Finite Group Databasebeta

← #33949547

Subgroups — #33949547

Order 18688.

4874 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989423C211yesmaximal, characteristic
2C22×#14556598C211yesmaximal, characteristic, complemented
3#22618461C218yesmaximal, complemented
4#22618454C218yesmaximal, complemented
5#22618448C216yesmaximal, complemented
6C2×#17988581C216yesmaximal, complemented
7C73×#3428C211yesmaximal, characteristic, complemented
8#17989423—28no—
9#17989423C4116yescomplemented
10C2×#14556639—21no—
11C2×#14556638C2214yes—
12C2×#14556637C2214yes—
13C2×#14556636—23no—
14C22×#11924028C2216yes—
15C2×#41×D73—24no—
16C2×#14556598—28no—
17C2×#14556598C2218yescomplemented
18C22×#11924012C2213yescomplemented
19C23×#10596831C2213yescomplemented
20#17988595C22112yescomplemented
21#17988591C22112yescomplemented
22#17988586C22124yescomplemented
23#17988581C2214yescomplemented
24#17988581C22112yescomplemented
25#17988581C22124yescomplemented
26C2×#14556514C22112yescomplemented
27C2×#14556511C22112yescomplemented
28C22×#41×C73C2211yescharacteristic
29#520×C73C2218yescomplemented
30#519×C73—21no—
31#518×C73—23no—
32C4×#41×C73—24no—
33C2×#117×C73C2216yescomplemented
34#14556639—216no—
35#14556638—216no—
36#14556638C2×C4132yescomplemented
37#14556637—216no—
38#14556637C2×C4132yescomplemented
39#14556636—248no—
40C2×#11924029—24no—
41C2×#11924028—26no—
42C2×#11924028—26no—
43C2×#11924028—224no—
44C2×#11924028C2×C4148yescomplemented
45C2×#11924028C2314yes—
46C2×#11924028C23112yes—
47C2×#11924027—212no—
48#41×D73—264no—
49C22×Dic146C2316yes—
50C2×Q8×D73—24no—
51C22×D292C2316yes—
52C22×C22⋊D73—26no—
53C2×D4×D73—212no—
54C22×C4×D73—26no—
55#14556598—232no—
56C2×#11924012—212no—
57C2×#11924012C23112yes—
58C22×#10596831—212no—
59C22×#10596831C23112yes—
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2313yes—
62#14556514C2318yes—
63#14556514C23124yes—
64#14556511C2318yes—
65#14556511C23124yes—
66C2×#41×C73C2×C418yes—
67C2×#41×C73D418yes—
68C2×#41×C73D414yes—
69C2×#41×C73C2314yes—
70C22×Q8×C73D411yescharacteristic
71C22×D4×C73D413yes—
72C23×C4×C73C2313yes—
73C4×Q8×C73—28no—
74C4×D4×C73—224no—
75#117×C73—26no—
76#117×C73—26no—
77#117×C73C2314yes—
78#117×C73C23112yes—
79C2×C4⋊C4×C73C23112yes—
80C2×C22⋊C4×C73—26no—
81C2×C42×C73—26no—
82C23×D73—26no—
83#11924029—232no—
84#11924028—248no—
85#11924028—248no—
86#11924028—28no—
87#11924028—224no—
88#11924028C22×C4116yes—
89#11924028C22×C4148yes—
90#11924027—296no—
91C2×Dic146—26no—
92C2×Dic146—212no—
93C2×Dic146C22×C4124yes—
94C2×Dic146C2414yes—
95Q8×D73—232no—
96C2×D292—26no—
97C2×D292—212no—
98C2×D292C22×C4124yes—
99C2×D292C2414yes—
100C2×C22⋊D73—248no—
101C2×C22⋊D73—212no—
102C2×C22⋊D73—24no—
103D4×D73—296no—
104C22×Dic73—26no—
105C2×C4×D73—248no—
106C2×C4×D73—24no—
107C2×C4×D73—212no—
108#11924012—224no—
109C2×#10596831—22no—
110C2×#10596831—26no—
111C2×#10596831—224no—
112C2×#10596831C2412yes—
113C2×#10596831C2416yes—
114C24×C73C2×D413yes—
115#41×C73C22⋊C4132yes—
116#41×C73C22×C4116yes—
117#41×C73C2×D4116yes—
118C2×Q8×C73C22⋊C418yes—
119C2×Q8×C73C2×D414yes—
120C2×D4×C73C22⋊C4124yes—
121C2×D4×C73C2×D4112yes—
122C22×C4×C73—212no—
123C22×C4×C73C22×C4112yes—
124C22×C4×C73C2×D416yes—
125C22×C4×C73C2×D413yes—
126C22×C4×C73C2×D4112yes—
127C22×C4×C73C2416yes—
128C22×C4×C73C2411yescharacteristic
129C4⋊C4×C73—212no—
130C4⋊C4×C73C2418yes—
131C22⋊C4×C73—24no—
132C22⋊C4×C73—212no—
133C42×C73—24no—
134C42×C73—212no—
135C23×C73—46no—
136C23×C73C2×C22⋊C4112yes—
137C23×C73C22×D411yescharacteristic
138C23×C73C22×D416yes—
139C22×D73—24no—
140C22×D73—224no—
141C22×D73—812no—
142Q8×C73C2×C22⋊C4116yes—
143D4×C73C2×C22⋊C4148yes—
144C2×C4×C73—28no—
145C2×C4×C73—812no—
146C2×C4×C73C2×C22⋊C4112yes—
147C2×C4×C73C2×C22⋊C4124yes—
148C2×C4×C73C23×C412yes—
149C2×C4×C73C23×C4112yes—
150C2×C4×C73C22×D416yes—
151C2×C4×C73C22×D411yescharacteristic
152C2×C4×C73C22×D412yes—
153C2×C4×C73C22×D4112yes—
154C2×C4×C73C2511yescharacteristic
155C22⋊D73—248no—
156C22⋊D73—216no—
157C2×Dic73—224no—
158C2×Dic73—24no—
159D292—224no—
160D292—24no—
161D292C23×C418yes—
162C4×D73—216no—
163C4×D73—248no—
164Dic146—224no—
165Dic146—24no—
166Dic146C23×C418yes—
167#10596831—22no—
168#10596831—26no—
169C22×C73—212no—
170C22×C73—22no—
171C22×C73—412no—
172C22×C73C22×C22⋊C4112yes—
173C22×C73C22×C22⋊C412yes—
174C22×C73C23×D411yescharacteristic
175C2×D73—28no—
176C2×D73—48no—
177C2×D73—824no—
178C4×C73—48no—
179C4×C73C22×C22⋊C4112yes—
180C4×C73C22×C22⋊C412yes—
181C4×C73C24×C411yescharacteristic
182C4×C73C23×D411yescharacteristic
183Dic73—28no—
184#60086—731nomaximal, Sylow
185C2×C73—212no—
186C2×C73—22no—
187C2×C73C23×C22⋊C411yescharacteristic
188C2×C73#358412yes—
189D73—48no—
190C2×#585—731no—
191C22×#131—731no—
192#3428D7311yescharacteristic, complemented
193#2898—738no—
194#2897—738no—
195#2896—736no—
196C2×#416—736no—
197C73#6008611yescharacteristic, complemented, Sylow
198#585—7316no—
199#585—1468no—
200C2×2−1+4—734no—
201C2×2−1+4—1461no—
202C2×H(5,𝔽2)—734no—
203C2×H(5,𝔽2)—1463no—
204C22×#41—736no—
205C22×#41—1464no—
206C22×#41C2×D7311yescharacteristic
207C2×#131—738no—
208C2×#131—1468no—
209C22×M16—733no—
210C23×C8—733no—
211#520C2×D7318yescomplemented
212#519—21no—
213#518—23no—
214C4×#41—24no—
215C2×#117C2×D7316yescomplemented
216#419—7312no—
217#418—7312no—
218#417—7324no—
219#416—734no—
220#416—7312no—
221#416—7324no—
222C2×#103—7312no—
223C2×#102—7312no—
2242−1+4—7332no—
2252−1+4—14616no—
2262−1+4—14616no—
227H(5,𝔽2)—7332no—
228H(5,𝔽2)—14616no—
229H(5,𝔽2)—14648no—
230C2×#41—7348no—
231C2×#41—734no—
232C2×#41—7312no—
233C2×#41—14664no—
234C2×#41—1464no—
235C2×#41—14612no—
236C2×#41—1466no—
237C2×#41—1466no—
238C2×#41—14624no—
239C2×#41C4×D7318yes—
240C2×#41D29218yes—
241C2×#41C22⋊D7314yes—
242C2×#41C22×D7314yes—
243C22×Q8—736no—
244C22×Q8—1464no—
245C22×Q8C22⋊D7311yescharacteristic
246C22×D4—736no—
247C22×D4—1466no—
248C22×D4—14612no—
249C22×D4C22⋊D7313yes—
250C23×C4—1466no—
251C23×C4C22×D7313yes—
252#131—14632no—
253C2×M16—7312no—
254C2×M16—14612no—
255C22×C8—731no—
256C22×C8—733no—
257C22×C8—7312no—
258C22×C8—14612no—
259C4×Q8—28no—
260C4×D4—224no—
261#117—26no—
262#117—26no—
263#117C22×D7314yes—
264#117C22×D73112yes—
265C2×C4⋊C4C22×D73112yes—
266C2×C22⋊C4—26no—
267C2×C42—26no—
268#103—738no—
269#103—7324no—
270#102—738no—
271#102—7324no—
272C24—1466no—
273C24C2×C22⋊D7313yes—
274#41—7316no—
275#41—7348no—
276#41—14632no—
277#41—14696no—
278#41—14648no—
279#41—14648no—
280#41—1468no—
281#41—14624no—
282#41#11924005132yes—
283#41C2×C4×D73116yes—
284#41C2×D292116yes—
285C2×Q8—7324no—
286C2×Q8—734no—
287C2×Q8—14632no—
288C2×Q8—1466no—
289C2×Q8—14612no—
290C2×Q8#1192400518yes—
291C2×Q8C2×C22⋊D7314yes—
292C2×D4—7324no—
293C2×D4—734no—
294C2×D4—14648no—
295C2×D4—1464no—
296C2×D4—14612no—
297C2×D4—14696no—
298C2×D4—1466no—
299C2×D4—14612no—
300C2×D4#11924005124yes—
301C2×D4C2×C22⋊D73112yes—
302C22×C4—212no—
303C22×C4—1466no—
304C22×C4—14648no—
305C22×C4—14612no—
306C22×C4—1464no—
307C22×C4C2×C4×D73112yes—
308C22×C4C2×C22⋊D7313yes—
309C22×C4C2×C22⋊D7316yes—
310C22×C4C2×D292112yes—
311C22×C4C23×D7316yes—
312C22×C4C23×D7311yescharacteristic
313M16—14624no—
314C2×C8—732no—
315C2×C8—736no—
316C2×C8—1462no—
317C2×C8—1466no—
318C2×C8—14624no—
319C4⋊C4—212no—
320C4⋊C4C23×D7318yes—
321C22⋊C4—24no—
322C22⋊C4—212no—
323C42—212no—
324C42—24no—
325C23—46no—
326C23—58412no—
327C23—1464no—
328C23—14624no—
329C23C2×#11924005112yes—
330C23C22×C22⋊D7311yescharacteristic
331C23C22×C22⋊D7316yes—
332Q8—738no—
333Q8—1464no—
334Q8—14624no—
335Q8C2×#11924005116yes—
336D4—738no—
337D4—14648no—
338D4—14616no—
339D4—1464no—
340D4—14624no—
341D4C2×#11924005148yes—
342C2×C4—28no—
343C2×C4—812no—
344C2×C4—1464no—
345C2×C4—14624no—
346C2×C4—14616no—
347C2×C4—14648no—
348C2×C4C2×#11924005124yes—
349C2×C4C2×#11924005112yes—
350C2×C4C22×C4×D7312yes—
351C2×C4C22×C4×D73112yes—
352C2×C4C22×C22⋊D7316yes—
353C2×C4C22×C22⋊D7311yescharacteristic
354C2×C4C22×D29212yes—
355C2×C4C22×D292112yes—
356C2×C4C24×D7311yescharacteristic
357C8—1462no—
358C8—1466no—
359C22—212no—
360C22—22no—
361C22—412no—
362C22—2928no—
363C22—58424no—
364C22—1468no—
365C22C22×#11924005112yes—
366C22C22×#1192400512yes—
367C22C23×C22⋊D7311yescharacteristic
368C4—48no—
369C4—1468no—
370C4C22×#1192400512yes—
371C4C22×#11924005112yes—
372C4C23×C4×D7311yescharacteristic
373C4C23×D29211yescharacteristic
374C2—22no—
375C2—212no—
376C2—2928no—
377C2C23×#1192400511yescharacteristic
378C2#2262470212yes—
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC4×C73C24×C4
FrattiniC2×C4C24×D73
FittingC73×#3428C2
socleC22×C73C23×D4
prime radicalC2×#17989423C2
cosocleC2×C4×C73C25
hypercenter#3428D73
hypocenterC73#60086
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×#41 ◁ #3428

Complements

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

NH ≅ G/Ntype
2. C22×#14556598C2semidirect
3. #22618461C2semidirect
4. #22618454C2semidirect
5. #22618448C2semidirect
6. C2×#17988581C2semidirect
7. C73×#3428C2semidirect
9. #17989423C4semidirect
17. C2×#14556598C22semidirect
18. C22×#11924012C22semidirect
19. C23×#10596831C22semidirect
20. #17988595C22semidirect
21. #17988591C22semidirect
22. #17988586C22semidirect
23. #17988581C22semidirect
24. #17988581C22semidirect
25. #17988581C22semidirect
26. C2×#14556514C22semidirect
27. C2×#14556511C22semidirect
29. #520×C73C22semidirect
33. C2×#117×C73C22semidirect
36. #14556638C2×C4semidirect
38. #14556637C2×C4semidirect
44. C2×#11924028C2×C4semidirect
192. #3428D73semidirect
197. C73#60086semidirect
211. #520C2×D73semidirect
215. C2×#117C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179894232
C22×#145565982
#226184612
#226184542
#226184482
C2×#179885812
C73×#34282
#6008673