Finite Group Databasebeta

← #33098293

Subgroups — #33098293

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#22618866C211yesmaximal, characteristic, complemented
2#22618575C211yesmaximal, characteristic
3#22610876C211yesmaximal, characteristic, complemented
4#22610707C211yesmaximal, characteristic, complemented
5#22610680C212yesmaximal, complemented
6#22610637C212yesmaximal
7#22610581C212yesmaximal
8#22610532C212yesmaximal, complemented
9#22609951C211yesmaximal, characteristic
10#22609796C211yesmaximal, characteristic, complemented
11C73×#3076C211yesmaximal, characteristic
12#17988718—21no—
13#17988718—21no—
14Q8×#10596831—21no—
15Q8×#10596831—21no—
16#17988648—21no—
17#17988648—21no—
18#17988640—21no—
19#17988640—21no—
20C2×#14556527C2212yes—
21#17988614C2211yescharacteristic
22#17988614C2212yes—
23#17988614C2211yescharacteristic
24#17988611C2211yescharacteristic
25C2×#14556518C2212yes—
26#17988554C2211yescharacteristic
27#17988554C2211yescharacteristic
28#17988543C2212yes—
29C4×#11924012C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31C2×C8×Dic73C2212yes—
32C4×C4⋊Dic73C2211yescharacteristic
33#17988352C2212yes—
34#17988346C2212yes—
35#17988346C2212yes—
36#17988346C2212yes—
37#17988341C2212yes—
38#533×C73C2211yescharacteristic
39C4×#41×C73C2211yescharacteristic
40#479×C73C412yescomplemented
41#478×C73C412yescomplemented
42#477×C73C411yescharacteristic, complemented
43#476×C73C411yescharacteristic, complemented
44#475×C73C411yescharacteristic, complemented
45#474×C73C411yescharacteristic, complemented
46#427×C73C2212yes—
47C2×C4⋊C8×C73C2211yescharacteristic
48#416×C73C2212yes—
49C2×#11924012C2311yescharacteristic
50C2×#11924012C2312yes—
51C2×#11924012C2311yescharacteristic
52C22×#10596831C2311yescharacteristic
53C22×#10596831C2311yescharacteristic
54C2×C4⋊Dic73—21no—
55C2×C4×Dic73C2312yes—
56#14556527—42no—
57C4⋊#10596831—21no—
58C4⋊#10596831—21no—
59C4⋊#10596831—21no—
60C4⋊#10596831—21no—
61C4⋊#10596831—22no—
62C4⋊#10596831—22no—
63C4⋊#10596831—21no—
64C4⋊#10596831—21no—
65#14556518—42no—
66C22⋊#10596831—21no—
67C22⋊#10596831—21no—
68C22⋊#10596831—21no—
69C22⋊#10596831—21no—
70#14556502—22no—
71#14556502—21no—
72C4×#10596831—22no—
73C4×#10596831—21no—
74C4×#10596831D411yescharacteristic
75C4×#10596831D411yescharacteristic
76C4×#10596831D411yescharacteristic
77C4×#10596831D411yescharacteristic
78C8×Dic73—42no—
79#14556498—21no—
80C2×#41×C73C2311yescharacteristic
81C22×C8×C73C2312yes—
82C4⋊Q8×C73C2×C411yescharacteristic
83C4⋊Q8×C73C2×C411yescharacteristic
84#125×C73C2×C412yes—
85C22⋊Q8×C73—21no—
86C22⋊Q8×C73—21no—
87C4×Q8×C73—21no—
88C4×Q8×C73—21no—
89C4×Q8×C73C2×C411yescharacteristic
90C4×Q8×C73C2×C411yescharacteristic
91C4×D4×C73—21no—
92C4×D4×C73—21no—
93C4×D4×C73C2×C411yescharacteristic
94C4×D4×C73C2×C411yescharacteristic
95#117×C73—21no—
96#117×C73C2311yescharacteristic
97#117×C73C2312yes—
98C2×C42×C73—21no—
99C2×C42×C73C2311yescharacteristic
100#107×C73C2×C412yes—
101#107×C73C2×C412yes—
102#106×C73C2×C412yes—
103#106×C73C2×C412yes—
104C4⋊C8×C73C2×C411yescharacteristic
105C4⋊C8×C73C2×C411yescharacteristic
106C4⋊C8×C73C2×C412yes—
107#103×C73C2×C412yes—
108#103×C73C2×C412yes—
109#102×C73C2×C412yes—
110#102×C73C2×C412yes—
111C22×Dic73—22no—
112#11924012—22no—
113#11924012—21no—
114#11924012—21no—
115#11924012—21no—
116#11924012—22no—
117#11924012—21no—
118C2×#10596831—21no—
119C2×#10596831—21no—
120C2×#10596831—21no—
121C2×#10596831—21no—
122C2×#10596831—21no—
123C2×#10596831—21no—
124C2×#10596831—22no—
125C2×#10596831C2×D411yescharacteristic
126C2×#10596831C2×D411yescharacteristic
127C2×#10596831C2×D411yescharacteristic
128C2×#10596831C2×D411yescharacteristic
129C4⋊Dic73—24no—
130C4×Dic73—22no—
131C4×Dic73—22no—
132#41×C73—21no—
133#41×C73—21no—
134#41×C73—21no—
135#41×C73—21no—
136C2×Q8×C73—21no—
137C2×Q8×C73—21no—
138C2×Q8×C73C22×C411yescharacteristic
139C2×D4×C73—21no—
140C2×D4×C73C22×C411yescharacteristic
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73#4111yescharacteristic
145C22×C4×C73#4111yescharacteristic
146C22×C4×C73C2411yescharacteristic
147C2×C8×C73—22no—
148C2×C8×C73—22no—
149C2×C8×C73C22×C412yes—
150C2×C8×C73C22×C412yes—
151C4⋊C4×C73—21no—
152C4⋊C4×C73—21no—
153C4⋊C4×C73—22no—
154C4⋊C4×C73—22no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73C22×C412yes—
158C4⋊C4×C73C22×C412yes—
159C4⋊C4×C73C22×C411yescharacteristic
160C4⋊C4×C73C22×C411yescharacteristic
161C22⋊C4×C73—21no—
162C22⋊C4×C73—22no—
163C22⋊C4×C73—21no—
164C22⋊C4×C73—21no—
165C42×C73—21no—
166C42×C73—21no—
167C42×C73—21no—
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—
177Q8×C73—42no—
178Q8×C73—42no—
179D4×C73—21no—
180D4×C73—21no—
181D4×C73—41no—
182D4×C73—41no—
183C2×C4×C73—22no—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—21no—
187C2×C4×C73—22no—
188C2×C4×C73—22no—
189C2×C4×C73—41no—
190C2×C4×C73—42no—
191C2×C4×C73—41no—
192C2×C4×C73—41no—
193C2×C4×C73—42no—
194C2×C4×C73C4×D411yescharacteristic
195C2×C4×C73C4×D411yescharacteristic
196C2×C4×C73C4×D411yescharacteristic
197C2×C4×C73C4×D411yescharacteristic
198C2×C4×C73C23×C411yescharacteristic
199C2×C4×C73C22×D411yescharacteristic
200C2×C4×C73H(5,𝔽2)11yescharacteristic
201C2×C4×C732−1+411yescharacteristic
202C2×C4×C732−1+411yescharacteristic
203C2×C4×C732−1+411yescharacteristic
204C2×Dic73—24no—
205C2×Dic73—42no—
206C2×Dic73—42no—
207C8×C73—22no—
208C8×C73—22no—
209#10596831—21no—
210#10596831—21no—
211#10596831—21no—
212#10596831—21no—
213#10596831—41no—
214#10596831—41no—
215#10596831—42no—
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—41no—
228C4×C73—41no—
229C4×C73—42no—
230C4×C73—41no—
231C4×C73—42no—
232C4×C73#44312yes—
233C4×C73#51911yescharacteristic
234C4×C73#52011yescharacteristic
235C4×C73#53611yescharacteristic
236C4×C73#53711yescharacteristic
237Dic73—44no—
238#22130—731nomaximal, Sylow
239C2×C73—21no—
240C2×C73—21no—
241C2×C73—81no—
242C2×C73#232311yescharacteristic
243C2×C73C2×#44311yescharacteristic
244C2×C73#295511yescharacteristic
245#3076D7311yescharacteristic
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#1806—731no—
255C8×C4⋊C4—731no—
256C73#2213011yescharacteristic, complemented, Sylow
257#533C2×D7311yescharacteristic
258C4×#41C2×D7311yescharacteristic
259#479Dic7312yescomplemented
260#478Dic7312yescomplemented
261#477Dic7311yescharacteristic, complemented
262#476Dic7311yescharacteristic, complemented
263#475Dic7311yescharacteristic, complemented
264#474Dic7311yescharacteristic, complemented
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#427C2×D7312yes—
275#424—731no—
276#424—731no—
277#424—732no—
278#423—731no—
279C2×C4⋊C8C2×D7311yescharacteristic
280C2×#104—732no—
281#416C2×D7312yes—
282#408—731no—
283#408—731no—
284#405—732no—
285C4×M16—731no—
286C2×C4×C8—732no—
287C2×C4×C8—731no—
288C4×C4⋊C4—731no—
289#341—732no—
290#339—732no—
291#339—732no—
292#339—732no—
293#336—732no—
294C2×#41C22×D7311yescharacteristic
295C2×M16—731no—
296C2×M16—731no—
297C2×M16—732no—
298C22×C8—731no—
299C22×C8—731no—
300C22×C8C22×D7312yes—
301C4⋊Q8C2×Dic7311yescharacteristic
302C4⋊Q8C2×Dic7311yescharacteristic
303#125C2×Dic7312yes—
304C22⋊Q8—21no—
305C22⋊Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8C2×Dic7311yescharacteristic
309C4×Q8C2×Dic7311yescharacteristic
310C4×D4—21no—
311C4×D4—21no—
312C4×D4C2×Dic7311yescharacteristic
313C4×D4C2×Dic7311yescharacteristic
314#117—21no—
315#117C22×D7311yescharacteristic
316#117C22×D7312yes—
317C2×C4⋊C4—1461no—
318C2×C42—21no—
319C2×C42—732no—
320C2×C42C22×D7311yescharacteristic
321#108—2922no—
322#107C2×Dic7312yes—
323#107C2×Dic7312yes—
324#106C2×Dic7312yes—
325#106C2×Dic7312yes—
326C4⋊C8—1461no—
327C4⋊C8—1461no—
328C4⋊C8—1461no—
329C4⋊C8—1461no—
330C4⋊C8—1462no—
331C4⋊C8—1462no—
332C4⋊C8—1461no—
333C4⋊C8—1461no—
334C4⋊C8C2×Dic7311yescharacteristic
335C4⋊C8C2×Dic7311yescharacteristic
336C4⋊C8C2×Dic7312yes—
337#104—2922no—
338#103C2×Dic7312yes—
339#103C2×Dic7312yes—
340#102C2×Dic7312yes—
341#102C2×Dic7312yes—
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—1461no—
356#41—21no—
357#41—21no—
358#41—21no—
359#41—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8C22×Dic7311yescharacteristic
363C2×D4—21no—
364C2×D4C22×Dic7311yescharacteristic
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—1462no—
369C22×C4#1192402711yescharacteristic
370C22×C4#1192402711yescharacteristic
371C22×C4C23×D7311yescharacteristic
372M16—1462no—
373M16—1461no—
374M16—1461no—
375M16—1461no—
376M16—1461no—
377M16—1462no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—731no—
381C2×C8—731no—
382C2×C8—731no—
383C2×C8—731no—
384C2×C8—1461no—
385C2×C8—1461no—
386C2×C8—1462no—
387C2×C8—1461no—
388C2×C8—1461no—
389C2×C8—1461no—
390C2×C8—1461no—
391C2×C8C22×Dic7312yes—
392C2×C8C22×Dic7312yes—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—1464no—
400C4⋊C4C22×Dic7312yes—
401C4⋊C4C22×Dic7312yes—
402C4⋊C4C22×Dic7311yescharacteristic
403C4⋊C4C22×Dic7311yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—22no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C42—21no—
409C42—21no—
410C42—22no—
411C42—21no—
412C42—1462no—
413C42—1462no—
414C42C22×Dic7311yescharacteristic
415C42C22×Dic7311yescharacteristic
416C42D4×D7311yescharacteristic
417C42D4×D7311yescharacteristic
418C23—21no—
419C23C2×#1192402711yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—42no—
423Q8—42no—
424D4—21no—
425D4—21no—
426D4—41no—
427D4—41no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—2922no—
440C2×C4—2922no—
441C2×C4—1464no—
442C2×C4D4×Dic7311yescharacteristic
443C2×C4D4×Dic7311yescharacteristic
444C2×C4D4×Dic7311yescharacteristic
445C2×C4D4×Dic7311yescharacteristic
446C2×C4C23×Dic7311yescharacteristic
447C2×C4C2×D4×D7311yescharacteristic
448C2×C4#1455663711yescharacteristic
449C2×C4#1455663811yescharacteristic
450C2×C4#1455663911yescharacteristic
451C2×C4#1455663911yescharacteristic
452C8—22no—
453C8—22no—
454C8—2921no—
455C8—2921no—
456C8—2922no—
457C8—1461no—
458C8—1461no—
459C8—1461no—
460C8—1461no—
461C22—21no—
462C22—21no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22C2×D4×Dic7311yescharacteristic
467C22#1798919711yescharacteristic
468C22#1798921011yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—41no—
473C4—41no—
474C4—42no—
475C4—41no—
476C4—42no—
477C4—2924no—
478C4#1798870412yes—
479C4#1798906511yescharacteristic
480C4#1798907011yescharacteristic
481C4#1798916011yescharacteristic
482C4#1798916711yescharacteristic
483C2—21no—
484C2—21no—
485C2—81no—
486C2#2261261511yescharacteristic
487C2C2×#1798870411yescharacteristic
488C2#2261883411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#3076C2
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C23
cosocleC22×C4×C73C24
hypercenter#3076D73
hypocenterC73#22130
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #22618866C2semidirect
3. #22610876C2semidirect
4. #22610707C2semidirect
5. #22610680C2semidirect
8. #22610532C2semidirect
10. #22609796C2semidirect
40. #479×C73C4semidirect
41. #478×C73C4semidirect
42. #477×C73C4semidirect
43. #476×C73C4semidirect
44. #475×C73C4semidirect
45. #474×C73C4semidirect
256. C73#22130semidirect
259. #479Dic73semidirect
260. #478Dic73semidirect
261. #477Dic73semidirect
262. #476Dic73semidirect
263. #475Dic73semidirect
264. #474Dic73semidirect

Maximal subgroups

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

subgroupindex
#226188662
#226185752
#226108762
#226107072
#226106802
#226106372
#226105812
#226105322
#226099512
#226097962
C73×#30762
#2213073