteristic
281#367—731no—
282#367—731no—
283#366—731no—
284#366—731no—
285C4⋊C16—731no—
286C4⋊C16—731no—
287#358—731no—
288#358—731no—
289#358—731no—
290#358—731no—
291#358—731no—
292#358—731no—
293#357—731no—
294#357—731no—
295C22⋊C16—731no—
296C22⋊C16—731no—
297#346—731no—
298C4×C16—731no—
299C22×Q8—1461no—
300C2×Q16—731no—
301C2×Q16—731no—
302C2×Q16—1462no—
303C2×Q16—1462no—
304C2×Q16—1461no—
305C2×Q16C22×D7311yescharacteristic
306C2×SD16—21no—
307C2×D8C22×D7311yescharacteristic
308C2×M16C22⋊D7311yescharacteristic
309C22×C8C22×D7311yescharacteristic
310C4⋊Q8—21no—
311#126—41no—
312#124—21no—
313#124—41no—
314#123—41no—
315C22⋊Q8—21no—
316C22⋊Q8—41no—
317#121—21no—
318C4×Q8—21no—
319C4×Q8—2921no—
320C4×Q8—2921no—
321C4×Q8—1461no—
322C4×Q8—1461no—
323C4×Q8—1461no—
324C4×Q8—1461no—
325C4×D4—21no—
326#117—2921no—
327#117—2921no—
328#117—1461no—
329#117C22⋊D7311yescharacteristic
330Q32—1462no—
331Q32—1462no—
332SD32—1462no—
333SD32—1462no—
334C2×C16—731no—
335C2×C16—731no—
336C2×C16—731no—
337C2×C16—731no—
338#107—21no—
339#107—731no—
340#107—731no—
341#107C22×D7311yescharacteristic
342#107C22×D7311yescharacteristic
343#106—1461no—
344#103—21no—
345#103—21no—
346#103—1461no—
347#103—1461no—
348#103—1461no—
349#103—1461no—
350#102—21no—
351#102—21no—
352#97C22⋊D7311yescharacteristic
353#97C22⋊D7311yescharacteristic
354C4×C8C22×D7311yescharacteristic
355C4×C8C22×D7311yescharacteristic
356C2×Q8—21no—
357C2×Q8—41no—
358C2×Q8—2922no—
359C2×Q8—2921no—
360C2×Q8—1464no—
361C2×Q8—1461no—
362C2×Q8—1461no—
363C2×D4—21no—
364C2×D4—41no—
365C22×C4—41no—
366C22×C4—2921no—
367C22×C4C2×C22⋊D7311yescharacteristic
368Q16—21no—
369Q16—21no—
370Q16—732no—
371Q16—732no—
372Q16—1464no—
373Q16—1461no—
374Q16—1461no—
375SD16—42no—
376D8—21no—
377D8—21no—
378M16—41no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8C2×C22⋊D7311yescharacteristic
382C2×C8C2×C22⋊D7311yescharacteristic
383C2×C8C2×C22⋊D7311yescharacteristic
384C2×C8C23×D7311yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—2921no—
391C4⋊C4—2921no—
392C4⋊C4—2921no—
393C4⋊C4—2921no—
394C4⋊C4—1461no—
395C4⋊C4—1461no—
396C4⋊C4#1192401311yescharacteristic
397C4⋊C4#1192401311yescharacteristic
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—2921no—
404C22⋊C4#1192401311yescharacteristic
405C22⋊C4#1192401311yescharacteristic
406C42—41no—
407C42—2921no—
408C42—2921no—
409C42—2921no—
410C42—2921no—
411C42—2921no—
412C42C2×C22⋊D7311yescharacteristic
413C42C2×C22⋊D7311yescharacteristic
414C16—1461no—
415C16—1461no—
416C16—1461no—
417C16—1461no—
418C23—41no—
419C23C2×#1192401311yescharacteristic
420Q8—41no—
421Q8—41no—
422Q8—161no—
423Q8—2924no—
424Q8—2921no—
425Q8—2921no—
426Q8—1462no—
427Q8—1462no—
428D4—41no—
429D4—41no—
430D4—161no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—161no—
437C2×C4—2924no—
438C2×C4—2921no—
439C2×C4—2921no—
440C2×C4—2921no—
441C2×C4—2921no—
442C2×C4C2×#1192401311yescharacteristic
443C2×C4C2×#1192401311yescharacteristic
444C2×C4C2×#1192401311yescharacteristic
445C2×C4C2×#1192401311yescharacteristic
446C2×C4C2×#1192401311yescharacteristic
447C2×C4C22×C22⋊D7311yescharacteristic
448C8—21no—
449C8—21no—
450C8—21no—
451C8#41×D7311yescharacteristic
452C8#41×D7311yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22C22×#1192401311yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—2922no—
469C4—2922no—
470C4—5841no—
471C4—5841no—
472C4#1798909411yescharacteristic
473C4#1798938511yescharacteristic
474C2—41no—
475C2—161no—
476C2#2261992711yescharacteristic
477C2#2262307811yescharacteristic
478C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3175C2
socleC22×C73C22×D8
prime radical#466×C73C22
hypercenter#3175D73
hypocenterC73#33681
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #22618800C2semidirect
2. #22612136C2semidirect
4. #22612080C2semidirect
5. #22612070C2semidirect
6. #22611990C2semidirect
7. #22611990C2semidirect
8. #22611864C2semidirect
10. #22611800C2semidirect
11. #22611795C2semidirect
12. C4×#14556537C2semidirect
13. C4×#14556535C2semidirect
14. #22611738C2semidirect
255. C73#33681semidirect

Maximal subgroups

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

subgroupindex
#226188002
#226121362
#226121192
#226120802
#226120702
#226119902
#226118642
#226118472
#226118002
#226117952
C4×#145565372
C4×#145565352
#226117382
C73×#31752
#3368173