Finite Group Databasebeta

← #33093253

Subgroups — #33093253

Order 18688.

844 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988924C211yesmaximal, characteristic
2#22612611C211yesmaximal, characteristic
3#22612285C211yesmaximal, characteristic
4#22610572C211yesmaximal, characteristic, complemented
5#22610566C211yesmaximal, characteristic, complemented
6#22610520C211yesmaximal, characteristic, complemented
7#22610242C211yesmaximal, characteristic
8#22610131C211yesmaximal, characteristic
9#22610016C211yesmaximal, characteristic
10#22609830C211yesmaximal, characteristic, complemented
11#22609823C211yesmaximal, characteristic, complemented
12#22609760C211yesmaximal, characteristic, complemented
13#22609750C211yesmaximal, characteristic, complemented
14#22609696C211yesmaximal, characteristic
15C73×#1918C211yesmaximal, characteristic, complemented
16C2×#14556571C2211yescharacteristic
17C2×#14556556C2211yescharacteristic
18C2×#14556549C2211yescharacteristic
19C2×#14556548C2211yescharacteristic
20C2×C4⋊C4×D73C2211yescharacteristic
21#17988924C418yescomplemented
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556503C2211yescharacteristic
30C2×#14556503C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988483—21no—
35#17988477—21no—
36#17988476—21no—
37#17988470—21no—
38#17988444C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4×#11924007—21no—
43C4×C4⋊Dic73C2211yescharacteristic
44#17988438—21no—
45C4×#11924005—21no—
46#17988434—21no—
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988342C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55C4×C4⋊C4×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×Dic146C2311yescharacteristic
61C22×C4×D73—21no—
62C22×C4×D73C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556571—22no—
66#14556571C2×C414yes—
67#14556556—22no—
68#14556556—22no—
69#14556549—22no—
70#14556549—22no—
71#14556548—22no—
72#14556548—22no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007D411yescharacteristic
80C2×#11924007D411yescharacteristic
81C4⋊C4×D73—22no—
82C4⋊C4×D73C2×C414yes—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×#11924005D411yescharacteristic
87C2×#11924005D411yescharacteristic
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73C2311yescharacteristic
95#14556524—22no—
96#14556524C2×C414yes—
97#14556523—22no—
98#14556516—22no—
99#14556516C2×C414yes—
100#14556514—22no—
101#14556514C2×C414yes—
102#14556503—22no—
103#14556503C2×C414yes—
104#14556501—22no—
105C8×Dic73—22no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C42×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#107×C73C2×C414yes—
123#95×C73—21no—
124#95×C73—21no—
125C23×D73C2×D411yescharacteristic
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146C22×C412yes—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C2×C4×D73—21no—
140C2×C4×D73—21no—
141C2×C4×D73—21no—
142C2×C4×D73—21no—
143C2×C4×D73—24no—
144C2×C4×D73—81no—
145C2×C4×D73—81no—
146C2×C4×D73C22×C412yes—
147C2×C4×D73#4112yes—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831C22×C412yes—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—24no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73C22×C412yes—
160C4⋊Dic73#4112yes—
161#11924007—24no—
162#11924007—22no—
163#11924007—22no—
164#11924007—42no—
165#11924005—24no—
166#11924005—24no—
167#11924005—22no—
168#11924005—22no—
169#11924005—81no—
170C4×Dic73—24no—
171C4×Dic73—42no—
172C4×Dic73—81no—
173C4×Dic73—81no—
174C4×Dic73—81no—
175C4×Dic73—81no—
176C4×Dic73#4112yes—
177C4×Dic73#4112yes—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2411yescharacteristic
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73C22×C412yes—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—81no—
197C4⋊C4×C73C22×C412yes—
198C4⋊C4×C73C22×C412yes—
199C4⋊C4×C73#4112yes—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—81no—
203C23×C73C22×D411yescharacteristic
204C22×D73—21no—
205C22×D73—21no—
206C22×D73—21no—
207C22×D73—21no—
208C22×D73—21no—
209C22×D73—21no—
210C22×D73C4×D412yes—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—22no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73C23×C411yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
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×Dic73C4×D412yes—
254C2×Dic73#12312yes—
255C2×Dic73#12312yes—
256C4×D73—22no—
257C4×D73—22no—
258C4×D73—22no—
259C4×D73—22no—
260C4×D73—84no—
261Dic146—22no—
262Dic146—22no—
263Dic146—41no—
264Dic146—41no—
265C8×C73—22no—
266C8×C73—41no—
267#10596831—22no—
268#10596831—41no—
269C22×C73C2×C4×D411yescharacteristic
270C22×C73C2×#12311yescharacteristic
271C22×C73#52511yescharacteristic
272C22×C73#57611yescharacteristic
273C22×C73#57711yescharacteristic
274C22×C73#57811yescharacteristic
275C22×C73#57811yescharacteristic
276C2×D73—22no—
277C2×D73—22no—
278C2×D73—22no—
279C2×D73—41no—
280C2×D73—41no—
281C2×D73—41no—
282C2×D73—41no—
283C4×C73—41no—
284C4×C73—41no—
285C4×C73—42no—
286C4×C73—42no—
287C4×C73—42no—
288C4×C73—81no—
289C4×C73C4×#4111yescharacteristic
290C4×C73C4×#4111yescharacteristic
291C4×C73#52911yescharacteristic
292C4×C73#52911yescharacteristic
293Dic73—22no—
294Dic73—22no—
295Dic73—22no—
296Dic73—42no—
297Dic73—42no—
298Dic73—42no—
299Dic73—81no—
300#21585—731nomaximal, Sylow
301C2×C73C4×#12311yescharacteristic
302C2×C73#294611yescharacteristic
303C2×C73#295011yescharacteristic
304C2×C73#309711yescharacteristic
305C2×C73#309811yescharacteristic
306C2×C73#312011yescharacteristic
307C2×C73#312411yescharacteristic
308D73—42no—
309C2×#484—731no—
310#2323—731no—
311#2274—731no—
312#1928—731no—
313#1927—731no—
314#1921—731no—
315#1918D7311yescharacteristic, complemented
316#1873—731no—
317#1854—731no—
318#1823—731no—
319#1776—731no—
320C4×#107—731no—
321#1763—731no—
322C4×#103—731no—
323#1750—731no—
324C73#2158511yescharacteristic, complemented, Sylow
325C2×C22⋊Q8—731no—
326C2×C4×Q8—731no—
327C2×#117—731no—
328C2×#117—731no—
329C22×C4⋊C4—731no—
330#484—738no—
331C2×#107—731no—
332C2×#107—731no—
333C2×#107C2×D7311yescharacteristic
334C2×#106—731no—
335C2×#103—731no—
336C2×#103—731no—
337C2×#103—731no—
338C2×#103—731no—
339C2×C22⋊C8—731no—
340C2×C22⋊C8—731no—
341C2×#97—731no—
342C2×C4×C8—731no—
343#391—731no—
344#389—1461no—
345#389C2×D7311yescharacteristic
346#388—1461no—
347#388—1461no—
348#387—1461no—
349#380—731no—
350C4×C4⋊C4—731no—
351C4×C4⋊C4—731no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—1461no—
355C4×C4⋊C4—1461no—
356C4×C4⋊C4C2×D7311yescharacteristic
357C4×C22⋊C4—1461no—
358C4×C22⋊C4—1461no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340C2×D7311yescharacteristic
365#340C2×D7311yescharacteristic
366#340C2×D7311yescharacteristic
367#336—731no—
368#336C2×D7311yescharacteristic
369C22×Q8—731no—
370C23×C4—731no—
371C23×C4—1461no—
372C22×C8—731no—
373C22×C8—731no—
374C22×C8C22×D7311yescharacteristic
375C22×C8C22×D7311yescharacteristic
376C22⋊Q8—734no—
377C22⋊Q8—1462no—
378C4×Q8—1462no—
379C4×Q8—1462no—
380#117—1462no—
381#117—1462no—
382#117—1462no—
383#117—1462no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—734no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—1462no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4C22×D7311yescharacteristic
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C22⋊C4—731no—
400C2×C22⋊C4—731no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4—1461no—
404C2×C42—21no—
405C2×C42—731no—
406C2×C42—1461no—
407C2×C42—1461no—
408C2×C42—1461no—
409C2×C42—1461no—
410C2×C42—1461no—
411C2×C42—1461no—
412C2×C42C22×D7311yescharacteristic
413#107—734no—
414#107—1462no—
415#107C4×D7314yes—
416#106—1462no—
417#103—734no—
418#103—734no—
419#103—1462no—
420#103—1462no—
421C22⋊C8—734no—
422C22⋊C8—1462no—
423#97—1462no—
424C4×C8—1462no—
425#95—21no—
426#95—21no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434C24—731no—
435C2×Q8—732no—
436C2×Q8—1461no—
437C2×Q8—1461no—
438C2×Q8—1461no—
439C2×Q8—1461no—
440C2×Q8—1461no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—5841no—
446C22×C4—5841no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—732no—
451C22×C4—732no—
452C22×C4—1464no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4D4×D7311yescharacteristic
462C22×C4D4×D7311yescharacteristic
463C22×C4C23×D7311yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—732no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8C2×C4×D7312yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—81no—
482C4⋊C4—2922no—
483C4⋊C4—732no—
484C4⋊C4—732no—
485C4⋊C4—1464no—
486C4⋊C4—1462no—
487C4⋊C4—1462no—
488C4⋊C4—1464no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4C2×C4×D7312yes—
494C4⋊C4C2×C4×D7312yes—
495C4⋊C4#1192402912yes—
496C22⋊C4—5841no—
497C22⋊C4—1462no—
498C22⋊C4—1462no—
499C22⋊C4—1464no—
500C22⋊C4—1464no—
501C42—21no—
502C42—21no—
503C42—81no—
504C42—2922no—
505C42—5841no—
506C42—5841no—
507C42—5841no—
508C42—5841no—
509C42—732no—
510C42—732no—
511C42—1464no—
512C23—732no—
513C23—1461no—
514C23—1461no—
515C23—1461no—
516C23—1461no—
517C23—1461no—
518C23—1461no—
519C23C2×D4×D7311yescharacteristic
520Q8—2921no—
521Q8—2921no—
522Q8—1462no—
523Q8—1462no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
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—5844no—
552C2×C4—732no—
553C2×C4—732no—
554C2×C4—732no—
555C2×C4—1462no—
556C2×C4—1462no—
557C2×C4—1462no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1462no—
565C2×C4—1462no—
566C2×C4—1462no—
567C2×C4—1462no—
568C2×C4C22×C4×D7311yescharacteristic
569C2×C4#41×D7311yescharacteristic
570C2×C4#41×D7311yescharacteristic
571C2×C4#41×D7311yescharacteristic
572C2×C4#41×D7311yescharacteristic
573C2×C4C2×#1192402911yescharacteristic
574C8—22no—
575C8—41no—
576C8—2921no—
577C8—1462no—
578C22—2921no—
579C22—2921no—
580C22—2921no—
581C22—2921no—
582C22—1462no—
583C22—1462no—
584C22—1462no—
585C22C4×D4×D7311yescharacteristic
586C22#123×D7311yescharacteristic
587C22#1798910011yescharacteristic
588C22#1798938811yescharacteristic
589C22#1798939411yescharacteristic
590C22#1798940311yescharacteristic
591C22#1798940311yescharacteristic
592C4—41no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—2922no—
599C4—2922no—
600C4—2922no—
601C4—5841no—
602C4—1462no—
603C4—1462no—
604C4—1462no—
605C4#1798905511yescharacteristic
606C4C4×#1192402911yescharacteristic
607C4#1798912011yescharacteristic
608C4#1798912211yescharacteristic
609C2—2922no—
610C2#2261245211yescharacteristic
611C2#2261875611yescharacteristic
612C2#2261879611yescharacteristic
613C2#2261989711yescharacteristic
614C2#2261990411yescharacteristic
615C2#2262015211yescharacteristic
616C2#2262019811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1918C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1918D73
hypocenterC73#21585
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
4. #22610572C2semidirect
5. #22610566C2semidirect
6. #22610520C2semidirect
10. #22609830C2semidirect
11. #22609823C2semidirect
12. #22609760C2semidirect
13. #22609750C2semidirect
15. C73×#1918C2semidirect
21. #17988924C4semidirect
315. #1918D73semidirect
324. C73#21585semidirect

Maximal subgroups

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

subgroupindex
C2×#179889242
#226126112
#226122852
#226105722
#226105662
#226105202
#226102422
#226101312
#226100162
#226098302
#226098232
#226097602
#226097502
#226096962
C73×#19182
#2158573