Finite Group Databasebeta

← #33069958

Subgroups — #33069958

Order 18688.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988935C211yesmaximal, characteristic, complemented
2#22619411C211yesmaximal, characteristic
3#41×#10596831C211yesmaximal, characteristic, complemented
4#22609377C214yesmaximal, complemented
5#22609370C214yesmaximal, complemented
6#22608689C211yesmaximal, characteristic, complemented
7#22608603C211yesmaximal, characteristic, complemented
8#22608600C211yesmaximal, characteristic, complemented
9C73×#3039C211yesmaximal, characteristic, complemented
10C2×#14556616—21no—
11C2×#14556588C2211yescharacteristic
12#17988935—21no—
13#17988935—21no—
14#17988935C2214yes—
15#17988787—21no—
16#17988787C2212yes—
17#17988782—21no—
18#17988782C2212yes—
19Q8×#10596831C2212yescomplemented
20D4×#10596831—22no—
21D4×#10596831C2212yescomplemented
22#17988640—21no—
23#17988640C2211yescharacteristic
24#17988611C2211yescharacteristic
25#17988591—21no—
26C2×#14556511—22no—
27C2×C4×#10596831—21no—
28C2×C4×#10596831C2211yescharacteristic
29#17988339C2214yescomplemented
30#17988333C2214yescomplemented
31#17988330C2214yes—
32C2×#137×C73—21no—
33C2×C4⋊Q8×C73C2211yescharacteristic
34C4×#41×C73C2211yescharacteristic
35#462×C73—21no—
36#462×C73C2212yescomplemented
37#461×C73—21no—
38#461×C73C2212yescomplemented
39#423×C73C2211yescharacteristic
40#419×C73—21no—
41C22×D292—21no—
42C22×D292—21no—
43C22×D292—21no—
44#14556616—44no—
45C2×#11924015—22no—
46C2×#11924013—22no—
47C2×#11924012—21no—
48C22×#10596831—22no—
49C22×#10596831D412yes—
50#14556588—21no—
51#14556588—21no—
52#14556588—21no—
53#14556588C2312yes—
54C4⋊#10596831—22no—
55C4⋊#10596831—21no—
56C4⋊#10596831C2312yes—
57C4⋊#10596831C2312yes—
58#14556511—24no—
59#14556511—24no—
60#14556511—22no—
61#14556511—22no—
62C22⋊#10596831—22no—
63C22⋊#10596831—21no—
64C4×#10596831—22no—
65C4×#10596831—21no—
66C4×#10596831D411yescharacteristic
67C4×#10596831D411yescharacteristic
68C4×#10596831C2312yes—
69C2×#41×C73D411yescharacteristic
70C22×Q8×C73—21no—
71#137×C73—44no—
72C2×Q16×C73—22no—
73C2×SD16×C73—22no—
74C2×M16×C73—21no—
75C4⋊Q8×C73—21no—
76C4⋊Q8×C73—21no—
77C4⋊Q8×C73—21no—
78C4⋊Q8×C73C2312yes—
79C4×Q8×C73D411yescharacteristic
80C4×Q8×C73C2311yescharacteristic
81C4×D4×C73—21no—
82C4×D4×C73—21no—
83C4×D4×C73D411yescharacteristic
84C4×D4×C73C2311yescharacteristic
85#117×C73—21no—
86#117×C73D411yescharacteristic
87C2×C4⋊C4×C73—21no—
88C2×C4⋊C4×C73—21no—
89C2×C42×C73—21no—
90C2×C42×C73C2311yescharacteristic
91C4⋊C8×C73—21no—
92C4⋊C8×C73C2312yes—
93#103×C73—22no—
94#103×C73—22no—
95C23×D73—41no—
96C2×D292—22no—
97C2×D292—22no—
98C2×D292—22no—
99C2×D292—44no—
100C2×D292—42no—
101C2×D292—42no—
102C2×D292—41no—
103C2×D292—41no—
104C2×D292—41no—
105C2×D292—82no—
106#11924015—44no—
107#11924013—44no—
108#11924012—42no—
109C2×#10596831—22no—
110C2×#10596831—22no—
111C2×#10596831—42no—
112C2×#10596831—44no—
113C2×#10596831C2×D412yes—
114C2×#10596831C2×D412yes—
115#41×C73—22no—
116#41×C73D814yes—
117C2×Q8×C73—22no—
118C2×Q8×C73—41no—
119C2×Q8×C73—42no—
120C2×Q8×C73—42no—
121C2×Q8×C73C2×D411yescharacteristic
122C2×D4×C73—21no—
123C2×D4×C73C2×D411yescharacteristic
124C22×C4×C73—21no—
125C22×C4×C73—21no—
126C22×C4×C73—21no—
127C22×C4×C73—41no—
128C22×C4×C73C2×D411yescharacteristic
129C22×C4×C73C2×D411yescharacteristic
130C22×C4×C73#4111yescharacteristic
131Q16×C73—44no—
132SD16×C73—44no—
133M16×C73—42no—
134C2×C8×C73—22no—
135C4⋊C4×C73—21no—
136C4⋊C4×C73—21no—
137C4⋊C4×C73—22no—
138C4⋊C4×C73—22no—
139C4⋊C4×C73—41no—
140C4⋊C4×C73—41no—
141C4⋊C4×C73C2×D411yescharacteristic
142C4⋊C4×C73C2×D411yescharacteristic
143C22⋊C4×C73—21no—
144C22⋊C4×C73—21no—
145C22⋊C4×C73—21no—
146C42×C73—21no—
147C42×C73—21no—
148C42×C73—21no—
149C42×C73C2×D411yescharacteristic
150C42×C73C2×D411yescharacteristic
151C42×C73#4111yescharacteristic
152C42×C73C2411yescharacteristic
153C23×C73—21no—
154C23×C73#12111yescharacteristic
155C22×D73—42no—
156C22×D73—82no—
157C22×D73—82no—
158C22×D73—82no—
159Q8×C73—21no—
160Q8×C73—42no—
161Q8×C73—42no—
162Q8×C73—82no—
163Q8×C73C2×D812yes—
164D4×C73—24no—
165D4×C73—21no—
166D4×C73C2×D812yes—
167C2×C4×C73—21no—
168C2×C4×C73—22no—
169C2×C4×C73—24no—
170C2×C4×C73—22no—
171C2×C4×C73—21no—
172C2×C4×C73—21no—
173C2×C4×C73—42no—
174C2×C4×C73—42no—
175C2×C4×C73—42no—
176C2×C4×C73—82no—
177C2×C4×C73#12111yescharacteristic
178C2×C4×C73#12111yescharacteristic
179C2×C4×C73#12111yescharacteristic
180C2×C4×C73C2×D812yes—
181C2×C4×C73#13512yes—
182C2×C4×C73C22×D411yescharacteristic
183C2×C4×C73C22×D411yescharacteristic
184C2×C4×C73C2×#4111yescharacteristic
185D292—42no—
186D292—42no—
187D292—44no—
188D292—82no—
189D292—82no—
190D292—82no—
191D292—82no—
192C8×C73—42no—
193#10596831—22no—
194#10596831—22no—
195#10596831—42no—
196#10596831—42no—
197C22×C73—21no—
198C22×C73—21no—
199C22×C73—22no—
200C22×C73—41no—
201C22×C73—41no—
202C22×C73#46612yes—
203C22×C73C2×#12111yescharacteristic
204C2×D73—82no—
205C2×D73—82no—
206C2×D73—82no—
207C2×D73—164no—
208C4×C73—21no—
209C4×C73—22no—
210C4×C73—21no—
211C4×C73—41no—
212C4×C73—41no—
213C4×C73—82no—
214C4×C73#46612yes—
215C4×C73#47112yes—
216C4×C73C22×D811yescharacteristic
217C4×C73C2×#13511yescharacteristic
218C4×C73#57512yes—
219#19276—731nomaximal, Sylow
220C2×C73—22no—
221C2×C73—42no—
222C2×C73#304011yescharacteristic
223C2×C73C2×#46611yescharacteristic
224C2×C73C2×#47111yescharacteristic
225D73—162no—
226C2×#486—731no—
227#3039D7311yescharacteristic, complemented
228#3038—731no—
229C8×#41—731no—
230#1682—734no—
231#1681—734no—
232#1567—731no—
233#1546—731no—
234#1545—731no—
235C73#1927611yescharacteristic, complemented, Sylow
236C2×#137—21no—
237C2×AGL(1,ℤ/8)—1461no—
238C2×C4⋊Q8C2×D7311yescharacteristic
239C2×#127—731no—
240C4×#41C2×D7311yescharacteristic
241#486—734no—
242#486—1461no—
243#486—1461no—
244#462—21no—
245#462C2×D7312yescomplemented
246#461—21no—
247#461C2×D7312yescomplemented
248#460—732no—
249#460—1461no—
250#459—732no—
251#459—1461no—
252C8×Q8—732no—
253C8×D4—732no—
254C8×D4—1462no—
255#431—731no—
256#431—1461no—
257#423—731no—
258#423C2×D7311yescharacteristic
259#419—21no—
260#418—1461no—
261C2×#102—1462no—
262C2×C4×C8—731no—
263C2×C4×C8—1461no—
264#335—734no—
265#332—734no—
266#331—734no—
267C2×#41C22⋊D7311yescharacteristic
268C22×Q8—21no—
269C22×D4—1461no—
270C22×D4—1461no—
271C22×D4—1461no—
272#137—44no—
273AGL(1,ℤ/8)—2924no—
274C2×Q16—22no—
275C2×SD16—22no—
276C2×SD16—1462no—
277C2×D8—1462no—
278C2×M16—21no—
279C2×M16—1461no—
280C22×C8—732no—
281C22×C8—1462no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8C22×D7312yes—
286#127—732no—
287#127—1461no—
288#127—1461no—
289#127—1461no—
290C4×Q8C22⋊D7311yescharacteristic
291C4×Q8C22×D7311yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4C22⋊D7311yescharacteristic
295C4×D4C22×D7311yescharacteristic
296#117—21no—
297#117C22⋊D7311yescharacteristic
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C42—21no—
301C2×C42C22×D7311yescharacteristic
302C4⋊C8—21no—
303C4⋊C8—732no—
304C4⋊C8—732no—
305C4⋊C8—1462no—
306C4⋊C8—1461no—
307C4⋊C8C22×D7312yes—
308#103—22no—
309#103—22no—
310#102—1464no—
311#102—1464no—
312#102—1462no—
313#102—1462no—
314C22⋊C8—1462no—
315C22⋊C8—1461no—
316C4×C8—731no—
317C4×C8—731no—
318C4×C8—732no—
319C4×C8—1462no—
320C4×C8—1461no—
321C24—2921no—
322#41—22no—
323#41#1192401314yes—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8C2×C22⋊D7311yescharacteristic
329C2×D4—21no—
330C2×D4—2924no—
331C2×D4—2921no—
332C2×D4—2921no—
333C2×D4—2921no—
334C2×D4—2922no—
335C2×D4—2922no—
336C2×D4—5842no—
337C2×D4—1462no—
338C2×D4—1462no—
339C2×D4—1462no—
340C2×D4C2×C22⋊D7311yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—41no—
345C22×C4D4×D7311yescharacteristic
346C22×C4C2×C22⋊D7311yescharacteristic
347C22×C4#1192402711yescharacteristic
348Q16—44no—
349SD16—44no—
350SD16—2924no—
351D8—2924no—
352M16—42no—
353M16—2922no—
354C2×C8—22no—
355C2×C8—2924no—
356C2×C8—2922no—
357C2×C8—732no—
358C2×C8—732no—
359C2×C8—1462no—
360C2×C8—1462no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4C2×C22⋊D7311yescharacteristic
368C4⋊C4C2×C22⋊D7311yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42D4×D7311yescharacteristic
376C42C2×C22⋊D7311yescharacteristic
377C42#1192402711yescharacteristic
378C42C23×D7311yescharacteristic
379C23—21no—
380C23—2922no—
381C23—5842no—
382C23—5842no—
383C23—5842no—
384C23#1455656311yescharacteristic
385Q8—21no—
386Q8—42no—
387Q8—42no—
388Q8—82no—
389Q8C2×#1192401312yes—
390D4—24no—
391D4—21no—
392D4—2924no—
393D4—2922no—
394D4—2922no—
395D4—5842no—
396D4—5842no—
397D4—5842no—
398D4—5842no—
399D4C2×#1192401312yes—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—82no—
410C2×C4#1455656311yescharacteristic
411C2×C4#1455656311yescharacteristic
412C2×C4#1455656311yescharacteristic
413C2×C4C2×#1192401312yes—
414C2×C4#1455661112yes—
415C2×C4C2×D4×D7311yescharacteristic
416C2×C4C22×C22⋊D7311yescharacteristic
417C2×C4C2×#1192402711yescharacteristic
418C8—42no—
419C8—2922no—
420C8—2922no—
421C8—1462no—
422C8—1462no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—5842no—
429C22—5842no—
430C22—5842no—
431C22—11684no—
432C22#1798882112yes—
433C22C2×#1455656311yescharacteristic
434C4—21no—
435C4—22no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—82no—
440C4#1798882112yes—
441C4#1798885212yes—
442C4C22×#1192401311yescharacteristic
443C4C2×#1455661111yescharacteristic
444C4#1798938612yes—
445C2—22no—
446C2—42no—
447C2—11682no—
448C2#2261942711yescharacteristic
449C2C2×#1798882111yescharacteristic
450C2C2×#1798885211yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3039C2
socleC22×C73C2×#121
prime radical#22619411C2
hypercenter#3039D73
hypocenterC73#19276
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3039

Complements

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

NH ≅ G/Ntype
1. C2×#17988935C2semidirect
3. #41×#10596831C2semidirect
4. #22609377C2semidirect
5. #22609370C2semidirect
6. #22608689C2semidirect
7. #22608603C2semidirect
8. #22608600C2semidirect
9. C73×#3039C2semidirect
19. Q8×#10596831C22semidirect
21. D4×#10596831C22semidirect
29. #17988339C22semidirect
30. #17988333C22semidirect
36. #462×C73C22semidirect
38. #461×C73C22semidirect
227. #3039D73semidirect
235. C73#19276semidirect
245. #462C2×D73semidirect
247. #461C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179889352
#226194112
#41×#105968312
#226093772
#226093702
#226086892
#226086032
#226086002
C73×#30392
#1927673