Finite Group Databasebeta

← #32980798

Subgroups — #32980798

Order 18688.

680 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988789C211yesmaximal, characteristic
2#22612831C211yesmaximal, characteristic
3#22610884C211yesmaximal, characteristic, complemented
4C2×#17988316C211yesmaximal, characteristic
5#22607814C211yesmaximal, characteristic, complemented
6#22607771C211yesmaximal, characteristic, complemented
7C73×#1990C211yesmaximal, characteristic, complemented
8C22×#11924015—21no—
9C2×C4×D292C2211yescharacteristic
10C2×#14556553—21no—
11C2×C4⋊C4×D73—21no—
12#17988789—22no—
13#17988789C414yescomplemented
14C2×C4⋊#10596831C2211yescharacteristic
15C2×C4⋊#10596831C2211yescharacteristic
16C2×#14556515—21no—
17#17988483—21no—
18#17988458—22no—
19#17988444—21no—
20C4×C4⋊Dic73C2211yescharacteristic
21#17988349—21no—
22#17988349—21no—
23#17988316—22no—
24C2×C4⋊Q8×C73C2211yescharacteristic
25C2×C4×C8×C73C2211yescharacteristic
26#384×C73C2211yescharacteristic
27#340×C73—22no—
28C22×D292D411yescharacteristic
29C22×C4×D73—22no—
30C22×C4×D73—21no—
31C2×#11924015—24no—
32C2×#11924015—42no—
33C2×#11924015—42no—
34C22×#10596831—21no—
35C22×#10596831—21no—
36C4×D292—21no—
37C4×D292—21no—
38C4×D292C2×C412yes—
39C4×D292D412yes—
40#14556553—42no—
41#14556553—42no—
42C2×C4⋊Dic73D411yescharacteristic
43C2×C4⋊Dic73D411yescharacteristic
44C2×C4⋊Dic73D411yescharacteristic
45C2×#11924007—22no—
46C4⋊C4×D73—42no—
47C4⋊C4×D73—42no—
48C2×#11924005—22no—
49C2×#11924005—21no—
50C2×C4×Dic73—21no—
51C4⋊#10596831—21no—
52C4⋊#10596831—21no—
53C4⋊#10596831—21no—
54C4⋊#10596831C2×C412yescomplemented
55#14556515—24no—
56#14556498—22no—
57#14556498—21no—
58C22×Q8×C73—21no—
59C22×C8×C73—21no—
60C4⋊Q8×C73—21no—
61C4⋊Q8×C73—21no—
62C4⋊Q8×C73—21no—
63C4⋊Q8×C73C2×C412yescomplemented
64C2×C4⋊C4×C73—21no—
65C2×C4⋊C4×C73—21no—
66C2×C4⋊C4×C73—21no—
67C2×C4⋊C4×C73—21no—
68C2×C4⋊C4×C73D412yes—
69C2×C42×C73C2311yescharacteristic
70C4×C8×C73—21no—
71C4×C8×C73—21no—
72C23×D73—21no—
73C2×D292—21no—
74C2×D292—21no—
75C2×D292—21no—
76C2×D292C22⋊C412yes—
77C2×D292D812yes—
78C2×D292SD1612yes—
79C22×Dic73—21no—
80C22×Dic73—21no—
81C22×Dic73—21no—
82C2×C4×D73—22no—
83C2×C4×D73—22no—
84C2×C4×D73—41no—
85C2×C4×D73—41no—
86C2×C4×D73—41no—
87C2×C4×D73—41no—
88C2×C4×D73—44no—
89C2×C4×D73—44no—
90C2×C4×D73—82no—
91#11924015—48no—
92C2×#10596831—22no—
93C2×#10596831—41no—
94C2×#10596831—41no—
95C2×#10596831—41no—
96C2×#10596831—41no—
97C2×#10596831—41no—
98C4⋊Dic73—21no—
99C4⋊Dic73—21no—
100C4⋊Dic73—21no—
101C4⋊Dic73—21no—
102C4⋊Dic73C22⋊C412yes—
103C4⋊Dic73SD1612yescomplemented
104#11924007—44no—
105#11924005—22no—
106#11924005—22no—
107#11924005—44no—
108C4×Dic73—41no—
109C4×Dic73—41no—
110C2×Q8×C73—22no—
111C2×Q8×C73—41no—
112C2×Q8×C73—41no—
113C2×Q8×C73—41no—
114C2×Q8×C73—41no—
115C2×Q8×C73—41no—
116C22×C4×C73—22no—
117C22×C4×C73—41no—
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73C2×D411yescharacteristic
120C22×C4×C73C2×D411yescharacteristic
121C2×C8×C73—22no—
122C2×C8×C73—41no—
123C2×C8×C73—41no—
124C4⋊C4×C73—22no—
125C4⋊C4×C73—22no—
126C4⋊C4×C73—42no—
127C4⋊C4×C73—41no—
128C4⋊C4×C73—41no—
129C4⋊C4×C73—41no—
130C4⋊C4×C73—41no—
131C4⋊C4×C73—41no—
132C4⋊C4×C73—41no—
133C42×C73C22×C411yescharacteristic
134C42×C73C2×D411yescharacteristic
135C42×C73#4111yescharacteristic
136C42×C73#4111yescharacteristic
137C23×C73#12011yescharacteristic
138C22×D73—22no—
139C22×D73—22no—
140C22×D73—22no—
141C22×D73—41no—
142C22×D73—41no—
143C22×D73—41no—
144C22×D73—41no—
145Q8×C73—42no—
146Q8×C73—42no—
147Q8×C73—81no—
148Q8×C73—81no—
149C2×C4×C73—21no—
150C2×C4×C73—21no—
151C2×C4×C73—42no—
152C2×C4×C73—42no—
153C2×C4×C73—42no—
154C2×C4×C73—42no—
155C2×C4×C73—81no—
156C2×C4×C73—81no—
157C2×C4×C73C2×C22⋊C411yescharacteristic
158C2×C4×C73C4×D411yescharacteristic
159C2×C4×C73C4×D411yescharacteristic
160C2×C4×C73#12111yescharacteristic
161C2×C4×C73#12111yescharacteristic
162C2×C4×C73#12311yescharacteristic
163C2×C4×C73C2×D811yescharacteristic
164C2×C4×C73C2×SD1611yescharacteristic
165C2×C4×C73C2×SD1611yescharacteristic
166C2×C4×C73#13511yescharacteristic
167C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C73#13711yescharacteristic
170C2×C4×C73#13711yescharacteristic
171C2×Dic73—22no—
172C2×Dic73—22no—
173C2×Dic73—41no—
174C2×Dic73—41no—
175C2×Dic73—41no—
176C2×Dic73—41no—
177C2×Dic73—41no—
178C2×Dic73—41no—
179C2×Dic73—41no—
180D292—22no—
181D292—22no—
182D292—22no—
183D292#10214yes—
184C4×D73—42no—
185C4×D73—42no—
186C4×D73—42no—
187C4×D73—42no—
188C4×D73—88no—
189C8×C73—42no—
190#10596831—42no—
191#10596831—81no—
192C22×C73#38611yescharacteristic
193C22×C73#44711yescharacteristic
194C22×C73#44811yescharacteristic
195C22×C73#45011yescharacteristic
196C22×C73#45211yescharacteristic
197C22×C73#45511yescharacteristic
198C22×C73#45611yescharacteristic
199C2×D73—24no—
200C2×D73—42no—
201C2×D73—42no—
202C2×D73—42no—
203C2×D73—42no—
204C2×D73—42no—
205C2×D73—42no—
206C4×C73—21no—
207C4×C73—21no—
208C4×C73—82no—
209C4×C73—82no—
210C4×C73C2×#10211yescharacteristic
211C4×C73#41711yescharacteristic
212C4×C73C4×SD1611yescharacteristic
213C4×C73#44011yescharacteristic
214C4×C73#46111yescharacteristic
215C4×C73#46311yescharacteristic
216C4×C73#48011yescharacteristic
217C4×C73#48111yescharacteristic
218Dic73—42no—
219Dic73—42no—
220Dic73—81no—
221#8985—731nomaximal, Sylow
222C2×C73#163911yescharacteristic
223C2×C73#164311yescharacteristic
224C2×C73#164711yescharacteristic
225C2×C73#165111yescharacteristic
226C2×C73#187411yescharacteristic
227C2×C73#187711yescharacteristic
228C2×C73#188211yescharacteristic
229D73—44no—
230C2×#461—731no—
231D4×C4⋊C4—731no—
232#1990D7311yescharacteristic, complemented
233#1988—731no—
234C2×D4⋊C8—731no—
235#1296—731no—
236#1279—731no—
237C73#898511yescharacteristic, complemented, Sylow
238C22×SD16—1461no—
239C2×C4⋊Q8C2×D7311yescharacteristic
240C2×C4×D4—731no—
241C2×C4×D4—1461no—
242C22×C4⋊C4—1461no—
243#461—734no—
244#461—1462no—
245C2×C4⋊C8—731no—
246C2×C4⋊C8—731no—
247C2×#103—1461no—
248C2×C4×C8C2×D7311yescharacteristic
249#389—1461no—
250#385—1462no—
251#384C2×D7311yescharacteristic
252#380—1461no—
253C4×C4⋊C4—731no—
254#340—22no—
255#340—1461no—
256#340—1461no—
257D4⋊C8—1462no—
258C22×Q8—21no—
259C22×D4—731no—
260C23×C4—1462no—
261C23×C4—1461no—
262C2×SD16—2922no—
263C2×SD16—2922no—
264C2×SD16—1464no—
265C22×C8—21no—
266C22×C8—1461no—
267C22×C8—1461no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8C4×D7312yescomplemented
272C4×D4—2922no—
273C4×D4—2922no—
274C4×D4—732no—
275C4×D4—732no—
276C4×D4—1461no—
277C4×D4—1461no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—2922no—
283C2×C4⋊C4—2922no—
284C2×C4⋊C4—731no—
285C2×C4⋊C4—731no—
286C2×C4⋊C4—731no—
287C2×C4⋊C4—1462no—
288C2×C4⋊C4C22⋊D7312yes—
289C2×C22⋊C4—1462no—
290C2×C22⋊C4—1461no—
291C2×C42—1461no—
292C2×C42C22×D7311yescharacteristic
293C4⋊C8—732no—
294C4⋊C8—1461no—
295C4⋊C8—1461no—
296C4⋊C8—1461no—
297#103—1464no—
298C4×C8—21no—
299C4×C8—21no—
300#95—1462no—
301#95—1461no—
302C24—1461no—
303C2×Q8—22no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×D4—732no—
310C2×D4—732no—
311C2×D4—732no—
312C2×D4—1461no—
313C2×D4—1461no—
314C2×D4—1461no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—2924no—
318C22×C4—2924no—
319C22×C4—2921no—
320C22×C4—2921no—
321C22×C4—2921no—
322C22×C4—2921no—
323C22×C4—5842no—
324C22×C4—1461no—
325C22×C4—1461no—
326C22×C4—1461no—
327C22×C4—1462no—
328C22×C4—1462no—
329C22×C4D4×D7311yescharacteristic
330C22×C4D4×D7311yescharacteristic
331C22×C4C2×C22⋊D7311yescharacteristic
332SD16—2928no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—2921no—
337C2×C8—2921no—
338C2×C8—2921no—
339C2×C8—2921no—
340C2×C8—2921no—
341C2×C8—1462no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—2924no—
352C4⋊C4—732no—
353C4⋊C4—732no—
354C4⋊C4—1461no—
355C4⋊C4—1461no—
356C4⋊C4—1461no—
357C4⋊C4—1461no—
358C22⋊C4—2924no—
359C22⋊C4—1462no—
360C22⋊C4—1462no—
361C42—2921no—
362C42—2921no—
363C42C2×C4×D7311yescharacteristic
364C42D4×D7311yescharacteristic
365C42#1192402711yescharacteristic
366C42#1192402811yescharacteristic
367C23—2921no—
368C23—2921no—
369C23—2921no—
370C23—2921no—
371C23—1462no—
372C23—1462no—
373C23—1462no—
374C23#1455655911yescharacteristic
375Q8—42no—
376Q8—42no—
377Q8—81no—
378Q8—81no—
379D4—734no—
380D4—1462no—
381D4—1462no—
382D4—1462no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—2921no—
392C2×C4—2921no—
393C2×C4—2921no—
394C2×C4—2921no—
395C2×C4—2922no—
396C2×C4—2922no—
397C2×C4—2922no—
398C2×C4—2922no—
399C2×C4—2921no—
400C2×C4—2921no—
401C2×C4—2921no—
402C2×C4—5848no—
403C2×C4—1462no—
404C2×C4—1462no—
405C2×C4C22⋊C4×D7311yescharacteristic
406C2×C4#1455655111yescharacteristic
407C2×C4C4×C22⋊D7311yescharacteristic
408C2×C4#1455656211yescharacteristic
409C2×C4#1455656411yescharacteristic
410C2×C4#1455657311yescharacteristic
411C2×C4D8×D7311yescharacteristic
412C2×C4SD16×D7311yescharacteristic
413C2×C4SD16×D7311yescharacteristic
414C2×C4#1455661411yescharacteristic
415C2×C4#1455661511yescharacteristic
416C2×C4#1455661811yescharacteristic
417C2×C4#1455662011yescharacteristic
418C2×C4#1455662311yescharacteristic
419C8—42no—
420C8—2922no—
421C8—5841no—
422C22—2922no—
423C22—2922no—
424C22—2922no—
425C22—2922no—
426C22—2922no—
427C22—2922no—
428C22—1464no—
429C22#1798846311yescharacteristic
430C22#1798871911yescharacteristic
431C22#1798872411yescharacteristic
432C22#1798873611yescharacteristic
433C22#1798874811yescharacteristic
434C22#1798876411yescharacteristic
435C22#1798876911yescharacteristic
436C4—21no—
437C4—21no—
438C4—82no—
439C4—82no—
440C4—2922no—
441C4—2922no—
442C4—5841no—
443C4#102×D7311yescharacteristic
444C4#1798858511yescharacteristic
445C4#1798867211yescharacteristic
446C4#1798869011yescharacteristic
447C4#1798879111yescharacteristic
448C4#1798880511yescharacteristic
449C4#1798889911yescharacteristic
450C4#1798890411yescharacteristic
451C2—2924no—
452C2#2260910711yescharacteristic
453C2#2260913511yescharacteristic
454C2#2260916311yescharacteristic
455C2#2260919111yescharacteristic
456C2#2261024511yescharacteristic
457C2#2261026211yescharacteristic
458C2#2261030311yescharacteristic
subgroupG/H
centerC23#14556559
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1990C2
socleC23×C73#120
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenter#1990D73
hypocenterC73#8985
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
3. #22610884C2semidirect
5. #22607814C2semidirect
6. #22607771C2semidirect
7. C73×#1990C2semidirect
13. #17988789C4semidirect
54. C4⋊#10596831C2×C4semidirect
63. C4⋊Q8×C73C2×C4semidirect
103. C4⋊Dic73SD16semidirect
232. #1990D73semidirect
237. C73#8985semidirect
271. C4⋊Q8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887892
#226128312
#226108842
C2×#179883162
#226078142
#226077712
C73×#19902
#898573