Finite Group Databasebeta

← #33107825

Subgroups — #33107825

Order 18688.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988887C211yesmaximal, characteristic
2#22613027C211yesmaximal, characteristic, complemented
3#22612841C211yesmaximal, characteristic
4#22610896C211yesmaximal, characteristic, complemented
5#22610699C211yesmaximal, characteristic, complemented
6#22610663C211yesmaximal, characteristic, complemented
7#22610658C211yesmaximal, characteristic, complemented
8#22610526C211yesmaximal, characteristic, complemented
9#22610519C211yesmaximal, characteristic, complemented
10#22610190C211yesmaximal, characteristic
11#22610176C211yesmaximal, characteristic
12C8×C4⋊Dic73C211yesmaximal, characteristic, complemented
13#102×Dic73C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C4×#102×C73C211yesmaximal, characteristic
16C2×#14556583C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19#17988887C418yescomplemented
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×C8×Dic73C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988486C2211yescharacteristic
30#17988486C2211yescharacteristic
31#17988484—21no—
32#17988483C2211yescharacteristic
33#17988483C2211yescharacteristic
34#17988474—21no—
35#17988471—21no—
36#17988462—21no—
37#17988462—21no—
38#17988453C2211yescharacteristic
39#17988447—21no—
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4×#11924007—21no—
43C4×C4⋊Dic73C2211yescharacteristic
44C2×#14556498—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C23×Dic73—21no—
61C23×Dic73—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64#14556583—22no—
65#14556583C2×C414yescomplemented
66D4×Dic73—22no—
67D4×Dic73—22no—
68D4×Dic73—41no—
69D4×Dic73—41no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73D411yescharacteristic
73C2×C4⋊Dic73D411yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556524—41no—
89#14556524C2×C414yescomplemented
90#14556522—41no—
91#14556522C2×C414yescomplemented
92C4⋊#10596831C2×C414yescomplemented
93#14556512C2×C414yes—
94#14556512C2×C414yes—
95C8×Dic73—41no—
96C8×Dic73—41no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498D411yescharacteristic
106#14556498D411yescharacteristic
107C22×D4×C73C2311yescharacteristic
108C23×C4×C73—21no—
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C4×D4×C73—22no—
112C4×D4×C73C2×C414yes—
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C22⋊C4×C73—21no—
117C2×C42×C73—21no—
118C2×C42×C73C2311yescharacteristic
119#102×C73—22no—
120#102×C73Q814yes—
121C4×C8×C73—22no—
122#95×C73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—42no—
128C22×Dic73—42no—
129C22×Dic73—42no—
130C22×Dic73—42no—
131C22×Dic73—81no—
132C22×Dic73—81no—
133C22×Dic73C2×D411yescharacteristic
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831C22×C412yes—
142C2×#10596831C22×C412yes—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—22no—
146C4⋊Dic73—22no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—81no—
152C4⋊Dic73—81no—
153C4⋊Dic73C22×C412yes—
154C4⋊Dic73C22×C412yes—
155#11924007—24no—
156#11924007—24no—
157#11924007—42no—
158#11924007—81no—
159C22⋊Dic73—42no—
160C22⋊Dic73—42no—
161C4×Dic73—21no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—42no—
166C4×Dic73—81no—
167C24×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73C22×C412yes—
172C2×D4×C73C2×Q812yes—
173C2×D4×C73#4112yes—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—24no—
177C22×C4×C73—21no—
178C22×C4×C73—81no—
179C22×C4×C73—81no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2411yescharacteristic
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73C2×Q812yes—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73C22×C412yes—
194C4⋊C4×C73C2×Q812yes—
195C22⋊C4×C73—24no—
196C42×C73—81no—
197C42×C73C22×C412yes—
198C42×C73#4112yes—
199C23×C73—22no—
200C23×C73—22no—
201C23×C73—22no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73C22×D411yescharacteristic
207D4×C73—22no—
208D4×C73—22no—
209D4×C73—22no—
210D4×C73C4×Q814yes—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—84no—
225C2×C4×C73C4×D412yes—
226C2×C4×C73C4×D412yes—
227C2×C4×C73C23×C411yescharacteristic
228C2×C4×C73C22×Q811yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73H(5,𝔽2)11yescharacteristic
232C2×C4×C732−1+411yescharacteristic
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
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—84no—
256C2×Dic73—84no—
257C2×Dic73#13512yes—
258C2×Dic73#13512yes—
259C8×C73—22no—
260C8×C73—41no—
261#10596831—22no—
262#10596831—22no—
263C22×C73—24no—
264C22×C73—42no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73—42no—
268C22×C73—42no—
269C22×C73—42no—
270C22×C73C2×C4×D411yescharacteristic
271C22×C73#54611yescharacteristic
272C22×C73D4×Q811yescharacteristic
273C22×C73C2×#13511yescharacteristic
274C22×C73C2×#13511yescharacteristic
275C22×C73#57611yescharacteristic
276C22×C73#57711yescharacteristic
277C4×C73—22no—
278C4×C73—22no—
279C4×C73—42no—
280C4×C73—42no—
281C4×C73—81no—
282C4×C73C2×C4×Q811yescharacteristic
283C4×C73#52011yescharacteristic
284C4×C73#55211yescharacteristic
285C4×C73#55411yescharacteristic
286Dic73—22no—
287Dic73—22no—
288Dic73—42no—
289Dic73—42no—
290Dic73—41no—
291Dic73—41no—
292Dic73—81no—
293Dic73—81no—
294#22966—731nomaximal, Sylow
295C2×C73—44no—
296C2×C73#236011yescharacteristic
297C2×C73C4×#13511yescharacteristic
298C2×C73#294611yescharacteristic
299C2×C73#329611yescharacteristic
300C2×C73#329711yescharacteristic
301C2×C73#338011yescharacteristic
302C2×C73#338111yescharacteristic
303C2×#478—731no—
304#2370—731no—
305#2349—731no—
306#1991—731no—
307#1949—731no—
308#1942—731no—
309#1941—731no—
310#1921—731no—
311#1920—731no—
312#1864—731no—
313#1862—731no—
314C8×C4⋊C4—731no—
315C4×#102—731no—
316C4×#102—731no—
317C4×#102D7311yescharacteristic
318C73#2296611yescharacteristic, complemented, Sylow
319C2×#125—731no—
320C2×C4×D4—731no—
321C2×C4×D4—731no—
322C2×C4×D4C2×D7311yescharacteristic
323#478—738no—
324C2×#107—731no—
325C2×#107—731no—
326C2×#106—731no—
327C2×#106—731no—
328C2×C4⋊C8—731no—
329C2×#102—731no—
330C2×#102—731no—
331C2×#102C2×D7311yescharacteristic
332C2×#102C2×D7311yescharacteristic
333C2×C4×C8—731no—
334C2×C4×C8—731no—
335C2×C4×C8C2×D7311yescharacteristic
336#389—731no—
337#389—731no—
338#389—731no—
339#389—731no—
340#389—1461no—
341#387—1461no—
342#387—1461no—
343#385—1461no—
344#385—1461no—
345#382—731no—
346#380—1461no—
347C4×C4⋊C4—731no—
348C4×C4⋊C4—731no—
349C4×C4⋊C4—731no—
350C4×C4⋊C4—1461no—
351C4×C4⋊C4C2×D7311yescharacteristic
352C2×#95—1461no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340C2×D7311yescharacteristic
360#340C2×D7311yescharacteristic
361#336—731no—
362#336—731no—
363C22×D4C22×D7311yescharacteristic
364C23×C4—21no—
365C23×C4—1461no—
366C23×C4—1461no—
367C22×C8—731no—
368C22×C8—731no—
369C22×C8C22×D7311yescharacteristic
370C22×C8C22×D7311yescharacteristic
371#125—734no—
372#125—1462no—
373C4×D4—22no—
374C4×D4—2921no—
375C4×D4—2921no—
376C4×D4—1462no—
377C4×D4—1462no—
378C4×D4C4×D7314yes—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4—1461no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4C22×D7311yescharacteristic
392C2×C4⋊C4C22×D7311yescharacteristic
393C2×C4⋊C4C22×D7311yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—1461no—
396C2×C22⋊C4—1461no—
397C2×C42—21no—
398C2×C42—731no—
399C2×C42—731no—
400C2×C42—1461no—
401C2×C42—1461no—
402C2×C42C22×D7311yescharacteristic
403#107—2921no—
404#107—734no—
405#106—2921no—
406#106—734no—
407C4⋊C8—734no—
408#102—22no—
409#102—734no—
410#102—734no—
411#102Dic14614yes—
412C4×C8—22no—
413C4×C8—2921no—
414C4×C8—2921no—
415#95—21no—
416#95—731no—
417#95—731no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426C24—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×C4×D7312yes—
431C2×D4C2×Dic14612yes—
432C2×D4#1192402812yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—2922no—
440C22×C4—2922no—
441C22×C4—2922no—
442C22×C4—2922no—
443C22×C4—5841no—
444C22×C4—5841no—
445C22×C4—731no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4D4×D7311yescharacteristic
454C22×C4D4×D7311yescharacteristic
455C22×C4C23×D7311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—732no—
462C2×C8—732no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8C2×Dic14612yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—2922no—
473C4⋊C4—5841no—
474C4⋊C4—5841no—
475C4⋊C4—5841no—
476C4⋊C4—732no—
477C4⋊C4—732no—
478C4⋊C4—1462no—
479C4⋊C4—1462no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1464no—
485C4⋊C4—1464no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4C2×C4×D7312yes—
489C4⋊C4C2×Dic14612yes—
490C22⋊C4—24no—
491C22⋊C4—2922no—
492C22⋊C4—2922no—
493C42—81no—
494C42—2922no—
495C42—5841no—
496C42—1461no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C42C2×C4×D7312yes—
501C42#1192402712yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×D4×D7311yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4C4×Dic14614yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—84no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—5844no—
539C2×C4—5844no—
540C2×C4—732no—
541C2×C4—732no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4#1455655112yes—
555C2×C4#1455655112yes—
556C2×C4C22×C4×D7311yescharacteristic
557C2×C4C22×Dic14611yescharacteristic
558C2×C4C2×#1192402711yescharacteristic
559C2×C4C2×#1192402811yescharacteristic
560C2×C4#1455663711yescharacteristic
561C2×C4#1455663811yescharacteristic
562C8—22no—
563C8—41no—
564C8—1462no—
565C8—1462no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C2×#1455655111yescharacteristic
574C22#1798923311yescharacteristic
575C22D4×Dic14611yescharacteristic
576C22#135×D7311yescharacteristic
577C22#135×D7311yescharacteristic
578C22#1798938811yescharacteristic
579C22#1798939311yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—81no—
585C4—2922no—
586C4—2921no—
587C4—2921no—
588C4—2922no—
589C4—5841no—
590C4—5841no—
591C4—1462no—
592C4—1462no—
593C4C2×C4×Dic14611yescharacteristic
594C4#1798906711yescharacteristic
595C4#1798927711yescharacteristic
596C4#1798929411yescharacteristic
597C2—44no—
598C2#2261293411yescharacteristic
599C2#2261869911yescharacteristic
600C2#2261875711yescharacteristic
601C2#2262177111yescharacteristic
602C2#2262178611yescharacteristic
603C2#2262279811yescharacteristic
604C2#2262280711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×#102×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC4×#102D73
hypocenterC73#22966
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #22613027C2semidirect
4. #22610896C2semidirect
5. #22610699C2semidirect
6. #22610663C2semidirect
7. #22610658C2semidirect
8. #22610526C2semidirect
9. #22610519C2semidirect
12. C8×C4⋊Dic73C2semidirect
19. #17988887C4semidirect
65. #14556583C2×C4semidirect
89. #14556524C2×C4semidirect
91. #14556522C2×C4semidirect
92. C4⋊#10596831C2×C4semidirect
318. C73#22966semidirect

Maximal subgroups

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

subgroupindex
C2×#179888872
#226130272
#226128412
#226108962
#226106992
#226106632
#226106582
#226105262
#226105192
#226101902
#226101762
C8×C4⋊Dic732
#102×Dic732
C4×#102×C732
#2296673