Finite Group Databasebeta

← #33103679

Subgroups — #33103679

Order 18688.

866 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D73C211yesmaximal, characteristic
2#22612799C211yesmaximal, characteristic
3#22612796C211yesmaximal, characteristic
4#22610668C211yesmaximal, characteristic, complemented
5#22610666C211yesmaximal, characteristic, complemented
6#22610525C211yesmaximal, characteristic, complemented
7#22610524C211yesmaximal, characteristic, complemented
8#22610522C211yesmaximal, characteristic, complemented
9#22610016C211yesmaximal, characteristic
10#22610016C211yesmaximal, characteristic
11#22609955C211yesmaximal, characteristic
12#22609829C211yesmaximal, characteristic, complemented
13#22609827C211yesmaximal, characteristic, complemented
14#22609706C211yesmaximal, characteristic
15C73×#1921C211yesmaximal, characteristic, complemented
16C22×C8×D73C2211yescharacteristic
17C2×#14556549—21no—
18C2×#14556548—21no—
19C2×C4⋊C4×D73C2211yescharacteristic
20C2×C4⋊C4×D73C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24#107×D73C418yescomplemented
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×#14556503C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30#17988486C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988474—21no—
34#17988471—21no—
35#17988469—21no—
36#17988468—21no—
37#17988449—21no—
38#17988449—21no—
39#17988444C2211yescharacteristic
40#17988444C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4×C4⋊Dic73C2211yescharacteristic
44C4×#11924005C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55C4×C4⋊C4×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×C4×D73—21no—
61C22×C4×D73—21no—
62C22×C4×D73C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C2×C8×D73—22no—
66C2×C8×D73—22no—
67C2×C8×D73C2×C414yes—
68#14556549—44no—
69#14556548—44no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C4⋊C4×D73—22no—
80C4⋊C4×D73—22no—
81C4⋊C4×D73C2×C414yes—
82C4⋊C4×D73C2×C414yes—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005D411yescharacteristic
86C2×#11924005D411yescharacteristic
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73C2311yescharacteristic
91#14556525C2×C414yes—
92#14556524C2×C414yes—
93#14556524C2×C414yes—
94#14556522—22no—
95#14556522—22no—
96#14556503—22no—
97#14556502—22no—
98#14556501—22no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C42×C73—21no—
117C2×C42×C73C2311yescharacteristic
118#107×C73C2×C414yes—
119#95×C73—21no—
120#95×C73—21no—
121C23×D73C2×D411yescharacteristic
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C2×C4×D73—21no—
130C2×C4×D73—21no—
131C2×C4×D73—21no—
132C2×C4×D73—21no—
133C2×C4×D73—24no—
134C2×C4×D73—24no—
135C2×C4×D73—42no—
136C2×C4×D73—42no—
137C2×C4×D73—42no—
138C2×C4×D73—42no—
139C2×C4×D73C22×C412yes—
140C2×C4×D73#4112yes—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831C22×C412yes—
145C2×#10596831#4112yes—
146C2×#10596831#4112yes—
147C8×D73—28no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—42no—
155C4⋊Dic73C22×C412yes—
156C4⋊Dic73C22×C412yes—
157#11924007—24no—
158#11924007—24no—
159#11924007—42no—
160#11924007—42no—
161#11924005—22no—
162#11924005—22no—
163#11924005—42no—
164#11924005—42no—
165C4×Dic73—21no—
166C4×Dic73—21no—
167C4×Dic73—42no—
168C4×Dic73—42no—
169C4×Dic73—42no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2411yescharacteristic
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73C22×C412yes—
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×C73C22×C412yes—
191C4⋊C4×C73C22×C412yes—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—42no—
195C23×C73C22×D411yescharacteristic
196C22×D73—21no—
197C22×D73—21no—
198C22×D73—21no—
199C22×D73—21no—
200C22×D73C4×D412yes—
201C22×D73AGL(1,ℤ/8)12yes—
202C22×D73AGL(1,ℤ/8)12yes—
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—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73C23×C411yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C732−1+411yescharacteristic
226C2×C4×C732−1+411yescharacteristic
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
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×Dic73C4×D412yes—
248C4×D73—22no—
249C4×D73—22no—
250C4×D73—22no—
251C4×D73—22no—
252C4×D73—48no—
253C4×D73—48no—
254C8×C73—22no—
255C8×C73—41no—
256#10596831—41no—
257#10596831#11714yescomplemented
258C22×C73C2×C4×D411yescharacteristic
259C22×C73#54711yescharacteristic
260C22×C73#54711yescharacteristic
261C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C73#57811yescharacteristic
264C22×C73#57811yescharacteristic
265C2×D73—22no—
266C2×D73—22no—
267C2×D73—22no—
268C2×D73—22no—
269C2×D73—22no—
270C2×D73—22no—
271C2×D73#44214yes—
272C4×C73—41no—
273C4×C73—41no—
274C4×C73—42no—
275C4×C73—42no—
276C4×C73—81no—
277C4×C73—81no—
278C4×C73C2×#11711yescharacteristic
279C4×C73#51911yescharacteristic
280C4×C73#55211yescharacteristic
281C4×C73#55211yescharacteristic
282Dic73—22no—
283Dic73—41no—
284Dic73—41no—
285Dic73—42no—
286Dic73—42no—
287Dic73—81no—
288Dic73—81no—
289#22617—731nomaximal, Sylow
290C2×C73#234411yescharacteristic
291C2×C73C2×#44211yescharacteristic
292C2×C73#295011yescharacteristic
293C2×C73#334311yescharacteristic
294C2×C73#334311yescharacteristic
295C2×C73#334311yescharacteristic
296C2×C73#334311yescharacteristic
297D73—24no—
298C22×#107—731no—
299#2344—731no—
300#2344—731no—
301#1943—731no—
302#1943—731no—
303#1921—731no—
304#1921—731no—
305#1921—731no—
306#1921D7311yescharacteristic, complemented
307#1823—731no—
308#1823—731no—
309#1807—731no—
310#1776—731no—
311#1776—731no—
312#1752—731no—
313C73#2261711yescharacteristic, complemented, Sylow
314C23×C8—731no—
315C2×#117—1461no—
316C2×#117—1461no—
317C22×C4⋊C4—731no—
318C22×C4⋊C4—731no—
319C2×#107—731no—
320C2×#107—731no—
321C2×#107—731no—
322C2×#107—738no—
323C2×#107C2×D7311yescharacteristic
324C2×#106—731no—
325C2×#106—731no—
326C2×C22⋊C8—731no—
327C2×#97—731no—
328C2×#97—731no—
329#389—731no—
330#389—731no—
331#389—731no—
332#389C2×D7311yescharacteristic
333#387—1461no—
334#387—1461no—
335#387—1461no—
336#387—1461no—
337#381—1461no—
338#381—1461no—
339#380—731no—
340#380—731no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4—731no—
343C4×C4⋊C4—731no—
344C4×C4⋊C4C2×D7311yescharacteristic
345C4×C22⋊C4—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340C2×D7311yescharacteristic
354#340C2×D7311yescharacteristic
355#340C2×D7311yescharacteristic
356#336—731no—
357#336C2×D7311yescharacteristic
358C23×C4—731no—
359C23×C4—1461no—
360C23×C4—1461no—
361C22×C8—731no—
362C22×C8—734no—
363C22×C8—731no—
364C22×C8—1462no—
365C22×C8—1462no—
366C22×C8C22×D7311yescharacteristic
367C22×C8C22×D7311yescharacteristic
368#117—2924no—
369#117—2924no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—734no—
374C2×C4⋊C4—734no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—1462no—
378C2×C4⋊C4—1462no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C4⋊C4C22×D7311yescharacteristic
387C2×C4⋊C4C22×D7311yescharacteristic
388C2×C22⋊C4—731no—
389C2×C22⋊C4—731no—
390C2×C22⋊C4—1461no—
391C2×C22⋊C4—1461no—
392C2×C42—21no—
393C2×C42—731no—
394C2×C42—1461no—
395C2×C42—1461no—
396C2×C42—1461no—
397C2×C42C22×D7311yescharacteristic
398#107—734no—
399#107—734no—
400#107—734no—
401#107C4×D7314yes—
402#106—1462no—
403#106—1462no—
404C22⋊C8—1462no—
405#97—1462no—
406#97—1462no—
407#95—21no—
408#95—21no—
409#95—731no—
410#95—731no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419C24—731no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—2922no—
425C22×C4—2922no—
426C22×C4—2922no—
427C22×C4—2922no—
428C22×C4—731no—
429C22×C4—732no—
430C22×C4—732no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—1464no—
434C22×C4—1464no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4—1461no—
442C22×C4—1461no—
443C22×C4D4×D7311yescharacteristic
444C22×C4D4×D7311yescharacteristic
445C22×C4C23×D7311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—732no—
452C2×C8—732no—
453C2×C8—732no—
454C2×C8—1468no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8C2×C4×D7312yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—2922no—
467C4⋊C4—2922no—
468C4⋊C4—2922no—
469C4⋊C4—732no—
470C4⋊C4—732no—
471C4⋊C4—1464no—
472C4⋊C4—1464no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4C2×C4×D7312yes—
480C4⋊C4C2×C4×D7312yes—
481C22⋊C4—2922no—
482C22⋊C4—2922no—
483C22⋊C4—1462no—
484C22⋊C4—1462no—
485C42—21no—
486C42—21no—
487C42—42no—
488C42—2922no—
489C42—2922no—
490C42—2922no—
491C42—1461no—
492C42—1461no—
493C23—732no—
494C23—732no—
495C23—732no—
496C23—1461no—
497C23—1461no—
498C23—1461no—
499C23—1461no—
500C23C2×D4×D7311yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—2928no—
520C2×C4—2928no—
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—732no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1462no—
537C2×C4—1462no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1462no—
543C2×C4—1462no—
544C2×C4—1462no—
545C2×C4—1462no—
546C2×C4C22×C4×D7311yescharacteristic
547C2×C4#41×D7311yescharacteristic
548C2×C4#41×D7311yescharacteristic
549C2×C4#41×D7311yescharacteristic
550C2×C4#1455663811yescharacteristic
551C2×C4#1455663911yescharacteristic
552C8—22no—
553C8—41no—
554C8—2921no—
555C8—734no—
556C22—734no—
557C22—1462no—
558C22—1462no—
559C22—1462no—
560C22—1462no—
561C22—1462no—
562C22—1462no—
563C22C4×D4×D7311yescharacteristic
564C22#1798923511yescharacteristic
565C22#1798923911yescharacteristic
566C22AGL(1,ℤ/8)×D7311yescharacteristic
567C22AGL(1,ℤ/8)×D7311yescharacteristic
568C22#1798940011yescharacteristic
569C22#1798940111yescharacteristic
570C4—42no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—2922no—
577C4—2922no—
578C4—2921no—
579C4—2921no—
580C4—5841no—
581C4—5841no—
582C4—1462no—
583C4#117×D7311yescharacteristic
584C4#1798906311yescharacteristic
585C4#1798926911yescharacteristic
586C4#1798927311yescharacteristic
587C2—1464no—
588C2#2261279711yescharacteristic
589C2#442×D7311yescharacteristic
590C2#2261879411yescharacteristic
591C2#2262236411yescharacteristic
592C2#2262236811yescharacteristic
593C2#2262237411yescharacteristic
594C2#2262237811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1921C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1921D73
hypocenterC73#22617
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
4. #22610668C2semidirect
5. #22610666C2semidirect
6. #22610525C2semidirect
7. #22610524C2semidirect
8. #22610522C2semidirect
12. #22609829C2semidirect
13. #22609827C2semidirect
15. C73×#1921C2semidirect
24. #107×D73C4semidirect
257. #10596831#117semidirect
306. #1921D73semidirect
313. C73#22617semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D732
#226127992
#226127962
#226106682
#226106662
#226105252
#226105242
#226105222
#226100162
#226099552
#226098292
#226098272
#226097062
C73×#19212
#2261773