Finite Group Databasebeta

← #33239401

Subgroups — #33239401

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623065C211yesmaximal, characteristic
2#22618849C211yesmaximal, characteristic, complemented
3#22611815C214yesmaximal, complemented
4#22611810C214yesmaximal, complemented
5#22611725C211yesmaximal, characteristic, complemented
6C2×#17988394C211yesmaximal, characteristic, complemented
7C2×#17988393C211yesmaximal, characteristic, complemented
8#22611528C211yesmaximal, characteristic, complemented
9C73×#2957C211yesmaximal, characteristic, complemented
10C2×#14556613C2211yescharacteristic
11#17989353—21no—
12#17989028C412yescomplemented
13#17989027C412yescomplemented
14#17989023C412yescomplemented
15#17989022C412yescomplemented
16#17989017C2214yes—
17C2×#14556531C2211yescharacteristic
18C22×#11924003C2211yescharacteristic
19#17988710—22no—
20#17988705C2214yescomplemented
21C2×#14556527C2211yescharacteristic
22#17988605—21no—
23#17988416C2214yescomplemented
24#17988394C2212yescomplemented
25#17988394C2212yescomplemented
26#17988393C2212yescomplemented
27#17988393C2212yescomplemented
28#17988363C2212yescomplemented
29C4×#11924003C2212yescomplemented
30C2×#135×C73C2211yescharacteristic
31#568×C73—21no—
32#444×C73—22no—
33#443×C73C2214yescomplemented
34C2×#108×C73C2211yescharacteristic
35#421×C73—21no—
36#405×C73C2211yescharacteristic
37C2×#11924028—21no—
38#14556613—21no—
39#14556613—21no—
40#14556613C2×C412yes—
41#14556613C2312yes—
42C2×Dic292C2×C411yescharacteristic
43C2×#11924017—21no—
44C2×D584C2×C411yescharacteristic
45#14556599—42no—
46#14556597—24no—
47C2×#11924012—21no—
48M16×D73—42no—
49#14556537C2×C414yes—
50#14556535C2×C414yes—
51#14556534C2×C414yes—
52#14556532C2×C414yes—
53#14556531C2×C412yes—
54#14556531C2×C412yes—
55C2×#11924003—21no—
56C2×#11924003—21no—
57C2×#11924003C2311yescharacteristic
58C2×#11924003C2311yescharacteristic
59C2×#11924003C2312yes—
60#14556527—21no—
61#14556527C2312yes—
62#14556519—24no—
63C2×#41×C73—21no—
64#135×C73—21no—
65#135×C73—21no—
66#135×C73C2×C412yes—
67#135×C73C2312yes—
68C2×Q16×C73C2×C411yescharacteristic
69C2×SD16×C73—21no—
70C2×D8×C73C2×C411yescharacteristic
71#131×C73—24no—
72#131×C73—42no—
73C2×M16×C73—21no—
74C2×M16×C73—42no—
75C2×M16×C73D411yescharacteristic
76C22×C8×C73C2311yescharacteristic
77#117×C73D411yescharacteristic
78#108×C73—21no—
79#108×C73C2312yes—
80#104×C73—24no—
81#97×C73D412yes—
82C4×C8×C73C2312yes—
83#11924028—22no—
84#11924028—22no—
85#11924028—41no—
86#11924028—41no—
87C2×Dic146—21no—
88C2×D292—21no—
89C2×C22⋊D73—41no—
90C2×C4×D73—41no—
91Dic292—21no—
92Dic292C22×C412yes—
93#11924017—22no—
94#11924017—22no—
95D584—21no—
96D584C22×C412yes—
97#11924012—22no—
98#11924012—41no—
99#11924011—44no—
100C2×#10596831—41no—
101C8×D73—44no—
102#11924003—21no—
103#11924003—21no—
104#11924003C22×C414yes—
105#41×C73—22no—
106#41×C73—22no—
107#41×C73—41no—
108#41×C73—41no—
109C2×Q8×C73—21no—
110C2×D4×C73—21no—
111C2×D4×C73—41no—
112C22×C4×C73—41no—
113C22×C4×C73C2×D411yescharacteristic
114Q16×C73—21no—
115Q16×C73C22×C412yes—
116SD16×C73—22no—
117SD16×C73—22no—
118D8×C73—21no—
119D8×C73C22×C412yes—
120M16×C73—22no—
121M16×C73—41no—
122M16×C73—44no—
123M16×C73D812yes—
124M16×C73D812yes—
125C2×C8×C73—41no—
126C2×C8×C73—44no—
127C2×C8×C73C22×C411yescharacteristic
128C2×C8×C73C22×C411yescharacteristic
129C2×C8×C73C2×D411yescharacteristic
130C2×C8×C73C2×D412yes—
131C2×C8×C73#4111yescharacteristic
132C2×C8×C73#4111yescharacteristic
133C2×C8×C73C2411yescharacteristic
134C4⋊C4×C73—21no—
135C22⋊C4×C73—21no—
136C42×C73C2×D412yes—
137C23×C73—41no—
138C23×C73#13511yescharacteristic
139C22×D73—41no—
140Q8×C73—22no—
141Q8×C73—41no—
142D4×C73—22no—
143D4×C73—42no—
144D4×C73—42no—
145D4×C73—41no—
146C2×C4×C73—22no—
147C2×C4×C73—41no—
148C2×C4×C73—42no—
149C2×C4×C73—42no—
150C2×C4×C73C4×D411yescharacteristic
151C2×C4×C73C4×D411yescharacteristic
152C2×C4×C73C2×D811yescharacteristic
153C2×C4×C73C2×D811yescharacteristic
154C2×C4×C73#13511yescharacteristic
155C2×C4×C73C22×D411yescharacteristic
156C22⋊D73—42no—
157C22⋊D73—42no—
158C2×Dic73—41no—
159D292—22no—
160D292—41no—
161C4×D73—42no—
162C4×D73—42no—
163Dic146—22no—
164Dic146—41no—
165C8×C73—42no—
166C8×C73C4×D411yescharacteristic
167C8×C73C4×D411yescharacteristic
168C8×C73C2×D814yes—
169C8×C73C23×C411yescharacteristic
170C8×C73C2×#4111yescharacteristic
171#10596831—42no—
172C22×C73—41no—
173C22×C73—42no—
174C22×C73—161no—
175C22×C73C4×D811yescharacteristic
176C22×C73C4×D811yescharacteristic
177C22×C73C2×#13511yescharacteristic
178C2×D73—42no—
179C2×D73—161no—
180C4×C73—42no—
181C4×C73—42no—
182C4×C73C4×D811yescharacteristic
183C4×C73C4×D811yescharacteristic
184C4×C73C2×C4×D411yescharacteristic
185C4×C73C22×D811yescharacteristic
186Dic73—42no—
187#33464—731nomaximal, Sylow
188C2×C73—21no—
189C2×C73—21no—
190C2×C73—21no—
191C2×C73—82no—
192C2×C73C2×C4×D811yescharacteristic
193D73—82no—
194#3414—731no—
195#2957—731no—
196#2957D7311yescharacteristic, complemented
197#2179—734no—
198#2178—734no—
199#2161—731no—
200C2×#359—731no—
201C2×#359—731no—
202#2108—731no—
203C73#3346411yescharacteristic, complemented, Sylow
204C2×#135—731no—
205C2×#135C2×D7311yescharacteristic
206#568—21no—
207#568—1461no—
208#510—732no—
209#510—732no—
210#509—732no—
211#509—732no—
212#508—734no—
213C2×M32—731no—
214C22×C16—731no—
215#444—22no—
216#444—1462no—
217#443—734no—
218#443C2×D7314yescomplemented
219C2×#108—731no—
220C2×#108C2×D7311yescharacteristic
221#421—21no—
222#421—1461no—
223#405C2×D7311yescharacteristic
224#368—734no—
225#359—732no—
226#359—732no—
227#359—732no—
228#359—732no—
229#346—732no—
230C4×C16—732no—
231C2×#41—21no—
232C2×#41—1461no—
233#135—21no—
234#135—21no—
235#135—732no—
236#135—732no—
237#135—1461no—
238#135—1461no—
239#135C4×D7312yes—
240#135C22×D7312yes—
241C2×Q16—731no—
242C2×Q16C4×D7311yescharacteristic
243C2×SD16—21no—
244C2×SD16—1461no—
245C2×D8—731no—
246C2×D8C4×D7311yescharacteristic
247#131—24no—
248#131—42no—
249#131—2922no—
250#131—1464no—
251C2×M16—21no—
252C2×M16—42no—
253C2×M16—2922no—
254C2×M16—1461no—
255C2×M16C22⋊D7311yescharacteristic
256C22×C8C22×D7311yescharacteristic
257#117C22⋊D7311yescharacteristic
258Q32—734no—
259SD32—734no—
260SD32—734no—
261D16—734no—
262M32—732no—
263M32—732no—
264C2×C16—732no—
265C2×C16—731no—
266C2×C16—731no—
267C2×C16—1461no—
268C2×C16—1461no—
269#108—21no—
270#108—732no—
271#108—1461no—
272#108C22×D7312yes—
273#104—24no—
274#104—1464no—
275#97C22⋊D7312yes—
276C4×C8C22×D7312yes—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—2921no—
282#41—2921no—
283#41—1462no—
284#41—1462no—
285C2×Q8—21no—
286C2×Q8—1461no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—2921no—
290C2×D4—1461no—
291C22×C4—41no—
292C22×C4—2921no—
293C22×C4C2×C22⋊D7311yescharacteristic
294Q16—21no—
295Q16—732no—
296Q16—1461no—
297Q16C2×C4×D7312yes—
298SD16—22no—
299SD16—22no—
300SD16—1462no—
301SD16—1462no—
302D8—21no—
303D8—732no—
304D8—1461no—
305D8C2×C4×D7312yes—
306M16—22no—
307M16—41no—
308M16—44no—
309M16—2921no—
310M16—2924no—
311M16—1462no—
312M16#1192401312yes—
313M16#1192401312yes—
314C2×C8—41no—
315C2×C8—44no—
316C2×C8—2921no—
317C2×C8—2924no—
318C2×C8C2×C4×D7311yescharacteristic
319C2×C8C2×C4×D7311yescharacteristic
320C2×C8C2×C22⋊D7311yescharacteristic
321C2×C8C2×C22⋊D7312yes—
322C2×C8#1192402811yescharacteristic
323C2×C8#1192402811yescharacteristic
324C2×C8C23×D7311yescharacteristic
325C4⋊C4—21no—
326C22⋊C4—21no—
327C42C2×C22⋊D7312yes—
328C16—734no—
329C16—1461no—
330C16—1461no—
331C23—41no—
332C23—2921no—
333C23#1455661111yescharacteristic
334Q8—22no—
335Q8—41no—
336Q8—2921no—
337Q8—1462no—
338D4—22no—
339D4—42no—
340D4—42no—
341D4—41no—
342D4—2922no—
343D4—2922no—
344D4—2921no—
345D4—1462no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—2921no—
351C2×C4—2922no—
352C2×C4—2922no—
353C2×C4C4×C22⋊D7311yescharacteristic
354C2×C4C4×C22⋊D7311yescharacteristic
355C2×C4C2×#1192401311yescharacteristic
356C2×C4C2×#1192401311yescharacteristic
357C2×C4#1455661111yescharacteristic
358C2×C4C22×C22⋊D7311yescharacteristic
359C8—42no—
360C8—2922no—
361C8C4×C22⋊D7311yescharacteristic
362C8C4×C22⋊D7311yescharacteristic
363C8C2×#1192401314yes—
364C8C22×C4×D7311yescharacteristic
365C8C2×#1192402811yescharacteristic
366C22—41no—
367C22—42no—
368C22—161no—
369C22—2922no—
370C22—11681no—
371C22C4×#1192401311yescharacteristic
372C22C4×#1192401311yescharacteristic
373C22C2×#1455661111yescharacteristic
374C4—42no—
375C4—42no—
376C4—2922no—
377C4C4×#1192401311yescharacteristic
378C4C4×#1192401311yescharacteristic
379C4C2×C4×C22⋊D7311yescharacteristic
380C4C22×#1192401311yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—82no—
385C2—5842no—
386C2C2×C4×#1192401311yescharacteristic
subgroupG/H
centerC4C22×#11924013
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2957C2
socleC2×C73C2×C4×D8
prime radical#22623065C2
cosocleC2×C8×C73C24
hypercenter#2957D73
hypocenterC73#33464
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×M16 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
2. #22618849C2semidirect
3. #22611815C2semidirect
4. #22611810C2semidirect
5. #22611725C2semidirect
6. C2×#17988394C2semidirect
7. C2×#17988393C2semidirect
8. #22611528C2semidirect
9. C73×#2957C2semidirect
12. #17989028C4semidirect
13. #17989027C4semidirect
14. #17989023C4semidirect
15. #17989022C4semidirect
20. #17988705C22semidirect
23. #17988416C22semidirect
24. #17988394C22semidirect
25. #17988394C22semidirect
26. #17988393C22semidirect
27. #17988393C22semidirect
28. #17988363C22semidirect
29. C4×#11924003C22semidirect
33. #443×C73C22semidirect
196. #2957D73semidirect
203. C73#33464semidirect
218. #443C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226230652
#226188492
#226118152
#226118102
#226117252
C2×#179883942
C2×#179883932
#226115282
C73×#29572
#3346473