Finite Group Databasebeta

← #33069940

Subgroups — #33069940

Order 18688.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 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#22609349C214yesmaximal, complemented
5#22609342C214yesmaximal, complemented
6#22608686C211yesmaximal, characteristic, complemented
7#22608612C211yesmaximal, characteristic, complemented
8#22608600C211yesmaximal, characteristic, complemented
9C73×#3079C211yesmaximal, 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#17988337C2214yescomplemented
30#17988336C2214yescomplemented
31#17988330C2214yes—
32C2×C4⋊Q8×C73C2211yescharacteristic
33C4×#41×C73C2211yescharacteristic
34#477×C73C2212yescomplemented
35#476×C73C2212yescomplemented
36#475×C73—21no—
37#474×C73—21no—
38#428×C73—21no—
39#423×C73C2211yescharacteristic
40#417×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—
71C2×M16×C73—21no—
72C4⋊Q8×C73—21no—
73C4⋊Q8×C73—21no—
74C4⋊Q8×C73—21no—
75C4⋊Q8×C73C2312yes—
76C4×Q8×C73D411yescharacteristic
77C4×Q8×C73C2311yescharacteristic
78C4×D4×C73—21no—
79C4×D4×C73—21no—
80C4×D4×C73D411yescharacteristic
81C4×D4×C73C2311yescharacteristic
82#117×C73—21no—
83#117×C73D411yescharacteristic
84C2×C4⋊C4×C73—21no—
85C2×C4⋊C4×C73—21no—
86C2×C42×C73—21no—
87C2×C42×C73C2311yescharacteristic
88#107×C73—22no—
89#106×C73—22no—
90C4⋊C8×C73—21no—
91C4⋊C8×C73C2312yes—
92#103×C73—22no—
93#102×C73—22no—
94C23×D73—41no—
95C2×D292—22no—
96C2×D292—22no—
97C2×D292—22no—
98C2×D292—44no—
99C2×D292—42no—
100C2×D292—42no—
101C2×D292—41no—
102C2×D292—41no—
103C2×D292—41no—
104C2×D292—82no—
105#11924015—44no—
106#11924013—44no—
107#11924012—42no—
108C2×#10596831—22no—
109C2×#10596831—22no—
110C2×#10596831—42no—
111C2×#10596831—44no—
112C2×#10596831C2×D412yes—
113C2×#10596831C2×D412yes—
114#41×C73—22no—
115#41×C73SD1614yes—
116C2×Q8×C73—22no—
117C2×Q8×C73—41no—
118C2×Q8×C73—82no—
119C2×Q8×C73C2×D411yescharacteristic
120C2×D4×C73—21no—
121C2×D4×C73C2×D411yescharacteristic
122C22×C4×C73—21no—
123C22×C4×C73—21no—
124C22×C4×C73—21no—
125C22×C4×C73—41no—
126C22×C4×C73C2×D411yescharacteristic
127C22×C4×C73C2×D411yescharacteristic
128C22×C4×C73#4111yescharacteristic
129M16×C73—42no—
130C2×C8×C73—22no—
131C4⋊C4×C73—21no—
132C4⋊C4×C73—21no—
133C4⋊C4×C73—22no—
134C4⋊C4×C73—22no—
135C4⋊C4×C73—41no—
136C4⋊C4×C73—41no—
137C4⋊C4×C73C2×D411yescharacteristic
138C4⋊C4×C73C2×D411yescharacteristic
139C22⋊C4×C73—21no—
140C22⋊C4×C73—21no—
141C22⋊C4×C73—21no—
142C42×C73—21no—
143C42×C73—21no—
144C42×C73—21no—
145C42×C73C2×D411yescharacteristic
146C42×C73C2×D411yescharacteristic
147C42×C73#4111yescharacteristic
148C42×C73C2411yescharacteristic
149C23×C73—21no—
150C23×C73#12111yescharacteristic
151C22×D73—42no—
152C22×D73—82no—
153C22×D73—82no—
154C22×D73—82no—
155Q8×C73—21no—
156Q8×C73—82no—
157Q8×C73—82no—
158Q8×C73C2×SD1612yes—
159D4×C73—24no—
160D4×C73—21no—
161D4×C73C2×SD1612yes—
162C2×C4×C73—21no—
163C2×C4×C73—22no—
164C2×C4×C73—24no—
165C2×C4×C73—22no—
166C2×C4×C73—21no—
167C2×C4×C73—21no—
168C2×C4×C73—42no—
169C2×C4×C73—42no—
170C2×C4×C73—42no—
171C2×C4×C73—82no—
172C2×C4×C73#12111yescharacteristic
173C2×C4×C73#12111yescharacteristic
174C2×C4×C73#12111yescharacteristic
175C2×C4×C73C2×SD1612yes—
176C2×C4×C73#13512yes—
177C2×C4×C73C22×D411yescharacteristic
178C2×C4×C73C22×D411yescharacteristic
179C2×C4×C73C2×#4111yescharacteristic
180D292—42no—
181D292—42no—
182D292—44no—
183D292—82no—
184D292—82no—
185D292—82no—
186D292—82no—
187C8×C73—42no—
188#10596831—22no—
189#10596831—22no—
190#10596831—42no—
191#10596831—42no—
192C22×C73—21no—
193C22×C73—21no—
194C22×C73—22no—
195C22×C73—41no—
196C22×C73—41no—
197C22×C73#46512yes—
198C22×C73C2×#12111yescharacteristic
199C2×D73—82no—
200C2×D73—82no—
201C2×D73—82no—
202C2×D73—164no—
203C4×C73—21no—
204C4×C73—22no—
205C4×C73—21no—
206C4×C73—41no—
207C4×C73—41no—
208C4×C73—82no—
209C4×C73#46512yes—
210C4×C73#47212yes—
211C4×C73C22×SD1611yescharacteristic
212C4×C73C2×#13511yescharacteristic
213C4×C73#57512yes—
214#19274—731nomaximal, Sylow
215C2×C73—22no—
216C2×C73—42no—
217C2×C73#304011yescharacteristic
218C2×C73C2×#46511yescharacteristic
219C2×C73C2×#47211yescharacteristic
220D73—162no—
221C2×#486—731no—
222#3079D7311yescharacteristic, complemented
223#3038—731no—
224C8×#41—731no—
225#1678—734no—
226#1677—734no—
227#1566—731no—
228#1548—731no—
229#1545—731no—
230C73#1927411yescharacteristic, complemented, Sylow
231C2×AGL(1,ℤ/8)—1461no—
232C2×C4⋊Q8C2×D7311yescharacteristic
233C2×#127—731no—
234C4×#41C2×D7311yescharacteristic
235#486—734no—
236#486—1461no—
237#486—1461no—
238#477C2×D7312yescomplemented
239#476C2×D7312yescomplemented
240#475—21no—
241#474—21no—
242#460—732no—
243#460—1461no—
244#459—732no—
245#459—1461no—
246C8×Q8—732no—
247C8×D4—732no—
248C8×D4—1462no—
249#431—731no—
250#431—1461no—
251#428—21no—
252#423—731no—
253#423C2×D7311yescharacteristic
254#418—1461no—
255#417—21no—
256C2×#102—1462no—
257C2×C4×C8—731no—
258C2×C4×C8—1461no—
259#334—734no—
260#333—734no—
261#331—734no—
262C2×#41C22⋊D7311yescharacteristic
263C22×Q8—21no—
264C22×D4—1461no—
265C22×D4—1461no—
266C22×D4—1461no—
267AGL(1,ℤ/8)—2924no—
268C2×SD16—1462no—
269C2×D8—1462no—
270C2×M16—21no—
271C2×M16—1461no—
272C22×C8—732no—
273C22×C8—1462no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8C22×D7312yes—
278#127—732no—
279#127—1461no—
280#127—1461no—
281#127—1461no—
282C4×Q8C22⋊D7311yescharacteristic
283C4×Q8C22×D7311yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C22⋊D7311yescharacteristic
287C4×D4C22×D7311yescharacteristic
288#117—21no—
289#117C22⋊D7311yescharacteristic
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C42—21no—
293C2×C42C22×D7311yescharacteristic
294#107—22no—
295#106—22no—
296C4⋊C8—21no—
297C4⋊C8—732no—
298C4⋊C8—732no—
299C4⋊C8—1462no—
300C4⋊C8—1461no—
301C4⋊C8C22×D7312yes—
302#103—22no—
303#102—22no—
304#102—1464no—
305#102—1464no—
306#102—1462no—
307#102—1462no—
308C22⋊C8—1462no—
309C22⋊C8—1461no—
310C4×C8—731no—
311C4×C8—731no—
312C4×C8—732no—
313C4×C8—1462no—
314C4×C8—1461no—
315C24—2921no—
316#41—22no—
317#41#1192401514yes—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×Q8C2×C22⋊D7311yescharacteristic
322C2×D4—21no—
323C2×D4—2924no—
324C2×D4—2921no—
325C2×D4—2921no—
326C2×D4—2921no—
327C2×D4—2922no—
328C2×D4—2922no—
329C2×D4—5842no—
330C2×D4—1462no—
331C2×D4—1462no—
332C2×D4—1462no—
333C2×D4C2×C22⋊D7311yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4D4×D7311yescharacteristic
339C22×C4C2×C22⋊D7311yescharacteristic
340C22×C4#1192402711yescharacteristic
341SD16—2924no—
342D8—2924no—
343M16—42no—
344M16—2922no—
345C2×C8—22no—
346C2×C8—2924no—
347C2×C8—2922no—
348C2×C8—732no—
349C2×C8—732no—
350C2×C8—1462no—
351C2×C8—1462no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4C2×C22⋊D7311yescharacteristic
359C4⋊C4C2×C22⋊D7311yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42D4×D7311yescharacteristic
367C42C2×C22⋊D7311yescharacteristic
368C42#1192402711yescharacteristic
369C42C23×D7311yescharacteristic
370C23—21no—
371C23—2922no—
372C23—5842no—
373C23—5842no—
374C23—5842no—
375C23#1455656311yescharacteristic
376Q8—21no—
377Q8—82no—
378Q8—82no—
379Q8C2×#1192401512yes—
380D4—24no—
381D4—21no—
382D4—2924no—
383D4—2922no—
384D4—2922no—
385D4—5842no—
386D4—5842no—
387D4—5842no—
388D4—5842no—
389D4C2×#1192401512yes—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4#1455656311yescharacteristic
401C2×C4#1455656311yescharacteristic
402C2×C4#1455656311yescharacteristic
403C2×C4C2×#1192401512yes—
404C2×C4#1455661112yes—
405C2×C4C2×D4×D7311yescharacteristic
406C2×C4C22×C22⋊D7311yescharacteristic
407C2×C4C2×#1192402711yescharacteristic
408C8—42no—
409C8—2922no—
410C8—2922no—
411C8—1462no—
412C8—1462no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—5842no—
419C22—5842no—
420C22—5842no—
421C22—11684no—
422C22#1798881812yes—
423C22C2×#1455656311yescharacteristic
424C4—21no—
425C4—22no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—82no—
430C4#1798881812yes—
431C4#1798885512yes—
432C4C22×#1192401511yescharacteristic
433C4C2×#1455661111yescharacteristic
434C4#1798938612yes—
435C2—22no—
436C2—42no—
437C2—11682no—
438C2#2261942711yescharacteristic
439C2C2×#1798881811yescharacteristic
440C2C2×#1798885511yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3079C2
socleC22×C73C2×#121
prime radical#22619411C2
hypercenter#3079D73
hypocenterC73#19274
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3079

Complements

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

NH ≅ G/Ntype
1. C2×#17988935C2semidirect
3. #41×#10596831C2semidirect
4. #22609349C2semidirect
5. #22609342C2semidirect
6. #22608686C2semidirect
7. #22608612C2semidirect
8. #22608600C2semidirect
9. C73×#3079C2semidirect
19. Q8×#10596831C22semidirect
21. D4×#10596831C22semidirect
29. #17988337C22semidirect
30. #17988336C22semidirect
34. #477×C73C22semidirect
35. #476×C73C22semidirect
222. #3079D73semidirect
230. C73#19274semidirect
238. #477C2×D73semidirect
239. #476C2×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
#226093492
#226093422
#226086862
#226086122
#226086002
C73×#30792
#1927473