Finite Group Databasebeta

← #33098285

Subgroups — #33098285

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619713C211yesmaximal, characteristic
2#22618573C211yesmaximal, characteristic
3#22610876C211yesmaximal, characteristic, complemented
4#22610704C211yesmaximal, characteristic, complemented
5#22610676C212yesmaximal, complemented
6#22610634C212yesmaximal
7#22610579C212yesmaximal
8#22610530C212yesmaximal, complemented
9#22609948C211yesmaximal, characteristic
10C8×C4⋊Dic73C211yesmaximal, characteristic, complemented
11C73×#2960C211yesmaximal, characteristic, complemented
12C4×#11924028C2211yescharacteristic
13#17988891C412yescomplemented
14#17988884C412yescomplemented
15#17988879C411yescharacteristic, complemented
16#17988874C411yescharacteristic, complemented
17#17988869C411yescharacteristic, complemented
18#17988864C411yescharacteristic, complemented
19#17988646—21no—
20#17988646—21no—
21#17988638—21no—
22#17988638—21no—
23C2×#14556526C2212yes—
24#17988623C2212yes—
25#17988613C2211yescharacteristic
26#17988613C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556517C2212yes—
29#17988581C2212yes—
30#17988553C2211yescharacteristic
31#17988553C2211yescharacteristic
32C4×#11924012C2211yescharacteristic
33C2×C8×Dic73C2212yes—
34C4×C4⋊Dic73C2211yescharacteristic
35#17988351C2212yes—
36#17988348C2212yes—
37#17988346C2212yes—
38#17988346C2212yes—
39#17988341C2212yes—
40#533×C73C2211yescharacteristic
41#446×C73—21no—
42#446×C73—21no—
43C8×Q8×C73—21no—
44C8×Q8×C73—21no—
45#424×C73C2212yes—
46#423×C73C2211yescharacteristic
47#405×C73C2212yes—
48C2×C4×C8×C73C2211yescharacteristic
49C2×#11924028C2311yescharacteristic
50C2×#11924012C2311yescharacteristic
51C2×#11924012C2311yescharacteristic
52C22×#10596831C2312yes—
53C4×Dic146C2×C411yescharacteristic
54C4×Dic146C2×C411yescharacteristic
55C4×D292C2×C411yescharacteristic
56C4×D292C2×C411yescharacteristic
57C4×C22⋊D73—21no—
58C4×C22⋊D73—21no—
59#14556550C2311yescharacteristic
60#14556548—21no—
61C2×C4⋊Dic73—21no—
62C2×C4×Dic73C2312yes—
63C42×D73—21no—
64#14556526—42no—
65#14556524C2×C412yes—
66#14556524C2×C412yes—
67#14556522C2×C412yes—
68#14556522C2×C412yes—
69C4⋊#10596831C2×C411yescharacteristic
70C4⋊#10596831C2×C411yescharacteristic
71C4⋊#10596831C2×C412yes—
72#14556520—21no—
73#14556520—21no—
74#14556520—21no—
75#14556520—21no—
76#14556517—42no—
77#14556514C2×C412yes—
78#14556514C2×C412yes—
79#14556511C2×C412yes—
80#14556511C2×C412yes—
81#14556503—21no—
82#14556503—21no—
83#14556503—21no—
84#14556503—21no—
85#14556502—21no—
86C4×#10596831—21no—
87C8×Dic73—42no—
88#14556498—21no—
89C2×M16×C73C2312yes—
90C22×C8×C73C2311yescharacteristic
91C22×C8×C73C2311yescharacteristic
92C4⋊Q8×C73C2×C411yescharacteristic
93C4⋊Q8×C73C2×C411yescharacteristic
94#125×C73C2×C412yes—
95C22⋊Q8×C73—21no—
96C22⋊Q8×C73—21no—
97C4×Q8×C73—21no—
98C4×Q8×C73—21no—
99#117×C73C2312yes—
100C2×C42×C73C2311yescharacteristic
101C4⋊C8×C73—22no—
102C4⋊C8×C73—22no—
103C4⋊C8×C73—21no—
104C4⋊C8×C73—21no—
105#97×C73—22no—
106C4×C8×C73—22no—
107C4×C8×C73D411yescharacteristic
108C4×C8×C73D411yescharacteristic
109C4×C8×C73D411yescharacteristic
110C4×C8×C73D411yescharacteristic
111#11924028—21no—
112#11924028—21no—
113#11924028—21no—
114#11924028—21no—
115C2×Dic146C22×C411yescharacteristic
116C2×D292C22×C411yescharacteristic
117C2×C22⋊D73—21no—
118C22×Dic73—22no—
119C2×C4×D73—21no—
120C2×C4×D73—21no—
121C2×C4×D73—21no—
122#11924012—21no—
123#11924012—21no—
124#11924012—21no—
125#11924012—21no—
126C2×#10596831—22no—
127C2×#10596831—22no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831C22×C412yes—
131C2×#10596831C22×C412yes—
132C4⋊Dic73—24no—
133C4⋊Dic73C22×C411yescharacteristic
134C4⋊Dic73C22×C411yescharacteristic
135#11924007—21no—
136#11924007—21no—
137C22⋊Dic73—21no—
138#11924005—21no—
139#11924005—21no—
140C4×Dic73—21no—
141C4×Dic73—21no—
142C4×Dic73—21no—
143C4×Dic73—22no—
144C4×Dic73—22no—
145C2×Q8×C73—21no—
146C2×Q8×C73—21no—
147C22×C4×C73#4111yescharacteristic
148C22×C4×C73#4111yescharacteristic
149C22×C4×C73C2411yescharacteristic
150M16×C73—22no—
151M16×C73—22no—
152C2×C8×C73—21no—
153C2×C8×C73—21no—
154C2×C8×C73—21no—
155C2×C8×C73—21no—
156C2×C8×C73—22no—
157C2×C8×C73C2×D411yescharacteristic
158C2×C8×C73C2×D411yescharacteristic
159C2×C8×C73C2×D411yescharacteristic
160C2×C8×C73C2×D411yescharacteristic
161C4⋊C4×C73—22no—
162C4⋊C4×C73—22no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73—21no—
165C4⋊C4×C73C22×C412yes—
166C4⋊C4×C73C22×C412yes—
167C22⋊C4×C73—22no—
168C42×C73—22no—
169C42×C73C22×C411yescharacteristic
170C42×C73C22×C411yescharacteristic
171C42×C73C2×D411yescharacteristic
172C42×C73C2×D411yescharacteristic
173C23×C73C2×#4111yescharacteristic
174C22×D73—21no—
175Q8×C73—42no—
176Q8×C73—42no—
177C2×C4×C73—22no—
178C2×C4×C73—22no—
179C2×C4×C73—22no—
180C2×C4×C73—41no—
181C2×C4×C73C4×D411yescharacteristic
182C2×C4×C73C4×D411yescharacteristic
183C2×C4×C73C4×D411yescharacteristic
184C2×C4×C73C4×D411yescharacteristic
185C2×C4×C73C23×C411yescharacteristic
186C2×C4×C73C22×D411yescharacteristic
187C2×C4×C73H(5,𝔽2)11yescharacteristic
188C2×C4×C732−1+411yescharacteristic
189C2×C4×C732−1+411yescharacteristic
190C2×C4×C732−1+411yescharacteristic
191C22⋊D73—41no—
192C22⋊D73—41no—
193C2×Dic73—24no—
194C2×Dic73—21no—
195C2×Dic73—21no—
196C2×Dic73—21no—
197C2×Dic73—42no—
198C2×Dic73—42no—
199D292—21no—
200D292—21no—
201C4×D73—42no—
202C4×D73—41no—
203C4×D73—41no—
204C4×D73—42no—
205Dic146—21no—
206Dic146—21no—
207C8×C73—21no—
208C8×C73—21no—
209C8×C73—21no—
210C8×C73—21no—
211C8×C73—42no—
212#10596831—22no—
213#10596831—22no—
214#10596831—41no—
215#10596831—41no—
216C22×C73—21no—
217C22×C73—21no—
218C22×C73C2×C4×D411yescharacteristic
219C22×C73#54111yescharacteristic
220C22×C73#54211yescharacteristic
221C2×D73—41no—
222C2×D73—41no—
223C2×D73—41no—
224C4×C73—21no—
225C4×C73—21no—
226C4×C73—21no—
227C4×C73—42no—
228C4×C73—42no—
229C4×C73#44312yes—
230C4×C73#51911yescharacteristic
231C4×C73#52011yescharacteristic
232C4×C73#53611yescharacteristic
233C4×C73#53711yescharacteristic
234Dic73—44no—
235Dic73—41no—
236Dic73—41no—
237Dic73—41no—
238#22130—731nomaximal, Sylow
239C2×C73—21no—
240C2×C73—21no—
241C2×C73#232311yescharacteristic
242C2×C73C2×#44311yescharacteristic
243C2×C73#295511yescharacteristic
244D73—81no—
245#3076—731no—
246#2960D7311yescharacteristic, complemented
247#2922—731no—
248#1986—731no—
249#1950—731no—
250#1945—732no—
251#1937—732no—
252#1929—732no—
253#1922—732no—
254#1806—731no—
255C8×C4⋊C4—731no—
256C73#2213011yescharacteristic, complemented, Sylow
257#533C2×D7311yescharacteristic
258C4×#41—731no—
259#479—732no—
260#478—732no—
261#477—731no—
262#476—731no—
263#475—731no—
264#474—731no—
265#446—21no—
266#446—21no—
267C8×Q8—21no—
268C8×Q8—21no—
269#433—1461no—
270#433—1461no—
271#431—1461no—
272#431—1461no—
273C2×#108—732no—
274#427—732no—
275#424—731no—
276#424—731no—
277#424C2×D7312yes—
278#423C2×D7311yescharacteristic
279C2×C4⋊C8—731no—
280C2×#104—732no—
281#416—732no—
282#408—731no—
283#408—731no—
284#405C2×D7312yes—
285C4×M16—731no—
286C2×C4×C8—732no—
287C2×C4×C8C2×D7311yescharacteristic
288C4×C4⋊C4—731no—
289#341—732no—
290#339—732no—
291#339—732no—
292#339—732no—
293#336—732no—
294C2×#41—731no—
295C2×M16—731no—
296C2×M16—731no—
297C2×M16C22×D7312yes—
298C22×C8—732no—
299C22×C8C22×D7311yescharacteristic
300C22×C8C22×D7311yescharacteristic
301C4⋊Q8C4×D7311yescharacteristic
302C4⋊Q8C4×D7311yescharacteristic
303#125C4×D7312yes—
304C22⋊Q8—21no—
305C22⋊Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—731no—
309C4×Q8—731no—
310C4×D4—731no—
311C4×D4—731no—
312C4×D4—1461no—
313C4×D4—1461no—
314#117—731no—
315#117—1461no—
316#117C22×D7312yes—
317C2×C4⋊C4—1461no—
318C2×C42—732no—
319C2×C42—1461no—
320C2×C42C22×D7311yescharacteristic
321#108—2922no—
322#107—732no—
323#107—732no—
324#106—732no—
325#106—732no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—731no—
331C4⋊C8—731no—
332C4⋊C8—732no—
333C4⋊C8—1461no—
334C4⋊C8—1461no—
335C4⋊C8—1461no—
336C4⋊C8—1461no—
337#104—2922no—
338#103—732no—
339#103—732no—
340#102—732no—
341#102—732no—
342C22⋊C8—1461no—
343C22⋊C8—1461no—
344C22⋊C8—1461no—
345C22⋊C8—1461no—
346#97—22no—
347#97—1461no—
348C4×C8—22no—
349C4×C8—2922no—
350C4×C8—1461no—
351C4×C8C22⋊D7311yescharacteristic
352C4×C8C22⋊D7311yescharacteristic
353C4×C8C22⋊D7311yescharacteristic
354C4×C8C22⋊D7311yescharacteristic
355#95—1461no—
356#41—1461no—
357#41—1461no—
358#41—1461no—
359#41—1461no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—731no—
363C2×D4—731no—
364C2×D4—1461no—
365C22×C4—1462no—
366C22×C4—1461no—
367C22×C4—1461no—
368C22×C4—1461no—
369C22×C4#1192402811yescharacteristic
370C22×C4#1192402811yescharacteristic
371C22×C4C23×D7311yescharacteristic
372M16—22no—
373M16—22no—
374M16—1461no—
375M16—1461no—
376M16—1461no—
377M16—1461no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—732no—
384C2×C8—732no—
385C2×C8—1461no—
386C2×C8—1461no—
387C2×C8—1462no—
388C2×C8—1462no—
389C2×C8C2×C22⋊D7311yescharacteristic
390C2×C8C2×C22⋊D7311yescharacteristic
391C2×C8C2×C22⋊D7311yescharacteristic
392C2×C8C2×C22⋊D7311yescharacteristic
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—731no—
398C4⋊C4—731no—
399C4⋊C4—1464no—
400C4⋊C4—1461no—
401C4⋊C4—1461no—
402C4⋊C4C2×C4×D7312yes—
403C4⋊C4C2×C4×D7312yes—
404C22⋊C4—22no—
405C22⋊C4—1461no—
406C22⋊C4—1461no—
407C22⋊C4—1461no—
408C42—22no—
409C42—1462no—
410C42—1462no—
411C42—1461no—
412C42—1461no—
413C42—1461no—
414C42C2×C4×D7311yescharacteristic
415C42C2×C4×D7311yescharacteristic
416C42C2×C22⋊D7311yescharacteristic
417C42C2×C22⋊D7311yescharacteristic
418C23—1461no—
419C23C2×#1192402811yescharacteristic
420Q8—42no—
421Q8—42no—
422Q8—1461no—
423Q8—1461no—
424D4—2921no—
425D4—2921no—
426D4—1461no—
427D4—1461no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—2922no—
433C2×C4—2922no—
434C2×C4—2922no—
435C2×C4—2921no—
436C2×C4—2921no—
437C2×C4—2922no—
438C2×C4—1464no—
439C2×C4—1461no—
440C2×C4—1461no—
441C2×C4—1461no—
442C2×C4C4×C22⋊D7311yescharacteristic
443C2×C4C4×C22⋊D7311yescharacteristic
444C2×C4C4×C22⋊D7311yescharacteristic
445C2×C4C4×C22⋊D7311yescharacteristic
446C2×C4C22×C4×D7311yescharacteristic
447C2×C4C22×C22⋊D7311yescharacteristic
448C2×C4#1455663711yescharacteristic
449C2×C4#1455663811yescharacteristic
450C2×C4#1455663911yescharacteristic
451C2×C4#1455663911yescharacteristic
452C8—21no—
453C8—21no—
454C8—21no—
455C8—21no—
456C8—42no—
457C8—2921no—
458C8—2921no—
459C8—1462no—
460C8—1462no—
461C22—21no—
462C22—21no—
463C22—2921no—
464C22—2921no—
465C22—2921no—
466C22C2×C4×C22⋊D7311yescharacteristic
467C22#1798919511yescharacteristic
468C22#1798920911yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—42no—
473C4—42no—
474C4—2924no—
475C4—2921no—
476C4—2921no—
477C4—2921no—
478C4#1798870212yes—
479C4#1798906311yescharacteristic
480C4#1798906911yescharacteristic
481C4#1798916111yescharacteristic
482C4#1798916511yescharacteristic
483C2—21no—
484C2—21no—
485C2—5841no—
486C2#2261260711yescharacteristic
487C2C2×#1798870211yescharacteristic
488C2#2261883211yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2960C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C23
cosocleC22×C4×C73C24
hypercenter#2960D73
hypocenterC73#22130
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2960

Complements

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

NH ≅ G/Ntype
3. #22610876C2semidirect
4. #22610704C2semidirect
5. #22610676C2semidirect
8. #22610530C2semidirect
10. C8×C4⋊Dic73C2semidirect
11. C73×#2960C2semidirect
13. #17988891C4semidirect
14. #17988884C4semidirect
15. #17988879C4semidirect
16. #17988874C4semidirect
17. #17988869C4semidirect
18. #17988864C4semidirect
246. #2960D73semidirect
256. C73#22130semidirect

Maximal subgroups

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

subgroupindex
#226197132
#226185732
#226108762
#226107042
#226106762
#226106342
#226105792
#226105302
#226099482
C8×C4⋊Dic732
C73×#29602
#2213073