Finite Group Databasebeta

← #33144861

Subgroups — #33144861

Order 18688.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988997C211yesmaximal, characteristic, complemented
2C2×#17988627C211yesmaximal, characteristic
3#22613622C211yesmaximal, characteristic
4#22611413C211yesmaximal, characteristic, complemented
5#22611412C211yesmaximal, characteristic, complemented
6#22611266C211yesmaximal, characteristic, complemented
7#22611259C211yesmaximal, characteristic, complemented
8#22610666C211yesmaximal, characteristic, complemented
9#22610661C211yesmaximal, characteristic, complemented
10#22610128C211yesmaximal, characteristic
11#22610119C211yesmaximal, characteristic
12#22610093C211yesmaximal, characteristic, complemented
13#22609962C211yesmaximal, characteristic
14#22609715C211yesmaximal, characteristic
15C73×#2442C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556550C2211yescharacteristic
19#17988997—24no—
20#17988627—24no—
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556524C2211yescharacteristic
27C2×#14556522C2211yescharacteristic
28C2×#14556522C2211yescharacteristic
29C2×#14556522C2211yescharacteristic
30C2×#14556522C2211yescharacteristic
31C2×C22⋊#10596831C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33C2×C4×#10596831C2211yescharacteristic
34#17988510—21no—
35#17988499—21no—
36#17988491—21no—
37#17988485C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988474—21no—
40#17988453C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×#125×C73C2211yescharacteristic
51C22×C4⋊C4×C73C2211yescharacteristic
52#400×C73—21no—
53#398×C73—21no—
54#389×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#385×C73—21no—
57#382×C73C2211yescharacteristic
58#380×C73—21no—
59#380×C73C2211yescharacteristic
60C4×C22⋊C4×C73C2211yescharacteristic
61C22×Dic146—21no—
62C2×#11924012—22no—
63C2×#11924012—22no—
64C2×#11924012Q814yescomplemented
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69#14556591—22no—
70#14556591—41no—
71#14556550—22no—
72#14556550—41no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C4×Dic73—21no—
83#14556524—22no—
84#14556524—22no—
85#14556524—22no—
86#14556524—41no—
87#14556524—41no—
88#14556524D414yes—
89#14556522—22no—
90#14556522—22no—
91#14556522—22no—
92#14556522—22no—
93C22⋊#10596831Q814yescomplemented
94#14556502—22no—
95C4×#10596831—22no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101C23×C4×C73—21no—
102C23×C4×C73C2311yescharacteristic
103#125×C73—22no—
104#125×C73—41no—
105C2×C4⋊C4×C73—22no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73—41no—
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C42×C73C2311yescharacteristic
120C2×C42×C73C2311yescharacteristic
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73—21no—
125#95×C73—21no—
126#95×C73D411yescharacteristic
127#95×C73D411yescharacteristic
128C2×Dic146—42no—
129C2×Dic146—42no—
130C2×Dic146—81no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135#11924012—28no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831C2×D412yes—
143C2×#10596831C2×Q812yes—
144C2×#10596831C2×Q812yes—
145C2×#10596831C2×Q812yes—
146C2×#10596831C2×Q812yes—
147C2×#10596831#4112yes—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—42no—
156C4⋊Dic73C2×D412yes—
157#11924007—42no—
158#11924007—81no—
159C22⋊Dic73—42no—
160C22⋊Dic73—81no—
161C4×Dic73—42no—
162C24×C73C2×D411yescharacteristic
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—42no—
172C22×C4×C73—42no—
173C22×C4×C73—81no—
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×Q812yes—
180C22×C4×C73C2×Q812yes—
181C22×C4×C73C2411yescharacteristic
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—42no—
190C4⋊C4×C73—42no—
191C4⋊C4×C73—42no—
192C4⋊C4×C73—81no—
193C4⋊C4×C73—81no—
194C4⋊C4×C73C2×D412yes—
195C22⋊C4×C73—41no—
196C22⋊C4×C73—41no—
197C22⋊C4×C73—42no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73C4⋊Q812yes—
209C23×C73C22×D411yescharacteristic
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—22no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—84no—
241C2×C4×C73C4⋊Q812yes—
242C2×C4×C73C22×D411yescharacteristic
243C2×C4×C73C22×Q811yescharacteristic
244C2×C4×C73C22×Q811yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73H(5,𝔽2)11yescharacteristic
247C2×C4×C732−1+411yescharacteristic
248C2×Dic73—22no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260Dic146—84no—
261#10596831—22no—
262#10596831—22no—
263#10596831—22no—
264#10596831C22⋊Q814yes—
265C22×C73—22no—
266C22×C73—22no—
267C22×C73—22no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73C2×C4⋊Q811yescharacteristic
273C22×C73D4211yescharacteristic
274C22×C73#54711yescharacteristic
275C22×C73#57511yescharacteristic
276C22×C73#57511yescharacteristic
277C22×C73#57611yescharacteristic
278C22×C73#57811yescharacteristic
279C4×C73—22no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73—42no—
284C4×C73—41no—
285C4×C73—81no—
286C4×C73—81no—
287C4×C73—81no—
288C4×C73C2×C22⋊Q811yescharacteristic
289C4×C73#54411yescharacteristic
290C4×C73D4×Q811yescharacteristic
291C4×C73#55411yescharacteristic
292Dic73—42no—
293Dic73—81no—
294Dic73—81no—
295Dic73—81no—
296#26079—731nomaximal, Sylow
297C2×C73—42no—
298C2×C73#244011yescharacteristic
299C2×C73#316411yescharacteristic
300C2×C73#316611yescharacteristic
301C2×C73#327311yescharacteristic
302C2×C73#327711yescharacteristic
303C2×C73#334711yescharacteristic
304C2×C73#334911yescharacteristic
305C2×#501—731no—
306C2×#428—731no—
307#2442D7311yescharacteristic
308#2439—731no—
309#2085—731no—
310#2085—731no—
311#2059—731no—
312#2057—731no—
313#1943—731no—
314#1942—731no—
315#1854—731no—
316#1852—731no—
317#1846—731no—
318#1809—731no—
319#1753—731no—
320C73#2607911yescharacteristic, complemented, Sylow
321C22×M16—731no—
322C2×C4⋊Q8—731no—
323C2×#125C2×D7311yescharacteristic
324C2×#117—731no—
325C22×C4⋊C4C2×D7311yescharacteristic
326#501—1464no—
327#428—1464no—
328C2×#107—731no—
329C2×#107—731no—
330C2×#107—731no—
331C2×#107—731no—
332C2×#107—731no—
333C2×#107—731no—
334C2×#106—731no—
335C2×#106—731no—
336C2×#106—731no—
337C2×#106—731no—
338C2×C22⋊C8—731no—
339C2×#97—731no—
340C2×C4×C8—731no—
341#400—21no—
342#398—21no—
343#396—1461no—
344#393—1461no—
345#391—1461no—
346#389—731no—
347#389—731no—
348#389C2×D7311yescharacteristic
349#389C2×D7311yescharacteristic
350#387—1461no—
351#385—21no—
352#382—731no—
353#382C2×D7311yescharacteristic
354#380—21no—
355#380—731no—
356#380C2×D7311yescharacteristic
357C4×C22⋊C4C2×D7311yescharacteristic
358#340—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#336—731no—
365#336—731no—
366C22×Q8—1461no—
367C23×C4—21no—
368C23×C4C22×D7311yescharacteristic
369C2×M16—734no—
370C2×M16—1462no—
371C2×M16—1462no—
372C22×C8—731no—
373C22×C8—731no—
374C22×C8—731no—
375C22×C8—731no—
376C4⋊Q8—2921no—
377C4⋊Q8—1462no—
378#125—22no—
379#125—41no—
380#117—2921no—
381#117—1462no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—41no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4C22⋊D7311yescharacteristic
404C2×C22⋊C4C22⋊D7311yescharacteristic
405C2×C42—1461no—
406C2×C42C22×D7311yescharacteristic
407C2×C42C22×D7311yescharacteristic
408#107—2921no—
409#107—2921no—
410#107—734no—
411#107—1462no—
412#107—1462no—
413#107—1462no—
414#106—1462no—
415#106—1462no—
416#106—1462no—
417#106—1462no—
418C22⋊C8—734no—
419#97—1462no—
420C4×C8—1462no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95C22⋊D7311yescharacteristic
432#95C22⋊D7311yescharacteristic
433C24C2×C22⋊D7311yescharacteristic
434C2×Q8—2922no—
435C2×Q8—2922no—
436C2×Q8—5841no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4C2×C22⋊D7311yescharacteristic
453C22×C4C2×C22⋊D7311yescharacteristic
454C22×C4C2×C22⋊D7311yescharacteristic
455C22×C4C2×C22⋊D7311yescharacteristic
456C22×C4C2×C22⋊D7311yescharacteristic
457C22×C4Q8×D7312yes—
458C22×C4Q8×D7312yes—
459C22×C4C23×D7311yescharacteristic
460M16—1468no—
461C2×C8—732no—
462C2×C8—732no—
463C2×C8—732no—
464C2×C8—732no—
465C2×C8—732no—
466C2×C8—732no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—2922no—
486C4⋊C4—2922no—
487C4⋊C4—5841no—
488C4⋊C4—732no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4D4×D7312yes—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—42no—
500C22⋊C4—2922no—
501C22⋊C4—5841no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—2922no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1455658912yes—
514C23C22×C22⋊D7311yescharacteristic
515Q8—5844no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—84no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—1462no—
559C2×C4#1455658912yes—
560C2×C4C2×D4×D7311yescharacteristic
561C2×C4C2×Q8×D7311yescharacteristic
562C2×C4C2×Q8×D7311yescharacteristic
563C2×C4#41×D7311yescharacteristic
564C2×C4#1455663611yescharacteristic
565C2×C4#1455663811yescharacteristic
566C8—734no—
567C8—1462no—
568C8—1462no—
569C8—1462no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1455658911yescharacteristic
578C22D4×C22⋊D7311yescharacteristic
579C22#1798923611yescharacteristic
580C22#1798938511yescharacteristic
581C22#1798938511yescharacteristic
582C22#1798939011yescharacteristic
583C22#1798940211yescharacteristic
584C4—22no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—2922no—
594C4—5841no—
595C4—5841no—
596C4—5841no—
597C4C22⋊Q8×D7311yescharacteristic
598C4#1798921611yescharacteristic
599C4#1798925011yescharacteristic
600C4#1798929111yescharacteristic
601C2—42no—
602C2#2261363011yescharacteristic
603C2#2262046511yescharacteristic
604C2#2262048011yescharacteristic
605C2#2262153811yescharacteristic
606C2#2262158411yescharacteristic
607C2#2262242511yescharacteristic
608C2#2262246111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2442C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2442D73
hypocenterC73#26079
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C2×#17988997C2semidirect
4. #22611413C2semidirect
5. #22611412C2semidirect
6. #22611266C2semidirect
7. #22611259C2semidirect
8. #22610666C2semidirect
9. #22610661C2semidirect
12. #22610093C2semidirect
64. C2×#11924012Q8semidirect
93. C22⋊#10596831Q8semidirect
320. C73#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#179889972
C2×#179886272
#226136222
#226114132
#226114122
#226112662
#226112592
#226106662
#226106612
#226101282
#226101192
#226100932
#226099622
#226097152
C73×#24422
#2607973