Finite Group Databasebeta

← #33241116

Subgroups — #33241116

Order 18688.

534 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#22618808C211yesmaximal, characteristic
2#22612131C211yesmaximal, characteristic, complemented
3#22612126C211yesmaximal, characteristic, complemented
4#22612092C211yesmaximal, characteristic, complemented
5#22612085C211yesmaximal, characteristic, complemented
6#22611978C211yesmaximal, characteristic, complemented
7#22611973C211yesmaximal, characteristic, complemented
8#22611869C211yesmaximal, characteristic
9#22611869C211yesmaximal, characteristic
10#22611800C211yesmaximal, characteristic, complemented
11#22611793C211yesmaximal, characteristic, complemented
12C4×#14556537C211yesmaximal, characteristic, complemented
13C4×#14556534C211yesmaximal, characteristic, complemented
14#22611738C211yesmaximal, characteristic, complemented
15C73×#3177C211yesmaximal, characteristic, complemented
16C2×#14556613C2211yescharacteristic
17#17989067—21no—
18C2×#14556537C2211yescharacteristic
19C2×#14556537C2211yescharacteristic
20C2×#14556534C2211yescharacteristic
21C2×#14556534C2211yescharacteristic
22#17988689—21no—
23#17988689—21no—
24C4×Dic292C2211yescharacteristic
25C4×Dic292C2211yescharacteristic
26C4×D584C2211yescharacteristic
27C4×D584C2211yescharacteristic
28C2×#14556525C2211yescharacteristic
29#17988589—21no—
30#17988414C2211yescharacteristic
31#17988414C2211yescharacteristic
32#17988412C2211yescharacteristic
33#17988412C2211yescharacteristic
34C4⋊#11924003C2211yescharacteristic
35C4⋊#11924003C2211yescharacteristic
36#17988391C2211yescharacteristic
37#17988391C2211yescharacteristic
38#17988391C2211yescharacteristic
39#17988391C2211yescharacteristic
40#17988390C2211yescharacteristic
41#17988390C2211yescharacteristic
42#17988387C2211yescharacteristic
43#17988387C2211yescharacteristic
44C22⋊#11924003C2211yescharacteristic
45C22⋊#11924003C2211yescharacteristic
46#17988363C2211yescharacteristic
47C4×#11924003C2211yescharacteristic
48#541×C73—21no—
49#499×C73C2211yescharacteristic
50#494×C73C2211yescharacteristic
51#491×C73—21no—
52#470×C73—21no—
53#467×C73C2211yescharacteristic
54#467×C73C2211yescharacteristic
55#441×C73—21no—
56C4×Q16×C73C2211yescharacteristic
57C4×Q16×C73C2211yescharacteristic
58#405×C73C2211yescharacteristic
59C2×#11924028—21no—
60#14556613—22no—
61#14556613—22no—
62C2×Dic292C2311yescharacteristic
63C2×#11924017—21no—
64C2×D584C2311yescharacteristic
65C4×Dic146—21no—
66C4×Dic146—21no—
67C4×D292—21no—
68C4×D292—21no—
69#14556551—41no—
70#14556551—41no—
71#14556549—41no—
72C2×C4⋊Dic73—21no—
73C4⋊C4×D73—41no—
74#14556537—22no—
75#14556537—22no—
76#14556534—22no—
77#14556534—22no—
78C2×#11924003C2311yescharacteristic
79C2×#11924003C2311yescharacteristic
80C2×#11924003C2311yescharacteristic
81C2×#11924003C2311yescharacteristic
82#14556525—21no—
83#14556525C2311yescharacteristic
84#14556525C2311yescharacteristic
85#14556516—21no—
86#14556516—21no—
87#14556513—21no—
88#14556513—21no—
89C2×Q16×C73—21no—
90C2×Q16×C73C2311yescharacteristic
91C2×Q16×C73C2311yescharacteristic
92C2×M16×C73D411yescharacteristic
93C22×C8×C73C2311yescharacteristic
94C4⋊Q8×C73—21no—
95#126×C73—41no—
96#126×C73—41no—
97#125×C73—21no—
98#125×C73—41no—
99#125×C73—41no—
100C22⋊Q8×C73—21no—
101C22⋊Q8×C73—21no—
102C4×Q8×C73—21no—
103C4×Q8×C73—21no—
104#117×C73D411yescharacteristic
105#107×C73C2311yescharacteristic
106#107×C73C2311yescharacteristic
107#106×C73—21no—
108#103×C73—21no—
109#103×C73—21no—
110#103×C73—21no—
111#103×C73—21no—
112#97×C73D411yescharacteristic
113#97×C73D411yescharacteristic
114C4×C8×C73C2311yescharacteristic
115C4×C8×C73C2311yescharacteristic
116#11924028—24no—
117#11924028—42no—
118C2×Dic146—21no—
119C2×D292—21no—
120C2×C22⋊D73—41no—
121C22×Dic73—41no—
122C2×C4×D73—41no—
123C2×C4×D73—41no—
124C2×C4×D73—41no—
125Dic292—21no—
126Dic292#4112yes—
127#11924017—24no—
128D584—21no—
129D584#4112yes—
130C4⋊Dic73—21no—
131C4⋊Dic73—21no—
132C4⋊Dic73—41no—
133#11924007—41no—
134#11924007—41no—
135#11924005—41no—
136#11924005—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—21no—
145C22×C4×C73C2×D411yescharacteristic
146Q16×C73—21no—
147Q16×C73—21no—
148Q16×C73—21no—
149Q16×C73—21no—
150Q16×C73—42no—
151M16×C73—41no—
152C2×C8×C73—21no—
153C2×C8×C73—21no—
154C2×C8×C73C2×D411yescharacteristic
155C2×C8×C73C2×D411yescharacteristic
156C2×C8×C73C2×D411yescharacteristic
157C2×C8×C73C2411yescharacteristic
158C4⋊C4×C73—21no—
159C4⋊C4×C73—21no—
160C4⋊C4×C73—41no—
161C4⋊C4×C73—41no—
162C4⋊C4×C73—41no—
163C4⋊C4×C73—41no—
164C4⋊C4×C73—41no—
165C4⋊C4×C73—41no—
166C4⋊C4×C73—41no—
167C4⋊C4×C73—41no—
168C4⋊C4×C73D811yescharacteristic
169C4⋊C4×C73D811yescharacteristic
170C22⋊C4×C73—41no—
171C22⋊C4×C73—41no—
172C22⋊C4×C73D811yescharacteristic
173C22⋊C4×C73D811yescharacteristic
174C42×C73—41no—
175C42×C73—41no—
176C42×C73C2×D411yescharacteristic
177C42×C73C2×D411yescharacteristic
178C23×C73C2×D811yescharacteristic
179C22×D73—41no—
180Q8×C73—41no—
181Q8×C73—41no—
182Q8×C73—41no—
183Q8×C73—41no—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—41no—
187C2×C4×C73—41no—
188C2×C4×C73—41no—
189C2×C4×C73—41no—
190C2×C4×C73C2×D811yescharacteristic
191C2×C4×C73C2×D811yescharacteristic
192C2×C4×C73C2×D811yescharacteristic
193C2×C4×C73C2×D811yescharacteristic
194C2×C4×C73C2×D811yescharacteristic
195C2×C4×C73C22×D411yescharacteristic
196C22⋊D73—44no—
197C2×Dic73—41no—
198C2×Dic73—41no—
199C2×Dic73—41no—
200C2×Dic73—161no—
201D292—22no—
202D292—41no—
203C4×D73—44no—
204C4×D73—161no—
205C4×D73—161no—
206Dic146—22no—
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×C73C22×D811yescharacteristic
217C2×D73—42no—
218C2×D73—81no—
219C2×D73—81no—
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—81no—
229C4×C73#52511yescharacteristic
230C4×C73#57511yescharacteristic
231Dic73—42no—
232Dic73—81no—
233Dic73—81no—
234#33680—731nomaximal, Sylow
235C2×C73—41no—
236C2×C73#310011yescharacteristic
237C2×C73#341611yescharacteristic
238C2×C73#341711yescharacteristic
239D73—82no—
240#3177D7311yescharacteristic, complemented
241#2951—731no—
242#2236—731no—
243#2235—731no—
244#2229—731no—
245#2228—731no—
246#2209—731no—
247#2208—731no—
248#2189—731no—
249#2189—731no—
250#2176—731no—
251#2175—731no—
252C4×Q32—731no—
253C4×SD32—731no—
254#2164—731no—
255C73#3368011yescharacteristic, complemented, Sylow
256C2×#135—731no—
257#541—21no—
258#520—1461no—
259C2×Q32—731no—
260C2×Q32—731no—
261C2×SD32—731no—
262C2×SD32—731no—
263#499C2×D7311yescharacteristic
264#494C2×D7311yescharacteristic
265#491—21no—
266#470—21no—
267#467C2×D7311yescharacteristic
268#467C2×D7311yescharacteristic
269#441—21no—
270#440—1461no—
271#440—1461no—
272C4×Q16—731no—
273C4×Q16—731no—
274C4×Q16C2×D7311yescharacteristic
275C4×Q16C2×D7311yescharacteristic
276C4×D8—731no—
277C4×D8—731no—
278C2×#107—731no—
279#417—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—
299C2×#41—1461no—
300#135—1462no—
301#135—1462no—
302C2×Q16—21no—
303C2×Q16—731no—
304C2×Q16C22×D7311yescharacteristic
305C2×Q16C22×D7311yescharacteristic
306C2×SD16—1461no—
307C2×D8—731no—
308C2×M16C22⋊D7311yescharacteristic
309C22×C8C22×D7311yescharacteristic
310C4⋊Q8—21no—
311#126—41no—
312#126—41no—
313#125—21no—
314#125—41no—
315#125—41no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8—1461no—
321C4×Q8—1461no—
322C4×D4—2921no—
323C4×D4—2921no—
324C4×D4—1461no—
325C4×D4—1461no—
326#117—2921no—
327#117C22⋊D7311yescharacteristic
328C2×C4⋊C4—2921no—
329C2×C4⋊C4—1461no—
330Q32—1462no—
331Q32—1462no—
332SD32—1462no—
333SD32—1462no—
334C2×C16—731no—
335C2×C16—731no—
336C2×C16—731no—
337C2×C16—731no—
338#107—731no—
339#107—731no—
340#107—1461no—
341#107C22×D7311yescharacteristic
342#107C22×D7311yescharacteristic
343#106—21no—
344#103—21no—
345#103—21no—
346#103—21no—
347#103—21no—
348#103—1461no—
349#103—1461no—
350#102—1461no—
351#102—1461no—
352#97C22⋊D7311yescharacteristic
353#97C22⋊D7311yescharacteristic
354C4×C8C22×D7311yescharacteristic
355C4×C8C22×D7311yescharacteristic
356#41—2922no—
357#41—1464no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—1461no—
361C2×D4—2921no—
362C2×D4—1461no—
363C22×C4—2921no—
364C22×C4—2921no—
365C22×C4—2921no—
366C22×C4—2921no—
367C22×C4C2×C22⋊D7311yescharacteristic
368Q16—21no—
369Q16—21no—
370Q16—21no—
371Q16—21no—
372Q16—42no—
373Q16—732no—
374Q16—1461no—
375SD16—1464no—
376D8—732no—
377D8—1461no—
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—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—2921no—
396C4⋊C4—2921no—
397C4⋊C4—2921no—
398C4⋊C4—1461no—
399C4⋊C4—1461no—
400C4⋊C4#1192401311yescharacteristic
401C4⋊C4#1192401311yescharacteristic
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—2921no—
405C22⋊C4—2921no—
406C22⋊C4#1192401311yescharacteristic
407C22⋊C4#1192401311yescharacteristic
408C42—41no—
409C42—41no—
410C42—2921no—
411C42—2921no—
412C42C2×C22⋊D7311yescharacteristic
413C42C2×C22⋊D7311yescharacteristic
414C16—1461no—
415C16—1461no—
416C16—1461no—
417C16—1461no—
418C23—2921no—
419C23C2×#1192401311yescharacteristic
420Q8—41no—
421Q8—41no—
422Q8—41no—
423Q8—41no—
424Q8—2921no—
425Q8—1462no—
426D4—2924no—
427D4—2921no—
428D4—1462no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—2924no—
436C2×C4—2921no—
437C2×C4—2921no—
438C2×C4—2921no—
439C2×C4—11681no—
440C2×C4—11681no—
441C2×C4—11681no—
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—2922no—
457C22—5841no—
458C22—5841no—
459C22C22×#1192401311yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—2922no—
470C4—5841no—
471C4—5841no—
472C4#1798909411yescharacteristic
473C4#1798938511yescharacteristic
474C2—41no—
475C2—5842no—
476C2#2261992711yescharacteristic
477C2#2262307511yescharacteristic
478C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3177C2
socleC22×C73C22×D8
prime radicalC2×#14556613C22
hypercenter#3177D73
hypocenterC73#33680
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3177

Complements

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

NH ≅ G/Ntype
2. #22612131C2semidirect
3. #22612126C2semidirect
4. #22612092C2semidirect
5. #22612085C2semidirect
6. #22611978C2semidirect
7. #22611973C2semidirect
10. #22611800C2semidirect
11. #22611793C2semidirect
12. C4×#14556537C2semidirect
13. C4×#14556534C2semidirect
14. #22611738C2semidirect
15. C73×#3177C2semidirect
240. #3177D73semidirect
255. C73#33680semidirect

Maximal subgroups

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

subgroupindex
#226188082
#226121312
#226121262
#226120922
#226120852
#226119782
#226119732
#226118692
#226118002
#226117932
C4×#145565372
C4×#145565342
#226117382
C73×#31772
#3368073