Finite Group Databasebeta

← #33096530

Subgroups — #33096530

Order 18688.

806 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988639C211yesmaximal, characteristic
2#22612336C211yesmaximal, characteristic
3#22612327C211yesmaximal, characteristic
4#22610694C211yesmaximal, characteristic, complemented
5#22610692C211yesmaximal, characteristic, complemented
6#22610690C211yesmaximal, characteristic, complemented
7#22610133C212yesmaximal
8#22610127C212yesmaximal
9#22610094C212yesmaximal, complemented
10#22610060C212yesmaximal, complemented
11C73×#1947C211yesmaximal, characteristic, complemented
12C2×#14556549C2212yes—
13C2×C42×D73C2211yescharacteristic
14#17988639C418yescomplemented
15C2×#14556525C2212yes—
16C2×#14556522C2212yes—
17C2×#14556522C2212yes—
18C2×C4⋊#10596831C2211yescharacteristic
19C2×#14556503C2212yes—
20C2×C8×Dic73C2212yes—
21#17988457C2211yescharacteristic
22#17988454—21no—
23#17988452C2211yescharacteristic
24#17988452C2211yescharacteristic
25#17988452C2212yes—
26#17988449—21no—
27#17988444—21no—
28#17988444—21no—
29#17988444C2211yescharacteristic
30#17988444C2211yescharacteristic
31C4⋊C4×Dic73—21no—
32C4⋊C4×Dic73—21no—
33C4×C4⋊Dic73C2211yescharacteristic
34C4×#11924005—21no—
35#17988433—21no—
36#17988350C2212yes—
37#17988349C2212yes—
38#17988349C2212yes—
39C2×#107×C73C2212yes—
40C2×C4⋊C8×C73C2211yescharacteristic
41#384×C73C2211yescharacteristic
42C4×C4⋊C4×C73C2211yescharacteristic
43#340×C73C2212yes—
44C22×C4×D73D411yescharacteristic
45C22×C4×D73D411yescharacteristic
46C22×C4×D73C2311yescharacteristic
47C22×#10596831C2312yes—
48#14556549—42no—
49#14556549—42no—
50C2×C4⋊Dic73—21no—
51C2×C4⋊Dic73—21no—
52C2×C4⋊Dic73C2311yescharacteristic
53C2×C4⋊Dic73C2312yes—
54C2×C4⋊Dic73C2311yescharacteristic
55C2×#11924005—21no—
56C2×#11924005—21no—
57C2×#11924005—22no—
58C2×C4×Dic73—22no—
59C2×C4×Dic73—21no—
60C2×C4×Dic73D411yescharacteristic
61C2×C4×Dic73D411yescharacteristic
62C2×C4×Dic73C2311yescharacteristic
63C42×D73C2×C414yes—
64C42×D73Q814yes—
65#14556525—42no—
66#14556522—42no—
67#14556522—42no—
68C4⋊#10596831C2×C414yes—
69#14556503C2×C418yes—
70C8×Dic73C2×C418yes—
71#14556498—21no—
72#14556498—21no—
73#14556498—21no—
74#14556498—21no—
75#14556498—21no—
76#14556498—21no—
77#14556498—22no—
78#14556498—22no—
79#14556498—21no—
80C22×C8×C73C2312yes—
81C2×C4⋊C4×C73—21no—
82C2×C4⋊C4×C73—21no—
83C2×C4⋊C4×C73C2311yescharacteristic
84C2×C4⋊C4×C73C2311yescharacteristic
85C2×C4⋊C4×C73C2312yes—
86C2×C42×C73—21no—
87C2×C42×C73C2311yescharacteristic
88#107×C73—42no—
89C4⋊C8×C73C2×C414yes—
90#95×C73—21no—
91C23×D73C2×D411yescharacteristic
92C22×Dic73—21no—
93C22×Dic73—21no—
94C22×Dic73—22no—
95C22×Dic73C2×D411yescharacteristic
96C22×Dic73C2×D411yescharacteristic
97C22×Dic73C2×D411yescharacteristic
98C2×C4×D73—22no—
99C2×C4×D73—22no—
100C2×C4×D73—21no—
101C2×C4×D73—21no—
102C2×C4×D73—21no—
103C2×C4×D73—21no—
104C2×C4×D73—41no—
105C2×C4×D73—41no—
106C2×C4×D73C4⋊C414yes—
107C2×C4×D73C4⋊C414yes—
108C2×C4×D73C22×C412yes—
109C2×C4×D73C2×Q812yes—
110C2×#10596831—22no—
111C2×#10596831—22no—
112C2×#10596831C22×C414yes—
113C4⋊Dic73—21no—
114C4⋊Dic73—22no—
115C4⋊Dic73—21no—
116C4⋊Dic73—22no—
117C4⋊Dic73—21no—
118C4⋊Dic73—21no—
119C4⋊Dic73—42no—
120C4⋊Dic73—42no—
121#11924005—42no—
122#11924005—42no—
123#11924005—44no—
124C4×Dic73—44no—
125C4×Dic73—42no—
126C4×Dic73C4⋊C414yes—
127C4×Dic73C4⋊C414yes—
128C4×Dic73C22×C412yes—
129C4×Dic73C2×Q812yes—
130C22×C4×C73—21no—
131C22×C4×C73—21no—
132C22×C4×C73—22no—
133C22×C4×C73C2×D411yescharacteristic
134C22×C4×C73C2×D411yescharacteristic
135C22×C4×C73C2411yescharacteristic
136C2×C8×C73—22no—
137C2×C8×C73—22no—
138C2×C8×C73C22×C414yes—
139C4⋊C4×C73—21no—
140C4⋊C4×C73—22no—
141C4⋊C4×C73—21no—
142C4⋊C4×C73—22no—
143C4⋊C4×C73—21no—
144C4⋊C4×C73—21no—
145C4⋊C4×C73—42no—
146C4⋊C4×C73—42no—
147C42×C73—42no—
148C42×C73C22×C412yes—
149C42×C73C2×Q812yes—
150C23×C73C22×D411yescharacteristic
151C22×D73—21no—
152C22×D73—21no—
153C22×D73—21no—
154C22×D73—21no—
155C22×D73—21no—
156C22×D73—21no—
157C22×D73C2×C4⋊C412yes—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—41no—
161C2×C4×C73—42no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—42no—
166C2×C4×C73—42no—
167C2×C4×C73—41no—
168C2×C4×C73—41no—
169C2×C4×C73C2×C4⋊C412yes—
170C2×C4×C73C2×C4⋊C412yes—
171C2×C4×C73#13512yes—
172C2×C4×C73#13512yes—
173C2×C4×C73C23×C411yescharacteristic
174C2×C4×C73C22×Q811yescharacteristic
175C2×C4×C73H(5,𝔽2)11yescharacteristic
176C2×C4×C73H(5,𝔽2)11yescharacteristic
177C2×C4×C732−1+411yescharacteristic
178C2×C4×C732−1+411yescharacteristic
179C2×Dic73—21no—
180C2×Dic73—21no—
181C2×Dic73—21no—
182C2×Dic73—21no—
183C2×Dic73—21no—
184C2×Dic73—21no—
185C2×Dic73—42no—
186C2×Dic73—41no—
187C2×Dic73—41no—
188C2×Dic73—41no—
189C2×Dic73—41no—
190C2×Dic73—42no—
191C2×Dic73—41no—
192C2×Dic73—41no—
193C2×Dic73—42no—
194C2×Dic73C2×C4⋊C412yes—
195C2×Dic73C2×C4⋊C412yes—
196C2×Dic73C2×C4⋊C412yes—
197C4×D73—22no—
198C4×D73—22no—
199C4×D73—22no—
200C4×D73—22no—
201C4×D73—28no—
202C4×D73—44no—
203C8×C73—24no—
204#10596831—24no—
205C22×C73C22×C4⋊C411yescharacteristic
206C22×C73#53511yescharacteristic
207C22×C73#53611yescharacteristic
208C22×C73C2×#13511yescharacteristic
209C22×C73C2×#13511yescharacteristic
210C22×C73#57511yescharacteristic
211C22×C73#57511yescharacteristic
212C2×D73—22no—
213C2×D73—22no—
214C2×D73—22no—
215C2×D73—41no—
216C2×D73—41no—
217C2×D73—41no—
218C2×D73—41no—
219C4×C73—22no—
220C4×C73—81no—
221C4×C73—82no—
222C4×C73—81no—
223C4×C73#42714yes—
224C4×C73#52011yescharacteristic
225C4×C73#52011yescharacteristic
226C4×C73#54411yescharacteristic
227C4×C73#54411yescharacteristic
228Dic73—22no—
229Dic73—22no—
230Dic73—22no—
231Dic73—82no—
232Dic73—81no—
233Dic73—81no—
234#21985—731nomaximal, Sylow
235C2×C73#231711yescharacteristic
236C2×C73C2×#42711yescharacteristic
237C2×C73#291111yescharacteristic
238C2×C73#321211yescharacteristic
239C2×C73#321211yescharacteristic
240C2×C73#323311yescharacteristic
241C2×C73#323311yescharacteristic
242D73—42no—
243C2×#431—731no—
244#2285—731no—
245#2283—731no—
246#1948—731no—
247#1948—731no—
248#1947—731no—
249#1947D7311yescharacteristic, complemented
250#1854—732no—
251#1854—732no—
252#1846—732no—
253#1835—732no—
254C73#2198511yescharacteristic, complemented, Sylow
255C2×#117—732no—
256C22×C42—731no—
257#431—738no—
258C2×#107—732no—
259C2×#107C2×D7312yes—
260C2×#106—732no—
261C2×#106—732no—
262C2×C4⋊C8—731no—
263C2×C4⋊C8C2×D7311yescharacteristic
264C2×C22⋊C8—732no—
265C2×C4×C8—732no—
266#384—731no—
267#384C2×D7311yescharacteristic
268#383—1461no—
269#382—731no—
270#382—732no—
271#382—731no—
272#381—1461no—
273#380—731no—
274#380—731no—
275#380—1461no—
276#380—1461no—
277C4×C4⋊C4—731no—
278C4×C4⋊C4—1461no—
279C4×C4⋊C4—1461no—
280C4×C4⋊C4C2×D7311yescharacteristic
281C4×C22⋊C4—1461no—
282#376—1461no—
283#340—732no—
284#340—732no—
285#340—732no—
286#340C2×D7312yes—
287C23×C4—731no—
288C23×C4—731no—
289C23×C4—731no—
290C22×C8—732no—
291C22×C8C22×D7312yes—
292#117—2922no—
293#117—2922no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—732no—
297C2×C4⋊C4—731no—
298C2×C4⋊C4—731no—
299C2×C4⋊C4—1461no—
300C2×C4⋊C4—1461no—
301C2×C4⋊C4C22×D7312yes—
302C2×C4⋊C4C22×D7311yescharacteristic
303C2×C4⋊C4C22×D7311yescharacteristic
304C2×C22⋊C4—1462no—
305C2×C22⋊C4—1461no—
306C2×C22⋊C4—1461no—
307C2×C42—21no—
308C2×C42—731no—
309C2×C42—731no—
310C2×C42—731no—
311C2×C42—734no—
312C2×C42—734no—
313C2×C42—1462no—
314C2×C42—1461no—
315C2×C42C22×D7311yescharacteristic
316#107—42no—
317#107—2922no—
318#106—2922no—
319#106—2922no—
320C4⋊C8—734no—
321C4⋊C8C4×D7314yes—
322C22⋊C8—738no—
323C4×C8—738no—
324#95—21no—
325#95—1461no—
326#95—1461no—
327#95—1461no—
328#95—1461no—
329#95—1461no—
330#95—1461no—
331#95—1462no—
332#95—1462no—
333#95—1461no—
334C24—731no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—2921no—
339C22×C4—2921no—
340C22×C4—734no—
341C22×C4—731no—
342C22×C4—731no—
343C22×C4—731no—
344C22×C4—734no—
345C22×C4—732no—
346C22×C4—732no—
347C22×C4—1461no—
348C22×C4—1461no—
349C22×C4—1462no—
350C22×C4—1462no—
351C22×C4—1462no—
352C22×C4—1461no—
353C22×C4—1461no—
354C22×C4—1461no—
355C22×C4—1461no—
356C22×C4D4×D7311yescharacteristic
357C22×C4D4×D7311yescharacteristic
358C22×C4C23×D7311yescharacteristic
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—734no—
362C2×C8—1462no—
363C2×C8—1462no—
364C2×C8C2×C4×D7314yes—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—2922no—
374C4⋊C4—2922no—
375C4⋊C4—1461no—
376C4⋊C4—1462no—
377C4⋊C4—1461no—
378C4⋊C4—1462no—
379C4⋊C4—1461no—
380C4⋊C4—1461no—
381C22⋊C4—2922no—
382C22⋊C4—2922no—
383C22⋊C4—2924no—
384C42—42no—
385C42—2924no—
386C42—2922no—
387C42—734no—
388C42—734no—
389C42—732no—
390C42—732no—
391C42C2×C4×D7312yes—
392C42Q8×D7312yes—
393C23—732no—
394C23—1461no—
395C23—1461no—
396C23—1461no—
397C23—1461no—
398C23—1461no—
399C23—1461no—
400C23C2×D4×D7311yescharacteristic
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—2924no—
413C2×C4—2921no—
414C2×C4—2921no—
415C2×C4—2922no—
416C2×C4—2922no—
417C2×C4—2921no—
418C2×C4—2921no—
419C2×C4—2921no—
420C2×C4—2921no—
421C2×C4—2922no—
422C2×C4—732no—
423C2×C4—732no—
424C2×C4—732no—
425C2×C4—1461no—
426C2×C4—1461no—
427C2×C4—1461no—
428C2×C4—1461no—
429C2×C4—1461no—
430C2×C4—1461no—
431C2×C4—1468no—
432C2×C4—1462no—
433C2×C4—1462no—
434C2×C4—1462no—
435C2×C4—1462no—
436C2×C4C4⋊C4×D7312yes—
437C2×C4C4⋊C4×D7312yes—
438C2×C4#1455661012yes—
439C2×C4#1455661412yes—
440C2×C4C22×C4×D7311yescharacteristic
441C2×C4C2×Q8×D7311yescharacteristic
442C2×C4#1455663611yescharacteristic
443C2×C4#1455663711yescharacteristic
444C2×C4#1455663811yescharacteristic
445C2×C4#1455663911yescharacteristic
446C8—24no—
447C8—1464no—
448C22—2921no—
449C22—2921no—
450C22—2921no—
451C22—2921no—
452C22—1462no—
453C22—1462no—
454C22—1462no—
455C22C2×C4⋊C4×D7311yescharacteristic
456C22#1798915311yescharacteristic
457C22#1798916011yescharacteristic
458C22C2×#1455661011yescharacteristic
459C22C2×#1455661411yescharacteristic
460C22#1798938311yescharacteristic
461C22#1798938611yescharacteristic
462C4—22no—
463C4—81no—
464C4—81no—
465C4—82no—
466C4—5841no—
467C4—5841no—
468C4—5842no—
469C4—1462no—
470C4—1462no—
471C4—1462no—
472C4#1798862414yes—
473C4#1798906611yescharacteristic
474C4#1798906911yescharacteristic
475C4#1798921411yescharacteristic
476C4#1798921711yescharacteristic
477C2—2922no—
478C2#2261255311yescharacteristic
479C2C2×#1798862411yescharacteristic
480C2#2261851811yescharacteristic
481C2#2262090511yescharacteristic
482C2#2262091211yescharacteristic
483C2#2262110811yescharacteristic
484C2#2262111511yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1947C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1947D73
hypocenterC73#21985
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
4. #22610694C2semidirect
5. #22610692C2semidirect
6. #22610690C2semidirect
9. #22610094C2semidirect
10. #22610060C2semidirect
11. C73×#1947C2semidirect
14. #17988639C4semidirect
249. #1947D73semidirect
254. C73#21985semidirect

Maximal subgroups

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

subgroupindex
C2×#179886392
#226123362
#226123272
#226106942
#226106922
#226106902
#226101332
#226101272
#226100942
#226100602
C73×#19472
#2198573