Finite Group Databasebeta

← #33144856

Subgroups — #33144856

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×#17988996C211yesmaximal, characteristic, complemented
2C2×#17988629C211yesmaximal, characteristic
3#22613655C211yesmaximal, characteristic
4#22613622C211yesmaximal, characteristic
5#22611414C211yesmaximal, characteristic, complemented
6#22611414C211yesmaximal, characteristic, complemented
7#22611270C211yesmaximal, characteristic, complemented
8#22611260C211yesmaximal, characteristic, complemented
9#22610669C211yesmaximal, characteristic, complemented
10#22610664C211yesmaximal, characteristic, complemented
11#22610132C211yesmaximal, characteristic
12#22610121C211yesmaximal, characteristic
13#22610095C211yesmaximal, characteristic, complemented
14#22609964C211yesmaximal, characteristic
15C73×#1753C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17C2×#14556583C2211yescharacteristic
18C2×#14556550C2211yescharacteristic
19C22×C4⋊Dic73C2211yescharacteristic
20#17988996—24no—
21#17988629—24no—
22C2×#14556525C2211yescharacteristic
23C2×#14556525C2211yescharacteristic
24C2×#14556525C2211yescharacteristic
25C2×#14556525C2211yescharacteristic
26C2×#14556525C2211yescharacteristic
27C2×#14556525C2211yescharacteristic
28C2×#14556523C2211yescharacteristic
29C2×#14556523C2211yescharacteristic
30C2×#14556523C2211yescharacteristic
31C2×#14556523C2211yescharacteristic
32#17988529—21no—
33#17988521—21no—
34#17988510—21no—
35#17988499—21no—
36#17988491—21no—
37#17988485C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988485C2211yescharacteristic
41#17988474—21no—
42#17988459—21no—
43#17988453C2211yescharacteristic
44#17988453C2211yescharacteristic
45#17988447C2211yescharacteristic
46#17988447C2211yescharacteristic
47#17988445—21no—
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988350C2211yescharacteristic
54C22×M16×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C2×#97×C73C2211yescharacteristic
57C2×C4×C8×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61C22×Dic146—21no—
62C23×Dic73—21no—
63#14556591—22no—
64#14556591—41no—
65#14556583—22no—
66#14556583—41no—
67#14556550—22no—
68#14556550—41no—
69C2×C4⋊Dic73—22no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—41no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×C22⋊Dic73—21no—
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C4×Dic73—21no—
91#14556525—22no—
92#14556525—22no—
93#14556525—22no—
94#14556525—41no—
95#14556525—41no—
96#14556525D414yes—
97#14556523—22no—
98#14556523—22no—
99#14556523—22no—
100#14556523—22no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111C23×C4×C73C2311yescharacteristic
112C2×M16×C73—22no—
113C2×M16×C73—22no—
114C2×M16×C73Q814yescomplemented
115C22×C8×C73C2311yescharacteristic
116C22×C8×C73C2311yescharacteristic
117C22×C8×C73C2311yescharacteristic
118C22×C8×C73C2311yescharacteristic
119C2×C22⋊C4×C73D411yescharacteristic
120C2×C22⋊C4×C73D411yescharacteristic
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123C22⋊C8×C73Q814yescomplemented
124#97×C73—22no—
125C4×C8×C73—22no—
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—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—42no—
140C22×Dic73—42no—
141C22×Dic73—81no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—42no—
157C4⋊Dic73—42no—
158C4⋊Dic73C2×D412yes—
159C4⋊Dic73C2×D412yes—
160#11924007—42no—
161#11924007—42no—
162#11924007—42no—
163#11924007—81no—
164#11924007—81no—
165#11924007—81no—
166C22⋊Dic73—42no—
167C22⋊Dic73—42no—
168C22⋊Dic73—81no—
169C4×Dic73—42no—
170C24×C73C2×D411yescharacteristic
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×Q812yes—
181C22×C4×C73C2×Q812yes—
182C22×C4×C73C2411yescharacteristic
183M16×C73—28no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73C2×D412yes—
191C2×C8×C73C2×Q812yes—
192C2×C8×C73C2×Q812yes—
193C2×C8×C73C2×Q812yes—
194C2×C8×C73C2×Q812yes—
195C2×C8×C73#4112yes—
196C22⋊C4×C73—41no—
197C22⋊C4×C73—41no—
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—21no—
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×C73C4⋊Q812yes—
229C2×C4×C73C22×D411yescharacteristic
230C2×C4×C73C22×Q811yescharacteristic
231C2×C4×C73C22×Q811yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73H(5,𝔽2)11yescharacteristic
234C2×C4×C732−1+411yescharacteristic
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
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—84no—
260Dic146—84no—
261C8×C73—22no—
262C8×C73—22no—
263C8×C73—22no—
264C8×C73C22⋊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—41no—
284C4×C73C2×C22⋊Q811yescharacteristic
285C4×C73#54411yescharacteristic
286C4×C73D4×Q811yescharacteristic
287C4×C73#55411yescharacteristic
288Dic73—42no—
289Dic73—42no—
290Dic73—81no—
291Dic73—81no—
292Dic73—81no—
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#2442—731no—
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#1753D7311yescharacteristic
320C73#2607911yescharacteristic, complemented, Sylow
321C22×M16C2×D7311yescharacteristic
322C2×C4⋊Q8—731no—
323C2×#125—731no—
324C2×#117—731no—
325C22×C4⋊C4—731no—
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⋊C8C2×D7311yescharacteristic
339C2×#97C2×D7311yescharacteristic
340C2×C4×C8C2×D7311yescharacteristic
341#400—1461no—
342#398—1461no—
343#396—1461no—
344#393—1461no—
345#391—1461no—
346#389—731no—
347#389—731no—
348#389—731no—
349#389—731no—
350#387—1461no—
351#385—1461no—
352#382—731no—
353#382—731no—
354#380—731no—
355#380—731no—
356#380—1461no—
357C4×C22⋊C4C2×D7311yescharacteristic
358#340—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#336C2×D7311yescharacteristic
365#336C2×D7311yescharacteristic
366C22×Q8—1461no—
367C23×C4—1461no—
368C23×C4C22×D7311yescharacteristic
369C2×M16—22no—
370C2×M16—22no—
371C2×M16Dic14614yescomplemented
372C22×C8C22×D7311yescharacteristic
373C22×C8C22×D7311yescharacteristic
374C22×C8C22×D7311yescharacteristic
375C22×C8C22×D7311yescharacteristic
376C4⋊Q8—2921no—
377C4⋊Q8—1462no—
378#125—2921no—
379#125—1462no—
380#117—2921no—
381#117—1462no—
382C2×C4⋊C4—2921no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1462no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4D29211yescharacteristic
404C2×C22⋊C4D29211yescharacteristic
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⋊C8Dic14614yescomplemented
419#97—22no—
420C4×C8—22no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95D29211yescharacteristic
432#95D29211yescharacteristic
433C24C2×D29211yescharacteristic
434C2×Q8—2922no—
435C2×Q8—2922no—
436C2×Q8—5841no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—2922no—
442C22×C4—2922no—
443C22×C4—5841no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4C2×D29211yescharacteristic
453C22×C4C2×D29211yescharacteristic
454C22×C4C2×D29211yescharacteristic
455C22×C4C2×D29211yescharacteristic
456C22×C4C2×D29211yescharacteristic
457C22×C4C2×Dic14612yes—
458C22×C4C2×Dic14612yes—
459C22×C4C23×D7311yescharacteristic
460M16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8D4×D7312yes—
468C2×C8C2×Dic14612yes—
469C2×C8C2×Dic14612yes—
470C2×C8C2×Dic14612yes—
471C2×C8C2×Dic14612yes—
472C2×C8#1192402712yes—
473C4⋊C4—2922no—
474C4⋊C4—2922no—
475C4⋊C4—2922no—
476C4⋊C4—2922no—
477C4⋊C4—2922no—
478C4⋊C4—5841no—
479C4⋊C4—5841no—
480C4⋊C4—5841no—
481C4⋊C4—732no—
482C4⋊C4—732no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—2922no—
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#1455659112yes—
514C23C22×D29211yescharacteristic
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—21no—
533C2×C4—21no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
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—5844no—
557C2×C4—1462no—
558C2×C4—1462no—
559C2×C4#1455659112yes—
560C2×C4C2×D4×D7311yescharacteristic
561C2×C4C22×Dic14611yescharacteristic
562C2×C4C22×Dic14611yescharacteristic
563C2×C4C2×#1192402711yescharacteristic
564C2×C4#1455663711yescharacteristic
565C2×C4#1455663811yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8#1455656814yes—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1455659111yescharacteristic
578C22D4×D29211yescharacteristic
579C22#1798924111yescharacteristic
580C22#1798938711yescharacteristic
581C22#1798938711yescharacteristic
582C22#1798939211yescharacteristic
583C22#1798940411yescharacteristic
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—2922no—
590C4—2922no—
591C4—5841no—
592C4—5841no—
593C4—5841no—
594C4—5841no—
595C4—5841no—
596C4—5841no—
597C4C2×#1455656811yescharacteristic
598C4#1798921511yescharacteristic
599C4D4×Dic14611yescharacteristic
600C4#1798929411yescharacteristic
601C2—42no—
602C2#2261363311yescharacteristic
603C2#2262047011yescharacteristic
604C2#2262048311yescharacteristic
605C2#2262154311yescharacteristic
606C2#2262158911yescharacteristic
607C2#2262243311yescharacteristic
608C2#2262246311yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1753C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1753D73
hypocenterC73#26079
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#17988996C2semidirect
5. #22611414C2semidirect
6. #22611414C2semidirect
7. #22611270C2semidirect
8. #22611260C2semidirect
9. #22610669C2semidirect
10. #22610664C2semidirect
13. #22610095C2semidirect
114. C2×M16×C73Q8semidirect
123. C22⋊C8×C73Q8semidirect
320. C73#26079semidirect
371. C2×M16Dic146semidirect
418. C22⋊C8Dic146semidirect

Maximal subgroups

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

subgroupindex
C2×#179889962
C2×#179886292
#226136552
#226136222
#226114142
#226112702
#226112602
#226106692
#226106642
#226101322
#226101212
#226100952
#226099642
C73×#17532
#2607973