Finite Group Databasebeta

← #33148918

Subgroups — #33148918

Order 18688.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988895C211yesmaximal, characteristic, complemented
2C2×#17988629C211yesmaximal, characteristic
3#22613655C211yesmaximal, characteristic
4#22612956C211yesmaximal, characteristic
5#22611478C211yesmaximal, characteristic, complemented
6#22611424C211yesmaximal, characteristic, complemented
7#22611325C211yesmaximal, characteristic, complemented
8#22611270C211yesmaximal, characteristic, complemented
9#22610701C211yesmaximal, characteristic, complemented
10#22610510C211yesmaximal, characteristic, complemented
11#22610364C211yesmaximal, characteristic
12#22610236C211yesmaximal, characteristic
13#22610202C211yesmaximal, characteristic, complemented
14#22610132C211yesmaximal, characteristic
15C73×#1851C211yesmaximal, characteristic
16C2×#14556583C2211yescharacteristic
17C2×#14556570C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×#14556550C2211yescharacteristic
20C22×C4⋊Dic73C2211yescharacteristic
21#17988895—24no—
22#17988629—24no—
23C2×#14556525C2211yescharacteristic
24C2×#14556525C2211yescharacteristic
25C2×#14556525C2211yescharacteristic
26C2×#14556523C2211yescharacteristic
27C2×#14556523C2211yescharacteristic
28C2×#14556523C2211yescharacteristic
29C2×#14556516C2211yescharacteristic
30C2×#14556516C2211yescharacteristic
31C2×#14556516C2211yescharacteristic
32C2×#14556516C2211yescharacteristic
33#17988529—21no—
34#17988521—21no—
35#17988493C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988485C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988481—21no—
40#17988475—21no—
41#17988469—21no—
42#17988459—21no—
43#17988453C2211yescharacteristic
44#17988447C2211yescharacteristic
45#17988445—21no—
46C4×C4⋊Dic73C2211yescharacteristic
47#17988438—21no—
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988350C2211yescharacteristic
54C22×M16×C73C2211yescharacteristic
55C2×C4⋊C8×C73C2211yescharacteristic
56C2×C4⋊C8×C73C2211yescharacteristic
57C2×C22⋊C8×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C23×Dic73—21no—
63#14556583—22no—
64#14556583—41no—
65#14556570—22no—
66#14556570Q814yes—
67C4×Dic146—22no—
68C4×Dic146—41no—
69#14556550—22no—
70#14556550—41no—
71C2×C4⋊Dic73—22no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—41no—
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×#11924007—21no—
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94#14556525—22no—
95#14556525—22no—
96#14556525—22no—
97#14556523—22no—
98#14556523—41no—
99#14556523D414yes—
100#14556516—22no—
101#14556516—22no—
102#14556516—22no—
103#14556516—41no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113C23×C4×C73C2311yescharacteristic
114C2×M16×C73—22no—
115C2×M16×C73—22no—
116C2×M16×C73—22no—
117C22×C8×C73C2311yescharacteristic
118C22×C8×C73C2311yescharacteristic
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C2×C22⋊C4×C73D411yescharacteristic
122C2×C22⋊C4×C73D411yescharacteristic
123C2×C42×C73C2311yescharacteristic
124C2×C42×C73C2311yescharacteristic
125C4⋊C8×C73—22no—
126C4⋊C8×C73—22no—
127C22⋊C8×C73—22no—
128#95×C73D411yescharacteristic
129#95×C73D411yescharacteristic
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146C2×Q812yes—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—42no—
145C22×Dic73—42no—
146C22×Dic73—81no—
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—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—42no—
159C4⋊Dic73C2×D412yes—
160C4⋊Dic73C2×D412yes—
161C4⋊Dic73C2×Q812yes—
162#11924007—24no—
163#11924007—42no—
164#11924007—42no—
165#11924007—42no—
166#11924007—42no—
167#11924007—81no—
168#11924007—81no—
169C22⋊Dic73—24no—
170C22⋊Dic73—42no—
171C22⋊Dic73—42no—
172C4×Dic73—42no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C24×C73C2×D411yescharacteristic
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×Q812yes—
186C22×C4×C73#4112yes—
187C22×C4×C73C2411yescharacteristic
188M16×C73—28no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73C2×D412yes—
200C2×C8×C73#4112yes—
201C22⋊C4×C73—41no—
202C22⋊C4×C73—41no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73C22⋊Q812yes—
214C23×C73C22×D411yescharacteristic
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
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—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73C22⋊Q812yes—
234C2×C4×C73C22×D411yescharacteristic
235C2×C4×C73C22×Q811yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73H(5,𝔽2)11yescharacteristic
239C2×C4×C732−1+411yescharacteristic
240C2×Dic73—22no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—22no—
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—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—84no—
265Dic146—22no—
266Dic146—22no—
267Dic146—41no—
268Dic146—41no—
269C8×C73—22no—
270C8×C73—22no—
271C8×C73—41no—
272C8×C73—41no—
273C22×C73—22no—
274C22×C73—22no—
275C22×C73—22no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73C2×C22⋊Q811yescharacteristic
281C22×C73D4211yescharacteristic
282C22×C73#54711yescharacteristic
283C22×C73#57511yescharacteristic
284C22×C73#57511yescharacteristic
285C22×C73#57711yescharacteristic
286C22×C73#57811yescharacteristic
287C4×C73—22no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—41no—
291C4×C73—41no—
292C4×C73#52511yescharacteristic
293C4×C73#53911yescharacteristic
294C4×C73D4×Q811yescharacteristic
295C4×C73#55411yescharacteristic
296Dic73—42no—
297Dic73—42no—
298Dic73—42no—
299Dic73—42no—
300Dic73—81no—
301Dic73—81no—
302Dic73—81no—
303Dic73—81no—
304#26367—731nomaximal, Sylow
305C2×C73—42no—
306C2×C73#244911yescharacteristic
307C2×C73#308011yescharacteristic
308C2×C73#308211yescharacteristic
309C2×C73#326911yescharacteristic
310C2×C73#327511yescharacteristic
311C2×C73#334911yescharacteristic
312C2×C73#335111yescharacteristic
313C2×#479—731no—
314C2×#428—731no—
315#2442—731no—
316#2362—731no—
317#2097—731no—
318#2087—731no—
319#2068—731no—
320#2059—731no—
321#1949—731no—
322#1919—731no—
323#1894—731no—
324#1872—731no—
325#1865—731no—
326#1854—731no—
327#1851D7311yescharacteristic
328C73#2636711yescharacteristic, complemented, Sylow
329C22×M16C2×D7311yescharacteristic
330C2×#125—731no—
331C2×C22⋊Q8—731no—
332C2×C4×Q8—731no—
333C2×#117—731no—
334C22×C4⋊C4—731no—
335#479—1464no—
336#428—1464no—
337C2×#107—731no—
338C2×#107—731no—
339C2×#107—731no—
340C2×#106—731no—
341C2×#106—731no—
342C2×#106—731no—
343C2×C4⋊C8C2×D7311yescharacteristic
344C2×C4⋊C8C2×D7311yescharacteristic
345C2×#103—731no—
346C2×#103—731no—
347C2×#103—731no—
348C2×#103—731no—
349C2×C22⋊C8C2×D7311yescharacteristic
350#400—1461no—
351#398—1461no—
352#391—731no—
353#389—731no—
354#389—731no—
355#389—731no—
356#388—1461no—
357#388—1461no—
358#387—1461no—
359#385—1461no—
360#382—731no—
361#380—731no—
362#380—1461no—
363C4×C4⋊C4—731no—
364C4×C4⋊C4—1461no—
365C4×C22⋊C4C2×D7311yescharacteristic
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#336C2×D7311yescharacteristic
373#336C2×D7311yescharacteristic
374C22×Q8—731no—
375C23×C4—1461no—
376C23×C4C22×D7311yescharacteristic
377C2×M16—22no—
378C2×M16—22no—
379C2×M16—22no—
380C22×C8C22×D7311yescharacteristic
381C22×C8C22×D7311yescharacteristic
382C22×C8C22×D7311yescharacteristic
383C22×C8C22×D7311yescharacteristic
384#125—2921no—
385#125—1462no—
386C22⋊Q8—734no—
387C22⋊Q8—1462no—
388C4×Q8—2921no—
389C4×Q8—1462no—
390#117—2921no—
391#117—1462no—
392C2×C4⋊C4—2921no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1462no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C22⋊C4—1461no—
410C2×C22⋊C4—1461no—
411C2×C22⋊C4—1461no—
412C2×C22⋊C4D29211yescharacteristic
413C2×C22⋊C4D29211yescharacteristic
414C2×C42—1461no—
415C2×C42—1461no—
416C2×C42—1461no—
417C2×C42C22×D7311yescharacteristic
418C2×C42C22×D7311yescharacteristic
419#107—1462no—
420#107—1462no—
421#107—1462no—
422#106—2921no—
423#106—734no—
424#106—1462no—
425C4⋊C8—22no—
426C4⋊C8—22no—
427#103—2921no—
428#103—1462no—
429#103—1462no—
430#103—1462no—
431C22⋊C8—22no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95D29211yescharacteristic
442#95D29211yescharacteristic
443C24C2×D29211yescharacteristic
444C2×Q8—732no—
445C2×Q8—1461no—
446C2×Q8—1461no—
447C2×Q8—1461no—
448C2×Q8—1461no—
449C2×Q8—1461no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—2922no—
455C22×C4—2922no—
456C22×C4—5841no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4C2×D29211yescharacteristic
466C22×C4C2×D29211yescharacteristic
467C22×C4C2×D29211yescharacteristic
468C22×C4C2×D29211yescharacteristic
469C22×C4C2×D29211yescharacteristic
470C22×C4Q8×D7312yes—
471C22×C4#1192402712yes—
472C22×C4C23×D7311yescharacteristic
473M16—28no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8D4×D7312yes—
485C2×C8#1192402712yes—
486C4⋊C4—2922no—
487C4⋊C4—2922no—
488C4⋊C4—2922no—
489C4⋊C4—2922no—
490C4⋊C4—2922no—
491C4⋊C4—5841no—
492C4⋊C4—5841no—
493C4⋊C4—732no—
494C4⋊C4—732no—
495C4⋊C4—732no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1464no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—2922no—
511C22⋊C4—2922no—
512C22⋊C4—1464no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—2922no—
518C42—2922no—
519C42—2922no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1455657112yes—
527C23C22×D29211yescharacteristic
528Q8—2921no—
529Q8—2921no—
530Q8—1462no—
531Q8—1462no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
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—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—5844no—
571C2×C4—1462no—
572C2×C4—1462no—
573C2×C4—1462no—
574C2×C4—1462no—
575C2×C4#1455657112yes—
576C2×C4C2×D4×D7311yescharacteristic
577C2×C4C2×Q8×D7311yescharacteristic
578C2×C4C2×#1192402711yescharacteristic
579C2×C4C2×#1192402711yescharacteristic
580C2×C4#1455663711yescharacteristic
581C2×C4#1455663811yescharacteristic
582C8—22no—
583C8—22no—
584C8—41no—
585C8—41no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1455657111yescharacteristic
594C22D4×D29211yescharacteristic
595C22#1798924111yescharacteristic
596C22#1798938711yescharacteristic
597C22#1798938711yescharacteristic
598C22#1798939911yescharacteristic
599C22#1798940411yescharacteristic
600C4—22no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—2922no—
606C4—2922no—
607C4—2922no—
608C4—2922no—
609C4—5841no—
610C4—5841no—
611C4—5841no—
612C4—5841no—
613C4#1798909311yescharacteristic
614C4#1798918111yescharacteristic
615C4#1798925011yescharacteristic
616C4#1798929511yescharacteristic
617C2—42no—
618C2#2261373811yescharacteristic
619C2#2261975411yescharacteristic
620C2#2261977411yescharacteristic
621C2#2262149111yescharacteristic
622C2#2262156711yescharacteristic
623C2#2262246311yescharacteristic
624C2#2262249311yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1851C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1851D73
hypocenterC73#26367
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. C2×#17988895C2semidirect
5. #22611478C2semidirect
6. #22611424C2semidirect
7. #22611325C2semidirect
8. #22611270C2semidirect
9. #22610701C2semidirect
10. #22610510C2semidirect
13. #22610202C2semidirect
328. C73#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#179888952
C2×#179886292
#226136552
#226129562
#226114782
#226114242
#226113252
#226112702
#226107012
#226105102
#226103642
#226102362
#226102022
#226101322
C73×#18512
#2636773