Finite Group Databasebeta

← #33098225

Subgroups — #33098225

Order 18688.

732 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988884C211yesmaximal, characteristic
2#22612575C211yesmaximal, characteristic
3#22610889C211yesmaximal, characteristic, complemented
4#22610687C211yesmaximal, characteristic, complemented
5#22610671C211yesmaximal, characteristic, complemented
6#22610661C211yesmaximal, characteristic, complemented
7#22610627C211yesmaximal, characteristic
8#22610613C211yesmaximal, characteristic
9#22610551C211yesmaximal, characteristic
10#22610551C211yesmaximal, characteristic
11#22610523C211yesmaximal, characteristic, complemented
12#22610509C211yesmaximal, characteristic, complemented
13#22609928C211yesmaximal, characteristic
14#22609788C211yesmaximal, characteristic, complemented
15C73×#2330C211yesmaximal, characteristic, complemented
16C2×C4×D292C2211yescharacteristic
17#17988884C418yescomplemented
18C2×#14556524C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556511C2211yescharacteristic
25C2×#14556511C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30#17988489C2211yescharacteristic
31#17988489C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988476—21no—
35#17988476—21no—
36#17988449—21no—
37#17988446—21no—
38C4×C4⋊Dic73C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C2×#125×C73C2211yescharacteristic
50#389×C73—21no—
51#389×C73—21no—
52#389×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#387×C73—21no—
55#387×C73—21no—
56#384×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59#376×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C4×D292—22no—
67C4×D292C2×C414yes—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924005—21no—
74C2×#11924005—21no—
75C2×#11924005—21no—
76C2×#11924005—21no—
77C2×#11924005—21no—
78C2×C4×Dic73—21no—
79#14556524—41no—
80#14556524—41no—
81#14556524C2×C414yes—
82#14556522C2×C414yes—
83C4⋊#10596831—22no—
84C4⋊#10596831C2×C414yes—
85#14556511—41no—
86#14556511—41no—
87#14556511C2×C414yes—
88#14556511C2×C414yes—
89#14556502—41no—
90C4×#10596831—41no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#125×C73—22no—
95#125×C73C2×C414yes—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C42×C73—21no—
107C2×C42×C73—21no—
108C2×C42×C73C2311yescharacteristic
109C2×C42×C73C2311yescharacteristic
110C2×C42×C73C2311yescharacteristic
111#95×C73—21no—
112#95×C73—21no—
113#95×C73—21no—
114#95×C73—21no—
115#95×C73D411yescharacteristic
116#95×C73D411yescharacteristic
117#95×C73D411yescharacteristic
118#95×C73D411yescharacteristic
119C23×D73—21no—
120C2×D292—21no—
121C2×D292—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292C22×C412yes—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C2×C4×D73—24no—
130C2×C4×D73—81no—
131C2×C4×D73—81no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831C22×C412yes—
143C2×#10596831C22×C412yes—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73C22×C412yes—
150#11924007—81no—
151#11924007—81no—
152#11924005—24no—
153#11924005—81no—
154#11924005—81no—
155#11924005—81no—
156#11924005—81no—
157C4×Dic73—81no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73C2×D411yescharacteristic
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2411yescharacteristic
169C4⋊C4×C73—24no—
170C4⋊C4×C73—24no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—81no—
178C4⋊C4×C73—81no—
179C4⋊C4×C73—81no—
180C4⋊C4×C73—81no—
181C4⋊C4×C73C22×C412yes—
182C4⋊C4×C73C22×C412yes—
183C42×C73—21no—
184C42×C73—21no—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—81no—
188C42×C73—81no—
189C42×C73C22×C412yes—
190C42×C73#4112yes—
191C23×C73C22×D411yescharacteristic
192C22×D73—22no—
193C22×D73—41no—
194C22×D73—41no—
195C22×D73—41no—
196C22×D73—41no—
197C22×D73—41no—
198C22×D73—41no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73C4×D412yes—
228C2×C4×C73C4×D412yes—
229C2×C4×C73C23×C411yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73H(5,𝔽2)11yescharacteristic
232C2×C4×C732−1+411yescharacteristic
233C2×C4×C732−1+411yescharacteristic
234C2×C4×C732−1+411yescharacteristic
235C2×Dic73—22no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244D292—22no—
245D292—22no—
246D292—41no—
247D292—41no—
248C4×D73—84no—
249#10596831—22no—
250#10596831—22no—
251#10596831—41no—
252#10596831—41no—
253C22×C73C2×C4×D411yescharacteristic
254C22×C73#53611yescharacteristic
255C22×C73#53711yescharacteristic
256C22×C73#57611yescharacteristic
257C22×C73#57611yescharacteristic
258C22×C73#57711yescharacteristic
259C22×C73#57811yescharacteristic
260C2×D73—42no—
261C2×D73—42no—
262C2×D73—42no—
263C2×D73—81no—
264C2×D73—81no—
265C2×D73—81no—
266C2×D73—81no—
267C4×C73—22no—
268C4×C73—22no—
269C4×C73—42no—
270C4×C73—42no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73—41no—
275C4×C73—81no—
276C4×C73—81no—
277C4×C73#51911yescharacteristic
278C4×C73#52011yescharacteristic
279C4×C73#54111yescharacteristic
280C4×C73#54211yescharacteristic
281Dic73—42no—
282Dic73—81no—
283Dic73—81no—
284#22126—731nomaximal, Sylow
285C2×C73#232311yescharacteristic
286C2×C73#294611yescharacteristic
287C2×C73#295111yescharacteristic
288C2×C73#324111yescharacteristic
289C2×C73#324311yescharacteristic
290C2×C73#325811yescharacteristic
291C2×C73#325911yescharacteristic
292D73—82no—
293C2×#478—731no—
294#2330D7311yescharacteristic, complemented
295#2319—731no—
296#1989—731no—
297#1947—731no—
298#1944—731no—
299#1942—731no—
300#1936—731no—
301#1934—731no—
302#1925—731no—
303#1925—731no—
304#1921—731no—
305#1919—731no—
306#1802—731no—
307#1768—731no—
308C73#2212611yescharacteristic, complemented, Sylow
309C2×#125C2×D7311yescharacteristic
310C2×C4×D4—731no—
311#478—738no—
312C2×#107—731no—
313C2×#107—731no—
314C2×#107—731no—
315C2×#106—731no—
316C2×C4⋊C8—731no—
317C2×C4⋊C8—731no—
318C2×#102—731no—
319C2×#102—731no—
320C2×#102—731no—
321C2×#102—731no—
322C2×#97—731no—
323C2×C4×C8—731no—
324#390—731no—
325#390—731no—
326#389—21no—
327#389—21no—
328#389—731no—
329#389—731no—
330#389C2×D7311yescharacteristic
331#389C2×D7311yescharacteristic
332#388—1461no—
333#388—1461no—
334#387—21no—
335#387—21no—
336#384C2×D7311yescharacteristic
337#381—1461no—
338#380—1461no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4C2×D7311yescharacteristic
341C4×C4⋊C4C2×D7311yescharacteristic
342#376C2×D7311yescharacteristic
343#340—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#336—731no—
352#336—731no—
353C22×D4—731no—
354C23×C4—1461no—
355C22×C8—731no—
356C22×C8—731no—
357C22×C8—731no—
358C22×C8—731no—
359#125—22no—
360#125C4×D7314yes—
361C4×D4—734no—
362C4×D4—1462no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4C22×D7311yescharacteristic
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C4⋊C4C22×D7311yescharacteristic
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C22⋊C4—1461no—
379C2×C22⋊C4—1461no—
380C2×C22⋊C4—1461no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1461no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—1461no—
386C2×C42C22×D7311yescharacteristic
387C2×C42C22×D7311yescharacteristic
388C2×C42C22×D7311yescharacteristic
389#107—2921no—
390#107—2921no—
391#107—734no—
392#106—734no—
393C4⋊C8—734no—
394C4⋊C8—1462no—
395#102—2921no—
396#102—2921no—
397#102—734no—
398#102—734no—
399#97—2921no—
400C4×C8—2921no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95C22⋊D7311yescharacteristic
409#95C22⋊D7311yescharacteristic
410#95C22⋊D7311yescharacteristic
411#95C22⋊D7311yescharacteristic
412C24—1461no—
413C2×D4—732no—
414C2×D4—1461no—
415C2×D4—1461no—
416C2×D4—1461no—
417C2×D4—1461no—
418C2×D4—1461no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—5841no—
424C22×C4—5841no—
425C22×C4—1461no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1464no—
429C22×C4C2×C22⋊D7311yescharacteristic
430C22×C4C2×C22⋊D7311yescharacteristic
431C22×C4C2×C22⋊D7311yescharacteristic
432C22×C4C2×C22⋊D7311yescharacteristic
433C22×C4C2×C22⋊D7311yescharacteristic
434C22×C4C2×C22⋊D7311yescharacteristic
435C22×C4C23×D7311yescharacteristic
436C2×C8—732no—
437C2×C8—732no—
438C2×C8—1461no—
439C2×C8—1461no—
440C2×C8—1461no—
441C2×C8—1461no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—5841no—
461C4⋊C4—5841no—
462C4⋊C4—732no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4C2×C4×D7312yes—
469C4⋊C4C2×C4×D7312yes—
470C22⋊C4—5841no—
471C22⋊C4—5841no—
472C22⋊C4—5841no—
473C22⋊C4—5841no—
474C22⋊C4—1464no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—81no—
480C42—81no—
481C42—5841no—
482C42C2×C4×D7312yes—
483C42#1192402812yes—
484C23—2921no—
485C23—2921no—
486C23—2921no—
487C23—2921no—
488C23—2921no—
489C23—2921no—
490C23—1462no—
491C23C22×C22⋊D7311yescharacteristic
492D4—2921no—
493D4—2921no—
494D4—1462no—
495D4—1462no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—5844no—
533C2×C4—1462no—
534C2×C4C4×C22⋊D7312yes—
535C2×C4C4×C22⋊D7312yes—
536C2×C4C22×C4×D7311yescharacteristic
537C2×C4C2×#1192402811yescharacteristic
538C2×C4#1455663711yescharacteristic
539C2×C4#1455663811yescharacteristic
540C2×C4#1455663911yescharacteristic
541C2×C4#1455663911yescharacteristic
542C8—2921no—
543C8—2921no—
544C8—1462no—
545C8—1462no—
546C22—2922no—
547C22—2922no—
548C22—2922no—
549C22—5841no—
550C22—5841no—
551C22—5841no—
552C22—5841no—
553C22C2×C4×C22⋊D7311yescharacteristic
554C22#1798916111yescharacteristic
555C22#1798916511yescharacteristic
556C22#1798939011yescharacteristic
557C22#1798939011yescharacteristic
558C22#1798939511yescharacteristic
559C22#1798940211yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—81no—
569C4—81no—
570C4—2922no—
571C4—5841no—
572C4—5841no—
573C4#1798906311yescharacteristic
574C4#1798906911yescharacteristic
575C4#1798919511yescharacteristic
576C4#1798920911yescharacteristic
577C2—5842no—
578C2#2261260711yescharacteristic
579C2#2261875911yescharacteristic
580C2#2261880411yescharacteristic
581C2#2262118411yescharacteristic
582C2#2262121411yescharacteristic
583C2#2262135111yescharacteristic
584C2#2262136611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2330C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#2330D73
hypocenterC73#22126
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ #376 ◁ #2330

Complements

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

NH ≅ G/Ntype
3. #22610889C2semidirect
4. #22610687C2semidirect
5. #22610671C2semidirect
6. #22610661C2semidirect
11. #22610523C2semidirect
12. #22610509C2semidirect
14. #22609788C2semidirect
15. C73×#2330C2semidirect
17. #17988884C4semidirect
294. #2330D73semidirect
308. C73#22126semidirect

Maximal subgroups

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

subgroupindex
C2×#179888842
#226125752
#226108892
#226106872
#226106712
#226106612
#226106272
#226106132
#226105512
#226105232
#226105092
#226099282
#226097882
C73×#23302
#2212673