Finite Group Databasebeta

← #33003509

Subgroups — #33003509

Order 18688.

782 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988561C211yesmaximal, characteristic, complemented
2C2×#17988380C212yesmaximal, complemented
3#22609849C211yesmaximal, characteristic, complemented
4#22608056C212yesmaximal, complemented
5C73×#2431C211yesmaximal, characteristic
6C2×#14556550—21no—
7#17988561—24no—
8C2×#14556506—21no—
9C2×#14556506—21no—
10C2×#14556506C2212yes—
11C2×#14556506C2212yes—
12#17988450—21no—
13#17988380—24no—
14C22×#41×C73C2211yescharacteristic
15C2×#127×C73C2211yescharacteristic
16C2×#124×C73C411yescharacteristic, complemented
17C2×#121×C73C411yescharacteristic, complemented
18#392×C73—21no—
19#392×C73—21no—
20#391×C73C411yescharacteristic, complemented
21#390×C73C2211yescharacteristic
22#386×C73—21no—
23#386×C73—21no—
24C4×C22⋊C4×C73C411yescharacteristic, complemented
25#14556550—24no—
26#14556550—42no—
27C2×C4⋊Dic73—41no—
28C2×C22⋊Dic73—22no—
29C2×C22⋊Dic73—22no—
30C2×C4×Dic73—41no—
31#14556506—24no—
32#14556506—22no—
33#14556506—22no—
34#14556506—22no—
35#14556506—42no—
36#14556506D414yescomplemented
37#14556498—41no—
38#14556498—41no—
39C2×#41×C73—22no—
40C2×#41×C73—22no—
41C2×#41×C73—41no—
42C2×#41×C73—41no—
43C2×#41×C73—42no—
44C22×Q8×C73C2×C411yescharacteristic
45C22×D4×C73—21no—
46C22×D4×C73—21no—
47C22×D4×C73—21no—
48C22×D4×C73C2×C411yescharacteristic
49C22×D4×C73D411yescharacteristic
50C22×D4×C73D411yescharacteristic
51C22×D4×C73C2311yescharacteristic
52C23×C4×C73C2×C411yescharacteristic
53C23×C4×C73D411yescharacteristic
54C23×C4×C73D411yescharacteristic
55#127×C73—22no—
56#127×C73—22no—
57#124×C73—42no—
58#121×C73—41no—
59#121×C73—41no—
60C2×C4⋊C4×C73C2×C411yescharacteristic
61C2×C22⋊C4×C73—21no—
62C2×C22⋊C4×C73—21no—
63C2×C22⋊C4×C73—21no—
64C2×C22⋊C4×C73—21no—
65C2×C22⋊C4×C73—21no—
66C2×C22⋊C4×C73—21no—
67C2×C42×C73C2×C411yescharacteristic
68C2×C42×C73C2×C411yescharacteristic
69#95×C73—21no—
70#95×C73—21no—
71#95×C73—21no—
72C22×Dic73—42no—
73C22×Dic73—42no—
74C4⋊Dic73—44no—
75C22⋊Dic73—24no—
76C22⋊Dic73—42no—
77C22⋊Dic73—42no—
78C22⋊Dic73—42no—
79C4×Dic73—44no—
80C24×C73—21no—
81C24×C73C22⋊C411yescharacteristic
82C24×C73C2×D411yescharacteristic
83C24×C73C2×D411yescharacteristic
84#41×C73—44no—
85#41×C73—44no—
86#41×C73—44no—
87#41×C73—44no—
88C2×Q8×C73—22no—
89C2×Q8×C73—22no—
90C2×Q8×C73—41no—
91C2×D4×C73—21no—
92C2×D4×C73—22no—
93C2×D4×C73—21no—
94C2×D4×C73—22no—
95C2×D4×C73—22no—
96C2×D4×C73—41no—
97C2×D4×C73—41no—
98C2×D4×C73—41no—
99C2×D4×C73—41no—
100C2×D4×C73—41no—
101C2×D4×C73—41no—
102C2×D4×C73—41no—
103C2×D4×C73—42no—
104C2×D4×C73—42no—
105C2×D4×C73—42no—
106C2×D4×C73—41no—
107C2×D4×C73—42no—
108C2×D4×C73—41no—
109C2×D4×C73—81no—
110C2×D4×C73—81no—
111C2×D4×C73—81no—
112C2×D4×C73—81no—
113C2×D4×C73C2×D412yes—
114C2×D4×C73#4112yes—
115C22×C4×C73—21no—
116C22×C4×C73—21no—
117C22×C4×C73—21no—
118C22×C4×C73—21no—
119C22×C4×C73—22no—
120C22×C4×C73—21no—
121C22×C4×C73—22no—
122C22×C4×C73—21no—
123C22×C4×C73—21no—
124C22×C4×C73—21no—
125C22×C4×C73—21no—
126C22×C4×C73—22no—
127C22×C4×C73—21no—
128C22×C4×C73—21no—
129C22×C4×C73—41no—
130C22×C4×C73—41no—
131C22×C4×C73C22⋊C411yescharacteristic
132C22×C4×C73C22⋊C411yescharacteristic
133C22×C4×C73C22⋊C411yescharacteristic
134C22×C4×C73C22×C411yescharacteristic
135C4⋊C4×C73—41no—
136C22⋊C4×C73—42no—
137C22⋊C4×C73—42no—
138C22⋊C4×C73—41no—
139C22⋊C4×C73—41no—
140C22⋊C4×C73—42no—
141C22⋊C4×C73—81no—
142C22⋊C4×C73—81no—
143C42×C73—22no—
144C42×C73—41no—
145C23×C73—21no—
146C23×C73—21no—
147C23×C73—21no—
148C23×C73—21no—
149C23×C73—21no—
150C23×C73—21no—
151C23×C73—21no—
152C23×C73—41no—
153C23×C73—41no—
154C23×C73—41no—
155C23×C73—41no—
156C23×C73—41no—
157C23×C73—41no—
158C23×C73—41no—
159C23×C73—41no—
160C23×C73—41no—
161C23×C73—41no—
162C23×C73—41no—
163C23×C73—41no—
164C23×C73—41no—
165C23×C73C2×C22⋊C411yescharacteristic
166C23×C73#12011yescharacteristic
167C23×C73#12112yes—
168C23×C73#12311yescharacteristic
169Q8×C73—44no—
170D4×C73—42no—
171D4×C73—42no—
172D4×C73—44no—
173D4×C73—42no—
174D4×C73—42no—
175D4×C73—82no—
176D4×C73—82no—
177D4×C73—84no—
178D4×C73—82no—
179D4×C73—84no—
180C2×C4×C73—22no—
181C2×C4×C73—22no—
182C2×C4×C73—22no—
183C2×C4×C73—22no—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—21no—
187C2×C4×C73—21no—
188C2×C4×C73—21no—
189C2×C4×C73—21no—
190C2×C4×C73—22no—
191C2×C4×C73—21no—
192C2×C4×C73—21no—
193C2×C4×C73—21no—
194C2×C4×C73—42no—
195C2×C4×C73—42no—
196C2×C4×C73—42no—
197C2×C4×C73—42no—
198C2×C4×C73—41no—
199C2×C4×C73—41no—
200C2×C4×C73—41no—
201C2×C4×C73—41no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73#9911yescharacteristic
207C2×C4×C73#9911yescharacteristic
208C2×C4×C73#9912yes—
209C2×C4×C73C4×D412yes—
210C2×Dic73—44no—
211C2×Dic73—82no—
212C2×Dic73—82no—
213C2×Dic73—82no—
214C2×Dic73—82no—
215C2×Dic73—82no—
216C22×C73—21no—
217C22×C73—21no—
218C22×C73—21no—
219C22×C73—22no—
220C22×C73—21no—
221C22×C73—41no—
222C22×C73—42no—
223C22×C73—41no—
224C22×C73—41no—
225C22×C73—41no—
226C22×C73—41no—
227C22×C73—41no—
228C22×C73—41no—
229C22×C73—41no—
230C22×C73—41no—
231C22×C73—81no—
232C22×C73—81no—
233C22×C73—81no—
234C22×C73—81no—
235C22×C73—81no—
236C22×C73—81no—
237C22×C73—81no—
238C22×C73—81no—
239C22×C73—81no—
240C22×C73—81no—
241C22×C73—81no—
242C22×C73#38611yescharacteristic
243C22×C73C2×#9911yescharacteristic
244C22×C73C2×#9911yescharacteristic
245C4×C73—21no—
246C4×C73—21no—
247C4×C73—22no—
248C4×C73—22no—
249C4×C73—42no—
250C4×C73—41no—
251C4×C73—81no—
252C4×C73—81no—
253Dic73—84no—
254Dic73—162no—
255#13089—731nomaximal, Sylow
256C2×C73—21no—
257C2×C73—21no—
258C2×C73—41no—
259C2×C73—42no—
260C2×C73—41no—
261C2×C73—81no—
262C2×C73—161no—
263C2×C73#189811yescharacteristic
264C2×C73#212611yescharacteristic
265C2×C73#212811yescharacteristic
266C2×#410—731no—
267#2431D7311yescharacteristic
268C2×#353—732no—
269#1782—731no—
270#1393—732no—
271C73#1308911yescharacteristic, complemented, Sylow
272C22×#41C2×D7311yescharacteristic
273C2×#127C2×D7311yescharacteristic
274C2×#124Dic7311yescharacteristic, complemented
275C2×#121Dic7311yescharacteristic, complemented
276C2×#117—1461no—
277#410—1464no—
278C2×#99—732no—
279C2×#99—732no—
280C2×#99—1461no—
281C2×#99—1461no—
282#392—21no—
283#392—21no—
284#391Dic7311yescharacteristic, complemented
285#390C2×D7311yescharacteristic
286#386—21no—
287#386—21no—
288#381—1461no—
289C4×C22⋊C4Dic7311yescharacteristic, complemented
290#353—1464no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—41no—
294C2×#41—41no—
295C2×#41—42no—
296C22×Q8C2×Dic7311yescharacteristic
297C22×D4—21no—
298C22×D4—21no—
299C22×D4—21no—
300C22×D4C2×Dic7311yescharacteristic
301C22×D4C22⋊D7311yescharacteristic
302C22×D4C22⋊D7311yescharacteristic
303C22×D4C22×D7311yescharacteristic
304C23×C4C2×Dic7311yescharacteristic
305C23×C4C22⋊D7311yescharacteristic
306C23×C4C22⋊D7311yescharacteristic
307#127—22no—
308#127—22no—
309#124—42no—
310#121—41no—
311#121—41no—
312#117—2922no—
313#117—1464no—
314C2×C4⋊C4—2921no—
315C2×C4⋊C4C2×Dic7311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—1462no—
323C2×C22⋊C4—1462no—
324C2×C42—2921no—
325C2×C42C2×Dic7311yescharacteristic
326C2×C42C2×Dic7311yescharacteristic
327#99—2922no—
328#99—734no—
329#99—1462no—
330#99—1462no—
331#99—1464no—
332#99—1462no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—2921no—
337#95—2921no—
338C24—21no—
339C24C22⋊Dic7311yescharacteristic
340C24C2×C22⋊D7311yescharacteristic
341C24C2×C22⋊D7311yescharacteristic
342#41—44no—
343#41—44no—
344#41—44no—
345#41—44no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—41no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4D4×D7312yes—
372C2×D4#1192402712yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—2922no—
390C22×C4—2922no—
391C22×C4C22⋊Dic7311yescharacteristic
392C22×C4C22⋊Dic7311yescharacteristic
393C22×C4C22⋊Dic7311yescharacteristic
394C22×C4C22×Dic7311yescharacteristic
395C4⋊C4—41no—
396C4⋊C4—2924no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—42no—
402C22⋊C4—81no—
403C22⋊C4—81no—
404C22⋊C4—2922no—
405C22⋊C4—2922no—
406C22⋊C4—2922no—
407C22⋊C4—1464no—
408C42—22no—
409C42—41no—
410C42—2924no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23C2×C22⋊Dic7311yescharacteristic
432C23#1455655911yescharacteristic
433C23#1455656312yes—
434C23#1455657211yescharacteristic
435Q8—44no—
436D4—42no—
437D4—42no—
438D4—44no—
439D4—42no—
440D4—42no—
441D4—82no—
442D4—82no—
443D4—84no—
444D4—82no—
445D4—84no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—2924no—
473C2×C4—5842no—
474C2×C4—5842no—
475C2×C4—5842no—
476C2×C4—5842no—
477C2×C4—5842no—
478C2×C4#1455650612yes—
479C2×C4#1455650611yescharacteristic
480C2×C4#1455650611yescharacteristic
481C2×C4D4×Dic7312yes—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—41no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22#1798846711yescharacteristic
509C22C2×#1455650611yescharacteristic
510C22C2×#1455650611yescharacteristic
511C4—21no—
512C4—21no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—41no—
517C4—81no—
518C4—81no—
519C4—5844no—
520C4—11682no—
521C2—21no—
522C2—21no—
523C2—41no—
524C2—42no—
525C2—41no—
526C2—81no—
527C2—161no—
528C2#2261038511yescharacteristic
529C2#2261159611yescharacteristic
530C2#2261160311yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC22×D4C22×D73
Fitting, prime radicalC73×#2431C2
socleC22×C73C2×#99
cosocleC22×D4×C73C23
hypercenter#2431D73
hypocenterC73#13089
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2431

Complements

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

NH ≅ G/Ntype
1. C2×#17988561C2semidirect
2. C2×#17988380C2semidirect
3. #22609849C2semidirect
4. #22608056C2semidirect
16. C2×#124×C73C4semidirect
17. C2×#121×C73C4semidirect
20. #391×C73C4semidirect
24. C4×C22⋊C4×C73C4semidirect
36. #14556506D4semidirect
271. C73#13089semidirect
274. C2×#124Dic73semidirect
275. C2×#121Dic73semidirect
284. #391Dic73semidirect
289. C4×C22⋊C4Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179885612
C2×#179883802
#226098492
#226080562
C73×#24312
#1308973