Finite Group Databasebeta

← #33003514

Subgroups — #33003514

Order 18688.

910 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×#14556506C211yesmaximal, characteristic
2C2×#17988382C212yesmaximal, complemented
3#22609846C211yesmaximal, characteristic
4#22608056C212yesmaximal, complemented
5C73×#2435C211yesmaximal, characteristic
6C22×C22⋊Dic73—21no—
7C2×#14556506—24no—
8C2×#14556506—21no—
9C2×#14556506—21no—
10C2×#14556506C2212yes—
11C2×#14556506C2212yes—
12#17988443—21no—
13#17988382—24no—
14C23×D4×C73C2211yescharacteristic
15C2×#124×C73C411yescharacteristic, complemented
16C2×#124×C73C2211yescharacteristic
17C2×C22⋊Q8×C73C411yescharacteristic, complemented
18#394×C73—21no—
19#394×C73—21no—
20#390×C73C411yescharacteristic, complemented
21#390×C73C2211yescharacteristic
22#379×C73—21no—
23#379×C73—21no—
24C4×C22⋊C4×C73C411yescharacteristic, complemented
25C23×Dic73—41no—
26C2×C22⋊Dic73—24no—
27C2×C22⋊Dic73—22no—
28C2×C22⋊Dic73—22no—
29C2×C22⋊Dic73—42no—
30C2×C22⋊Dic73—41no—
31C2×C22⋊Dic73—41no—
32C2×C22⋊Dic73—41no—
33#14556506—24no—
34#14556506—22no—
35#14556506—22no—
36#14556506—22no—
37#14556506—42no—
38#14556506D414yes—
39C25×C73D411yescharacteristic
40C25×C73D411yescharacteristic
41C22×Q8×C73C2×C411yescharacteristic, complemented
42C22×D4×C73—22no—
43C22×D4×C73—22no—
44C22×D4×C73—41no—
45C22×D4×C73—41no—
46C22×D4×C73—42no—
47C22×D4×C73C2×C411yescharacteristic
48C22×D4×C73D411yescharacteristic
49C22×D4×C73D411yescharacteristic
50C22×D4×C73C2311yescharacteristic
51C23×C4×C73C2×C411yescharacteristic
52#124×C73—22no—
53#124×C73—22no—
54#124×C73—42no—
55C22⋊Q8×C73—41no—
56C22⋊Q8×C73—41no—
57C2×C4⋊C4×C73—21no—
58C2×C4⋊C4×C73C2×C411yescharacteristic, complemented
59C2×C22⋊C4×C73—21no—
60C2×C22⋊C4×C73—21no—
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×C22⋊C4×C73—21no—
68C2×C22⋊C4×C73—21no—
69C2×C42×C73C2×C411yescharacteristic, complemented
70C2×C42×C73C2×C411yescharacteristic, complemented
71#95×C73—21no—
72C22×Dic73—44no—
73C22×Dic73—42no—
74C22×Dic73—42no—
75C22×Dic73—82no—
76C22×Dic73—82no—
77C22⋊Dic73—24no—
78C22⋊Dic73—44no—
79C22⋊Dic73—42no—
80C22⋊Dic73—42no—
81C22⋊Dic73—42no—
82C22⋊Dic73—82no—
83C22⋊Dic73—82no—
84C24×C73—21no—
85C24×C73—22no—
86C24×C73—21no—
87C24×C73—21no—
88C24×C73—21no—
89C24×C73—21no—
90C24×C73—21no—
91C24×C73—22no—
92C24×C73—41no—
93C24×C73—42no—
94C24×C73—42no—
95C24×C73—81no—
96C24×C73—81no—
97C24×C73C22⋊C411yescharacteristic
98C24×C73C22⋊C411yescharacteristic
99C24×C73C22⋊C411yescharacteristic
100C24×C73C2×D411yescharacteristic
101C24×C73C2×D411yescharacteristic
102C2×Q8×C73—22no—
103C2×Q8×C73—41no—
104C2×Q8×C73—41no—
105C2×D4×C73—22no—
106C2×D4×C73—22no—
107C2×D4×C73—21no—
108C2×D4×C73—22no—
109C2×D4×C73—21no—
110C2×D4×C73—22no—
111C2×D4×C73—41no—
112C2×D4×C73—41no—
113C2×D4×C73—41no—
114C2×D4×C73—41no—
115C2×D4×C73—41no—
116C2×D4×C73—41no—
117C2×D4×C73—44no—
118C2×D4×C73—44no—
119C2×D4×C73—44no—
120C2×D4×C73—44no—
121C2×D4×C73C2×D412yes—
122C2×D4×C73#4112yes—
123C22×C4×C73—21no—
124C22×C4×C73—21no—
125C22×C4×C73—22no—
126C22×C4×C73—21no—
127C22×C4×C73—21no—
128C22×C4×C73—21no—
129C22×C4×C73—21no—
130C22×C4×C73—41no—
131C22×C4×C73C22⋊C411yescharacteristic
132C22×C4×C73C22×C411yescharacteristic
133C4⋊C4×C73—41no—
134C4⋊C4×C73—42no—
135C22⋊C4×C73—42no—
136C22⋊C4×C73—42no—
137C22⋊C4×C73—41no—
138C22⋊C4×C73—41no—
139C22⋊C4×C73—42no—
140C22⋊C4×C73—41no—
141C22⋊C4×C73—42no—
142C22⋊C4×C73—41no—
143C22⋊C4×C73—81no—
144C22⋊C4×C73—81no—
145C22⋊C4×C73—81no—
146C22⋊C4×C73—81no—
147C42×C73—22no—
148C42×C73—41no—
149C23×C73—22no—
150C23×C73—22no—
151C23×C73—22no—
152C23×C73—22no—
153C23×C73—22no—
154C23×C73—21no—
155C23×C73—21no—
156C23×C73—21no—
157C23×C73—21no—
158C23×C73—21no—
159C23×C73—21no—
160C23×C73—21no—
161C23×C73—21no—
162C23×C73—21no—
163C23×C73—21no—
164C23×C73—21no—
165C23×C73—41no—
166C23×C73—42no—
167C23×C73—42no—
168C23×C73—42no—
169C23×C73—41no—
170C23×C73—41no—
171C23×C73—41no—
172C23×C73—41no—
173C23×C73—41no—
174C23×C73—41no—
175C23×C73—41no—
176C23×C73—41no—
177C23×C73—42no—
178C23×C73—42no—
179C23×C73—42no—
180C23×C73—81no—
181C23×C73—81no—
182C23×C73—81no—
183C23×C73—81no—
184C23×C73—84no—
185C23×C73—82no—
186C23×C73—82no—
187C23×C73—84no—
188C23×C73—84no—
189C23×C73#9911yescharacteristic
190C23×C73#9911yescharacteristic
191C23×C73#9912yes—
192C23×C73C2×C22⋊C411yescharacteristic
193C23×C73#12011yescharacteristic
194C23×C73#12112yes—
195C23×C73#12311yescharacteristic
196Q8×C73—82no—
197D4×C73—42no—
198D4×C73—42no—
199D4×C73—44no—
200D4×C73—44no—
201D4×C73—42no—
202D4×C73—42no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—41no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73C4×D412yes—
223C2×Dic73—44no—
224C2×Dic73—88no—
225C2×Dic73—82no—
226C2×Dic73—82no—
227C2×Dic73—82no—
228C2×Dic73—82no—
229C2×Dic73—82no—
230C22×C73—21no—
231C22×C73—21no—
232C22×C73—21no—
233C22×C73—21no—
234C22×C73—21no—
235C22×C73—21no—
236C22×C73—21no—
237C22×C73—22no—
238C22×C73—21no—
239C22×C73—22no—
240C22×C73—21no—
241C22×C73—42no—
242C22×C73—42no—
243C22×C73—42no—
244C22×C73—41no—
245C22×C73—42no—
246C22×C73—41no—
247C22×C73—44no—
248C22×C73—44no—
249C22×C73—42no—
250C22×C73—41no—
251C22×C73—41no—
252C22×C73—41no—
253C22×C73—41no—
254C22×C73—41no—
255C22×C73—41no—
256C22×C73—41no—
257C22×C73—41no—
258C22×C73—41no—
259C22×C73—81no—
260C22×C73—81no—
261C22×C73—81no—
262C22×C73—84no—
263C22×C73—81no—
264C22×C73—82no—
265C22×C73—82no—
266C22×C73—84no—
267C22×C73#35412yescomplemented
268C22×C73#38611yescharacteristic
269C22×C73C2×#9911yescharacteristic
270C22×C73C2×#9911yescharacteristic
271C4×C73—22no—
272C4×C73—41no—
273C4×C73—81no—
274C4×C73—81no—
275C4×C73—81no—
276C4×C73—81no—
277Dic73—84no—
278Dic73—162no—
279#13090—731nomaximal, Sylow
280C2×C73—21no—
281C2×C73—21no—
282C2×C73—22no—
283C2×C73—41no—
284C2×C73—42no—
285C2×C73—41no—
286C2×C73—42no—
287C2×C73—41no—
288C2×C73—81no—
289C2×C73#189811yescharacteristic
290C2×C73C2×#35411yescharacteristic
291C2×C73#212711yescharacteristic
292C22×#99—731no—
293#2435D7311yescharacteristic
294C2×#354—732no—
295#1781—731no—
296#1393—732no—
297C73#1309011yescharacteristic, complemented, Sylow
298C23×D4C2×D7311yescharacteristic
299C2×#124Dic7311yescharacteristic, complemented
300C2×#124C2×D7311yescharacteristic
301C2×C22⋊Q8Dic7311yescharacteristic, complemented
302C22×C22⋊C4—1461no—
303C2×#99—732no—
304C2×#99—732no—
305C2×#99—1461no—
306C2×#99—1464no—
307C2×#99—1461no—
308#394—21no—
309#394—21no—
310#390Dic7311yescharacteristic, complemented
311#390C2×D7311yescharacteristic
312#379—21no—
313#379—21no—
314#379—1461no—
315C4×C22⋊C4Dic7311yescharacteristic, complemented
316#354—1464no—
317C25C22⋊D7311yescharacteristic
318C25C22⋊D7311yescharacteristic
319C22×Q8C2×Dic7311yescharacteristic, complemented
320C22×D4—22no—
321C22×D4—22no—
322C22×D4—41no—
323C22×D4—41no—
324C22×D4—42no—
325C22×D4C2×Dic7311yescharacteristic
326C22×D4C22⋊D7311yescharacteristic
327C22×D4C22⋊D7311yescharacteristic
328C22×D4C22×D7311yescharacteristic
329C23×C4—2921no—
330C23×C4C2×Dic7311yescharacteristic
331#124—22no—
332#124—22no—
333#124—42no—
334C22⋊Q8—41no—
335C22⋊Q8—41no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4C2×Dic7311yescharacteristic, complemented
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—2921no—
349C2×C22⋊C4—2922no—
350C2×C22⋊C4—2921no—
351C2×C22⋊C4—2921no—
352C2×C22⋊C4—1462no—
353C2×C22⋊C4—1464no—
354C2×C22⋊C4—1462no—
355C2×C42C2×Dic7311yescharacteristic, complemented
356C2×C42C2×Dic7311yescharacteristic, complemented
357#99—2922no—
358#99—734no—
359#99—1462no—
360#99—1462no—
361#99—1464no—
362#99—1462no—
363#95—21no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—41no—
373C24—42no—
374C24—42no—
375C24—81no—
376C24—81no—
377C24C22⋊Dic7311yescharacteristic
378C24C22⋊Dic7311yescharacteristic
379C24C22⋊Dic7311yescharacteristic
380C24C2×C22⋊D7311yescharacteristic
381C24C2×C22⋊D7311yescharacteristic
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4D4×D7312yes—
402C2×D4#1192402712yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4—2924no—
412C22×C4—2922no—
413C22×C4—2922no—
414C22×C4—5842no—
415C22×C4—5842no—
416C22×C4C22⋊Dic7311yescharacteristic
417C22×C4C22×Dic7311yescharacteristic
418C4⋊C4—41no—
419C4⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—42no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C22⋊C4—2924no—
433C22⋊C4—2922no—
434C22⋊C4—2922no—
435C22⋊C4—2922no—
436C22⋊C4—5842no—
437C22⋊C4—5842no—
438C22⋊C4—1464no—
439C42—22no—
440C42—41no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—84no—
477C23—82no—
478C23—82no—
479C23—84no—
480C23—84no—
481C23#1455650611yescharacteristic
482C23#1455650611yescharacteristic
483C23#1455650612yes—
484C23C2×C22⋊Dic7311yescharacteristic
485C23#1455655911yescharacteristic
486C23#1455656312yes—
487C23#1455657211yescharacteristic
488Q8—82no—
489D4—42no—
490D4—42no—
491D4—44no—
492D4—44no—
493D4—42no—
494D4—42no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—2924no—
515C2×C4—5848no—
516C2×C4—5842no—
517C2×C4—5842no—
518C2×C4—5842no—
519C2×C4—5842no—
520C2×C4—5842no—
521C2×C4D4×Dic7312yes—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—22no—
532C22—21no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—84no—
555C22—81no—
556C22—82no—
557C22—82no—
558C22—84no—
559C22#1798838212yescomplemented
560C22#1798846711yescharacteristic
561C22C2×#1455650611yescharacteristic
562C22C2×#1455650611yescharacteristic
563C4—22no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—5844no—
570C4—11682no—
571C2—21no—
572C2—21no—
573C2—22no—
574C2—41no—
575C2—42no—
576C2—41no—
577C2—42no—
578C2—41no—
579C2—81no—
580C2#2261038511yescharacteristic
581C2C2×#1798838211yescharacteristic
582C2#2261160011yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#2435C2
socleC22×C73C2×#99
prime radicalC23×D4×C73C22
cosocleC22×D4×C73C23
hypercenter#2435D73
hypocenterC73#13090
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2435

Complements

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

NH ≅ G/Ntype
2. C2×#17988382C2semidirect
4. #22608056C2semidirect
15. C2×#124×C73C4semidirect
17. C2×C22⋊Q8×C73C4semidirect
20. #390×C73C4semidirect
24. C4×C22⋊C4×C73C4semidirect
41. C22×Q8×C73C2×C4semidirect
58. C2×C4⋊C4×C73C2×C4semidirect
69. C2×C42×C73C2×C4semidirect
70. C2×C42×C73C2×C4semidirect
267. C22×C73#354semidirect
297. C73#13090semidirect
299. C2×#124Dic73semidirect
301. C2×C22⋊Q8Dic73semidirect
310. #390Dic73semidirect
315. C4×C22⋊C4Dic73semidirect
319. C22×Q8C2×Dic73semidirect
337. C2×C4⋊C4C2×Dic73semidirect
355. C2×C42C2×Dic73semidirect
356. C2×C42C2×Dic73semidirect
559. C22#17988382semidirect

Maximal subgroups

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

subgroupindex
C22×#145565062
C2×#179883822
#226098462
#226080562
C73×#24352
#1309073