Finite Group Databasebeta

← #33095502

Subgroups — #33095502

Order 18688.

1054 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989143C211yesmaximal, characteristic
2C2×#17988642C211yesmaximal, characteristic
3#22612336C211yesmaximal, characteristic
4#22610901C211yesmaximal, characteristic, complemented
5#22610900C211yesmaximal, characteristic, complemented
6#22610899C211yesmaximal, characteristic, complemented
7#22610881C212yesmaximal, complemented
8#22610877C212yesmaximal, complemented
9#22610025C212yesmaximal
10#22610024C212yesmaximal
11C73×#1992C211yesmaximal, characteristic, complemented
12#17989143—24no—
13#17989143C418yescomplemented
14C2×#14556590C2211yescharacteristic
15C2×#14556590C2211yescharacteristic
16C2×#14556583C2211yescharacteristic
17C2×#14556569—21no—
18C2×#14556569—21no—
19C2×#14556556—21no—
20C2×#14556556—21no—
21C2×#14556549—21no—
22C2×#14556549C2211yescharacteristic
23C2×#14556549C2211yescharacteristic
24C2×C42×D73C2211yescharacteristic
25#17988642—24no—
26C2×#14556520C2212yes—
27C2×#14556503C2212yes—
28C2×#14556502C2211yescharacteristic
29#17988457C2211yescharacteristic
30#17988452—21no—
31#17988444—21no—
32#17988444C2212yes—
33C4⋊C4×Dic73—21no—
34C4⋊C4×Dic73C2212yes—
35#17988350C2212yes—
36#17988350C2212yes—
37#17988349C2212yes—
38#17988349C2212yes—
39#17988349C2212yes—
40#17988349C2212yes—
41C2×#125×C73C2211yescharacteristic
42C2×#97×C73C2211yescharacteristic
43#384×C73C2211yescharacteristic
44#340×C73C2212yes—
45#340×C73C2212yes—
46C22×Dic146—21no—
47C22×Dic146—21no—
48C22×C4×D73D411yescharacteristic
49C22×C4×D73D411yescharacteristic
50C22×C4×D73C2311yescharacteristic
51C22×#10596831C2312yes—
52#14556590—22no—
53#14556590—22no—
54#14556590C2×C414yes—
55#14556590C2×C414yes—
56#14556583—22no—
57#14556583C2×C414yes—
58#14556569—28no—
59#14556569—28no—
60#14556556—28no—
61#14556556—28no—
62#14556549—22no—
63#14556549—22no—
64#14556549—44no—
65#14556549C2×C414yes—
66#14556549C2×C414yes—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2312yes—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924005—21no—
77C2×#11924005—21no—
78C2×#11924005—22no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—22no—
82C2×C4×Dic73D411yescharacteristic
83C2×C4×Dic73D411yescharacteristic
84C2×C4×Dic73C2311yescharacteristic
85C42×D73C2×C414yes—
86C42×D73D414yes—
87#14556520—24no—
88#14556503—24no—
89#14556502—22no—
90#14556498—22no—
91#14556498—22no—
92C22×C8×C73C2312yes—
93#125×C73—22no—
94#125×C73C2×C414yes—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73C2312yes—
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C42×C73C2311yescharacteristic
103#97×C73—22no—
104C23×D73C2×D411yescharacteristic
105C2×Dic146—24no—
106C2×Dic146—24no—
107C2×Dic146—42no—
108C2×Dic146—42no—
109C2×Dic146—42no—
110C2×Dic146—42no—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C22×Dic73—22no—
114C22×Dic73C2×D411yescharacteristic
115C22×Dic73C2×D411yescharacteristic
116C22×Dic73C2×D411yescharacteristic
117C2×C4×D73—22no—
118C2×C4×D73—22no—
119C2×C4×D73—22no—
120C2×C4×D73—22no—
121C2×C4×D73—41no—
122C2×C4×D73—41no—
123C2×C4×D73C22⋊C414yes—
124C2×C4×D73C22⋊C414yes—
125C2×C4×D73C22×C412yes—
126C2×C4×D73C2×D412yes—
127C2×#10596831—22no—
128C2×#10596831—22no—
129C2×#10596831—22no—
130C4⋊Dic73—21no—
131C4⋊Dic73—21no—
132C4⋊Dic73—24no—
133C4⋊Dic73—42no—
134C4⋊Dic73—42no—
135C4⋊Dic73C22×C412yes—
136C4⋊Dic73C22×C412yes—
137#11924007—24no—
138#11924007—24no—
139#11924007—24no—
140#11924007—24no—
141#11924005—24no—
142#11924005—24no—
143#11924005—44no—
144C4×Dic73—24no—
145C4×Dic73—24no—
146C4×Dic73—44no—
147C4×Dic73C22⋊C414yes—
148C4×Dic73C22⋊C414yes—
149C4×Dic73C22×C412yes—
150C4×Dic73C2×D412yes—
151C22×C4×C73—21no—
152C22×C4×C73—21no—
153C22×C4×C73—22no—
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2411yescharacteristic
157C2×C8×C73—22no—
158C2×C8×C73—22no—
159C2×C8×C73—22no—
160C4⋊C4×C73—24no—
161C4⋊C4×C73—24no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73—24no—
165C4⋊C4×C73—42no—
166C4⋊C4×C73—42no—
167C4⋊C4×C73C22×C412yes—
168C4⋊C4×C73C22×C412yes—
169C42×C73C22×C412yes—
170C42×C73C2×D412yes—
171C23×C73C22×D411yescharacteristic
172C22×D73—21no—
173C22×D73—21no—
174C22×D73—21no—
175C22×D73—21no—
176C22×D73—21no—
177C22×D73—21no—
178C22×D73C2×C22⋊C412yes—
179C2×C4×C73—21no—
180C2×C4×C73—21no—
181C2×C4×C73—21no—
182C2×C4×C73—21no—
183C2×C4×C73—22no—
184C2×C4×C73—22no—
185C2×C4×C73—41no—
186C2×C4×C73—41no—
187C2×C4×C73—41no—
188C2×C4×C73—41no—
189C2×C4×C73—42no—
190C2×C4×C73—42no—
191C2×C4×C73—42no—
192C2×C4×C73C2×C22⋊C412yes—
193C2×C4×C73C2×C22⋊C412yes—
194C2×C4×C73C23×C411yescharacteristic
195C2×C4×C73C22×D411yescharacteristic
196C2×C4×C732−1+412yes—
197C2×C4×C732−1+412yes—
198C2×Dic73—21no—
199C2×Dic73—21no—
200C2×Dic73—21no—
201C2×Dic73—21no—
202C2×Dic73—22no—
203C2×Dic73—22no—
204C2×Dic73—41no—
205C2×Dic73—41no—
206C2×Dic73—41no—
207C2×Dic73—41no—
208C2×Dic73—42no—
209C2×Dic73—42no—
210C2×Dic73—42no—
211C2×Dic73C2×C22⋊C412yes—
212C2×Dic73C2×C22⋊C412yes—
213C2×Dic73C2×C22⋊C412yes—
214C2×Dic73AGL(1,ℤ/8)12yes—
215C2×Dic73#13712yes—
216C4×D73—24no—
217C4×D73—24no—
218C4×D73—28no—
219C4×D73—44no—
220Dic146—48no—
221Dic146—48no—
222C8×C73—42no—
223#10596831—42no—
224C22×C73C22×C22⋊C411yescharacteristic
225C22×C73#53611yescharacteristic
226C22×C73#53611yescharacteristic
227C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
228C22×C73C2×#13711yescharacteristic
229C22×C73#57511yescharacteristic
230C22×C73#57511yescharacteristic
231C2×D73—22no—
232C2×D73—22no—
233C2×D73—22no—
234C2×D73—41no—
235C2×D73—41no—
236C2×D73—41no—
237C2×D73—41no—
238C4×C73—22no—
239C4×C73—22no—
240C4×C73—42no—
241C4×C73—42no—
242C4×C73—82no—
243C4×C73#51912yes—
244C4×C73#53612yes—
245Dic73—22no—
246Dic73—22no—
247Dic73—42no—
248Dic73—42no—
249Dic73—82no—
250Dic73#41714yes—
251#21839—731nomaximal, Sylow
252C2×C73#231011yescharacteristic
253C2×C73C2×#41711yescharacteristic
254C2×C73#289611yescharacteristic
255C2×C73#323511yescharacteristic
256C2×C73#323511yescharacteristic
257C2×C73#323711yescharacteristic
258C2×C73#323711yescharacteristic
259D73—42no—
260C2×#533—731no—
261C2×#432—731no—
262#2285—731no—
263#1992—731no—
264#1992—731no—
265#1992—731no—
266#1992D7311yescharacteristic, complemented
267#1988—732no—
268#1987—732no—
269#1825—732no—
270#1825—732no—
271C73#2183911yescharacteristic, complemented, Sylow
272#533—738no—
273#533—1464no—
274C2×C4⋊Q8—731no—
275C2×C4⋊Q8—731no—
276C2×#125—731no—
277C2×#125C2×D7311yescharacteristic
278C2×C22⋊Q8—1461no—
279C2×C22⋊Q8—1461no—
280C2×C4×Q8—1461no—
281C2×C4×Q8—1461no—
282C2×#117—731no—
283C2×#117—731no—
284C2×#117—1461no—
285C22×C42—731no—
286#432—1464no—
287C2×C4⋊C8—732no—
288C2×C22⋊C8—732no—
289C2×#97—731no—
290C2×#97C2×D7311yescharacteristic
291#384—731no—
292#384C2×D7311yescharacteristic
293#382—1461no—
294#380—732no—
295#380—1461no—
296C4×C4⋊C4—732no—
297C4×C4⋊C4—1461no—
298#340—732no—
299#340—732no—
300#340—732no—
301#340—732no—
302#340—732no—
303#340—732no—
304#340C2×D7312yes—
305#340C2×D7312yes—
306C22×Q8—1461no—
307C22×Q8—1461no—
308C23×C4—731no—
309C23×C4—731no—
310C23×C4—731no—
311C22×C8—732no—
312C22×C8C22×D7312yes—
313C4⋊Q8—734no—
314C4⋊Q8—734no—
315C4⋊Q8—1462no—
316C4⋊Q8—1462no—
317#125—22no—
318#125—734no—
319#125—1462no—
320#125C4×D7314yes—
321C22⋊Q8—1468no—
322C22⋊Q8—1468no—
323C4×Q8—1468no—
324C4×Q8—1468no—
325#117—2924no—
326#117—734no—
327#117—734no—
328#117—1462no—
329#117—1462no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—731no—
335C2×C4⋊C4—731no—
336C2×C4⋊C4—732no—
337C2×C4⋊C4—1461no—
338C2×C4⋊C4—1461no—
339C2×C4⋊C4—1461no—
340C2×C4⋊C4—1461no—
341C2×C4⋊C4—1461no—
342C2×C4⋊C4—1461no—
343C2×C4⋊C4C22×D7312yes—
344C2×C4⋊C4C22×D7311yescharacteristic
345C2×C4⋊C4C22×D7311yescharacteristic
346C2×C22⋊C4—1462no—
347C2×C22⋊C4—1461no—
348C2×C22⋊C4—1461no—
349C2×C42—731no—
350C2×C42—731no—
351C2×C42—731no—
352C2×C42—734no—
353C2×C42—734no—
354C2×C42—1462no—
355C2×C42—1461no—
356C2×C42—1461no—
357C2×C42C22×D7311yescharacteristic
358C4⋊C8—1464no—
359C22⋊C8—1464no—
360#97—22no—
361#97—1462no—
362#95—1462no—
363#95—1462no—
364C24—731no—
365C2×Q8—2922no—
366C2×Q8—2922no—
367C2×Q8—2922no—
368C2×Q8—2922no—
369C2×Q8—1464no—
370C2×Q8—1464no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—2921no—
375C22×C4—2921no—
376C22×C4—734no—
377C22×C4—731no—
378C22×C4—731no—
379C22×C4—731no—
380C22×C4—734no—
381C22×C4—732no—
382C22×C4—732no—
383C22×C4—1462no—
384C22×C4—1462no—
385C22×C4—1462no—
386C22×C4—1461no—
387C22×C4—1461no—
388C22×C4—1462no—
389C22×C4—1462no—
390C22×C4D4×D7311yescharacteristic
391C22×C4D4×D7311yescharacteristic
392C22×C4C23×D7311yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—1462no—
397C2×C8—1462no—
398C2×C8—1462no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—24no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—2922no—
407C4⋊C4—2922no—
408C4⋊C4—732no—
409C4⋊C4—732no—
410C4⋊C4—1464no—
411C4⋊C4—1464no—
412C4⋊C4—1464no—
413C4⋊C4—1464no—
414C4⋊C4—1464no—
415C4⋊C4—1461no—
416C4⋊C4—1461no—
417C4⋊C4C2×C4×D7312yes—
418C4⋊C4C2×C4×D7312yes—
419C22⋊C4—2924no—
420C22⋊C4—1464no—
421C22⋊C4—1464no—
422C42—2924no—
423C42—734no—
424C42—734no—
425C42—732no—
426C42—732no—
427C42—1464no—
428C42—1464no—
429C42C2×C4×D7312yes—
430C42D4×D7312yes—
431C23—732no—
432C23—1461no—
433C23—1461no—
434C23—1461no—
435C23—1461no—
436C23—1461no—
437C23—1461no—
438C23C2×D4×D7311yescharacteristic
439Q8—2928no—
440Q8—2928no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—2924no—
455C2×C4—2921no—
456C2×C4—2921no—
457C2×C4—2921no—
458C2×C4—2921no—
459C2×C4—2922no—
460C2×C4—2922no—
461C2×C4—2922no—
462C2×C4—732no—
463C2×C4—732no—
464C2×C4—732no—
465C2×C4—732no—
466C2×C4—732no—
467C2×C4—1468no—
468C2×C4—1462no—
469C2×C4—1462no—
470C2×C4—1461no—
471C2×C4—1461no—
472C2×C4—1461no—
473C2×C4—1461no—
474C2×C4—1464no—
475C2×C4—1464no—
476C2×C4C22⋊C4×D7312yes—
477C2×C4C22⋊C4×D7312yes—
478C2×C4C22×C4×D7311yescharacteristic
479C2×C4C2×D4×D7311yescharacteristic
480C2×C4#1455663812yes—
481C2×C4#1455663912yes—
482C8—42no—
483C8—2922no—
484C22—2921no—
485C22—2921no—
486C22—2921no—
487C22—2921no—
488C22—1462no—
489C22—1462no—
490C22—1462no—
491C22C2×C22⋊C4×D7311yescharacteristic
492C22#1798915711yescharacteristic
493C22#1798916211yescharacteristic
494C22AGL(1,ℤ/8)×D7311yescharacteristic
495C22#137×D7311yescharacteristic
496C22#1798938311yescharacteristic
497C22#1798938611yescharacteristic
498C4—22no—
499C4—22no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—2922no—
504C4—2922no—
505C4—5842no—
506C4—734no—
507C4—1462no—
508C4—1462no—
509C4#1798906312yes—
510C4#1798916012yes—
511C2—2922no—
512C2#2261251411yescharacteristic
513C2#417×D7311yescharacteristic
514C2#2261844711yescharacteristic
515C2#2262112411yescharacteristic
516C2#2262113011yescharacteristic
517C2#2262114011yescharacteristic
518C2#2262114611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1992C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1992D73
hypocenterC73#21839
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
4. #22610901C2semidirect
5. #22610900C2semidirect
6. #22610899C2semidirect
7. #22610881C2semidirect
8. #22610877C2semidirect
11. C73×#1992C2semidirect
13. #17989143C4semidirect
266. #1992D73semidirect
271. C73#21839semidirect

Maximal subgroups

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

subgroupindex
C2×#179891432
C2×#179886422
#226123362
#226109012
#226109002
#226108992
#226108812
#226108772
#226100252
#226100242
C73×#19922
#2183973