Finite Group Databasebeta

← #33248863

Subgroups — #33248863

Order 18688.

2238 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D73C211yesmaximal, characteristic
2#22613112C211yesmaximal, characteristic, complemented
3#22613111C211yesmaximal, characteristic, complemented
4#22613111C211yesmaximal, characteristic, complemented
5#22612986C214yesmaximal, complemented
6#22612805C214yesmaximal
7#22612765C214yesmaximal
8#22612674C214yesmaximal, complemented
9#22612646C214yesmaximal, complemented
10#22612530C214yesmaximal
11#22612328C211yesmaximal, characteristic
12#22612328C211yesmaximal, characteristic
13C73×#2381C211yesmaximal, characteristic, complemented
14C22×Q8×D73C2212yes—
15C2×#14556591C2211yescharacteristic
16C2×#14556590C2212yes—
17C2×#14556590C2212yes—
18C2×#14556589C2212yes—
19C4⋊Q8×D73C4116yescomplemented
20C2×C4×Dic146C2214yes—
21C2×#14556556C2214yes—
22C2×C4⋊C4×D73C2212yes—
23C2×C4⋊C4×D73C2212yes—
24C2×C42×D73C2211yescharacteristic
25#17988494C2214yes—
26#17988491C2214yes—
27#17988491C2214yes—
28#17988491C2214yes—
29#17988491C2214yes—
30#17988484C2214yes—
31#17988482C2214yes—
32#17988474C2214yes—
33#17988473C2214yes—
34#17988473C2214yes—
35#17988472C2214yes—
36#17988472C2214yes—
37#17988471C2214yes—
38#17988470C2214yes—
39#17988470C2214yes—
40#17988469C2214yes—
41#17988468C2214yes—
42#17988455C2211yescharacteristic
43#17988455C2211yescharacteristic
44#17988451C2212yes—
45#17988451C2212yes—
46#17988449C2212yes—
47#17988449C2212yes—
48#17988446C2212yes—
49#17988446C2214yes—
50#17988446C2212yes—
51#17988444C2214yes—
52C4×#11924007C2214yes—
53#17988438C2214yes—
54#17988438C2212yes—
55#17988438C2212yes—
56C4×#11924005C2214yes—
57#17988434C2212yes—
58#17988434C2214yes—
59#17988434C2212yes—
60#17988432C2212yes—
61#17988432C2212yes—
62C2×C4⋊Q8×C73C2211yescharacteristic
63#391×C73C2214yes—
64#387×C73C2214yes—
65#387×C73C2214yes—
66#383×C73C2211yescharacteristic
67#383×C73C2211yescharacteristic
68C22×Dic146C2314yes—
69C22×Dic146C2312yes—
70C2×Q8×D73—44no—
71C2×Q8×D73—44no—
72C2×Q8×D73C2×C4116yes—
73C22×C4×D73C2311yescharacteristic
74C22×C4×D73C2311yescharacteristic
75C22×C4×D73C2311yescharacteristic
76C22×C4×D73C2314yes—
77#14556591C2×C418yes—
78#14556590C2×C4116yes—
79#14556590C2×C4116yes—
80#14556589C2×C4116yes—
81C4×Dic146—48no—
82#14556556—48no—
83C2×C4⋊Dic73C2314yes—
84C2×C4⋊Dic73C2312yes—
85C2×C4⋊Dic73C2312yes—
86C2×#11924007C2314yes—
87C2×#11924007C2314yes—
88C2×#11924007C2314yes—
89C2×#11924007C2314yes—
90C2×#11924007C2314yes—
91C2×#11924007C2314yes—
92C4⋊C4×D73C2×C4116yes—
93C4⋊C4×D73C2×C4116yes—
94C2×#11924005C2314yes—
95C2×#11924005C2314yes—
96C2×C4×Dic73C2314yes—
97C2×C4×Dic73C2314yes—
98C2×C4×Dic73C2314yes—
99C2×C4×Dic73C2314yes—
100C2×C4×Dic73C2314yes—
101C2×C4×Dic73C2311yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103C2×C4×Dic73C2311yescharacteristic
104C42×D73C2×C418yes—
105#14556498C2312yes—
106#14556498C2312yes—
107#14556498C2314yes—
108#14556498C2314yes—
109#14556498C2312yes—
110#14556498C2312yes—
111#14556498C2314yes—
112#14556498C2314yes—
113#14556498C2314yes—
114#14556498C2314yes—
115#14556498C2312yes—
116#14556498C2312yes—
117#14556498C2314yes—
118#14556498C2314yes—
119C22×Q8×C73C2312yes—
120C4⋊Q8×C73C2×C418yes—
121C2×C4⋊C4×C73C2314yes—
122C2×C4⋊C4×C73C2312yes—
123C2×C4⋊C4×C73C2312yes—
124C2×C42×C73C2314yes—
125C2×C42×C73C2311yescharacteristic
126#95×C73C2312yes—
127#95×C73C2312yes—
128#95×C73C2312yes—
129#95×C73C2312yes—
130#95×C73C2312yes—
131#95×C73C2312yes—
132#95×C73C2314yes—
133#95×C73C2314yes—
134C23×D73C2411yescharacteristic
135C2×Dic146—28no—
136C2×Dic146—28no—
137C2×Dic146—42no—
138C2×Dic146—42no—
139C2×Dic146C22×C4116yes—
140C2×Dic146C22×C418yes—
141Q8×D73—432no—
142C22×Dic73C2414yes—
143C22×Dic73C2414yes—
144C22×Dic73C2411yescharacteristic
145C22×Dic73C2414yes—
146C22×Dic73C2411yescharacteristic
147C22×Dic73C2411yescharacteristic
148C2×C4×D73—24no—
149C2×C4×D73—24no—
150C2×C4×D73—24no—
151C2×C4×D73—44no—
152C2×C4×D73—44no—
153C2×C4×D73C22×C414yes—
154C2×C4×D73C22×C414yes—
155C2×C4×D73C22×C414yes—
156C2×C4×D73C22×C4116yes—
157C4⋊Dic73—44no—
158C4⋊Dic73C22×C418yes—
159C4⋊Dic73C22×C418yes—
160#11924007—28no—
161#11924007—28no—
162#11924007—44no—
163#11924007—44no—
164#11924007C22×C4116yes—
165#11924007C22×C4116yes—
166#11924005—28no—
167#11924005—28no—
168C4×Dic73—28no—
169C4×Dic73—28no—
170C4×Dic73—44no—
171C4×Dic73—44no—
172C4×Dic73—44no—
173C4×Dic73C22×C414yes—
174C4×Dic73C22×C414yes—
175C4×Dic73C22×C414yes—
176C2×Q8×C73—42no—
177C2×Q8×C73—42no—
178C2×Q8×C73C22×C418yes—
179C22×C4×C73C2414yes—
180C22×C4×C73C2414yes—
181C22×C4×C73C2411yescharacteristic
182C22×C4×C73C2411yescharacteristic
183C22×C4×C73C2411yescharacteristic
184C22×C4×C73C2414yes—
185C4⋊C4×C73—44no—
186C4⋊C4×C73C22×C418yes—
187C4⋊C4×C73C22×C418yes—
188C42×C73—44no—
189C42×C73C22×C414yes—
190C23×C73C2511yescharacteristic
191C22×D73—22no—
192C22×D73—22no—
193C22×D73C23×C412yes—
194C22×D732−1+414yes—
195Q8×C73—48no—
196C2×C4×C73—24no—
197C2×C4×C73—24no—
198C2×C4×C73—22no—
199C2×C4×C73—22no—
200C2×C4×C73—22no—
201C2×C4×C73—24no—
202C2×C4×C73—24no—
203C2×C4×C73—24no—
204C2×C4×C73—24no—
205C2×C4×C73—24no—
206C2×C4×C73—24no—
207C2×C4×C73C23×C412yes—
208C2×C4×C73C23×C412yes—
209C2×C4×C73C23×C412yes—
210C2×C4×C73C23×C418yes—
211C2×Dic73—24no—
212C2×Dic73—24no—
213C2×Dic73—22no—
214C2×Dic73—24no—
215C2×Dic73—24no—
216C2×Dic73—24no—
217C2×Dic73—24no—
218C2×Dic73—24no—
219C2×Dic73—24no—
220C2×Dic73C23×C412yes—
221C2×Dic73C23×C418yes—
222C2×Dic73C23×C412yes—
223C2×Dic73C23×C412yes—
224C2×Dic73H(5,𝔽2)14yes—
225C2×Dic732−1+414yes—
226C4×D73—28no—
227C4×D73—28no—
228C4×D73—28no—
229C4×D73—416no—
230Dic146—232no—
231Dic146—48no—
232C22×C73C24×C411yescharacteristic
233C22×C73C2×H(5,𝔽2)12yes—
234C22×C73C2×2−1+412yes—
235C22×C73C2×2−1+412yes—
236C2×D73—22no—
237C2×D73—22no—
238C2×D73—28no—
239C2×D73#51914yes—
240C4×C73—28no—
241C4×C73—22no—
242C4×C73—22no—
243C4×C73—22no—
244C4×C73—44no—
245C4×C73—44no—
246Dic73—28no—
247Dic73—22no—
248Dic73—44no—
249Dic73—44no—
250Dic73#51814yes—
251Dic73#51914yes—
252#34371—731nomaximal, Sylow
253C2×C73C2×#51811yescharacteristic
254C2×C73C2×#51911yescharacteristic
255C2×C73C2×#51911yescharacteristic
256C2×C73#355214yes—
257D73—24no—
258C22×C4⋊Q8—731no—
259#2381—731no—
260#2381—731no—
261#2381—731no—
262#2381D7311yescharacteristic, complemented
263#2365—734no—
264#2345—734no—
265Q8×C22⋊C4—734no—
266#2331—734no—
267#2327—734no—
268#2313—734no—
269#2283—731no—
270#2283—731no—
271C73#3437111yescharacteristic, complemented, Sylow
272C23×Q8—732no—
273C2×C4⋊Q8—732no—
274C2×C4⋊Q8—7316no—
275C2×C4⋊Q8—732no—
276C2×C4⋊Q8—732no—
277C2×C4⋊Q8—731no—
278C2×C4⋊Q8C2×D7311yescharacteristic
279C2×C4×Q8—734no—
280C2×C4×Q8—734no—
281C22×C4⋊C4—732no—
282C22×C4⋊C4—732no—
283C22×C42—731no—
284#391—734no—
285#391—734no—
286#391—734no—
287#391—734no—
288#391—734no—
289#391C2×D7314yes—
290#389—734no—
291#389—734no—
292#387—734no—
293#387—734no—
294#387—734no—
295#387—734no—
296#387—734no—
297#387—734no—
298#387—734no—
299#387—734no—
300#387—734no—
301#387—734no—
302#387C2×D7314yes—
303#387C2×D7314yes—
304#383—731no—
305#383—731no—
306#383C2×D7311yescharacteristic
307#383C2×D7311yescharacteristic
308#382—732no—
309#382—732no—
310#381—732no—
311#381—732no—
312#380—734no—
313#380—734no—
314#380—732no—
315#380—732no—
316C4×C4⋊C4—734no—
317C4×C4⋊C4—734no—
318C4×C4⋊C4—732no—
319C4×C4⋊C4—732no—
320C4×C22⋊C4—734no—
321C4×C22⋊C4—734no—
322C4×C22⋊C4—732no—
323C4×C22⋊C4—732no—
324#376—732no—
325#376—732no—
326C22×Q8—2924no—
327C22×Q8—2924no—
328C22×Q8—7316no—
329C22×Q8—732no—
330C22×Q8—734no—
331C22×Q8C22×D7312yes—
332C23×C4—734no—
333C23×C4—731no—
334C23×C4—731no—
335C23×C4—731no—
336C4⋊Q8—7316no—
337C4⋊Q8—7316no—
338C4⋊Q8—7316no—
339C4⋊Q8—738no—
340C4⋊Q8C4×D7318yes—
341C4×Q8—2928no—
342C4×Q8—2928no—
343C2×C4⋊C4—734no—
344C2×C4⋊C4—734no—
345C2×C4⋊C4—734no—
346C2×C4⋊C4—734no—
347C2×C4⋊C4—734no—
348C2×C4⋊C4—732no—
349C2×C4⋊C4—732no—
350C2×C4⋊C4—7316no—
351C2×C4⋊C4—7316no—
352C2×C4⋊C4—734no—
353C2×C4⋊C4—734no—
354C2×C4⋊C4C22×D7314yes—
355C2×C4⋊C4C22×D7312yes—
356C2×C4⋊C4C22×D7312yes—
357C2×C22⋊C4—734no—
358C2×C22⋊C4—734no—
359C2×C42—734no—
360C2×C42—734no—
361C2×C42—734no—
362C2×C42—731no—
363C2×C42—731no—
364C2×C42—731no—
365C2×C42—738no—
366C2×C42—734no—
367C2×C42—734no—
368C2×C42C22×D7314yes—
369C2×C42C22×D7311yescharacteristic
370#95—732no—
371#95—732no—
372#95—734no—
373#95—734no—
374#95—732no—
375#95—732no—
376#95—734no—
377#95—734no—
378#95—734no—
379#95—734no—
380#95—734no—
381#95—734no—
382#95—732no—
383#95—732no—
384#95C22×D7312yes—
385#95C22×D7312yes—
386#95C22×D7312yes—
387#95C22×D7312yes—
388#95C22×D7314yes—
389#95C22×D7314yes—
390#95C22×D7312yes—
391#95C22×D7312yes—
392C24—731no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—2922no—
396C2×Q8—2922no—
397C2×Q8—29232no—
398C2×Q8—738no—
399C2×Q8—7316no—
400C2×Q8—1468no—
401C2×Q8—1468no—
402C2×Q8C2×C4×D7318yes—
403C22×C4—2924no—
404C22×C4—2924no—
405C22×C4—7316no—
406C22×C4—731no—
407C22×C4—731no—
408C22×C4—731no—
409C22×C4—734no—
410C22×C4—734no—
411C22×C4—734no—
412C22×C4—734no—
413C22×C4—734no—
414C22×C4—734no—
415C22×C4—1464no—
416C22×C4—1464no—
417C22×C4—1464no—
418C22×C4C23×D7314yes—
419C22×C4C23×D7314yes—
420C22×C4C23×D7311yescharacteristic
421C22×C4C23×D7311yescharacteristic
422C22×C4C23×D7311yescharacteristic
423C22×C4C23×D7314yes—
424C4⋊C4—44no—
425C4⋊C4—2924no—
426C4⋊C4—2924no—
427C4⋊C4—2924no—
428C4⋊C4—7316no—
429C4⋊C4—7316no—
430C4⋊C4—738no—
431C4⋊C4—738no—
432C4⋊C4—1468no—
433C4⋊C4—1468no—
434C4⋊C4C2×C4×D7318yes—
435C4⋊C4C2×C4×D7318yes—
436C22⋊C4—1468no—
437C22⋊C4—1468no—
438C42—44no—
439C42—2924no—
440C42—2924no—
441C42—2924no—
442C42—734no—
443C42—734no—
444C42—734no—
445C42—1468no—
446C42—1468no—
447C42C2×C4×D7314yes—
448C23—732no—
449C23—734no—
450C23—1462no—
451C23—1462no—
452C23C24×D7311yescharacteristic
453Q8—48no—
454Q8—2928no—
455Q8—14632no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—29216no—
468C2×C4—732no—
469C2×C4—732no—
470C2×C4—734no—
471C2×C4—734no—
472C2×C4—732no—
473C2×C4—738no—
474C2×C4—1468no—
475C2×C4—1468no—
476C2×C4—1468no—
477C2×C4—1464no—
478C2×C4—1464no—
479C2×C4—1464no—
480C2×C4—1464no—
481C2×C4—1464no—
482C2×C4—1464no—
483C2×C4—1464no—
484C2×C4—1464no—
485C2×C4—1462no—
486C2×C4C22×C4×D7312yes—
487C2×C4C22×C4×D7312yes—
488C2×C4C22×C4×D7312yes—
489C2×C4C22×C4×D7318yes—
490C22—734no—
491C22—1468no—
492C22—1462no—
493C22—1462no—
494C22C23×C4×D7311yescharacteristic
495C22H(5,𝔽2)×D7312yes—
496C222−1+4×D7312yes—
497C222−1+4×D7312yes—
498C4—28no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—44no—
503C4—44no—
504C4—2924no—
505C4—2924no—
506C4—734no—
507C4—734no—
508C4—1468no—
509C4—1462no—
510C2—1464no—
511C2#518×D7311yescharacteristic
512C2#519×D7311yescharacteristic
513C2#519×D7311yescharacteristic
514C2#2262436514yes—
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2381C2
socle, cosocleC23×C73C25
prime radicalC23×D73C24
hypercenter#2381D73
hypocenterC73#34371
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2381

Complements

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

NH ≅ G/Ntype
2. #22613112C2semidirect
3. #22613111C2semidirect
4. #22613111C2semidirect
5. #22612986C2semidirect
8. #22612674C2semidirect
9. #22612646C2semidirect
13. C73×#2381C2semidirect
19. C4⋊Q8×D73C4semidirect
262. #2381D73semidirect
271. C73#34371semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D732
#226131122
#226131112
#226129862
#226128052
#226127652
#226126742
#226126462
#226125302
#226123282
C73×#23812
#3437173