Finite Group Databasebeta

← #33103757

Subgroups — #33103757

Order 18688.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988629C211yesmaximal, characteristic
2#22612814C211yesmaximal, characteristic
3#22612798C211yesmaximal, characteristic
4#22610674C211yesmaximal, characteristic, complemented
5#22610659C211yesmaximal, characteristic, complemented
6#22610524C211yesmaximal, characteristic, complemented
7#22610517C211yesmaximal, characteristic, complemented
8#22610510C211yesmaximal, characteristic, complemented
9#22610503C211yesmaximal, characteristic, complemented
10#22610027C211yesmaximal, characteristic
11#22610022C211yesmaximal, characteristic
12#22609964C211yesmaximal, characteristic
13#22609828C211yesmaximal, characteristic, complemented
14C4×#14556525C211yesmaximal, characteristic, complemented
15C73×#1753C211yesmaximal, characteristic
16C2×#14556550C2211yescharacteristic
17C2×#14556547—21no—
18C22×C4⋊Dic73C2211yescharacteristic
19#17988629C418yescomplemented
20C2×#14556525C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556525C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556523C2211yescharacteristic
26#17988485C2211yescharacteristic
27#17988485C2211yescharacteristic
28#17988485C2211yescharacteristic
29#17988485C2211yescharacteristic
30#17988475—21no—
31#17988475—21no—
32#17988472—21no—
33#17988472—21no—
34#17988451—21no—
35#17988448—21no—
36#17988447C2211yescharacteristic
37#17988447C2211yescharacteristic
38C4×C4⋊Dic73C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42#17988432—21no—
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C22×M16×C73C2211yescharacteristic
54C2×C22⋊C8×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57C4×C22⋊C4×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C23×Dic73—21no—
61#14556550—22no—
62#14556550C2×C414yes—
63#14556547—44no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—22no—
66C2×C4⋊Dic73C2×C414yes—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87#14556525—22no—
88#14556525C2×C414yes—
89#14556525C2×C414yes—
90#14556523—22no—
91#14556523C2×C414yes—
92#14556523C2×C414yes—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103C23×C4×C73C2311yescharacteristic
104C2×M16×C73—22no—
105C2×M16×C73—22no—
106C2×M16×C73C2×C414yes—
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C2×C22⋊C4×C73D411yescharacteristic
112C2×C22⋊C4×C73D411yescharacteristic
113C2×C42×C73C2311yescharacteristic
114C2×C42×C73C2311yescharacteristic
115C22⋊C8×C73—22no—
116#97×C73—22no—
117C4×C8×C73—22no—
118#95×C73D411yescharacteristic
119#95×C73D411yescharacteristic
120C22×Dic73—24no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—42no—
130C22×Dic73—42no—
131C4⋊Dic73—24no—
132C4⋊Dic73—21no—
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73C22×C412yes—
145C4⋊Dic73C22×C412yes—
146C4⋊Dic73C22×C412yes—
147C4⋊Dic73C22×C412yes—
148#11924007—42no—
149#11924007—42no—
150#11924007—81no—
151#11924007—81no—
152C22⋊Dic73—24no—
153C22⋊Dic73—42no—
154C22⋊Dic73—81no—
155C4×Dic73—24no—
156C4×Dic73—42no—
157C4×Dic73—42no—
158C4×Dic73—42no—
159C4×Dic73—42no—
160C24×C73C2×D411yescharacteristic
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73C22×C412yes—
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73#4112yes—
172C22×C4×C73C2411yescharacteristic
173M16×C73—28no—
174C2×C8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73C22×C412yes—
183C2×C8×C73C22×C412yes—
184C2×C8×C73#4112yes—
185C2×C8×C73#4112yes—
186C22⋊C4×C73—22no—
187C22⋊C4×C73—41no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—21no—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73C4×D412yes—
199C23×C73C22×D411yescharacteristic
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
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×C73C4×D412yes—
218C2×C4×C73AGL(1,ℤ/8)12yes—
219C2×C4×C73C23×C411yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C732−1+411yescharacteristic
224C2×C4×C732−1+411yescharacteristic
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—48no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
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—
250C8×C73—22no—
251C8×C73—41no—
252C8×C73—41no—
253C8×C73#11714yescomplemented
254C22×C73—22no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—41no—
258C22×C73—41no—
259C22×C73—41no—
260C22×C73—41no—
261C22×C73C2×C4×D411yescharacteristic
262C22×C73#54711yescharacteristic
263C22×C73#54711yescharacteristic
264C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C73#57511yescharacteristic
266C22×C73#57711yescharacteristic
267C22×C73#57811yescharacteristic
268C4×C73—22no—
269C4×C73—22no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73C2×#11711yescharacteristic
274C4×C73#51911yescharacteristic
275C4×C73#55211yescharacteristic
276C4×C73#55211yescharacteristic
277Dic73—42no—
278Dic73—42no—
279Dic73—42no—
280Dic73—42no—
281Dic73—81no—
282Dic73—81no—
283Dic73—81no—
284Dic73—81no—
285#22624—731nomaximal, Sylow
286C2×C73—42no—
287C2×C73#234411yescharacteristic
288C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
289C2×C73#294711yescharacteristic
290C2×C73#334311yescharacteristic
291C2×C73#334811yescharacteristic
292C2×C73#335211yescharacteristic
293C2×C73#335411yescharacteristic
294C2×#428—731no—
295#2346—731no—
296#2344—731no—
297#1944—731no—
298#1941—731no—
299#1921—731no—
300#1920—731no—
301#1919—731no—
302#1918—731no—
303#1825—731no—
304#1824—731no—
305#1809—731no—
306#1776—731no—
307C4×#107—731no—
308#1753D7311yescharacteristic
309C73#2262411yescharacteristic, complemented, Sylow
310C22×M16C2×D7311yescharacteristic
311C2×#117—731no—
312C2×#117—1461no—
313C22×C4⋊C4—731no—
314#428—738no—
315C2×#107—731no—
316C2×#107—731no—
317C2×#107—731no—
318C2×#106—731no—
319C2×#106—731no—
320C2×#106—731no—
321C2×C22⋊C8C2×D7311yescharacteristic
322C2×#97C2×D7311yescharacteristic
323C2×C4×C8C2×D7311yescharacteristic
324#389—731no—
325#389—731no—
326#389—731no—
327#389—731no—
328#388—1461no—
329#388—1461no—
330#387—1461no—
331#387—1461no—
332#382—1461no—
333#381—1461no—
334#380—731no—
335#380—731no—
336C4×C4⋊C4—731no—
337C4×C4⋊C4—731no—
338C4×C4⋊C4—731no—
339C4×C4⋊C4—731no—
340C4×C22⋊C4C2×D7311yescharacteristic
341#376—1461no—
342#340—731no—
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#340—731no—
352#336C2×D7311yescharacteristic
353#336C2×D7311yescharacteristic
354C23×C4—1461no—
355C23×C4C22×D7311yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C4×D7314yes—
359C22×C8C22×D7311yescharacteristic
360C22×C8C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362C22×C8C22×D7311yescharacteristic
363#117—2924no—
364#117—734no—
365#117—1462no—
366C2×C4⋊C4—734no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1462no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1461no—
383C2×C22⋊C4—1461no—
384C2×C22⋊C4D29211yescharacteristic
385C2×C22⋊C4D29211yescharacteristic
386C2×C42—1461no—
387C2×C42—1461no—
388C2×C42—1461no—
389C2×C42—1461no—
390C2×C42—1461no—
391C2×C42C22×D7311yescharacteristic
392C2×C42C22×D7311yescharacteristic
393#107—734no—
394#107—734no—
395#107—1462no—
396#106—734no—
397#106—734no—
398#106—1462no—
399C22⋊C8—22no—
400#97—22no—
401C4×C8—22no—
402#95—1461no—
403#95—1461no—
404#95—1461no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95D29211yescharacteristic
413#95D29211yescharacteristic
414C24C2×D29211yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—2922no—
420C22×C4—2922no—
421C22×C4—1461no—
422C22×C4—1461no—
423C22×C4—1461no—
424C22×C4—1461no—
425C22×C4—1464no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1461no—
429C22×C4—1461no—
430C22×C4C2×C4×D7312yes—
431C22×C4C2×D29211yescharacteristic
432C22×C4C2×D29211yescharacteristic
433C22×C4C2×D29211yescharacteristic
434C22×C4C2×D29211yescharacteristic
435C22×C4C2×D29211yescharacteristic
436C22×C4#1192402812yes—
437C22×C4C23×D7311yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8C2×C4×D7312yes—
448C2×C8C2×C4×D7312yes—
449C2×C8#1192402712yes—
450C2×C8#1192402712yes—
451C4⋊C4—2922no—
452C4⋊C4—2922no—
453C4⋊C4—5841no—
454C4⋊C4—5841no—
455C4⋊C4—732no—
456C4⋊C4—732no—
457C4⋊C4—732no—
458C4⋊C4—732no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1464no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—2922no—
475C22⋊C4—5841no—
476C22⋊C4—1464no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—2922no—
482C42—2922no—
483C42—2922no—
484C42—2922no—
485C42—1464no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×D29212yes—
493C23C22×D29211yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
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—2928no—
512C2×C4—2921no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
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—1462no—
533C2×C4—1462no—
534C2×C4—1462no—
535C2×C4—1462no—
536C2×C4C4×D29212yes—
537C2×C4#1455661912yes—
538C2×C4C22×C4×D7311yescharacteristic
539C2×C4C2×#1192402711yescharacteristic
540C2×C4C2×#1192402711yescharacteristic
541C2×C4C2×#1192402811yescharacteristic
542C2×C4#1455663811yescharacteristic
543C2×C4#1455663811yescharacteristic
544C8—22no—
545C8—41no—
546C8—41no—
547C8#1455654714yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×D29211yescharacteristic
556C22#1798924111yescharacteristic
557C22#1798924111yescharacteristic
558C22C2×#1455661911yescharacteristic
559C22#1798938711yescharacteristic
560C22#1798939911yescharacteristic
561C22#1798940411yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—2922no—
568C4—2922no—
569C4—2922no—
570C4—2922no—
571C4—5841no—
572C4—5841no—
573C4—5841no—
574C4—5841no—
575C4C2×#1455654711yescharacteristic
576C4#1798906411yescharacteristic
577C4#1798927711yescharacteristic
578C4#1798927711yescharacteristic
579C2—42no—
580C2#2261279811yescharacteristic
581C2C4×#1455661911yescharacteristic
582C2#2261877611yescharacteristic
583C2#2262237311yescharacteristic
584C2#2262244811yescharacteristic
585C2#2262250511yescharacteristic
586C2#2262252711yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1753C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#1753D73
hypocenterC73#22624
LCSG ▷ C2×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
4. #22610674C2semidirect
5. #22610659C2semidirect
6. #22610524C2semidirect
7. #22610517C2semidirect
8. #22610510C2semidirect
9. #22610503C2semidirect
13. #22609828C2semidirect
14. C4×#14556525C2semidirect
19. #17988629C4semidirect
253. C8×C73#117semidirect
309. C73#22624semidirect
547. C8#14556547semidirect

Maximal subgroups

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

subgroupindex
C2×#179886292
#226128142
#226127982
#226106742
#226106592
#226105242
#226105172
#226105102
#226105032
#226100272
#226100222
#226099642
#226098282
C4×#145565252
C73×#17532
#2262473