Finite Group Databasebeta

← #33108013

Subgroups — #33108013

Order 18688.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988865C211yesmaximal, characteristic
2#22613018C211yesmaximal, characteristic, complemented
3#22612836C211yesmaximal, characteristic
4#22610902C211yesmaximal, characteristic, complemented
5#22610698C211yesmaximal, characteristic, complemented
6#22610669C211yesmaximal, characteristic, complemented
7#22610663C211yesmaximal, characteristic, complemented
8#22610522C211yesmaximal, characteristic, complemented
9#22610502C211yesmaximal, characteristic, complemented
10#22610194C211yesmaximal, characteristic
11#22610178C211yesmaximal, characteristic
12#22609807C211yesmaximal, characteristic, complemented
13#22609758C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#1762C211yesmaximal, characteristic
16C2×#14556590C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19#17988865C418yescomplemented
20C2×#14556525C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988493—21no—
30#17988491—21no—
31#17988486C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988484—21no—
35#17988483C2211yescharacteristic
36#17988460—21no—
37#17988459—21no—
38#17988452C2211yescharacteristic
39#17988445—21no—
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4×#11924007—21no—
43C4×#11924007C2211yescharacteristic
44C2×#14556498—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988342C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×#97×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×Dic146—21no—
61C23×Dic73—21no—
62C23×Dic73—21no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556590—22no—
66#14556590C2×C414yescomplemented
67D4×Dic73—22no—
68D4×Dic73—22no—
69D4×Dic73—41no—
70D4×Dic73C2×C414yes—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007D411yescharacteristic
81C2×#11924007D411yescharacteristic
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556525C2×C414yescomplemented
89#14556524—41no—
90#14556524C2×C414yescomplemented
91#14556523—41no—
92#14556520C2×C414yescomplemented
93#14556512—22no—
94#14556512C2×C414yes—
95#14556501—22no—
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×C73—41no—
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×C73C2×C414yes—
120#102×C73Q814yescomplemented
121#97×C73—41no—
122#95×C73—21no—
123C2×Dic146—24no—
124C2×Dic146—81no—
125C2×Dic146—81no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—24no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—42no—
132C22×Dic73—42no—
133C22×Dic73—81no—
134C22×Dic73—81no—
135C22×Dic73—81no—
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831C22×C412yes—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—81no—
153C4⋊Dic73C22×C412yes—
154C4⋊Dic73C22×C412yes—
155#11924007—24no—
156#11924007—22no—
157#11924007—41no—
158#11924007—81no—
159#11924007—81no—
160#11924007—81no—
161C22⋊Dic73—24no—
162C22⋊Dic73—42no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—42no—
166C4×Dic73—81no—
167C4×Dic73C22×C412yes—
168C4×Dic73#4112yes—
169C24×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73C22×C412yes—
174C2×D4×C73C2×Q812yes—
175C2×D4×C73#4112yes—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—42no—
180C22×C4×C73—42no—
181C22×C4×C73—81no—
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×C73C22×C412yes—
190C2×C8×C73C2×Q812yes—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73C22×C412yes—
196C4⋊C4×C73C2×Q812yes—
197C22⋊C4×C73—42no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—81no—
201C23×C73—22no—
202C23×C73—22no—
203C23×C73—22no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73C22×D411yescharacteristic
209D4×C73—22no—
210D4×C73—22no—
211D4×C73—22no—
212D4×C73C4×Q814yes—
213C2×C4×C73—22no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—22no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
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—84no—
229C2×C4×C73C23×C411yescharacteristic
230C2×C4×C73C22×Q811yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73H(5,𝔽2)11yescharacteristic
234C2×C4×C732−1+411yescharacteristic
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—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
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—41no—
256C2×Dic73—84no—
257C2×Dic73—84no—
258C2×Dic73C4×D412yes—
259C2×Dic73C4×D412yes—
260C2×Dic73#13712yes—
261Dic146—84no—
262C8×C73—22no—
263C8×C73—22no—
264#10596831—22no—
265#10596831—41no—
266C22×C73—24no—
267C22×C73—42no—
268C22×C73—42no—
269C22×C73—42no—
270C22×C73—42no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73C2×C4×D411yescharacteristic
274C22×C73#54611yescharacteristic
275C22×C73D4×Q811yescharacteristic
276C22×C73C2×#13711yescharacteristic
277C22×C73#57511yescharacteristic
278C22×C73#57611yescharacteristic
279C22×C73#57611yescharacteristic
280C4×C73—42no—
281C4×C73—42no—
282C4×C73—41no—
283C4×C73—41no—
284C4×C73—81no—
285C4×C73C2×C4×Q811yescharacteristic
286C4×C73#52011yescharacteristic
287C4×C73#55211yescharacteristic
288C4×C73#55411yescharacteristic
289Dic73—22no—
290Dic73—22no—
291Dic73—22no—
292Dic73—42no—
293Dic73—42no—
294Dic73—41no—
295Dic73—81no—
296Dic73—81no—
297#22979—731nomaximal, Sylow
298C2×C73—44no—
299C2×C73#236011yescharacteristic
300C2×C73C4×#13711yescharacteristic
301C2×C73#294811yescharacteristic
302C2×C73#331411yescharacteristic
303C2×C73#331911yescharacteristic
304C2×C73#338511yescharacteristic
305C2×C73#338911yescharacteristic
306C2×#474—731no—
307#2369—731no—
308#2349—731no—
309#1993—731no—
310#1949—731no—
311#1943—731no—
312#1942—731no—
313#1921—731no—
314#1918—731no—
315#1864—731no—
316#1862—731no—
317#1771—731no—
318#1762—731no—
319#1762D7311yescharacteristic
320C4×#102—731no—
321C73#2297911yescharacteristic, complemented, Sylow
322C2×C4⋊Q8—731no—
323C2×C4×D4—731no—
324C2×C4×D4—731no—
325C2×C4×D4C2×D7311yescharacteristic
326#474—738no—
327C2×#107—731no—
328C2×#107—731no—
329C2×#107—731no—
330C2×#106—731no—
331C2×C4⋊C8—731no—
332C2×#102—731no—
333C2×#102—731no—
334C2×#102C2×D7311yescharacteristic
335C2×#102C2×D7311yescharacteristic
336C2×#97—731no—
337C2×#97C2×D7311yescharacteristic
338C2×C4×C8—731no—
339#391—1461no—
340#391—1461no—
341#389—731no—
342#389—731no—
343#389—731no—
344#389—731no—
345#389—1461no—
346#385—1461no—
347#385—1461no—
348#382—731no—
349#380—1461no—
350C4×C4⋊C4—731no—
351C4×C4⋊C4—731no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—1461no—
354C4×C4⋊C4C2×D7311yescharacteristic
355C2×#95—1461no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340C2×D7311yescharacteristic
363#340C2×D7311yescharacteristic
364#336—731no—
365#336—731no—
366C22×Q8—1461no—
367C22×D4C22×D7311yescharacteristic
368C23×C4—21no—
369C23×C4—1461no—
370C23×C4—1461no—
371C22×C8—731no—
372C22×C8—731no—
373C22×C8C22×D7311yescharacteristic
374C22×C8C22×D7311yescharacteristic
375C4⋊Q8—734no—
376C4⋊Q8—1462no—
377C4×D4—22no—
378C4×D4—41no—
379C4×D4—2921no—
380C4×D4—734no—
381C4×D4—1462no—
382C4×D4—1462no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4C22×D7311yescharacteristic
395C2×C4⋊C4C22×D7311yescharacteristic
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1461no—
399C2×C22⋊C4—1461no—
400C2×C42—21no—
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—1461no—
404C2×C42—1461no—
405C2×C42C22×D7311yescharacteristic
406#107—2921no—
407#107—734no—
408#107—734no—
409#106—2921no—
410C4⋊C8—734no—
411#102—734no—
412#102—1462no—
413#102Dic14614yescomplemented
414#102C4×D7314yes—
415#97—41no—
416#97—1462no—
417C4×C8—2921no—
418#95—21no—
419#95—731no—
420#95—731no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429C24—21no—
430C2×Q8—5841no—
431C2×Q8—5841no—
432C2×Q8—1464no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4C2×C4×D7312yes—
437C2×D4C2×Dic14612yes—
438C2×D4#1192402812yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—81no—
445C22×C4—2922no—
446C22×C4—2922no—
447C22×C4—5841no—
448C22×C4—5841no—
449C22×C4—5841no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—731no—
453C22×C4—731no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1464no—
459C22×C4D4×D7311yescharacteristic
460C22×C4D4×D7311yescharacteristic
461C22×C4C23×D7311yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—732no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8C2×C4×D7312yes—
473C2×C8C2×Dic14612yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—2921no—
479C4⋊C4—5841no—
480C4⋊C4—5841no—
481C4⋊C4—5841no—
482C4⋊C4—5841no—
483C4⋊C4—732no—
484C4⋊C4—732no—
485C4⋊C4—1462no—
486C4⋊C4—1464no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4C2×C4×D7312yes—
494C4⋊C4C2×Dic14612yes—
495C22⋊C4—42no—
496C22⋊C4—2922no—
497C22⋊C4—1464no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42—2922no—
502C42—5841no—
503C42—732no—
504C42—732no—
505C42—1461no—
506C42—1461no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×D4×D7311yescharacteristic
515Q8—5844no—
516D4—22no—
517D4—22no—
518D4—22no—
519D4C4×Dic14614yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—5844no—
547C2×C4—5844no—
548C2×C4—732no—
549C2×C4—732no—
550C2×C4—732no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1462no—
555C2×C4—1462no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4C22×C4×D7311yescharacteristic
563C2×C4C22×Dic14611yescharacteristic
564C2×C4#41×D7311yescharacteristic
565C2×C4C2×#1192402811yescharacteristic
566C2×C4#1455663611yescharacteristic
567C2×C4#1455663811yescharacteristic
568C8—22no—
569C8—22no—
570C8—2921no—
571C8—1462no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C4×D4×D7311yescharacteristic
580C22#1798922711yescharacteristic
581C22D4×Dic14611yescharacteristic
582C22#137×D7311yescharacteristic
583C22#1798938311yescharacteristic
584C22#1798939111yescharacteristic
585C22#1798939111yescharacteristic
586C4—41no—
587C4—41no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—2921no—
592C4—2922no—
593C4—2922no—
594C4—5841no—
595C4—5841no—
596C4—1462no—
597C4—1462no—
598C4—1462no—
599C4C2×C4×Dic14611yescharacteristic
600C4#1798906611yescharacteristic
601C4#1798927011yescharacteristic
602C4#1798929611yescharacteristic
603C2—44no—
604C2#2261293711yescharacteristic
605C2#2261875311yescharacteristic
606C2#2261878011yescharacteristic
607C2#2262201311yescharacteristic
608C2#2262208811yescharacteristic
609C2#2262284211yescharacteristic
610C2#2262289011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1762C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1762D73
hypocenterC73#22979
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #22613018C2semidirect
4. #22610902C2semidirect
5. #22610698C2semidirect
6. #22610669C2semidirect
7. #22610663C2semidirect
8. #22610522C2semidirect
9. #22610502C2semidirect
12. #22609807C2semidirect
19. #17988865C4semidirect
66. #14556590C2×C4semidirect
88. #14556525C2×C4semidirect
90. #14556524C2×C4semidirect
92. #14556520C2×C4semidirect
120. #102×C73Q8semidirect
321. C73#22979semidirect
413. #102Dic146semidirect

Maximal subgroups

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

subgroupindex
C2×#179888652
#226130182
#226128362
#226109022
#226106982
#226106692
#226106632
#226105222
#226105022
#226101942
#226101782
#226098072
#226097582
#102×Dic732
C73×#17622
#2297973