Finite Group Databasebeta

← #33240468

Subgroups — #33240468

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989028C211yesmaximal, characteristic, complemented
2C22×#14556531C211yesmaximal, characteristic, complemented
3#22619512C211yesmaximal, characteristic
4#22612046C212yesmaximal, complemented
5#22612037C212yesmaximal, complemented
6#22612029C212yesmaximal, complemented
7#22612020C212yesmaximal, complemented
8#22611717C211yesmaximal, characteristic, complemented
9#22611651C211yesmaximal, characteristic, complemented
10#22611641C211yesmaximal, characteristic, complemented
11C2×#467×C73C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13#17989098—21no—
14#17989028—24no—
15C2×#14556537C2212yescomplemented
16C2×#14556534C2212yescomplemented
17C2×#14556531—24no—
18C2×#14556531C2212yes—
19C2×#14556531C2211yescharacteristic
20C2×#14556531C2211yescharacteristic
21C22×#11924003C2212yes—
22#17988828C2211yescharacteristic
23#17988828C2211yescharacteristic
24#17988823C2211yescharacteristic
25#17988823C2211yescharacteristic
26#17988816—21no—
27#17988816—21no—
28#17988625C2211yescharacteristic
29#17988584—21no—
30#17988414C2212yescomplemented
31#17988412C2212yescomplemented
32#17988391C2212yescomplemented
33#17988391C2212yescomplemented
34#17988390C2212yescomplemented
35#17988387C2212yescomplemented
36C22×Q16×C73C2211yescharacteristic
37C23×C8×C73C2211yescharacteristic
38C2×C22⋊Q8×C73—21no—
39#467×C73—21no—
40#467×C73—21no—
41#467×C73C2212yes—
42#467×C73C2212yes—
43C2×#107×C73C2211yescharacteristic
44C2×#103×C73—21no—
45C2×#11924028—21no—
46#14556613—22no—
47#14556613—22no—
48C2×Dic292C2311yescharacteristic
49C2×#11924017—21no—
50C2×D584C2311yescharacteristic
51#14556574—41no—
52#14556570—21no—
53#14556570—21no—
54#14556565—21no—
55#14556565—21no—
56#14556560—41no—
57C4×C22⋊D73—41no—
58C4×C22⋊D73—41no—
59#14556550—21no—
60#14556537—24no—
61#14556534—24no—
62#14556531—24no—
63#14556531—22no—
64#14556531D414yescomplemented
65C2×#11924003—24no—
66C2×#11924003C2312yes—
67C2×#11924003C2312yes—
68#14556525C2311yescharacteristic
69#14556525C2311yescharacteristic
70#14556523—21no—
71#14556516—21no—
72#14556516—21no—
73#14556513—21no—
74#14556513—21no—
75C22×Q8×C73—21no—
76C23×C4×C73D411yescharacteristic
77C2×Q16×C73—22no—
78C2×Q16×C73—21no—
79C2×Q16×C73—22no—
80C2×Q16×C73C2312yes—
81C22×C8×C73—22no—
82C22×C8×C73—22no—
83C22×C8×C73D411yescharacteristic
84C22×C8×C73D411yescharacteristic
85C22×C8×C73D411yescharacteristic
86C22×C8×C73C2311yescharacteristic
87C22×C8×C73C2311yescharacteristic
88C22×C8×C73C2311yescharacteristic
89C22⋊Q8×C73—22no—
90C22⋊Q8×C73—22no—
91C22⋊Q8×C73—41no—
92C22⋊Q8×C73—41no—
93C2×C4⋊C4×C73—21no—
94C2×C4⋊C4×C73—41no—
95C2×C22⋊C4×C73—41no—
96#107×C73—21no—
97#107×C73C2312yes—
98#103×C73—22no—
99#103×C73—22no—
100#11924028—42no—
101#11924028—42no—
102C2×Dic146—21no—
103C2×D292—21no—
104C2×C22⋊D73—41no—
105C2×C22⋊D73—41no—
106C2×C22⋊D73—41no—
107C2×C4×D73—41no—
108Dic292—21no—
109Dic292—21no—
110#11924017—42no—
111D584—21no—
112D584—21no—
113C4⋊Dic73—21no—
114C4⋊Dic73—21no—
115#11924007—41no—
116#11924007—41no—
117C22⋊Dic73—41no—
118C22⋊Dic73—41no—
119C22⋊Dic73—41no—
120#11924005—41no—
121#11924005—41no—
122C4×Dic73—41no—
123#11924003—22no—
124#11924003C2×D414yes—
125C24×C73D811yescharacteristic
126C2×Q8×C73—22no—
127C2×Q8×C73—41no—
128C2×Q8×C73—42no—
129C2×Q8×C73—42no—
130C22×C4×C73—22no—
131C22×C4×C73—22no—
132C22×C4×C73—41no—
133C22×C4×C73—41no—
134C22×C4×C73D811yescharacteristic
135C22×C4×C73D811yescharacteristic
136C22×C4×C73D811yescharacteristic
137C22×C4×C73C2×D411yescharacteristic
138C22×C4×C73C2×D411yescharacteristic
139C22×C4×C73C2×D411yescharacteristic
140Q16×C73—22no—
141Q16×C73—22no—
142Q16×C73—42no—
143C2×C8×C73—22no—
144C2×C8×C73—22no—
145C2×C8×C73—22no—
146C2×C8×C73—21no—
147C2×C8×C73—21no—
148C2×C8×C73—21no—
149C2×C8×C73—21no—
150C2×C8×C73C2×D412yes—
151C2×C8×C73C2×D411yescharacteristic
152C2×C8×C73C2×D411yescharacteristic
153C2×C8×C73C2×D411yescharacteristic
154C2×C8×C73#4112yes—
155C2×C8×C73C2411yescharacteristic
156C4⋊C4×C73—22no—
157C4⋊C4×C73—41no—
158C4⋊C4×C73—42no—
159C4⋊C4×C73—42no—
160C22⋊C4×C73—42no—
161C22⋊C4×C73—42no—
162C23×C73—22no—
163C23×C73—22no—
164C23×C73—41no—
165C23×C73C2×D811yescharacteristic
166C23×C73C2×D811yescharacteristic
167C23×C73C2×D811yescharacteristic
168C22×D73—41no—
169Q8×C73—42no—
170Q8×C73—42no—
171Q8×C73—82no—
172C2×C4×C73—22no—
173C2×C4×C73—21no—
174C2×C4×C73—21no—
175C2×C4×C73—21no—
176C2×C4×C73—21no—
177C2×C4×C73—22no—
178C2×C4×C73—22no—
179C2×C4×C73—42no—
180C2×C4×C73—42no—
181C2×C4×C73—82no—
182C2×C4×C73—82no—
183C2×C4×C73#12112yes—
184C2×C4×C73C2×D811yescharacteristic
185C2×C4×C73C2×D811yescharacteristic
186C2×C4×C73C2×D811yescharacteristic
187C2×C4×C73#13512yes—
188C2×C4×C73C22×D411yescharacteristic
189C22⋊D73—82no—
190C22⋊D73—82no—
191C22⋊D73—82no—
192C2×Dic73—41no—
193C2×Dic73—41no—
194C2×Dic73—41no—
195D292—41no—
196D292—41no—
197C4×D73—82no—
198Dic146—41no—
199Dic146—41no—
200C8×C73—21no—
201C8×C73—21no—
202C8×C73#12112yes—
203C8×C73C22×D411yescharacteristic
204C8×C73C2×#4111yescharacteristic
205C22×C73—21no—
206C22×C73—21no—
207C22×C73—21no—
208C22×C73—21no—
209C22×C73—21no—
210C22×C73—21no—
211C22×C73—21no—
212C22×C73—44no—
213C22×C73#46612yes—
214C22×C73#51012yescomplemented
215C22×C73C22×D811yescharacteristic
216C2×D73—81no—
217C2×D73—81no—
218C2×D73—81no—
219C4×C73—21no—
220C4×C73—21no—
221C4×C73—82no—
222C4×C73—82no—
223C4×C73#46612yes—
224C4×C73C2×#12111yescharacteristic
225C4×C73C2×#13511yescharacteristic
226Dic73—81no—
227Dic73—81no—
228Dic73—81no—
229#33606—731nomaximal, Sylow
230C2×C73—22no—
231C2×C73—22no—
232C2×C73—41no—
233C2×C73C2×#46611yescharacteristic
234C2×C73C2×#51011yescharacteristic
235C2×C73#341411yescharacteristic
236D73—161no—
237C2×#510—731no—
238C22×M32—731no—
239#3050—731no—
240C2×#467D7311yescharacteristic, complemented
241#2220—732no—
242#2219—732no—
243#2217—732no—
244#2216—732no—
245#2159—731no—
246#2141—731no—
247#2139—731no—
248C73#3360611yescharacteristic, complemented, Sylow
249C2×#135—731no—
250C22×Q16C2×D7311yescharacteristic
251C23×C8C2×D7311yescharacteristic
252#525—1461no—
253C2×C22⋊Q8—21no—
254#510—1464no—
255C2×Q32—732no—
256C2×SD32—732no—
257C2×M32—731no—
258C2×M32—731no—
259C2×M32—732no—
260C2×M32—1464no—
261C22×C16—732no—
262#467—21no—
263#467—21no—
264#467—731no—
265#467—731no—
266#467C2×D7312yes—
267#467C2×D7312yes—
268#466—731no—
269#466—731no—
270#465—1461no—
271#465—1461no—
272#427—731no—
273C2×#107C2×D7311yescharacteristic
274#416—1461no—
275C2×#103—21no—
276#367—732no—
277#366—732no—
278#358—732no—
279#358—732no—
280#358—732no—
281#357—732no—
282C2×#41—1461no—
283C22×Q8—21no—
284C23×C4C22⋊D7311yescharacteristic
285#135—1462no—
286#135—1462no—
287C2×Q16—22no—
288C2×Q16—21no—
289C2×Q16—22no—
290C2×Q16—731no—
291C2×Q16C22×D7312yes—
292C2×SD16—1461no—
293C2×D8—731no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8C22⋊D7311yescharacteristic
297C22×C8C22⋊D7311yescharacteristic
298C22×C8C22⋊D7311yescharacteristic
299C22×C8C22×D7311yescharacteristic
300C22×C8C22×D7311yescharacteristic
301C22×C8C22×D7311yescharacteristic
302#123—2921no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—41no—
306C22⋊Q8—41no—
307C22⋊Q8—1461no—
308C22⋊Q8—1461no—
309#121—1461no—
310#121—1461no—
311#120—2921no—
312C4×D4—2921no—
313C4×D4—2921no—
314#117—1461no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—41no—
317C2×C22⋊C4—41no—
318Q32—1464no—
319SD32—1464no—
320M32—734no—
321M32—1464no—
322M32—1462no—
323C2×C16—732no—
324C2×C16—732no—
325C2×C16—1464no—
326#107—21no—
327#107—731no—
328#107—731no—
329#107C22×D7312yes—
330#106—1461no—
331#103—22no—
332#103—22no—
333#103—1461no—
334#103—1461no—
335#102—1461no—
336#102—1461no—
337C24#1192401311yescharacteristic
338#41—2922no—
339#41—2922no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—1461no—
345C2×D4—2921no—
346C2×D4—2921no—
347C2×D4—2921no—
348C2×D4—1461no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—2921no—
354C22×C4#1192401311yescharacteristic
355C22×C4#1192401311yescharacteristic
356C22×C4#1192401311yescharacteristic
357C22×C4C2×C22⋊D7311yescharacteristic
358C22×C4C2×C22⋊D7311yescharacteristic
359C22×C4C2×C22⋊D7311yescharacteristic
360Q16—22no—
361Q16—22no—
362Q16—42no—
363Q16—1461no—
364Q16—1461no—
365SD16—2922no—
366D8—1461no—
367D8—1461no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8D4×D7312yes—
376C2×C8C2×C22⋊D7311yescharacteristic
377C2×C8C2×C22⋊D7311yescharacteristic
378C2×C8C2×C22⋊D7311yescharacteristic
379C2×C8#1192402712yes—
380C2×C8C23×D7311yescharacteristic
381C4⋊C4—22no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—2921no—
386C4⋊C4—2921no—
387C4⋊C4—1461no—
388C4⋊C4—1461no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—2921no—
392C22⋊C4—2921no—
393C22⋊C4—2921no—
394C22⋊C4—2921no—
395C22⋊C4—2921no—
396C42—2921no—
397C16—734no—
398C16—1462no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—2921no—
403C23C2×#1192401311yescharacteristic
404C23C2×#1192401311yescharacteristic
405C23C2×#1192401311yescharacteristic
406Q8—42no—
407Q8—42no—
408Q8—82no—
409Q8—2921no—
410Q8—2921no—
411D4—2921no—
412D4—2921no—
413D4—5842no—
414D4—5842no—
415D4—5842no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—2921no—
428C2×C4—2921no—
429C2×C4—2921no—
430C2×C4—5842no—
431C2×C4#1455656312yes—
432C2×C4C2×#1192401311yescharacteristic
433C2×C4C2×#1192401311yescharacteristic
434C2×C4C2×#1192401311yescharacteristic
435C2×C4#1455661112yes—
436C2×C4C22×C22⋊D7311yescharacteristic
437C8—21no—
438C8—21no—
439C8#1455656312yes—
440C8C2×D4×D7311yescharacteristic
441C8C2×#1192402711yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—5841no—
451C22—5841no—
452C22—5841no—
453C22#1798882112yes—
454C22#1798902812yescomplemented
455C22C22×#1192401311yescharacteristic
456C4—21no—
457C4—21no—
458C4—82no—
459C4—82no—
460C4—5841no—
461C4—5841no—
462C4—5841no—
463C4#1798882112yes—
464C4C2×#1455656311yescharacteristic
465C4C2×#1455661111yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—11681no—
470C2C2×#1798882111yescharacteristic
471C2C2×#1798902811yescharacteristic
472C2#2262306511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC2×#467×C73C2
socleC22×C73C22×D8
prime radical#22619512C2
hypercenterC2×#467D73
hypocenterC73#33606
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×#17989028C2semidirect
2. C22×#14556531C2semidirect
4. #22612046C2semidirect
5. #22612037C2semidirect
6. #22612029C2semidirect
7. #22612020C2semidirect
8. #22611717C2semidirect
9. #22611651C2semidirect
10. #22611641C2semidirect
11. C2×#467×C73C2semidirect
15. C2×#14556537C22semidirect
16. C2×#14556534C22semidirect
30. #17988414C22semidirect
31. #17988412C22semidirect
32. #17988391C22semidirect
33. #17988391C22semidirect
34. #17988390C22semidirect
35. #17988387C22semidirect
64. #14556531D4semidirect
214. C22×C73#510semidirect
240. C2×#467D73semidirect
248. C73#33606semidirect
454. C22#17989028semidirect

Maximal subgroups

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

subgroupindex
C2×#179890282
C22×#145565312
#226195122
#226120462
#226120372
#226120292
#226120202
#226117172
#226116512
#226116412
C2×#467×C732
#3360673