Finite Group Databasebeta

← #33239393

Subgroups — #33239393

Order 18688.

652 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623065C211yesmaximal, characteristic
2C2×#17988704C211yesmaximal, characteristic, complemented
3C2×#17988703C211yesmaximal, characteristic, complemented
4#22611814C214yesmaximal, complemented
5#22611809C214yesmaximal, complemented
6#22611725C211yesmaximal, characteristic, complemented
7#22611660C211yesmaximal, characteristic, complemented
8M32×Dic73C211yesmaximal, characteristic, complemented
9C73×#2143C211yesmaximal, characteristic, complemented
10C2×#14556613C2211yescharacteristic
11C2×#14556599—21no—
12C2×#14556598—21no—
13#17989028C412yescomplemented
14#17989027C412yescomplemented
15#17989023C412yescomplemented
16#17989022C412yescomplemented
17#17989017C2214yes—
18C2×#14556531C2211yescharacteristic
19#17988704—21no—
20#17988704—21no—
21#17988704C2212yescomplemented
22#17988704C2212yescomplemented
23#17988703—21no—
24#17988703—21no—
25#17988703C2212yescomplemented
26#17988703C2212yescomplemented
27C2×#14556527C2211yescharacteristic
28C2×#14556518—21no—
29C2×#14556517—21no—
30C2×C8×Dic73C2211yescharacteristic
31#17988416C2214yescomplemented
32#17988395C2214yescomplemented
33#17988362C2212yescomplemented
34C16×Dic73C2212yescomplemented
35C2×#135×C73C2211yescharacteristic
36C2×M32×C73C2211yescharacteristic
37C2×#108×C73C2211yescharacteristic
38#359×C73C2214yescomplemented
39C2×#11924028—21no—
40#14556613—21no—
41#14556613—21no—
42#14556613C2×C412yes—
43#14556613C2312yes—
44C2×Dic292C2×C411yescharacteristic
45C2×#11924017—21no—
46C2×D584C2×C411yescharacteristic
47#14556599—22no—
48#14556599—22no—
49#14556599—41no—
50#14556599—41no—
51#14556598—22no—
52#14556598—22no—
53#14556598—41no—
54#14556598—41no—
55C2×#11924012—21no—
56C2×#11924012—41no—
57C2×#11924011—41no—
58C22×#10596831—41no—
59C22×#10596831D411yescharacteristic
60C2×C8×D73—41no—
61C2×C4×Dic73D411yescharacteristic
62#14556537C2×C414yescomplemented
63#14556535C2×C414yescomplemented
64#14556534C2×C414yescomplemented
65#14556532C2×C414yescomplemented
66#14556531C2×C412yes—
67#14556531C2×C412yes—
68C2×#11924003C2312yes—
69#14556527—21no—
70#14556527C2312yes—
71#14556518—22no—
72#14556518—22no—
73#14556517—22no—
74#14556517—22no—
75C8×Dic73D411yescharacteristic
76C8×Dic73D411yescharacteristic
77C8×Dic73C2311yescharacteristic
78C8×Dic73C2311yescharacteristic
79C2×#41×C73—21no—
80#135×C73—21no—
81#135×C73—21no—
82#135×C73C2×C412yes—
83#135×C73C2312yes—
84C2×Q16×C73C2×C411yescharacteristic
85C2×SD16×C73—21no—
86C2×D8×C73C2×C411yescharacteristic
87C2×M16×C73—21no—
88C22×C8×C73C2311yescharacteristic
89M32×C73—21no—
90M32×C73—21no—
91C2×C16×C73C2312yes—
92#108×C73—21no—
93#108×C73C2312yes—
94#11924028—22no—
95#11924028—22no—
96#11924028—41no—
97#11924028—41no—
98C2×Dic146—21no—
99C2×D292—21no—
100C2×C22⋊D73—41no—
101C22×Dic73—21no—
102C2×C4×D73—41no—
103Dic292—21no—
104Dic292C22×C412yes—
105#11924017—22no—
106#11924017—22no—
107D584—21no—
108D584C22×C412yes—
109#11924012—22no—
110#11924012—41no—
111#11924012—42no—
112#11924012—42no—
113#11924011—42no—
114#11924011—42no—
115C2×#10596831—41no—
116C2×#10596831—42no—
117C2×#10596831—42no—
118C2×#10596831D811yescharacteristic
119C2×#10596831D811yescharacteristic
120C2×#10596831D811yescharacteristic
121C2×#10596831D811yescharacteristic
122C2×#10596831C2×D411yescharacteristic
123C2×#10596831C2×D411yescharacteristic
124C8×D73—42no—
125C8×D73—42no—
126C4×Dic73—21no—
127C4×Dic73C2×D411yescharacteristic
128C4×Dic73C2×D411yescharacteristic
129#11924003C22×C414yes—
130#41×C73—22no—
131#41×C73—22no—
132#41×C73—41no—
133#41×C73—41no—
134C2×Q8×C73—21no—
135C2×D4×C73—21no—
136C2×D4×C73—41no—
137C22×C4×C73—41no—
138C22×C4×C73C2×D411yescharacteristic
139Q16×C73—21no—
140Q16×C73C22×C412yes—
141SD16×C73—22no—
142SD16×C73—22no—
143D8×C73—21no—
144D8×C73C22×C412yes—
145M16×C73—22no—
146M16×C73—41no—
147C2×C8×C73—41no—
148C2×C8×C73C22×C411yescharacteristic
149C2×C8×C73C22×C411yescharacteristic
150C2×C8×C73C2×D411yescharacteristic
151C2×C8×C73#4111yescharacteristic
152C2×C8×C73#4111yescharacteristic
153C2×C8×C73C2411yescharacteristic
154C16×C73—22no—
155C23×C73—41no—
156C23×C73#13511yescharacteristic
157C22×D73—41no—
158Q8×C73—22no—
159Q8×C73—41no—
160D4×C73—22no—
161D4×C73—42no—
162D4×C73—42no—
163D4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—42no—
166C2×C4×C73—42no—
167C2×C4×C73C4×D411yescharacteristic
168C2×C4×C73C4×D411yescharacteristic
169C2×C4×C73C2×D811yescharacteristic
170C2×C4×C73C2×D811yescharacteristic
171C2×C4×C73#13511yescharacteristic
172C2×C4×C73C22×D411yescharacteristic
173C22⋊D73—42no—
174C22⋊D73—42no—
175C2×Dic73—21no—
176C2×Dic73—21no—
177C2×Dic73—42no—
178C2×Dic73—41no—
179D292—22no—
180D292—41no—
181C4×D73—42no—
182C4×D73—42no—
183Dic146—22no—
184Dic146—41no—
185C8×C73—42no—
186C8×C73C4×D411yescharacteristic
187C8×C73C4×D411yescharacteristic
188C8×C73C23×C411yescharacteristic
189C8×C73C2×#4111yescharacteristic
190#10596831—42no—
191#10596831C2×D811yescharacteristic
192#10596831C2×D811yescharacteristic
193#10596831C2×D811yescharacteristic
194#10596831C2×D811yescharacteristic
195C22×C73—41no—
196C22×C73—42no—
197C22×C73—82no—
198C22×C73C4×D811yescharacteristic
199C22×C73C4×D811yescharacteristic
200C22×C73C2×#13511yescharacteristic
201C2×D73—42no—
202C2×D73—82no—
203C4×C73—42no—
204C4×C73C4×D811yescharacteristic
205C4×C73C4×D811yescharacteristic
206C4×C73C2×C4×D411yescharacteristic
207C4×C73C22×D811yescharacteristic
208Dic73—22no—
209Dic73—41no—
210Dic73—42no—
211#33463—731nomaximal, Sylow
212C2×C73—21no—
213C2×C73—21no—
214C2×C73—21no—
215C2×C73—82no—
216C2×C73C2×C4×D811yescharacteristic
217D73—82no—
218#3414—731no—
219C2×#443—731no—
220C2×#443—731no—
221#2179—734no—
222#2178—734no—
223#2161—731no—
224#2143—731no—
225#2143D7311yescharacteristic, complemented
226C4×M32—731no—
227C73#3346311yescharacteristic, complemented, Sylow
228C2×#135—731no—
229C2×#135C2×D7311yescharacteristic
230C2×#131—1461no—
231C2×#131—1461no—
232#510—732no—
233#510—732no—
234#509—732no—
235#509—732no—
236#508—734no—
237C2×M32—731no—
238C2×M32C2×D7311yescharacteristic
239#443—732no—
240#443—732no—
241#443—732no—
242#443—732no—
243#443—1461no—
244#443—1461no—
245#443—1461no—
246#443—1461no—
247C2×#108—731no—
248C2×#108C2×D7311yescharacteristic
249C2×#104—1461no—
250C2×#104—1461no—
251C2×C4×C8—731no—
252#368—734no—
253#359—734no—
254#359C2×D7314yescomplemented
255#346—732no—
256C4×C16—732no—
257C2×#41—21no—
258C2×#41—1461no—
259#135—21no—
260#135—21no—
261#135—732no—
262#135—732no—
263#135—1461no—
264#135—1461no—
265#135C4×D7312yes—
266#135C22×D7312yes—
267C2×Q16—731no—
268C2×Q16C4×D7311yescharacteristic
269C2×SD16—21no—
270C2×SD16—1461no—
271C2×D8—731no—
272C2×D8C4×D7311yescharacteristic
273#131—2921no—
274#131—2921no—
275#131—2921no—
276#131—2921no—
277#131—1462no—
278#131—1462no—
279#131—1462no—
280#131—1462no—
281C2×M16—21no—
282C2×M16—2921no—
283C2×M16—2921no—
284C2×M16—1461no—
285C22×C8—2921no—
286C22×C8—2921no—
287C22×C8—731no—
288C22×C8C22×D7311yescharacteristic
289C2×C42—731no—
290Q32—734no—
291SD32—734no—
292SD32—734no—
293D16—734no—
294M32—21no—
295M32—21no—
296M32—732no—
297M32—732no—
298C2×C16—732no—
299C2×C16C22×D7312yes—
300#108—21no—
301#108—732no—
302#108—1461no—
303#108C22×D7312yes—
304#104—1462no—
305#104—1462no—
306#104—1462no—
307#104—1462no—
308C4×C8—731no—
309C4×C8—731no—
310C4×C8—731no—
311C4×C8—731no—
312#41—22no—
313#41—22no—
314#41—41no—
315#41—41no—
316#41—2921no—
317#41—2921no—
318#41—1462no—
319#41—1462no—
320C2×Q8—21no—
321C2×Q8—1461no—
322C2×D4—21no—
323C2×D4—41no—
324C2×D4—2921no—
325C2×D4—1461no—
326C22×C4—41no—
327C22×C4—2921no—
328C22×C4—1461no—
329C22×C4D4×D7311yescharacteristic
330Q16—21no—
331Q16—732no—
332Q16—1461no—
333Q16C2×C4×D7312yes—
334SD16—22no—
335SD16—22no—
336SD16—1462no—
337SD16—1462no—
338D8—21no—
339D8—732no—
340D8—1461no—
341D8C2×C4×D7312yes—
342M16—22no—
343M16—41no—
344M16—2922no—
345M16—2922no—
346M16—2921no—
347M16—2922no—
348M16—2922no—
349M16—1462no—
350C2×C8—41no—
351C2×C8—2922no—
352C2×C8—2922no—
353C2×C8—2921no—
354C2×C8—2922no—
355C2×C8—2922no—
356C2×C8—731no—
357C2×C8—731no—
358C2×C8—731no—
359C2×C8—731no—
360C2×C8—731no—
361C2×C8—731no—
362C2×C8C2×C4×D7311yescharacteristic
363C2×C8C2×C4×D7311yescharacteristic
364C2×C8D4×D7311yescharacteristic
365C2×C8#1192402711yescharacteristic
366C2×C8#1192402711yescharacteristic
367C2×C8C23×D7311yescharacteristic
368C42—731no—
369C42—731no—
370C42—1461no—
371C16—22no—
372C16—734no—
373C23—41no—
374C23—2921no—
375C23#1455661211yescharacteristic
376Q8—22no—
377Q8—41no—
378Q8—2921no—
379Q8—1462no—
380D4—22no—
381D4—42no—
382D4—42no—
383D4—41no—
384D4—2922no—
385D4—2922no—
386D4—2921no—
387D4—1462no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—2921no—
392C2×C4—2922no—
393C2×C4—2922no—
394C2×C4—2922no—
395C2×C4—1461no—
396C2×C4—1461no—
397C2×C4#1455655111yescharacteristic
398C2×C4#1455655111yescharacteristic
399C2×C4D8×D7311yescharacteristic
400C2×C4D8×D7311yescharacteristic
401C2×C4#1455661211yescharacteristic
402C2×C4C2×D4×D7311yescharacteristic
403C8—42no—
404C8—2922no—
405C8—731no—
406C8—731no—
407C8—731no—
408C8—731no—
409C8#1455655111yescharacteristic
410C8#1455655111yescharacteristic
411C8C22×C4×D7311yescharacteristic
412C8C2×#1192402711yescharacteristic
413C22—41no—
414C22—42no—
415C22—82no—
416C22—2922no—
417C22—5842no—
418C22#1798866711yescharacteristic
419C22#1798866711yescharacteristic
420C22C2×#1455661211yescharacteristic
421C4—42no—
422C4—2921no—
423C4—2922no—
424C4—1462no—
425C4#1798866711yescharacteristic
426C4#1798866711yescharacteristic
427C4C2×#1455655111yescharacteristic
428C4C2×D8×D7311yescharacteristic
429C2—21no—
430C2—21no—
431C2—21no—
432C2—82no—
433C2—5842no—
434C2C2×#1798866711yescharacteristic
subgroupG/H
centerC4C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2143C2
socleC2×C73C2×C4×D8
prime radical#22623065C2
cosocleC2×C8×C73C24
hypercenter#2143D73
hypocenterC73#33463
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2143

Complements

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

NH ≅ G/Ntype
2. C2×#17988704C2semidirect
3. C2×#17988703C2semidirect
4. #22611814C2semidirect
5. #22611809C2semidirect
6. #22611725C2semidirect
7. #22611660C2semidirect
8. M32×Dic73C2semidirect
9. C73×#2143C2semidirect
13. #17989028C4semidirect
14. #17989027C4semidirect
15. #17989023C4semidirect
16. #17989022C4semidirect
21. #17988704C22semidirect
22. #17988704C22semidirect
25. #17988703C22semidirect
26. #17988703C22semidirect
31. #17988416C22semidirect
32. #17988395C22semidirect
33. #17988362C22semidirect
34. C16×Dic73C22semidirect
38. #359×C73C22semidirect
62. #14556537C2×C4semidirect
63. #14556535C2×C4semidirect
64. #14556534C2×C4semidirect
65. #14556532C2×C4semidirect
225. #2143D73semidirect
227. C73#33463semidirect
254. #359C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226230652
C2×#179887042
C2×#179887032
#226118142
#226118092
#226117252
#226116602
M32×Dic732
C73×#21432
#3346373