Finite Group Databasebeta

← #33104469

Subgroups — #33104469

Order 18688.

846 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988702C211yesmaximal, characteristic
2#22618810C212yesmaximal
3#22618531C211yesmaximal, characteristic, complemented
4#22618527C211yesmaximal, characteristic, complemented
5#22610710C212yesmaximal, complemented
6#22610167C212yesmaximal, complemented
7#22610162C212yesmaximal, complemented
8#22610094C211yesmaximal, characteristic, complemented
9#22609992C211yesmaximal, characteristic, complemented
10#22609813C211yesmaximal, characteristic, complemented
11C73×#1816C211yesmaximal, characteristic, complemented
12C2×#14556611C2211yescharacteristic
13C2×#14556597C2211yescharacteristic
14#17989070C2212yes—
15#17989068C2212yes—
16#17988702C418yescomplemented
17#17988696—22no—
18#17988691—22no—
19#17988678—22no—
20#17988672—22no—
21C2×#14556526C2211yescharacteristic
22#17988629—22no—
23#17988629C2212yes—
24#17988628—22no—
25#17988627C2212yes—
26#17988625—21no—
27#17988625C2211yescharacteristic
28#17988624—21no—
29C2×#14556525—21no—
30C2×#14556525C2211yescharacteristic
31#107×D73—21no—
32C2×#14556522C2212yes—
33C2×#14556518C2211yescharacteristic
34C2×#14556517C2211yescharacteristic
35#17988589C2212yes—
36#17988545C2212yes—
37C2×C8×Dic73C2211yescharacteristic
38#17988452C2212yes—
39#17988355C2212yes—
40#17988348C2212yes—
41C2×#131×C73C2211yescharacteristic
42C2×#108×C73C2211yescharacteristic
43C2×#107×C73C2211yescharacteristic
44#417×C73C2212yes—
45#343×C73C2212yes—
46C2×#11924028C2311yescharacteristic
47#14556611C2×C414yes—
48#14556611Q814yes—
49C2×#11924018D411yescharacteristic
50C2×#11924016D411yescharacteristic
51C2×#11924015D411yescharacteristic
52C2×#11924013D411yescharacteristic
53#14556597—22no—
54#14556597C2×C414yescomplemented
55C2×#11924012C2311yescharacteristic
56C2×#11924011—21no—
57C22×#10596831C2311yescharacteristic
58C2×C8×D73—21no—
59Q8×Dic73—22no—
60#14556556—22no—
61D4×Dic73—22no—
62D4×Dic73—42no—
63#14556553—22no—
64C4×C22⋊D73—42no—
65#14556550—22no—
66#14556550C2312yes—
67#14556549—22no—
68C2×C4⋊Dic73—22no—
69C2×C4⋊Dic73C2312yes—
70C4⋊C4×D73—22no—
71C2×C4×Dic73C2311yescharacteristic
72#14556526C2×C414yescomplemented
73#14556525—22no—
74#14556525—21no—
75#14556525—21no—
76#14556525D411yescharacteristic
77#14556525D411yescharacteristic
78#14556524—22no—
79#14556523—22no—
80#14556523D412yes—
81#14556522—22no—
82#14556522—22no—
83#14556522—22no—
84#14556518C2×C414yescomplemented
85#14556517C2×C414yescomplemented
86#14556516—22no—
87#14556513—22no—
88#14556501—22no—
89C8×Dic73—22no—
90C8×Dic73C2×C414yescomplemented
91#14556498—42no—
92C2×#41×C73C2311yescharacteristic
93#131×C73—22no—
94#131×C73C2×C414yescomplemented
95C2×M16×C73—21no—
96C2×M16×C73C2312yes—
97C2×M16×C73C2311yescharacteristic
98C22×C8×C73—21no—
99C22×C8×C73C2311yescharacteristic
100C2×C4⋊C4×C73C2312yes—
101#108×C73—22no—
102#107×C73—21no—
103#107×C73—21no—
104#103×C73—22no—
105#102×C73—22no—
106#11924028—21no—
107#11924028—21no—
108#11924028C22×C412yes—
109#11924028C2×Q812yes—
110C2×Dic146C2×D411yescharacteristic
111C2×D292C2×D411yescharacteristic
112C2×C22⋊D73—21no—
113C22×Dic73—21no—
114C22×Dic73—22no—
115C22×Dic73—42no—
116C2×C4×D73—21no—
117C2×C4×D73—42no—
118#11924018C4⋊C414yes—
119#11924016C4⋊C414yes—
120#11924015C4⋊C414yes—
121#11924013C4⋊C414yes—
122#11924012—21no—
123#11924012C22×C412yes—
124#11924011—24no—
125C2×#10596831—21no—
126C2×#10596831C22×C412yes—
127C2×#10596831C2×D411yescharacteristic
128C2×#10596831C2×D411yescharacteristic
129C2×#10596831C2×Q812yes—
130C8×D73—24no—
131C4⋊Dic73—22no—
132C4⋊Dic73—22no—
133C4⋊Dic73—42no—
134C4⋊Dic73C2×D412yes—
135C4⋊Dic73C2×D412yes—
136#11924007—42no—
137C22⋊Dic73—22no—
138C22⋊Dic73—42no—
139#11924005—42no—
140C4×Dic73—22no—
141C4×Dic73—21no—
142C4×Dic73—42no—
143C4×Dic73—42no—
144C4×Dic73C22×C412yes—
145#41×C73—21no—
146#41×C73—21no—
147#41×C73C22×C412yes—
148#41×C73C2×Q812yes—
149C2×Q8×C73C2×D411yescharacteristic
150C2×D4×C73—21no—
151C2×D4×C73C2×D411yescharacteristic
152C22×C4×C73—21no—
153C22×C4×C73—22no—
154C22×C4×C73C2411yescharacteristic
155M16×C73—21no—
156M16×C73—24no—
157M16×C73—22no—
158M16×C73—22no—
159M16×C73C22×C412yes—
160C2×C8×C73—21no—
161C2×C8×C73—24no—
162C2×C8×C73—22no—
163C2×C8×C73C22×C412yes—
164C2×C8×C73C2×D411yescharacteristic
165C2×C8×C73C2×D411yescharacteristic
166C2×C8×C73#4112yes—
167C4⋊C4×C73—22no—
168C4⋊C4×C73—22no—
169C23×C73—21no—
170C23×C73H(5,𝔽2)11yescharacteristic
171C22×D73—21no—
172Q8×C73—21no—
173Q8×C73C2×C4⋊C412yes—
174D4×C73—22no—
175D4×C73—22no—
176D4×C73—21no—
177D4×C73C2×C4⋊C412yes—
178C2×C4×C73—21no—
179C2×C4×C73—22no—
180C2×C4×C73—22no—
181C2×C4×C73—42no—
182C2×C4×C73—82no—
183C2×C4×C73C23×C411yescharacteristic
184C2×C4×C73C22×D411yescharacteristic
185C2×C4×C73C22×D411yescharacteristic
186C2×C4×C73C22×Q811yescharacteristic
187C2×C4×C73C2×#4111yescharacteristic
188C2×C4×C732−1+411yescharacteristic
189C22⋊D73—22no—
190C22⋊D73—22no—
191C2×Dic73—21no—
192C2×Dic73—22no—
193C2×Dic73—22no—
194C2×Dic73—22no—
195C2×Dic73—41no—
196C2×Dic73—41no—
197C2×Dic73—42no—
198C2×Dic73—82no—
199C2×Dic73—162no—
200D292—21no—
201D292C2×C4⋊C412yes—
202C4×D73—22no—
203C4×D73—22no—
204C4×D73—162no—
205Dic146—21no—
206Dic146C2×C4⋊C412yes—
207C8×C73—22no—
208C8×C73—42no—
209C8×C73C4×D412yes—
210C8×C73C4×D412yes—
211#10596831—22no—
212#10596831C2×C4⋊C412yes—
213#10596831C2×C4⋊C412yes—
214C22×C73—21no—
215C22×C73—21no—
216C22×C73—22no—
217C22×C73—41no—
218C22×C73—41no—
219C22×C73#52011yescharacteristic
220C22×C73D4211yescharacteristic
221C22×C73#55411yescharacteristic
222C2×D73—22no—
223C2×D73—41no—
224C2×D73—41no—
225C4×C73—22no—
226C4×C73—82no—
227C4×C73C22×C4⋊C411yescharacteristic
228C4×C73C2×C4×D411yescharacteristic
229C4×C73#54711yescharacteristic
230C4×C73D4×Q811yescharacteristic
231Dic73—22no—
232Dic73—42no—
233Dic73—42no—
234Dic73—42no—
235Dic73—82no—
236#22691—731nomaximal, Sylow
237C2×C73—21no—
238C2×C73—21no—
239C2×C73—42no—
240C2×C73D4×C4⋊C411yescharacteristic
241C2×C73#329011yescharacteristic
242C2×C73#329011yescharacteristic
243D73—42no—
244C2×#443—731no—
245#2952—732no—
246#2913—731no—
247#2913—731no—
248#1951—732no—
249#1860—732no—
250#1860—732no—
251#1846—731no—
252#1816—731no—
253#1816D7311yescharacteristic, complemented
254#1772—731no—
255C73#2269111yescharacteristic, complemented, Sylow
256C2×#135—731no—
257C2×#131—731no—
258C2×#131C2×D7311yescharacteristic
259#520—732no—
260#520—732no—
261#443—738no—
262#442—1462no—
263#441—1462no—
264C4×SD16—1462no—
265C4×SD16—1462no—
266C2×#108—731no—
267C2×#108C2×D7311yescharacteristic
268#428—732no—
269#428—732no—
270#428—1462no—
271#428—1462no—
272#427—731no—
273#427—1461no—
274#427—1461no—
275C2×#107—731no—
276C2×#107—1461no—
277C2×#107—1461no—
278C2×#107C2×D7311yescharacteristic
279C2×#106—732no—
280C2×#104—731no—
281C2×#104—731no—
282#417—732no—
283#417C2×D7312yes—
284#405—732no—
285C2×C4×C8—731no—
286#382—732no—
287#343—732no—
288#343C2×D7312yes—
289#339—732no—
290C2×#41—731no—
291C2×#41C22×D7311yescharacteristic
292#135—734no—
293#135—734no—
294C2×Q16—731no—
295C2×SD16—731no—
296C2×SD16—731no—
297C2×D8—731no—
298#131—22no—
299#131—734no—
300#131—1462no—
301#131C4×D7314yescomplemented
302C2×M16—21no—
303C2×M16—731no—
304C2×M16—1461no—
305C2×M16C22×D7312yes—
306C2×M16C22×D7311yescharacteristic
307C22×C8—21no—
308C22×C8—731no—
309C22×C8—1461no—
310C22×C8C22×D7311yescharacteristic
311C4×Q8—1462no—
312C4×Q8—1462no—
313C4×D4—2922no—
314C4×D4—2922no—
315C4×D4—1462no—
316C4×D4—1462no—
317#117—732no—
318#117—1462no—
319#117—1462no—
320C2×C4⋊C4—732no—
321C2×C4⋊C4—1462no—
322C2×C4⋊C4—1462no—
323C2×C4⋊C4C22×D7312yes—
324C2×C42—731no—
325#108—22no—
326#108—734no—
327#107—21no—
328#107—21no—
329#107—731no—
330#107—731no—
331#107—1462no—
332#107—1462no—
333#107—1461no—
334#107—1461no—
335#106—732no—
336#106—1462no—
337#106—1462no—
338#106—1462no—
339#106—1462no—
340#104—734no—
341#104—734no—
342#103—22no—
343#103—1462no—
344#102—22no—
345#102—1462no—
346#97—1462no—
347C4×C8—734no—
348C4×C8—1462no—
349#95—2922no—
350#41—21no—
351#41—21no—
352#41—732no—
353#41—732no—
354#41—1461no—
355#41—1461no—
356#41C2×C4×D7312yes—
357#41Q8×D7312yes—
358C2×Q8—731no—
359C2×Q8D4×D7311yescharacteristic
360C2×D4—21no—
361C2×D4—731no—
362C2×D4—1461no—
363C2×D4D4×D7311yescharacteristic
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—2922no—
367C22×C4—2922no—
368C22×C4—1461no—
369C22×C4—1462no—
370C22×C4—1461no—
371C22×C4C23×D7311yescharacteristic
372Q16—734no—
373SD16—734no—
374SD16—734no—
375D8—734no—
376M16—21no—
377M16—24no—
378M16—22no—
379M16—22no—
380M16—732no—
381M16—1461no—
382M16—1464no—
383M16C2×C4×D7312yes—
384C2×C8—21no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—732no—
388C2×C8—731no—
389C2×C8—731no—
390C2×C8—732no—
391C2×C8—1461no—
392C2×C8—1464no—
393C2×C8C2×C4×D7312yes—
394C2×C8D4×D7311yescharacteristic
395C2×C8D4×D7311yescharacteristic
396C2×C8#1192402712yes—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—2922no—
400C4⋊C4—2922no—
401C4⋊C4—732no—
402C4⋊C4—732no—
403C4⋊C4—1462no—
404C4⋊C4—1462no—
405C22⋊C4—2922no—
406C22⋊C4—2922no—
407C22⋊C4—1462no—
408C42—2922no—
409C42—2922no—
410C42—732no—
411C42—1462no—
412C42—1461no—
413C23—21no—
414C23—1461no—
415C23#1455663711yescharacteristic
416Q8—21no—
417Q8—732no—
418Q8—1461no—
419Q8C4⋊C4×D7312yes—
420D4—22no—
421D4—22no—
422D4—21no—
423D4—732no—
424D4—1462no—
425D4—1462no—
426D4—1461no—
427D4C4⋊C4×D7312yes—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—2921no—
434C2×C4—2921no—
435C2×C4—2922no—
436C2×C4—5842no—
437C2×C4—11682no—
438C2×C4—11682no—
439C2×C4—1462no—
440C2×C4—1462no—
441C2×C4—1462no—
442C2×C4—1461no—
443C2×C4—1462no—
444C2×C4—1462no—
445C2×C4C22×C4×D7311yescharacteristic
446C2×C4C2×D4×D7311yescharacteristic
447C2×C4C2×D4×D7311yescharacteristic
448C2×C4C2×Q8×D7311yescharacteristic
449C2×C4C2×#1192402711yescharacteristic
450C2×C4#1455663811yescharacteristic
451C8—22no—
452C8—42no—
453C8—732no—
454C8—732no—
455C8—1462no—
456C8#1455655112yes—
457C8#1455655112yes—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—2921no—
464C22—2921no—
465C22—1462no—
466C22#1798906711yescharacteristic
467C22#1798922111yescharacteristic
468C22#1798929511yescharacteristic
469C4—22no—
470C4—82no—
471C4—2922no—
472C4—2922no—
473C4—2922no—
474C4—5842no—
475C4—1462no—
476C4C2×C4⋊C4×D7311yescharacteristic
477C4C2×#1455655111yescharacteristic
478C4#1798923711yescharacteristic
479C4#1798925011yescharacteristic
480C2—21no—
481C2—21no—
482C2—42no—
483C2—2922no—
484C2#2261282711yescharacteristic
485C2#2262171211yescharacteristic
486C2#2262172011yescharacteristic
subgroupG/H
centerC22#17989221
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1816C2
socleC22×C73D42
prime radicalC2×#14556611C22
cosocleC22×C4×C73C24
hypercenter#1816D73
hypocenterC73#22691
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1816

Complements

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

NH ≅ G/Ntype
3. #22618531C2semidirect
4. #22618527C2semidirect
5. #22610710C2semidirect
6. #22610167C2semidirect
7. #22610162C2semidirect
8. #22610094C2semidirect
9. #22609992C2semidirect
10. #22609813C2semidirect
11. C73×#1816C2semidirect
16. #17988702C4semidirect
54. #14556597C2×C4semidirect
72. #14556526C2×C4semidirect
84. #14556518C2×C4semidirect
85. #14556517C2×C4semidirect
90. C8×Dic73C2×C4semidirect
94. #131×C73C2×C4semidirect
253. #1816D73semidirect
255. C73#22691semidirect
301. #131C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887022
#226188102
#226185312
#226185272
#226107102
#226101672
#226101622
#226100942
#226099922
#226098132
C73×#18162
#2269173