Finite Group Databasebeta

← #33098287

Subgroups — #33098287

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619719C211yesmaximal, characteristic
2#22618861C211yesmaximal, characteristic, complemented
3#22618575C211yesmaximal, characteristic
4#22610874C211yesmaximal, characteristic, complemented
5#22610704C211yesmaximal, characteristic, complemented
6#22610678C212yesmaximal, complemented
7#22610632C212yesmaximal
8#22610582C212yesmaximal
9#22610533C212yesmaximal, complemented
10C4⋊C4×#10596831C211yesmaximal, characteristic, complemented
11C73×#1806C211yesmaximal, characteristic
12#17989144C2211yescharacteristic
13#17988894C412yescomplemented
14#17988887C412yescomplemented
15#17988882C411yescharacteristic, complemented
16#17988877C411yescharacteristic, complemented
17#17988872C411yescharacteristic, complemented
18#17988867C411yescharacteristic, complemented
19#17988714—21no—
20#17988714—21no—
21#17988711—21no—
22#17988711—21no—
23#17988648—21no—
24#17988648—21no—
25#17988640—21no—
26#17988640—21no—
27C2×#14556526C2212yes—
28#17988623C2212yes—
29#17988615C2212yes—
30#17988614C2211yescharacteristic
31#17988614C2211yescharacteristic
32C2×C4⋊#10596831C2211yescharacteristic
33#17988583C2212yes—
34#17988554C2211yescharacteristic
35#17988554C2211yescharacteristic
36#17988545C2212yes—
37C4×#11924012C2211yescharacteristic
38C2×C4×#10596831C2212yes—
39C2×C4×#10596831C2211yescharacteristic
40#17988351C2212yes—
41#17988347C2212yes—
42#17988347C2212yes—
43#17988342C2212yes—
44C4×#41×C73C2211yescharacteristic
45#423×C73C2211yescharacteristic
46C2×#104×C73C2212yes—
47C4×C4⋊C4×C73C2211yescharacteristic
48#339×C73C2212yes—
49C2×#11924012C2311yescharacteristic
50C2×#11924012C2311yescharacteristic
51C22×#10596831C2312yes—
52C22×#10596831C2311yescharacteristic
53C22×#10596831C2311yescharacteristic
54#14556591C2×C411yescharacteristic
55#14556591C2×C411yescharacteristic
56#14556583C2×C412yes—
57#14556570—21no—
58#14556570—21no—
59C4×Dic146—21no—
60C4×Dic146—21no—
61#14556550C2312yes—
62#14556526—42no—
63#14556524C2×C412yes—
64#14556524C2×C412yes—
65#14556522C2×C412yes—
66#14556522C2×C412yes—
67C4⋊#10596831—21no—
68C4⋊#10596831—21no—
69C4⋊#10596831—21no—
70C4⋊#10596831—21no—
71C4⋊#10596831C2×C411yescharacteristic
72C4⋊#10596831C2×C411yescharacteristic
73C4⋊#10596831C2×C412yes—
74#14556520—22no—
75#14556520—22no—
76#14556515C2×C412yes—
77#14556515C2×C412yes—
78#14556512C2×C412yes—
79#14556512C2×C412yes—
80C22⋊#10596831—21no—
81C22⋊#10596831—21no—
82C22⋊#10596831—21no—
83C22⋊#10596831—21no—
84#14556502—21no—
85#14556501—22no—
86C4×#10596831—21no—
87C4×#10596831—42no—
88C4×#10596831D411yescharacteristic
89C4×#10596831D411yescharacteristic
90C4×#10596831D411yescharacteristic
91C4×#10596831D411yescharacteristic
92C8×Dic73—22no—
93C2×#41×C73C2311yescharacteristic
94C2×M16×C73C2312yes—
95C4×Q8×C73C2×C411yescharacteristic
96C4×Q8×C73C2×C411yescharacteristic
97C4×D4×C73—21no—
98C4×D4×C73—21no—
99C4×D4×C73C2×C411yescharacteristic
100C4×D4×C73C2×C411yescharacteristic
101#117×C73—21no—
102#117×C73C2311yescharacteristic
103C2×C4⋊C4×C73—21no—
104C2×C42×C73—21no—
105C2×C42×C73C2312yes—
106C2×C42×C73C2311yescharacteristic
107C4⋊C8×C73—21no—
108C4⋊C8×C73—21no—
109#104×C73—42no—
110#95×C73—21no—
111C2×Dic146—21no—
112C2×Dic146—21no—
113#11924012—21no—
114#11924012—21no—
115#11924012—21no—
116#11924012—21no—
117C2×#10596831—21no—
118C2×#10596831—22no—
119C2×#10596831—21no—
120C2×#10596831—21no—
121C2×#10596831—22no—
122C2×#10596831—21no—
123C2×#10596831—21no—
124C2×#10596831—21no—
125C2×#10596831C22×C412yes—
126C2×#10596831C22×C412yes—
127C2×#10596831C2×D411yescharacteristic
128C2×#10596831C2×D411yescharacteristic
129C2×#10596831C2×D411yescharacteristic
130C2×#10596831C2×D411yescharacteristic
131C4⋊Dic73—21no—
132C4⋊Dic73—21no—
133C4⋊Dic73C22×C412yes—
134C4⋊Dic73C22×C412yes—
135#11924007—22no—
136#11924007—22no—
137C22⋊Dic73—22no—
138C4×Dic73—22no—
139#41×C73—21no—
140#41×C73—21no—
141#41×C73—21no—
142#41×C73—21no—
143C2×Q8×C73C22×C411yescharacteristic
144C2×D4×C73—21no—
145C2×D4×C73C22×C411yescharacteristic
146C22×C4×C73—22no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73#4111yescharacteristic
151C22×C4×C73#4111yescharacteristic
152C22×C4×C73C2411yescharacteristic
153M16×C73—22no—
154M16×C73—22no—
155C2×C8×C73—22no—
156C4⋊C4×C73—24no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73C22×C411yescharacteristic
160C4⋊C4×C73C22×C411yescharacteristic
161C22⋊C4×C73—21no—
162C22⋊C4×C73—21no—
163C22⋊C4×C73—21no—
164C42×C73—21no—
165C42×C73—21no—
166C42×C73—21no—
167C42×C73—22no—
168C42×C73—22no—
169C42×C73C22×C411yescharacteristic
170C42×C73C22×C411yescharacteristic
171C42×C73C2×D411yescharacteristic
172C42×C73C2×D411yescharacteristic
173C23×C73—21no—
174C23×C73C2×#4111yescharacteristic
175Q8×C73—21no—
176Q8×C73—21no—
177D4×C73—21no—
178D4×C73—21no—
179D4×C73—41no—
180D4×C73—41no—
181C2×C4×C73—22no—
182C2×C4×C73—24no—
183C2×C4×C73—21no—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—41no—
187C2×C4×C73—42no—
188C2×C4×C73—42no—
189C2×C4×C73—42no—
190C2×C4×C73—41no—
191C2×C4×C73—41no—
192C2×C4×C73—42no—
193C2×C4×C73C4×D411yescharacteristic
194C2×C4×C73C4×D411yescharacteristic
195C2×C4×C73C4×D411yescharacteristic
196C2×C4×C73C4×D411yescharacteristic
197C2×C4×C73C23×C411yescharacteristic
198C2×C4×C73C22×D411yescharacteristic
199C2×C4×C73H(5,𝔽2)11yescharacteristic
200C2×C4×C732−1+411yescharacteristic
201C2×C4×C732−1+411yescharacteristic
202C2×C4×C732−1+411yescharacteristic
203C2×Dic73—22no—
204C2×Dic73—22no—
205Dic146—42no—
206Dic146—42no—
207C8×C73—42no—
208#10596831—22no—
209#10596831—22no—
210#10596831—21no—
211#10596831—21no—
212#10596831—21no—
213#10596831—21no—
214#10596831—41no—
215#10596831—41no—
216C22×C73—21no—
217C22×C73—21no—
218C22×C73—41no—
219C22×C73—41no—
220C22×C73—41no—
221C22×C73C2×C4×D411yescharacteristic
222C22×C73#54111yescharacteristic
223C22×C73#54211yescharacteristic
224C4×C73—21no—
225C4×C73—21no—
226C4×C73—21no—
227C4×C73—44no—
228C4×C73—41no—
229C4×C73—41no—
230C4×C73—41no—
231C4×C73#44312yes—
232C4×C73#51911yescharacteristic
233C4×C73#52011yescharacteristic
234C4×C73#53611yescharacteristic
235C4×C73#53711yescharacteristic
236Dic73—42no—
237Dic73—42no—
238#22130—731nomaximal, Sylow
239C2×C73—21no—
240C2×C73—21no—
241C2×C73—81no—
242C2×C73#232311yescharacteristic
243C2×C73C2×#44311yescharacteristic
244C2×C73#295511yescharacteristic
245#3076—731no—
246#2960—731no—
247#2922—731no—
248#1986—731no—
249#1950—731no—
250#1945—732no—
251#1937—732no—
252#1929—732no—
253#1922—732no—
254#1806D7311yescharacteristic
255C8×C4⋊C4—731no—
256C73#2213011yescharacteristic, complemented, Sylow
257#533—731no—
258C4×#41C2×D7311yescharacteristic
259#479—732no—
260#478—732no—
261#477—731no—
262#476—731no—
263#475—731no—
264#474—731no—
265#446—1461no—
266#446—1461no—
267C8×Q8—1461no—
268C8×Q8—1461no—
269#433—1461no—
270#433—1461no—
271#431—1461no—
272#431—1461no—
273C2×#108—732no—
274#427—732no—
275#424—732no—
276#424—731no—
277#424—731no—
278#423C2×D7311yescharacteristic
279C2×C4⋊C8—731no—
280C2×#104C2×D7312yes—
281#416—732no—
282#408—731no—
283#408—731no—
284#405—732no—
285C4×M16—731no—
286C2×C4×C8—732no—
287C2×C4×C8—731no—
288C4×C4⋊C4C2×D7311yescharacteristic
289#341—732no—
290#339—732no—
291#339—732no—
292#339C2×D7312yes—
293#336—732no—
294C2×#41C22×D7311yescharacteristic
295C2×M16—731no—
296C2×M16—731no—
297C2×M16C22×D7312yes—
298C22×C8—731no—
299C22×C8—732no—
300C22×C8—731no—
301C4⋊Q8—731no—
302C4⋊Q8—731no—
303#125—732no—
304C22⋊Q8—1461no—
305C22⋊Q8—1461no—
306C4×Q8—1461no—
307C4×Q8—1461no—
308C4×Q8C4×D7311yescharacteristic
309C4×Q8C4×D7311yescharacteristic
310C4×D4—21no—
311C4×D4—21no—
312C4×D4C4×D7311yescharacteristic
313C4×D4C4×D7311yescharacteristic
314#117—21no—
315#117—732no—
316#117C22×D7311yescharacteristic
317C2×C4⋊C4—21no—
318C2×C42—21no—
319C2×C42C22×D7312yes—
320C2×C42C22×D7311yescharacteristic
321#108—2922no—
322#107—732no—
323#107—732no—
324#106—732no—
325#106—732no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—731no—
329C4⋊C8—731no—
330C4⋊C8—732no—
331C4⋊C8—1461no—
332C4⋊C8—1461no—
333C4⋊C8—1461no—
334C4⋊C8—1461no—
335C4⋊C8—1462no—
336C4⋊C8—1462no—
337#104—42no—
338#103—732no—
339#103—732no—
340#102—732no—
341#102—732no—
342C22⋊C8—1461no—
343C22⋊C8—1461no—
344C22⋊C8—1461no—
345C22⋊C8—1461no—
346#97—1462no—
347#97—1461no—
348C4×C8—2922no—
349C4×C8—731no—
350C4×C8—731no—
351C4×C8—731no—
352C4×C8—731no—
353C4×C8—1462no—
354C4×C8—1461no—
355#95—21no—
356#41—21no—
357#41—21no—
358#41—21no—
359#41—21no—
360C2×Q8—1461no—
361C2×Q8—1461no—
362C2×Q8C2×C4×D7311yescharacteristic
363C2×D4—21no—
364C2×D4C2×C4×D7311yescharacteristic
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4#1192402711yescharacteristic
370C22×C4#1192402711yescharacteristic
371C22×C4C23×D7311yescharacteristic
372M16—22no—
373M16—22no—
374M16—1461no—
375M16—1461no—
376M16—1461no—
377M16—1461no—
378C2×C8—22no—
379C2×C8—732no—
380C2×C8—732no—
381C2×C8—731no—
382C2×C8—731no—
383C2×C8—731no—
384C2×C8—731no—
385C2×C8—1461no—
386C2×C8—1461no—
387C2×C8—1461no—
388C2×C8—1462no—
389C2×C8—1461no—
390C2×C8—1461no—
391C2×C8—1461no—
392C2×C8—1462no—
393C4⋊C4—24no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—732no—
397C4⋊C4—732no—
398C4⋊C4—1462no—
399C4⋊C4—1462no—
400C4⋊C4—1461no—
401C4⋊C4—1461no—
402C4⋊C4C2×C4×D7311yescharacteristic
403C4⋊C4C2×C4×D7311yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—1462no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—22no—
412C42—22no—
413C42—1462no—
414C42C2×C4×D7311yescharacteristic
415C42C2×C4×D7311yescharacteristic
416C42D4×D7311yescharacteristic
417C42D4×D7311yescharacteristic
418C23—21no—
419C23C2×#1192402711yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—2922no—
423Q8—2922no—
424D4—21no—
425D4—21no—
426D4—41no—
427D4—41no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—1462no—
441C2×C4—1462no—
442C2×C4#1455655111yescharacteristic
443C2×C4#1455655111yescharacteristic
444C2×C4#1455655111yescharacteristic
445C2×C4#1455655111yescharacteristic
446C2×C4C22×C4×D7311yescharacteristic
447C2×C4C2×D4×D7311yescharacteristic
448C2×C4#1455663611yescharacteristic
449C2×C4#1455663811yescharacteristic
450C2×C4#1455663811yescharacteristic
451C2×C4#1455663911yescharacteristic
452C8—42no—
453C8—2921no—
454C8—2921no—
455C8—1461no—
456C8—1461no—
457C8—1461no—
458C8—1461no—
459C8—1462no—
460C8—1462no—
461C22—21no—
462C22—21no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22C2×#1455655111yescharacteristic
467C22#1798919711yescharacteristic
468C22#1798920411yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—44no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—2922no—
477C4—2922no—
478C4#1798870112yes—
479C4#1798906411yescharacteristic
480C4#1798906811yescharacteristic
481C4#1798916011yescharacteristic
482C4#1798916611yescharacteristic
483C2—21no—
484C2—21no—
485C2—81no—
486C2#2261260911yescharacteristic
487C2C2×#1798870111yescharacteristic
488C2#2261883011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1806C2
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C23
cosocleC22×C4×C73C24
hypercenter#1806D73
hypocenterC73#22130
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #22618861C2semidirect
4. #22610874C2semidirect
5. #22610704C2semidirect
6. #22610678C2semidirect
9. #22610533C2semidirect
10. C4⋊C4×#10596831C2semidirect
13. #17988894C4semidirect
14. #17988887C4semidirect
15. #17988882C4semidirect
16. #17988877C4semidirect
17. #17988872C4semidirect
18. #17988867C4semidirect
256. C73#22130semidirect

Maximal subgroups

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

subgroupindex
#226197192
#226188612
#226185752
#226108742
#226107042
#226106782
#226106322
#226105822
#226105332
C4⋊C4×#105968312
C73×#18062
#2213073