Finite Group Databasebeta

← #33096561

Subgroups — #33096561

Order 18688.

786 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988642C211yesmaximal, characteristic
2#22612336C211yesmaximal, characteristic
3#22612324C211yesmaximal, characteristic
4#22610700C212yesmaximal, complemented
5#22610696C212yesmaximal, complemented
6#22610129C212yesmaximal
7#22610127C212yesmaximal
8#22610100C211yesmaximal, characteristic, complemented
9#22610066C211yesmaximal, characteristic, complemented
10#22610065C211yesmaximal, characteristic, complemented
11C73×#1847C211yesmaximal, characteristic, complemented
12C2×#14556549C2212yes—
13C2×C42×D73C2211yescharacteristic
14#17988642C418yescomplemented
15C2×#14556524C2212yes—
16C2×#14556522C2212yes—
17C2×#14556520C2212yes—
18C2×#14556503C2212yes—
19C2×#14556502C2211yescharacteristic
20#17988457C2211yescharacteristic
21#17988454—21no—
22#17988453C2211yescharacteristic
23#17988452C2212yes—
24#17988449—21no—
25#17988444—21no—
26#17988444—21no—
27#17988444C2211yescharacteristic
28#17988444C2211yescharacteristic
29C4⋊C4×Dic73—21no—
30C4⋊C4×Dic73—21no—
31C4⋊C4×Dic73C2211yescharacteristic
32C4⋊C4×Dic73C2211yescharacteristic
33#17988434—21no—
34#17988432—21no—
35#17988350C2212yes—
36#17988350C2212yes—
37#17988349C2212yes—
38#17988349C2212yes—
39C2×#107×C73C2212yes—
40C2×#106×C73C2212yes—
41C2×#97×C73C2211yescharacteristic
42#384×C73C2211yescharacteristic
43#382×C73C2211yescharacteristic
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⋊Dic73C2312yes—
53C2×C4⋊Dic73C2311yescharacteristic
54C2×C4⋊Dic73C2311yescharacteristic
55C2×#11924005—21no—
56C2×#11924005—21no—
57C2×#11924005—22no—
58C2×C4×Dic73—21no—
59C2×C4×Dic73—21no—
60C2×C4×Dic73—22no—
61C2×C4×Dic73D411yescharacteristic
62C2×C4×Dic73D411yescharacteristic
63C2×C4×Dic73C2311yescharacteristic
64C42×D73C2×C414yes—
65C42×D73Q814yes—
66#14556524—42no—
67#14556522—42no—
68#14556520C2×C418yes—
69#14556503C2×C418yes—
70#14556502C2×C414yes—
71#14556498—21no—
72#14556498—21no—
73#14556498—22no—
74#14556498—22no—
75#14556498—21no—
76#14556498—21no—
77#14556498—21no—
78#14556498—21no—
79C22×C8×C73C2312yes—
80C2×C4⋊C4×C73—21no—
81C2×C4⋊C4×C73—21no—
82C2×C4⋊C4×C73C2311yescharacteristic
83C2×C4⋊C4×C73C2311yescharacteristic
84C2×C4⋊C4×C73C2312yes—
85C2×C42×C73C2311yescharacteristic
86#107×C73—42no—
87#106×C73—42no—
88#97×C73C2×C414yes—
89#95×C73—21no—
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—41no—
101C2×C4×D73—41no—
102C2×C4×D73—41no—
103C2×C4×D73—41no—
104C2×C4×D73C4⋊C414yes—
105C2×C4×D73C4⋊C414yes—
106C2×C4×D73C22×C412yes—
107C2×C4×D73C2×Q812yes—
108C2×#10596831—22no—
109C2×#10596831—22no—
110C2×#10596831C22×C414yes—
111C4⋊Dic73—22no—
112C4⋊Dic73—22no—
113C4⋊Dic73—22no—
114C4⋊Dic73—22no—
115C4⋊Dic73—42no—
116C4⋊Dic73—42no—
117#11924005—42no—
118#11924005—42no—
119#11924005—44no—
120C4×Dic73—42no—
121C4×Dic73—42no—
122C4×Dic73—44no—
123C4×Dic73C4⋊C414yes—
124C4×Dic73C4⋊C414yes—
125C4×Dic73C22×C412yes—
126C4×Dic73C2×Q812yes—
127C22×C4×C73—21no—
128C22×C4×C73—21no—
129C22×C4×C73—22no—
130C22×C4×C73C2×D411yescharacteristic
131C22×C4×C73C2×D411yescharacteristic
132C22×C4×C73C2411yescharacteristic
133C2×C8×C73—22no—
134C2×C8×C73—22no—
135C2×C8×C73C22×C414yes—
136C4⋊C4×C73—22no—
137C4⋊C4×C73—22no—
138C4⋊C4×C73—22no—
139C4⋊C4×C73—22no—
140C4⋊C4×C73—42no—
141C4⋊C4×C73—42no—
142C42×C73C22×C412yes—
143C42×C73C2×Q812yes—
144C23×C73C22×D411yescharacteristic
145C22×D73—21no—
146C22×D73—21no—
147C22×D73—21no—
148C22×D73—21no—
149C22×D73—21no—
150C22×D73—21no—
151C22×D73C2×C4⋊C412yes—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—42no—
159C2×C4×C73—42no—
160C2×C4×C73—42no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73C2×C4⋊C412yes—
166C2×C4×C73C2×C4⋊C412yes—
167C2×C4×C73C23×C411yescharacteristic
168C2×C4×C73C22×Q811yescharacteristic
169C2×C4×C73H(5,𝔽2)12yes—
170C2×C4×C732−1+412yes—
171C2×Dic73—21no—
172C2×Dic73—21no—
173C2×Dic73—21no—
174C2×Dic73—21no—
175C2×Dic73—21no—
176C2×Dic73—21no—
177C2×Dic73—42no—
178C2×Dic73—41no—
179C2×Dic73—41no—
180C2×Dic73—41no—
181C2×Dic73—41no—
182C2×Dic73—41no—
183C2×Dic73—41no—
184C2×Dic73—42no—
185C2×Dic73—42no—
186C2×Dic73C2×C4⋊C412yes—
187C2×Dic73C2×C4⋊C412yes—
188C2×Dic73C2×C4⋊C412yes—
189C4×D73—24no—
190C4×D73—24no—
191C4×D73—44no—
192C4×D73—44no—
193C8×C73—24no—
194#10596831—24no—
195C22×C73C22×C4⋊C411yescharacteristic
196C22×C73#53511yescharacteristic
197C22×C73#53611yescharacteristic
198C22×C73#57511yescharacteristic
199C22×C73#57511yescharacteristic
200C22×C73#57511yescharacteristic
201C22×C73#57511yescharacteristic
202C2×D73—22no—
203C2×D73—22no—
204C2×D73—22no—
205C2×D73—41no—
206C2×D73—41no—
207C2×D73—41no—
208C2×D73—41no—
209C4×C73—22no—
210C4×C73—22no—
211C4×C73—81no—
212C4×C73—82no—
213C4×C73—81no—
214C4×C73#52012yes—
215C4×C73#54412yes—
216Dic73—22no—
217Dic73—22no—
218Dic73—22no—
219Dic73—82no—
220Dic73—81no—
221Dic73—81no—
222#21989—731nomaximal, Sylow
223C2×C73#231711yescharacteristic
224C2×C73#291111yescharacteristic
225C2×C73#291111yescharacteristic
226C2×C73#321911yescharacteristic
227C2×C73#321911yescharacteristic
228C2×C73#324011yescharacteristic
229C2×C73#324011yescharacteristic
230D73—42no—
231C2×#432—731no—
232#2285—731no—
233#2283—731no—
234#1949—732no—
235#1949—732no—
236#1854—732no—
237#1854—732no—
238#1847—731no—
239#1847D7311yescharacteristic, complemented
240#1836—731no—
241#1836—731no—
242C73#2198911yescharacteristic, complemented, Sylow
243C2×#117—732no—
244C22×C42—731no—
245#432—738no—
246C2×#107—732no—
247C2×#107C2×D7312yes—
248C2×#106—732no—
249C2×#106C2×D7312yes—
250C2×C4⋊C8—732no—
251C2×C22⋊C8—732no—
252C2×#97—731no—
253C2×#97C2×D7311yescharacteristic
254#384—731no—
255#384C2×D7311yescharacteristic
256#383—1461no—
257#382—732no—
258#382—731no—
259#382C2×D7311yescharacteristic
260#381—1461no—
261#380—731no—
262#380—731no—
263#380—1461no—
264#380—1461no—
265C4×C4⋊C4—731no—
266C4×C4⋊C4—731no—
267C4×C4⋊C4—1461no—
268C4×C4⋊C4—1461no—
269C4×C22⋊C4—1461no—
270#376—1461no—
271#340—732no—
272#340—732no—
273#340—732no—
274#340—732no—
275C23×C4—731no—
276C23×C4—731no—
277C23×C4—731no—
278C22×C8—732no—
279C22×C8C22×D7312yes—
280#117—2922no—
281#117—2922no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—732no—
285C2×C4⋊C4—731no—
286C2×C4⋊C4—731no—
287C2×C4⋊C4—1461no—
288C2×C4⋊C4—1461no—
289C2×C4⋊C4C22×D7312yes—
290C2×C4⋊C4C22×D7311yescharacteristic
291C2×C4⋊C4C22×D7311yescharacteristic
292C2×C22⋊C4—1462no—
293C2×C22⋊C4—1461no—
294C2×C22⋊C4—1461no—
295C2×C42—731no—
296C2×C42—731no—
297C2×C42—731no—
298C2×C42—734no—
299C2×C42—734no—
300C2×C42—1462no—
301C2×C42—1461no—
302C2×C42—1461no—
303C2×C42C22×D7311yescharacteristic
304#107—42no—
305#107—2922no—
306#106—42no—
307#106—2922no—
308C4⋊C8—738no—
309C22⋊C8—738no—
310#97—734no—
311#97C4×D7314yes—
312#95—21no—
313#95—21no—
314#95—1461no—
315#95—1461no—
316#95—1461no—
317#95—1461no—
318#95—1462no—
319#95—1462no—
320#95—1461no—
321#95—1461no—
322C24—731no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—2921no—
327C22×C4—2921no—
328C22×C4—2921no—
329C22×C4—2921no—
330C22×C4—734no—
331C22×C4—734no—
332C22×C4—731no—
333C22×C4—731no—
334C22×C4—731no—
335C22×C4—732no—
336C22×C4—732no—
337C22×C4—1461no—
338C22×C4—1461no—
339C22×C4—1462no—
340C22×C4—1462no—
341C22×C4—1462no—
342C22×C4D4×D7311yescharacteristic
343C22×C4D4×D7311yescharacteristic
344C22×C4C23×D7311yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—734no—
348C2×C8—1462no—
349C2×C8—1462no—
350C2×C8C2×C4×D7314yes—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—2922no—
358C4⋊C4—2922no—
359C4⋊C4—1462no—
360C4⋊C4—1462no—
361C4⋊C4—1462no—
362C4⋊C4—1462no—
363C22⋊C4—2922no—
364C22⋊C4—2922no—
365C22⋊C4—2924no—
366C42—2922no—
367C42—2922no—
368C42—2924no—
369C42—734no—
370C42—734no—
371C42—732no—
372C42—732no—
373C42C2×C4×D7312yes—
374C42Q8×D7312yes—
375C23—732no—
376C23—1461no—
377C23—1461no—
378C23—1461no—
379C23—1461no—
380C23—1461no—
381C23—1461no—
382C23C2×D4×D7311yescharacteristic
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—2924no—
397C2×C4—2924no—
398C2×C4—2921no—
399C2×C4—2921no—
400C2×C4—2921no—
401C2×C4—2921no—
402C2×C4—2922no—
403C2×C4—2922no—
404C2×C4—2922no—
405C2×C4—2921no—
406C2×C4—2921no—
407C2×C4—732no—
408C2×C4—732no—
409C2×C4—732no—
410C2×C4—1461no—
411C2×C4—1461no—
412C2×C4—1461no—
413C2×C4—1461no—
414C2×C4—1461no—
415C2×C4—1461no—
416C2×C4—1464no—
417C2×C4—1464no—
418C2×C4C4⋊C4×D7312yes—
419C2×C4C4⋊C4×D7312yes—
420C2×C4C22×C4×D7311yescharacteristic
421C2×C4C2×Q8×D7311yescharacteristic
422C2×C4#1455663612yes—
423C2×C4#1455663912yes—
424C8—24no—
425C8—1464no—
426C22—2921no—
427C22—2921no—
428C22—2921no—
429C22—2921no—
430C22—1462no—
431C22—1462no—
432C22—1462no—
433C22C2×C4⋊C4×D7311yescharacteristic
434C22#1798915011yescharacteristic
435C22#1798916211yescharacteristic
436C22#1798938311yescharacteristic
437C22#1798938311yescharacteristic
438C22#1798938611yescharacteristic
439C22#1798938611yescharacteristic
440C4—22no—
441C4—22no—
442C4—81no—
443C4—82no—
444C4—81no—
445C4—5842no—
446C4—5841no—
447C4—5841no—
448C4—1462no—
449C4—1462no—
450C4—1462no—
451C4#1798906812yes—
452C4#1798921412yes—
453C2—2922no—
454C2#2261255211yescharacteristic
455C2#2261851811yescharacteristic
456C2#2261851811yescharacteristic
457C2#2262096311yescharacteristic
458C2#2262096511yescharacteristic
459C2#2262117211yescharacteristic
460C2#2262117411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1847C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1847D73
hypocenterC73#21989
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #22610700C2semidirect
5. #22610696C2semidirect
8. #22610100C2semidirect
9. #22610066C2semidirect
10. #22610065C2semidirect
11. C73×#1847C2semidirect
14. #17988642C4semidirect
239. #1847D73semidirect
242. C73#21989semidirect

Maximal subgroups

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

subgroupindex
C2×#179886422
#226123362
#226123242
#226107002
#226106962
#226101292
#226101272
#226101002
#226100662
#226100652
C73×#18472
#2198973