Finite Group Databasebeta

← #33240461

Subgroups — #33240461

Order 18688.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989025C211yesmaximal, characteristic, complemented
2#22619512C211yesmaximal, characteristic, complemented
3C2×#17988823C211yesmaximal, characteristic, complemented
4#22612044C212yesmaximal, complemented
5#22612034C212yesmaximal, complemented
6#22612027C212yesmaximal, complemented
7#22612017C212yesmaximal, complemented
8#22611719C211yesmaximal, characteristic, complemented
9#22611648C211yesmaximal, characteristic, complemented
10#22611644C211yesmaximal, characteristic, complemented
11C22×M32×C73C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13C22×D584C2211yescharacteristic
14#17989098—21no—
15C2×#14556565—21no—
16#17989025—24no—
17C2×#14556536C2212yescomplemented
18C2×D1168C2212yescomplemented
19#17988828C2211yescharacteristic, complemented
20#17988828C2211yescharacteristic, complemented
21#17988823—21no—
22#17988823—21no—
23#17988823C2212yes—
24#17988823C2212yes—
25#17988823C2211yescharacteristic, complemented
26#17988823C2211yescharacteristic, complemented
27#17988816—21no—
28#17988816—21no—
29#17988625C2211yescharacteristic
30C2×#14556525C2211yescharacteristic
31#17988584—21no—
32C2×#14556513—21no—
33#17988415C2212yescomplemented
34#17988413C2212yescomplemented
35#17988392C2212yescomplemented
36#17988389C2212yescomplemented
37#17988389C2212yescomplemented
38#17988389C2212yescomplemented
39C23×C8×C73C2211yescharacteristic
40C2×M32×C73—24no—
41C2×M32×C73C2211yescharacteristic
42C2×M32×C73C2211yescharacteristic
43C2×M32×C73C2212yescomplemented
44C22×C16×C73C2212yescomplemented
45C2×#11924028—21no—
46C22×D292—21no—
47C22×C22⋊D73—41no—
48#14556613—22no—
49#14556613—22no—
50C2×Dic292C2311yescharacteristic
51C2×#11924017—21no—
52C2×D584—22no—
53C2×D584—21no—
54C2×D584—22no—
55C2×D584C2311yescharacteristic
56C2×D584C2312yes—
57#14556574—41no—
58#14556570—21no—
59#14556570—21no—
60#14556565—21no—
61#14556565—21no—
62#14556565—22no—
63#14556565—22no—
64#14556565—41no—
65#14556565—41no—
66#14556560—41no—
67C4×C22⋊D73—41no—
68C4×C22⋊D73—41no—
69#14556550—21no—
70C2×C4⋊Dic73—21no—
71C2×#11924005—41no—
72#14556536—24no—
73D1168—24no—
74#14556525—21no—
75#14556525C2312yes—
76#14556525C2311yescharacteristic
77#14556525C2311yescharacteristic
78#14556523—21no—
79#14556516—21no—
80#14556516—21no—
81#14556513—21no—
82#14556513—21no—
83#14556513—22no—
84#14556513—22no—
85C23×C4×C73D411yescharacteristic
86C22×C8×C73—22no—
87C22×C8×C73—22no—
88C22×C8×C73D411yescharacteristic
89C22×C8×C73D411yescharacteristic
90C22×C8×C73D411yescharacteristic
91C22×C8×C73C2311yescharacteristic
92C22×C8×C73C2311yescharacteristic
93C22×C8×C73C2311yescharacteristic
94M32×C73—22no—
95M32×C73—24no—
96M32×C73D414yescomplemented
97C2×C16×C73—24no—
98C2×C16×C73C2312yes—
99C2×C16×C73C2312yes—
100C23×D73—41no—
101#11924028—42no—
102#11924028—42no—
103C2×Dic146—21no—
104C2×D292—22no—
105C2×D292—21no—
106C2×D292—42no—
107C2×D292—42no—
108C2×D292—41no—
109C2×C22⋊D73—41no—
110C2×C22⋊D73—41no—
111C2×C22⋊D73—41no—
112C2×C22⋊D73—42no—
113C2×C22⋊D73—42no—
114C2×C22⋊D73—84no—
115C22×Dic73—41no—
116C2×C4×D73—41no—
117Dic292—21no—
118Dic292—21no—
119#11924017—42no—
120D584—21no—
121D584—22no—
122D584—22no—
123D584—21no—
124D584—42no—
125C4⋊Dic73—22no—
126C4⋊Dic73—21no—
127C4⋊Dic73—21no—
128C4⋊Dic73—41no—
129#11924007—41no—
130#11924007—41no—
131C22⋊Dic73—41no—
132C22⋊Dic73—41no—
133C22⋊Dic73—41no—
134#11924005—42no—
135#11924005—42no—
136#11924005—41no—
137#11924005—41no—
138C4×Dic73—41no—
139C24×C73D811yescharacteristic
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73D811yescharacteristic
143C22×C4×C73D811yescharacteristic
144C22×C4×C73D811yescharacteristic
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×D411yescharacteristic
147C22×C4×C73C2×D411yescharacteristic
148C2×C8×C73—22no—
149C2×C8×C73—22no—
150C2×C8×C73—22no—
151C2×C8×C73—21no—
152C2×C8×C73—21no—
153C2×C8×C73—21no—
154C2×C8×C73—21no—
155C2×C8×C73C2×D412yes—
156C2×C8×C73C2×D411yescharacteristic
157C2×C8×C73C2×D411yescharacteristic
158C2×C8×C73C2×D411yescharacteristic
159C2×C8×C73#4112yes—
160C2×C8×C73C2411yescharacteristic
161C16×C73—22no—
162C16×C73C2×D414yescomplemented
163C23×C73—22no—
164C23×C73—22no—
165C23×C73—41no—
166C23×C73C2×D811yescharacteristic
167C23×C73C2×D811yescharacteristic
168C23×C73C2×D811yescharacteristic
169C22×D73—41no—
170C22×D73—42no—
171C22×D73—82no—
172C22×D73—82no—
173C22×D73—82no—
174C2×C4×C73—22no—
175C2×C4×C73—21no—
176C2×C4×C73—21no—
177C2×C4×C73—21no—
178C2×C4×C73—21no—
179C2×C4×C73—22no—
180C2×C4×C73—22no—
181C2×C4×C73#12112yes—
182C2×C4×C73C2×D811yescharacteristic
183C2×C4×C73C2×D811yescharacteristic
184C2×C4×C73C2×D811yescharacteristic
185C2×C4×C73#13512yes—
186C2×C4×C73C22×D411yescharacteristic
187C22⋊D73—82no—
188C22⋊D73—84no—
189C22⋊D73—82no—
190C22⋊D73—82no—
191C22⋊D73—84no—
192C2×Dic73—41no—
193C2×Dic73—41no—
194C2×Dic73—42no—
195C2×Dic73—41no—
196C2×Dic73—82no—
197D292—42no—
198D292—41no—
199D292—42no—
200D292—41no—
201D292—82no—
202C4×D73—82no—
203Dic146—41no—
204Dic146—41no—
205C8×C73—21no—
206C8×C73—21no—
207C8×C73#12112yes—
208C8×C73C22×D411yescharacteristic
209C8×C73C2×#4111yescharacteristic
210C22×C73—21no—
211C22×C73—21no—
212C22×C73—21no—
213C22×C73—21no—
214C22×C73—21no—
215C22×C73—21no—
216C22×C73—21no—
217C22×C73—44no—
218C22×C73#46612yes—
219C22×C73#50912yescomplemented
220C22×C73C22×D811yescharacteristic
221C2×D73—82no—
222C2×D73—81no—
223C2×D73—81no—
224C2×D73—82no—
225C2×D73—81no—
226C2×D73—82no—
227C2×D73—164no—
228C4×C73—21no—
229C4×C73—21no—
230C4×C73#46612yes—
231C4×C73C2×#12111yescharacteristic
232C4×C73C2×#13511yescharacteristic
233Dic73—82no—
234Dic73—81no—
235Dic73—81no—
236Dic73—81no—
237#33605—731nomaximal, Sylow
238C2×C73—22no—
239C2×C73—22no—
240C2×C73—41no—
241C2×C73C2×#46611yescharacteristic
242C2×C73C2×#50911yescharacteristic
243C2×C73#341411yescharacteristic
244D73—161no—
245D73—162no—
246C2×#509—731no—
247C22×M32D7311yescharacteristic, complemented
248#3050—731no—
249C2×#466—731no—
250#2220—732no—
251#2218—732no—
252#2217—732no—
253#2215—732no—
254#2159—731no—
255#2140—731no—
256#2139—731no—
257C73#3360511yescharacteristic, complemented, Sylow
258C2×#135—731no—
259C22×D8—731no—
260C23×C8C2×D7311yescharacteristic
261#525—1461no—
262C2×#121—1461no—
263#509—1464no—
264C2×SD32—732no—
265C2×D16—732no—
266C2×M32—24no—
267C2×M32C2×D7311yescharacteristic
268C2×M32C2×D7311yescharacteristic
269C2×M32C2×D7312yescomplemented
270C22×C16C2×D7312yescomplemented
271#467—731no—
272#467—731no—
273#466—731no—
274#466—732no—
275#466—731no—
276#466—732no—
277#466—1461no—
278#466—1461no—
279#465—1461no—
280#465—1461no—
281#427—731no—
282C2×#107—731no—
283#416—1461no—
284C2×#102—1461no—
285#367—732no—
286#366—732no—
287#358—732no—
288#357—732no—
289#357—732no—
290#357—732no—
291C2×#41—1461no—
292C22×D4—2921no—
293C22×D4—1461no—
294C23×C4D29211yescharacteristic
295#135—1462no—
296#135—1462no—
297C2×Q16—731no—
298C2×SD16—1461no—
299C2×D8—732no—
300C2×D8—731no—
301C2×D8—1462no—
302C2×D8—1461no—
303C2×D8—1462no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8D29211yescharacteristic
307C22×C8D29211yescharacteristic
308C22×C8D29211yescharacteristic
309C22×C8C22×D7311yescharacteristic
310C22×C8C22×D7311yescharacteristic
311C22×C8C22×D7311yescharacteristic
312#123—2921no—
313C22⋊Q8—1461no—
314C22⋊Q8—1461no—
315#121—2921no—
316#121—2921no—
317#121—1462no—
318#121—1462no—
319#121—1461no—
320#121—1461no—
321#120—2921no—
322C4×D4—2921no—
323C4×D4—2921no—
324#117—1461no—
325C2×C4⋊C4—1461no—
326C2×C22⋊C4—2921no—
327SD32—1464no—
328D16—1464no—
329M32—22no—
330M32—24no—
331M32C22⋊D7314yescomplemented
332C2×C16—24no—
333C2×C16C22×D7312yes—
334C2×C16C22×D7312yes—
335#107—731no—
336#107—731no—
337#107—732no—
338#107—1461no—
339#106—1461no—
340#103—1461no—
341#103—1461no—
342#102—1462no—
343#102—1462no—
344#102—1461no—
345#102—1461no—
346C24—2921no—
347C24D58411yescharacteristic
348#41—2922no—
349#41—2922no—
350C2×Q8—1461no—
351C2×D4—2921no—
352C2×D4—2921no—
353C2×D4—2921no—
354C2×D4—2922no—
355C2×D4—2922no—
356C2×D4—2922no—
357C2×D4—2921no—
358C2×D4—2922no—
359C2×D4—5844no—
360C2×D4—1462no—
361C2×D4—1461no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—2921no—
365C22×C4—2921no—
366C22×C4D58411yescharacteristic
367C22×C4D58411yescharacteristic
368C22×C4D58411yescharacteristic
369C22×C4C2×D29211yescharacteristic
370C22×C4C2×D29211yescharacteristic
371C22×C4C2×D29211yescharacteristic
372Q16—1461no—
373Q16—1461no—
374SD16—2922no—
375D8—2922no—
376D8—1461no—
377D8—1462no—
378D8—1462no—
379D8—1461no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8C2×C22⋊D7312yes—
388C2×C8C2×D29211yescharacteristic
389C2×C8C2×D29211yescharacteristic
390C2×C8C2×D29211yescharacteristic
391C2×C8#1192402812yes—
392C2×C8C23×D7311yescharacteristic
393C4⋊C4—2921no—
394C4⋊C4—2921no—
395C4⋊C4—2921no—
396C4⋊C4—1462no—
397C4⋊C4—1461no—
398C4⋊C4—1461no—
399C22⋊C4—2921no—
400C22⋊C4—2921no—
401C22⋊C4—2921no—
402C22⋊C4—2922no—
403C22⋊C4—2922no—
404C22⋊C4—2921no—
405C22⋊C4—2921no—
406C42—2921no—
407C16—22no—
408C16C2×C22⋊D7314yescomplemented
409C23—22no—
410C23—22no—
411C23—41no—
412C23—2921no—
413C23—2922no—
414C23—5842no—
415C23—5842no—
416C23—5842no—
417C23C2×D58411yescharacteristic
418C23C2×D58411yescharacteristic
419C23C2×D58411yescharacteristic
420Q8—2921no—
421Q8—2921no—
422D4—2922no—
423D4—2921no—
424D4—2922no—
425D4—2921no—
426D4—5844no—
427D4—5842no—
428D4—5844no—
429D4—5842no—
430D4—5842no—
431D4—5842no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—2921no—
440C2×C4—2921no—
441C2×C4—2921no—
442C2×C4—2922no—
443C2×C4—5842no—
444C2×C4—5842no—
445C2×C4#1455656512yes—
446C2×C4C2×D58411yescharacteristic
447C2×C4C2×D58411yescharacteristic
448C2×C4C2×D58411yescharacteristic
449C2×C4#1455661312yes—
450C2×C4C22×D29211yescharacteristic
451C8—21no—
452C8—21no—
453C8#1455656512yes—
454C8C22×C22⋊D7311yescharacteristic
455C8C2×#1192402811yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—5842no—
465C22—5841no—
466C22—5841no—
467C22—5842no—
468C22—5841no—
469C22—5842no—
470C22—11684no—
471C22#1798882312yes—
472C22#1798902512yescomplemented
473C22C22×D58411yescharacteristic
474C4—21no—
475C4—21no—
476C4—5842no—
477C4—5841no—
478C4—5841no—
479C4—5841no—
480C4#1798882312yes—
481C4C2×#1455656511yescharacteristic
482C4C2×#1455661311yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—11682no—
487C2—11681no—
488C2C2×#1798882311yescharacteristic
489C2C2×#1798902511yescharacteristic
490C2#2262306711yescharacteristic
subgroupG/H
centerC22C22×D584
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC22×M32×C73C2
socleC22×C73C22×D8
hypercenterC22×M32D73
hypocenterC73#33605
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×M32

Complements

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

NH ≅ G/Ntype
1. C2×#17989025C2semidirect
2. #22619512C2semidirect
3. C2×#17988823C2semidirect
4. #22612044C2semidirect
5. #22612034C2semidirect
6. #22612027C2semidirect
7. #22612017C2semidirect
8. #22611719C2semidirect
9. #22611648C2semidirect
10. #22611644C2semidirect
11. C22×M32×C73C2semidirect
17. C2×#14556536C22semidirect
18. C2×D1168C22semidirect
19. #17988828C22semidirect
20. #17988828C22semidirect
25. #17988823C22semidirect
26. #17988823C22semidirect
33. #17988415C22semidirect
34. #17988413C22semidirect
35. #17988392C22semidirect
36. #17988389C22semidirect
37. #17988389C22semidirect
38. #17988389C22semidirect
43. C2×M32×C73C22semidirect
44. C22×C16×C73C22semidirect
96. M32×C73D4semidirect
162. C16×C73C2×D4semidirect
219. C22×C73#509semidirect
247. C22×M32D73semidirect
257. C73#33605semidirect
269. C2×M32C2×D73semidirect
270. C22×C16C2×D73semidirect
331. M32C22⋊D73semidirect
408. C16C2×C22⋊D73semidirect
472. C22#17989025semidirect

Maximal subgroups

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

subgroupindex
C2×#179890252
#226195122
C2×#179888232
#226120442
#226120342
#226120272
#226120172
#226117192
#226116482
#226116442
C22×M32×C732
#3360573