Finite Group Databasebeta

← #33093243

Subgroups — #33093243

Order 18688.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988906C211yesmaximal, characteristic
2#22612624C211yesmaximal, characteristic
3#22612287C211yesmaximal, characteristic, complemented
4#22610559C211yesmaximal, characteristic, complemented
5#22610552C211yesmaximal, characteristic, complemented
6#22610517C211yesmaximal, characteristic, complemented
7#22610510C211yesmaximal, characteristic, complemented
8#22610251C211yesmaximal, characteristic
9#22610132C211yesmaximal, characteristic, complemented
10#22610018C211yesmaximal, characteristic, complemented
11#22609828C211yesmaximal, characteristic, complemented
12C4×#14556523C211yesmaximal, characteristic, complemented
13#22609757C211yesmaximal, characteristic, complemented
14C4×#14556513C211yesmaximal, characteristic, complemented
15C73×#1750C211yesmaximal, characteristic, complemented
16C2×#14556565C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×#14556550C2211yescharacteristic
19C22×C4⋊Dic73C2211yescharacteristic
20#17988906C418yescomplemented
21C2×#14556525C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556513C2211yescharacteristic
27C2×#14556513C2211yescharacteristic
28C2×#14556513C2211yescharacteristic
29#17988489C2211yescharacteristic
30#17988487—21no—
31#17988485C2211yescharacteristic
32#17988477—21no—
33#17988463—21no—
34#17988458—21no—
35#17988447C2211yescharacteristic
36C4⋊C4×Dic73—21no—
37C4⋊C4×Dic73—21no—
38C4×C4⋊Dic73C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43C22⋊C4×Dic73—21no—
44C22⋊C4×Dic73—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C2×#117×C73C2211yescharacteristic
54C2×C22⋊C8×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C2×#97×C73C2211yescharacteristic
57C2×C4×C8×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C22⋊D73—21no—
62C23×Dic73—21no—
63C22×C4×D73—21no—
64#14556565—22no—
65#14556565C2×C414yes—
66C4×D292—22no—
67C4×D292—22no—
68#14556550—22no—
69#14556550—22no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—22no—
72C2×C4⋊Dic73C2×C414yescomplemented
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924007—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93#14556525—22no—
94#14556523—22no—
95#14556523C2×C414yescomplemented
96#14556523C2×C414yescomplemented
97#14556513—22no—
98#14556513—22no—
99#14556513C2×C414yes—
100#14556513C2×C414yes—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108C23×C4×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113#117×C73—22no—
114#117×C73—22no—
115C2×C4⋊C4×C73D411yescharacteristic
116C2×C4⋊C4×C73D411yescharacteristic
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C42×C73C2311yescharacteristic
120C2×C42×C73C2311yescharacteristic
121C22⋊C8×C73—22no—
122C22⋊C8×C73C2×C414yescomplemented
123#97×C73—22no—
124C4×C8×C73—22no—
125C23×D73—21no—
126C2×D292—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292—21no—
131C2×D292C22×C412yes—
132C2×C22⋊D73—24no—
133C2×C22⋊D73—81no—
134C2×C22⋊D73—81no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—24no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—81no—
144C22×Dic73—81no—
145C2×C4×D73—42no—
146C2×C4×D73—81no—
147C2×C4×D73—81no—
148C4⋊Dic73—24no—
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⋊Dic73C22×C412yes—
159C4⋊Dic73C22×C412yes—
160C4⋊Dic73C22×C412yes—
161C4⋊Dic73#4112yes—
162C4⋊Dic73#4112yes—
163#11924007—81no—
164C22⋊Dic73—24no—
165C22⋊Dic73—81no—
166#11924005—24no—
167#11924005—42no—
168#11924005—81no—
169#11924005—81no—
170C4×Dic73—24no—
171C4×Dic73—81no—
172C4×Dic73—81no—
173C4×Dic73—81no—
174C4×Dic73—81no—
175C4×Dic73—81no—
176C24×C73C2×D411yescharacteristic
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73C22×C412yes—
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73#4112yes—
188C22×C4×C73C2411yescharacteristic
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×C73C22×C412yes—
200C2×C8×C73C22×C412yes—
201C4⋊C4×C73—22no—
202C4⋊C4×C73—22no—
203C22⋊C4×C73—22no—
204C22⋊C4×C73—22no—
205C42×C73—21no—
206C42×C73—21no—
207C42×C73#4112yes—
208C42×C73#4112yes—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73C4×D412yes—
216C23×C73C22×D411yescharacteristic
217C22×D73—22no—
218C22×D73—41no—
219C22×D73—41no—
220C22×D73—41no—
221C22×D73—41no—
222C22×D73—41no—
223C22×D73—41no—
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—22no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73C4×D412yes—
241C2×C4×C73#12312yes—
242C2×C4×C73#12312yes—
243C2×C4×C73C23×C411yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C22⋊D73—84no—
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—22no—
253C2×Dic73—22no—
254C2×Dic73—22no—
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—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—84no—
272D292—22no—
273D292—22no—
274D292—41no—
275D292—41no—
276C4×D73—84no—
277C8×C73—22no—
278C8×C73—22no—
279C8×C73—41no—
280C8×C73—41no—
281C22×C73—22no—
282C22×C73—22no—
283C22×C73—22no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73C2×C4×D411yescharacteristic
289C22×C73C2×#12311yescharacteristic
290C22×C73#52511yescharacteristic
291C22×C73#57611yescharacteristic
292C22×C73#57711yescharacteristic
293C22×C73#57711yescharacteristic
294C22×C73#57711yescharacteristic
295C2×D73—42no—
296C2×D73—42no—
297C2×D73—42no—
298C2×D73—81no—
299C2×D73—81no—
300C2×D73—81no—
301C2×D73—81no—
302C4×C73—22no—
303C4×C73—22no—
304C4×C73—22no—
305C4×C73—41no—
306C4×C73—41no—
307C4×C73C4×#4111yescharacteristic
308C4×C73C4×#4111yescharacteristic
309C4×C73#52911yescharacteristic
310C4×C73#52911yescharacteristic
311Dic73—42no—
312Dic73—42no—
313Dic73—42no—
314Dic73—42no—
315Dic73—42no—
316Dic73—81no—
317Dic73—81no—
318#21584—731nomaximal, Sylow
319C2×C73—42no—
320C2×C73C4×#12311yescharacteristic
321C2×C73#294611yescharacteristic
322C2×C73#294911yescharacteristic
323C2×C73#309611yescharacteristic
324C2×C73#309911yescharacteristic
325C2×C73#311911yescharacteristic
326C2×C73#312511yescharacteristic
327D73—82no—
328C2×#481—731no—
329#2324—731no—
330#2274—731no—
331#1926—731no—
332#1925—731no—
333#1920—731no—
334#1919—731no—
335#1875—731no—
336#1854—731no—
337#1823—731no—
338#1776—731no—
339C4×#106—731no—
340#1762—731no—
341C4×#102—731no—
342#1750D7311yescharacteristic, complemented
343C73#2158411yescharacteristic, complemented, Sylow
344C2×#121—731no—
345C2×C4×D4—731no—
346C2×#117—731no—
347C2×#117C2×D7311yescharacteristic
348C22×C4⋊C4—731no—
349#481—738no—
350C2×#107—731no—
351C2×#106—731no—
352C2×#106—731no—
353C2×#106—731no—
354C2×#102—731no—
355C2×#102—731no—
356C2×#102—731no—
357C2×#102—731no—
358C2×C22⋊C8C2×D7311yescharacteristic
359C2×C22⋊C8C2×D7311yescharacteristic
360C2×#97C2×D7311yescharacteristic
361C2×C4×C8C2×D7311yescharacteristic
362#390—731no—
363#390—1461no—
364#389—731no—
365#388—1461no—
366#386—1461no—
367#385—1461no—
368#380—731no—
369C4×C4⋊C4—731no—
370C4×C4⋊C4—731no—
371C4×C4⋊C4—731no—
372C4×C4⋊C4—731no—
373C4×C4⋊C4—731no—
374C4×C4⋊C4—1461no—
375C4×C4⋊C4—1461no—
376C4×C22⋊C4—1461no—
377C4×C22⋊C4—1461no—
378#340—731no—
379#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#340—731no—
386#336C2×D7311yescharacteristic
387#336C2×D7311yescharacteristic
388C22×D4—731no—
389C22×D4—1461no—
390C23×C4—1461no—
391C23×C4—1461no—
392C23×C4C22×D7311yescharacteristic
393C22×C8C22×D7311yescharacteristic
394C22×C8C22×D7311yescharacteristic
395C22×C8C22×D7311yescharacteristic
396C22×C8C22×D7311yescharacteristic
397#121—734no—
398#121—1462no—
399C4×D4—1462no—
400C4×D4—1462no—
401#117—22no—
402#117—22no—
403#117—1462no—
404#117—1462no—
405C2×C4⋊C4—734no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—1461no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4—1462no—
416C2×C4⋊C4D29211yescharacteristic
417C2×C4⋊C4D29211yescharacteristic
418C2×C22⋊C4—1461no—
419C2×C22⋊C4—1461no—
420C2×C22⋊C4—1461no—
421C2×C22⋊C4—1461no—
422C2×C22⋊C4—1461no—
423C2×C22⋊C4—1461no—
424C2×C22⋊C4D29211yescharacteristic
425C2×C22⋊C4D29211yescharacteristic
426C2×C42—1461no—
427C2×C42—1461no—
428C2×C42—1461no—
429C2×C42—1461no—
430C2×C42—1461no—
431C2×C42—1461no—
432C2×C42C22×D7311yescharacteristic
433C2×C42C22×D7311yescharacteristic
434#107—1462no—
435#106—734no—
436#106—734no—
437#106—1462no—
438#102—734no—
439#102—734no—
440#102—1462no—
441#102—1462no—
442C22⋊C8—22no—
443C22⋊C8C4×D7314yescomplemented
444#97—22no—
445C4×C8—22no—
446#95—1461no—
447#95—1461no—
448#95—1461no—
449#95—1461no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453C24—1461no—
454C24C2×D29211yescharacteristic
455C2×D4—5841no—
456C2×D4—5841no—
457C2×D4—732no—
458C2×D4—1464no—
459C2×D4—1461no—
460C2×D4—1461no—
461C2×D4—1461no—
462C2×D4—1461no—
463C2×D4—1461no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—2922no—
469C22×C4—5841no—
470C22×C4—5841no—
471C22×C4—5841no—
472C22×C4—5841no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1464no—
481C22×C4C2×C4×D7312yes—
482C22×C4C2×D29211yescharacteristic
483C22×C4C2×D29211yescharacteristic
484C22×C4C2×D29211yescharacteristic
485C22×C4C2×D29211yescharacteristic
486C22×C4C2×D29211yescharacteristic
487C22×C4#1192402812yes—
488C22×C4C23×D7311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8C2×C4×D7312yes—
500C2×C8C2×C4×D7312yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—5841no—
504C4⋊C4—732no—
505C4⋊C4—732no—
506C4⋊C4—732no—
507C4⋊C4—732no—
508C4⋊C4—732no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4—1464no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—2922no—
522C22⋊C4—5841no—
523C22⋊C4—5841no—
524C22⋊C4—5841no—
525C22⋊C4—1464no—
526C22⋊C4—1464no—
527C42—21no—
528C42—21no—
529C42—5841no—
530C42—5841no—
531C42—5841no—
532C42—5841no—
533C42—5841no—
534C42—1464no—
535C42#1192402712yes—
536C42#1192402712yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—2921no—
544C23—2921no—
545C23—2921no—
546C23—2921no—
547C23—2921no—
548C23—2921no—
549C23—1462no—
550C23C4×D29212yes—
551C23C22×D29211yescharacteristic
552D4—2921no—
553D4—2921no—
554D4—5844no—
555D4—1462no—
556D4—1462no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—5844no—
590C2×C4—5844no—
591C2×C4—1462no—
592C2×C4—1462no—
593C2×C4—1462no—
594C2×C4—1462no—
595C2×C4—1462no—
596C2×C4C4×D29212yes—
597C2×C4#1455657512yes—
598C2×C4#1455657512yes—
599C2×C4C22×C4×D7311yescharacteristic
600C2×C4#41×D7311yescharacteristic
601C2×C4#41×D7311yescharacteristic
602C2×C4C2×#1192402711yescharacteristic
603C2×C4C2×#1192402711yescharacteristic
604C2×C4C2×#1192402811yescharacteristic
605C8—22no—
606C8—22no—
607C8—41no—
608C8—41no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—2922no—
617C22—2922no—
618C22—2922no—
619C22—5841no—
620C22—5841no—
621C22—5841no—
622C22—5841no—
623C22C2×C4×D29211yescharacteristic
624C22C2×#1455657511yescharacteristic
625C22#1798909911yescharacteristic
626C22#1798939211yescharacteristic
627C22#1798939911yescharacteristic
628C22#1798939911yescharacteristic
629C22#1798939911yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—2922no—
636C4—2922no—
637C4—2922no—
638C4—2922no—
639C4—2922no—
640C4—5841no—
641C4—5841no—
642C4C4×#1192402711yescharacteristic
643C4C4×#1192402711yescharacteristic
644C4#1798911711yescharacteristic
645C4#1798911711yescharacteristic
646C2—42no—
647C2—5842no—
648C2C4×#1455657511yescharacteristic
649C2#2261876511yescharacteristic
650C2#2261879211yescharacteristic
651C2#2261989011yescharacteristic
652C2#2261992211yescharacteristic
653C2#2262013211yescharacteristic
654C2#2262021011yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1750C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
cosocleC22×C4×C73C24
hypercenter#1750D73
hypocenterC73#21584
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
3. #22612287C2semidirect
4. #22610559C2semidirect
5. #22610552C2semidirect
6. #22610517C2semidirect
7. #22610510C2semidirect
9. #22610132C2semidirect
10. #22610018C2semidirect
11. #22609828C2semidirect
12. C4×#14556523C2semidirect
13. #22609757C2semidirect
14. C4×#14556513C2semidirect
15. C73×#1750C2semidirect
20. #17988906C4semidirect
72. C2×C4⋊Dic73C2×C4semidirect
95. #14556523C2×C4semidirect
96. #14556523C2×C4semidirect
122. C22⋊C8×C73C2×C4semidirect
342. #1750D73semidirect
343. C73#21584semidirect
443. C22⋊C8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179889062
#226126242
#226122872
#226105592
#226105522
#226105172
#226105102
#226102512
#226101322
#226100182
#226098282
C4×#145565232
#226097572
C4×#145565132
C73×#17502
#2158473