Finite Group Databasebeta

← #33096498

Subgroups — #33096498

Order 18688.

966 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988639C211yesmaximal, characteristic
2#22612335C211yesmaximal, characteristic
3#384×D73C211yesmaximal, characteristic
4#22610695C211yesmaximal, characteristic, complemented
5#22610694C211yesmaximal, characteristic, complemented
6#22610692C211yesmaximal, characteristic, complemented
7#22610125C212yesmaximal
8#22610122C212yesmaximal
9#22610086C212yesmaximal, complemented
10#22610057C212yesmaximal, complemented
11C73×#1948C211yesmaximal, characteristic, complemented
12C2×#14556548—21no—
13C2×C4⋊C4×D73—21no—
14C2×C4⋊C4×D73—21no—
15C2×C4⋊C4×D73C2212yes—
16C2×C42×D73C2211yescharacteristic
17#17988639C418yescomplemented
18C2×#14556523C2212yes—
19C2×#14556522C2212yes—
20C2×#14556522C2212yes—
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556503C2212yes—
23C2×C8×Dic73C2212yes—
24#17988457C2211yescharacteristic
25#17988456—21no—
26#17988456—21no—
27#17988456C2212yes—
28#17988456C2211yescharacteristic
29#17988456C2211yescharacteristic
30#17988451—21no—
31#17988446—21no—
32#17988444C2211yescharacteristic
33#17988444C2211yescharacteristic
34C4×#11924007—21no—
35C4×C4⋊Dic73C2211yescharacteristic
36#17988350C2212yes—
37#17988349C2212yes—
38#17988349C2212yes—
39C2×#106×C73C2212yes—
40C2×C4⋊C8×C73C2211yescharacteristic
41#384×C73C2211yescharacteristic
42C4×C4⋊C4×C73C2211yescharacteristic
43#340×C73C2212yes—
44C22×C4×D73—22no—
45C22×C4×D73D411yescharacteristic
46C22×C4×D73D411yescharacteristic
47C22×C4×D73C2311yescharacteristic
48C22×#10596831C2312yes—
49#14556548—44no—
50C2×C4⋊Dic73—21no—
51C2×C4⋊Dic73—21no—
52C2×C4⋊Dic73C2311yescharacteristic
53C2×C4⋊Dic73C2312yes—
54C2×C4⋊Dic73C2311yescharacteristic
55C2×#11924007—21no—
56C2×#11924007—21no—
57C2×#11924007—21no—
58C2×#11924007—21no—
59C2×#11924007—22no—
60C2×#11924007—22no—
61C2×#11924007—22no—
62C4⋊C4×D73—42no—
63C4⋊C4×D73—42no—
64C4⋊C4×D73—44no—
65C4⋊C4×D73—44no—
66C2×#11924005—21no—
67C2×#11924005—21no—
68C2×C4×Dic73—21no—
69C2×C4×Dic73D411yescharacteristic
70C2×C4×Dic73D411yescharacteristic
71C2×C4×Dic73C2311yescharacteristic
72C42×D73C2×C414yes—
73C42×D73Q814yes—
74#14556523—42no—
75#14556522—42no—
76#14556522—42no—
77C4⋊#10596831C2×C414yescomplemented
78#14556503C2×C418yes—
79C8×Dic73C2×C418yescomplemented
80#14556498—21no—
81#14556498—21no—
82#14556498—21no—
83C22×C8×C73C2312yes—
84C2×C4⋊C4×C73—21no—
85C2×C4⋊C4×C73—21no—
86C2×C4⋊C4×C73C2311yescharacteristic
87C2×C4⋊C4×C73C2311yescharacteristic
88C2×C4⋊C4×C73C2312yes—
89C2×C42×C73—21no—
90C2×C42×C73C2311yescharacteristic
91#106×C73—42no—
92C4⋊C8×C73C2×C414yescomplemented
93#95×C73—21no—
94C23×D73C2×D411yescharacteristic
95C22×Dic73—21no—
96C22×Dic73—21no—
97C22×Dic73—22no—
98C22×Dic73C2×D411yescharacteristic
99C22×Dic73C2×D411yescharacteristic
100C22×Dic73C2×D411yescharacteristic
101C2×C4×D73—21no—
102C2×C4×D73—21no—
103C2×C4×D73—21no—
104C2×C4×D73—21no—
105C2×C4×D73—22no—
106C2×C4×D73—22no—
107C2×C4×D73—44no—
108C2×C4×D73—44no—
109C2×C4×D73—44no—
110C2×C4×D73C4⋊C414yes—
111C2×C4×D73C4⋊C414yes—
112C2×C4×D73SD1614yes—
113C2×C4×D73SD1614yes—
114C2×C4×D73C22×C412yes—
115C2×C4×D73C2×Q812yes—
116C2×#10596831—22no—
117C2×#10596831—22no—
118C2×#10596831C22×C414yes—
119C4⋊Dic73—21no—
120C4⋊Dic73—22no—
121C4⋊Dic73—21no—
122C4⋊Dic73—22no—
123C4⋊Dic73—21no—
124C4⋊Dic73—21no—
125C4⋊Dic73—42no—
126C4⋊Dic73—42no—
127#11924007—44no—
128#11924007—44no—
129#11924007—42no—
130#11924007—42no—
131#11924007—44no—
132#11924007—42no—
133#11924007—42no—
134#11924005—42no—
135#11924005—42no—
136C4×Dic73—42no—
137C4×Dic73C4⋊C414yes—
138C4×Dic73C4⋊C414yes—
139C4×Dic73C22×C412yes—
140C4×Dic73C2×Q812yes—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—22no—
144C22×C4×C73C2×D411yescharacteristic
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2411yescharacteristic
147C2×C8×C73—22no—
148C2×C8×C73—22no—
149C2×C8×C73C22×C414yes—
150C4⋊C4×C73—21no—
151C4⋊C4×C73—22no—
152C4⋊C4×C73—21no—
153C4⋊C4×C73—22no—
154C4⋊C4×C73—21no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—42no—
157C4⋊C4×C73—42no—
158C42×C73—42no—
159C42×C73C22×C412yes—
160C42×C73C2×Q812yes—
161C23×C73C22×D411yescharacteristic
162C22×D73—21no—
163C22×D73—21no—
164C22×D73—21no—
165C22×D73—21no—
166C22×D73C2×C4⋊C412yes—
167C22×D73C2×SD1612yes—
168C22×D73C2×SD1612yes—
169C2×C4×C73—21no—
170C2×C4×C73—21no—
171C2×C4×C73—41no—
172C2×C4×C73—42no—
173C2×C4×C73—41no—
174C2×C4×C73—41no—
175C2×C4×C73—41no—
176C2×C4×C73—42no—
177C2×C4×C73—42no—
178C2×C4×C73—41no—
179C2×C4×C73—41no—
180C2×C4×C73C2×C4⋊C412yes—
181C2×C4×C73C2×C4⋊C412yes—
182C2×C4×C73C2×SD1612yes—
183C2×C4×C73C2×SD1612yes—
184C2×C4×C73C23×C411yescharacteristic
185C2×C4×C73C22×Q811yescharacteristic
186C2×C4×C73H(5,𝔽2)11yescharacteristic
187C2×C4×C73H(5,𝔽2)11yescharacteristic
188C2×C4×C732−1+411yescharacteristic
189C2×C4×C732−1+411yescharacteristic
190C2×Dic73—21no—
191C2×Dic73—21no—
192C2×Dic73—42no—
193C2×Dic73—41no—
194C2×Dic73—41no—
195C2×Dic73—41no—
196C2×Dic73—41no—
197C2×Dic73—42no—
198C2×Dic73—42no—
199C2×Dic73—41no—
200C2×Dic73—41no—
201C2×Dic73C2×C4⋊C412yes—
202C2×Dic73C2×C4⋊C412yes—
203C2×Dic73C2×C4⋊C412yes—
204C2×Dic73C2×SD1612yes—
205C2×Dic73C2×SD1612yes—
206C2×Dic73AGL(1,ℤ/8)12yes—
207C2×Dic73#13712yes—
208C4×D73—22no—
209C4×D73—22no—
210C4×D73—22no—
211C4×D73—22no—
212C4×D73—28no—
213C4×D73—88no—
214C4×D73#106116yescomplemented
215C8×C73—24no—
216#10596831—24no—
217C22×C73C22×C4⋊C411yescharacteristic
218C22×C73#53511yescharacteristic
219C22×C73#53611yescharacteristic
220C22×C73C22×SD1611yescharacteristic
221C22×C73C22×SD1611yescharacteristic
222C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
223C22×C73C2×#13711yescharacteristic
224C2×D73—22no—
225C2×D73—22no—
226C2×D73—22no—
227C2×D73—22no—
228C2×D73—22no—
229C2×D73—22no—
230C2×D73C2×#10614yes—
231C4×C73—22no—
232C4×C73—81no—
233C4×C73—82no—
234C4×C73—81no—
235C4×C73C2×#10614yes—
236C4×C73#52011yescharacteristic
237C4×C73#52011yescharacteristic
238C4×C73#54411yescharacteristic
239C4×C73#54411yescharacteristic
240Dic73—22no—
241Dic73—81no—
242Dic73—82no—
243Dic73—81no—
244Dic73C2×#10614yes—
245Dic73#42814yes—
246#21981—731nomaximal, Sylow
247C2×C73#231711yescharacteristic
248C2×C73C22×#10611yescharacteristic
249C2×C73C2×#42811yescharacteristic
250C2×C73#320611yescharacteristic
251C2×C73#320811yescharacteristic
252C2×C73#322711yescharacteristic
253C2×C73#322911yescharacteristic
254D73—24no—
255C2×#431—731no—
256#2285—731no—
257C2×#384—731no—
258#1948—731no—
259#1948—731no—
260#1948—731no—
261#1948D7311yescharacteristic, complemented
262#1853—732no—
263#1853—732no—
264#1844—732no—
265#1834—732no—
266C73#2198111yescharacteristic, complemented, Sylow
267C2×#117—1461no—
268C22×C4⋊C4—732no—
269C22×C4⋊C4—1461no—
270C22×C4⋊C4—1461no—
271C22×C42—731no—
272#431—738no—
273C2×#106—732no—
274C2×#106—732no—
275C2×#106—732no—
276C2×#106C2×D7312yes—
277C2×C4⋊C8—731no—
278C2×C4⋊C8C2×D7311yescharacteristic
279C2×C22⋊C8—732no—
280C2×C4×C8—732no—
281#384—731no—
282#384—732no—
283#384—731no—
284#384—731no—
285#384—1461no—
286#384—1461no—
287#384C2×D7311yescharacteristic
288#382—1461no—
289#380—731no—
290#380—731no—
291#380—1461no—
292C4×C4⋊C4—731no—
293C4×C4⋊C4—1461no—
294C4×C4⋊C4C2×D7311yescharacteristic
295#340—732no—
296#340—732no—
297#340—732no—
298#340C2×D7312yes—
299C23×C4—731no—
300C23×C4—731no—
301C23×C4—731no—
302C23×C4—1462no—
303C22×C8—732no—
304C22×C8C22×D7312yes—
305#117—2924no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—2922no—
309C2×C4⋊C4—2922no—
310C2×C4⋊C4—2924no—
311C2×C4⋊C4—2924no—
312C2×C4⋊C4—732no—
313C2×C4⋊C4—731no—
314C2×C4⋊C4—731no—
315C2×C4⋊C4—1462no—
316C2×C4⋊C4—1462no—
317C2×C4⋊C4—1462no—
318C2×C4⋊C4—1461no—
319C2×C4⋊C4—1461no—
320C2×C4⋊C4—1461no—
321C2×C4⋊C4—1461no—
322C2×C4⋊C4—1461no—
323C2×C4⋊C4—1461no—
324C2×C4⋊C4C22×D7312yes—
325C2×C4⋊C4C22×D7311yescharacteristic
326C2×C4⋊C4C22×D7311yescharacteristic
327C2×C22⋊C4—1461no—
328C2×C22⋊C4—1461no—
329C2×C42—21no—
330C2×C42—731no—
331C2×C42—731no—
332C2×C42—731no—
333C2×C42—734no—
334C2×C42—734no—
335C2×C42—1461no—
336C2×C42C22×D7311yescharacteristic
337#106—42no—
338#106—2922no—
339#106—2922no—
340#106—2922no—
341C4⋊C8—734no—
342C4⋊C8C4×D7314yescomplemented
343C22⋊C8—738no—
344C4×C8—738no—
345#95—21no—
346#95—1461no—
347#95—1461no—
348#95—1461no—
349C24—731no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—2924no—
354C22×C4—2924no—
355C22×C4—2924no—
356C22×C4—731no—
357C22×C4—731no—
358C22×C4—734no—
359C22×C4—734no—
360C22×C4—734no—
361C22×C4—731no—
362C22×C4—734no—
363C22×C4—732no—
364C22×C4—732no—
365C22×C4—1462no—
366C22×C4—1462no—
367C22×C4—1461no—
368C22×C4—1461no—
369C22×C4—1462no—
370C22×C4—1461no—
371C22×C4—1461no—
372C22×C4—1461no—
373C22×C4—1461no—
374C22×C4D4×D7311yescharacteristic
375C22×C4D4×D7311yescharacteristic
376C22×C4C23×D7311yescharacteristic
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—734no—
380C2×C8—1462no—
381C2×C8—1462no—
382C2×C8C2×C4×D7314yes—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—2922no—
392C4⋊C4—2924no—
393C4⋊C4—2924no—
394C4⋊C4—2922no—
395C4⋊C4—2922no—
396C4⋊C4—2922no—
397C4⋊C4—2922no—
398C4⋊C4—2924no—
399C4⋊C4—2922no—
400C4⋊C4—1461no—
401C4⋊C4—1462no—
402C4⋊C4—1461no—
403C4⋊C4—1462no—
404C4⋊C4—1461no—
405C4⋊C4—1461no—
406C22⋊C4—2922no—
407C22⋊C4—2922no—
408C42—42no—
409C42—2922no—
410C42—734no—
411C42—734no—
412C42—732no—
413C42—732no—
414C42C2×C4×D7312yes—
415C42Q8×D7312yes—
416C23—732no—
417C23—732no—
418C23—732no—
419C23—1461no—
420C23—1461no—
421C23—1461no—
422C23—1461no—
423C23C2×D4×D7311yescharacteristic
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—2921no—
436C2×C4—2922no—
437C2×C4—2921no—
438C2×C4—2921no—
439C2×C4—2921no—
440C2×C4—2921no—
441C2×C4—2922no—
442C2×C4—2922no—
443C2×C4—2921no—
444C2×C4—5848no—
445C2×C4—732no—
446C2×C4—732no—
447C2×C4—732no—
448C2×C4—732no—
449C2×C4—732no—
450C2×C4—732no—
451C2×C4—7316no—
452C2×C4—732no—
453C2×C4—1468no—
454C2×C4—1461no—
455C2×C4—1461no—
456C2×C4—1462no—
457C2×C4—1462no—
458C2×C4—1462no—
459C2×C4—1462no—
460C2×C4C4⋊C4×D7312yes—
461C2×C4C4⋊C4×D7312yes—
462C2×C4SD16×D7312yes—
463C2×C4SD16×D7312yes—
464C2×C4C22×C4×D7311yescharacteristic
465C2×C4C2×Q8×D7311yescharacteristic
466C2×C4#1455663611yescharacteristic
467C2×C4#1455663711yescharacteristic
468C2×C4#1455663811yescharacteristic
469C2×C4#1455663911yescharacteristic
470C8—24no—
471C8—1464no—
472C22—734no—
473C22—1462no—
474C22—1462no—
475C22—1462no—
476C22—1462no—
477C22—1462no—
478C22—1462no—
479C22C2×C4⋊C4×D7311yescharacteristic
480C22#1798915311yescharacteristic
481C22#1798916011yescharacteristic
482C22C2×SD16×D7311yescharacteristic
483C22C2×SD16×D7311yescharacteristic
484C22AGL(1,ℤ/8)×D7311yescharacteristic
485C22#137×D7311yescharacteristic
486C4—22no—
487C4—81no—
488C4—81no—
489C4—82no—
490C4—5841no—
491C4—5841no—
492C4—5842no—
493C4—734no—
494C4—734no—
495C4—1462no—
496C4#106×D7314yes—
497C4#1798906611yescharacteristic
498C4#1798906911yescharacteristic
499C4#1798921411yescharacteristic
500C4#1798921711yescharacteristic
501C2—1464no—
502C2#2261255311yescharacteristic
503C2C2×#106×D7311yescharacteristic
504C2#428×D7311yescharacteristic
505C2#2262085511yescharacteristic
506C2#2262087111yescharacteristic
507C2#2262105811yescharacteristic
508C2#2262107411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1948C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1948D73
hypocenterC73#21981
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. #22610695C2semidirect
5. #22610694C2semidirect
6. #22610692C2semidirect
9. #22610086C2semidirect
10. #22610057C2semidirect
11. C73×#1948C2semidirect
17. #17988639C4semidirect
77. C4⋊#10596831C2×C4semidirect
79. C8×Dic73C2×C4semidirect
92. C4⋊C8×C73C2×C4semidirect
214. C4×D73#106semidirect
261. #1948D73semidirect
266. C73#21981semidirect
342. C4⋊C8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886392
#226123352
#384×D732
#226106952
#226106942
#226106922
#226101252
#226101222
#226100862
#226100572
C73×#19482
#2198173