Finite Group Databasebeta

← #33090972

Subgroups — #33090972

Order 18688.

1186 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989115C211yesmaximal, characteristic
2C2×#17988638C211yesmaximal, characteristic, complemented
3C2×#17988539C211yesmaximal, characteristic, complemented
4#22610865C212yesmaximal, complemented
5#22610723C212yesmaximal, complemented
6#22610227C212yesmaximal, complemented
7#22610225C212yesmaximal, complemented
8#22609700C211yesmaximal, characteristic, complemented
9#22609698C211yesmaximal, characteristic, complemented
10#22609697C211yesmaximal, characteristic, complemented
11C73×#1750C211yesmaximal, characteristic, complemented
12C22×#11924011C2212yes—
13#17989115—24no—
14#17989115—24no—
15C2×#14556586C411yescharacteristic
16C2×#14556584C411yescharacteristic
17C2×#14556581C411yescharacteristic
18C2×#14556577C411yescharacteristic
19C2×#14556576C411yescharacteristic
20C2×#14556573C411yescharacteristic
21C2×#14556569C411yescharacteristic
22C2×#14556564C411yescharacteristic
23C2×C4×Dic146C2211yescharacteristic
24C2×C4×D292C2211yescharacteristic
25C2×C4×C22⋊D73C2211yescharacteristic
26C2×C4×C22⋊D73C2211yescharacteristic
27C2×#14556548C2211yescharacteristic
28C2×C42×D73C2211yescharacteristic
29#17988638—24no—
30C2×#14556520C2212yes—
31C2×C22⋊#10596831C2212yes—
32C2×#14556503C2212yes—
33C2×#14556503C2212yes—
34C2×#14556503C2212yes—
35#17988539—24no—
36C2×#14556502C2211yescharacteristic
37C2×C4×#10596831C2211yescharacteristic
38C2×C8×Dic73C2212yes—
39#17988342C2212yes—
40#17988341C2212yes—
41#17988341C2212yes—
42C2×#117×C73C2211yescharacteristic
43C2×C22⋊C8×C73C2212yes—
44C2×#97×C73C2211yescharacteristic
45C2×C4×C8×C73C2211yescharacteristic
46#336×C73C2212yes—
47C22×Dic146C2×C411yescharacteristic
48C22×D292C2×C411yescharacteristic
49C22×C22⋊D73C2×C411yescharacteristic
50C22×C22⋊D73C2×C411yescharacteristic
51C22×C4×D73C2×C411yescharacteristic
52C22×C4×D73C2×C411yescharacteristic
53C22×C4×D73C2311yescharacteristic
54C22×C4×D73C2311yescharacteristic
55C2×#11924011—24no—
56C2×#11924011—24no—
57C2×#11924011—24no—
58C22×#10596831C2312yes—
59C22×#10596831C2312yes—
60#14556586—24no—
61#14556584—24no—
62#14556581—24no—
63#14556577—24no—
64#14556576—24no—
65#14556573—24no—
66#14556569—24no—
67#14556564—24no—
68C4×Dic146—22no—
69C4×Dic146—22no—
70C4×D292—22no—
71C4×D292—22no—
72C4×C22⋊D73—22no—
73C4×C22⋊D73—22no—
74C4×C22⋊D73—22no—
75C4×C22⋊D73—22no—
76#14556548—22no—
77#14556548—22no—
78C2×C4⋊Dic73C2×C411yescharacteristic
79C2×C4⋊Dic73C2×C411yescharacteristic
80C2×#11924007C2×C411yescharacteristic
81C2×#11924007C2×C411yescharacteristic
82C2×#11924007C2×C411yescharacteristic
83C2×#11924007C2×C411yescharacteristic
84C2×#11924007C2×C411yescharacteristic
85C2×#11924007C2×C411yescharacteristic
86C2×C22⋊Dic73C2×C411yescharacteristic
87C2×C22⋊Dic73C2×C411yescharacteristic
88C2×#11924005C2×C411yescharacteristic
89C2×#11924005C2×C411yescharacteristic
90C2×#11924005C2×C411yescharacteristic
91C2×#11924005C2×C411yescharacteristic
92C2×#11924005C2×C411yescharacteristic
93C2×#11924005C2×C411yescharacteristic
94C2×C4×Dic73C2×C411yescharacteristic
95C2×C4×Dic73C2×C411yescharacteristic
96C2×C4×Dic73C2311yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98C42×D73D414yes—
99C42×D73D414yes—
100#14556520—24no—
101C22⋊#10596831—24no—
102#14556503—24no—
103#14556503—24no—
104#14556503—24no—
105#14556502—22no—
106C4×#10596831—22no—
107C8×Dic73—24no—
108C23×C4×C73C2311yescharacteristic
109C22×C8×C73C2312yes—
110C22×C8×C73C2312yes—
111#117×C73—22no—
112#117×C73—22no—
113C2×C4⋊C4×C73C2×C411yescharacteristic
114C2×C4⋊C4×C73C2×C411yescharacteristic
115C2×C22⋊C4×C73C2×C411yescharacteristic
116C2×C22⋊C4×C73C2×C411yescharacteristic
117C2×C42×C73C2311yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C22⋊C8×C73—24no—
120#97×C73—22no—
121C4×C8×C73—22no—
122C23×D73C22×C411yescharacteristic
123C23×D73C22×C411yescharacteristic
124C2×Dic146—22no—
125C2×Dic146—24no—
126C2×D292—22no—
127C2×D292—42no—
128C2×C22⋊D73—22no—
129C2×C22⋊D73—22no—
130C2×C22⋊D73—24no—
131C2×C22⋊D73—42no—
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
137C22×Dic73C22×C411yescharacteristic
138C2×C4×D73—22no—
139C2×C4×D73—22no—
140C2×C4×D73—24no—
141C2×C4×D73—21no—
142C2×C4×D73—21no—
143C2×C4×D73—22no—
144C2×C4×D73—22no—
145C2×C4×D73—42no—
146C2×C4×D73C22⋊C414yes—
147C2×C4×D73C22⋊C414yes—
148C2×C4×D73C2×D412yes—
149C2×C4×D73C2×D412yes—
150#11924011—216no—
151C2×#10596831—22no—
152C2×#10596831—22no—
153C2×#10596831—22no—
154C2×#10596831—22no—
155C2×#10596831—22no—
156C2×#10596831#4114yes—
157C4⋊Dic73—22no—
158C4⋊Dic73—22no—
159#11924007—22no—
160#11924007—22no—
161#11924007—22no—
162#11924007—22no—
163#11924007—22no—
164#11924007—22no—
165C22⋊Dic73—22no—
166C22⋊Dic73—22no—
167#11924005—22no—
168#11924005—22no—
169#11924005—22no—
170#11924005—22no—
171#11924005—22no—
172#11924005—22no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73C22⋊C414yes—
176C4×Dic73C22⋊C414yes—
177C4×Dic73C2×D412yes—
178C4×Dic73C2×D412yes—
179C24×C73C22×C411yescharacteristic
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—22no—
183C22×C4×C73—22no—
184C22×C4×C73C22×C411yescharacteristic
185C22×C4×C73C22×C411yescharacteristic
186C22×C4×C73C22×C411yescharacteristic
187C22×C4×C73C22×C411yescharacteristic
188C22×C4×C73C22×C411yescharacteristic
189C22×C4×C73C2411yescharacteristic
190C2×C8×C73—22no—
191C2×C8×C73—22no—
192C2×C8×C73—22no—
193C2×C8×C73—22no—
194C2×C8×C73—22no—
195C2×C8×C73#4114yes—
196C4⋊C4×C73—22no—
197C4⋊C4×C73—22no—
198C22⋊C4×C73—22no—
199C22⋊C4×C73—22no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73C2×D412yes—
203C42×C73C2×D412yes—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—22no—
208C23×C73—22no—
209C23×C73C23×C411yescharacteristic
210C22×D73—21no—
211C22×D73—21no—
212C22×D73—22no—
213C22×D73—22no—
214C22×D73—21no—
215C22×D73—21no—
216C22×D73—21no—
217C22×D73—22no—
218C22×D73—22no—
219C22×D73C2×C22⋊C412yes—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73C2×C22⋊C412yes—
232C2×C4×C73C2×C22⋊C412yes—
233C2×C4×C73C22×D411yescharacteristic
234C2×C4×C73C22×D411yescharacteristic
235C2×C4×C73C2×#4112yes—
236C2×C4×C73C2×#4112yes—
237C22⋊D73—44no—
238C22⋊D73—44no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—22no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73C2×C22⋊C412yes—
250C2×Dic73C2×C22⋊C412yes—
251C2×Dic73C2×C22⋊C412yes—
252C2×Dic73#13114yes—
253D292—44no—
254C4×D73—24no—
255C4×D73—24no—
256C4×D73—28no—
257C4×D73—44no—
258C4×D73—42no—
259C4×D73—42no—
260Dic146—44no—
261C8×C73—24no—
262C8×C73—42no—
263#10596831—24no—
264#10596831—42no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73—42no—
268C22×C73—42no—
269C22×C73—41no—
270C22×C73C22×C22⋊C411yescharacteristic
271C22×C73C4×#4111yescharacteristic
272C22×C73C4×#4111yescharacteristic
273C22×C73C2×#13112yes—
274C22×C73#56812yes—
275C2×D73—22no—
276C2×D73—22no—
277C2×D73—22no—
278C2×D73—42no—
279C2×D73—42no—
280C2×D73—41no—
281C2×D73—41no—
282C2×D73—42no—
283C2×D73—42no—
284C2×D73—41no—
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73#52512yes—
291C4×C73#53212yes—
292Dic73—22no—
293Dic73—22no—
294Dic73—41no—
295Dic73—41no—
296Dic73—41no—
297Dic73#40814yes—
298#21378—731nomaximal, Sylow
299C2×C73—81no—
300C2×C73#229711yescharacteristic
301C2×C73C2×#40811yescharacteristic
302C2×C73#287911yescharacteristic
303C2×C73#296912yes—
304C2×C73#296912yes—
305D73—42no—
306D73—81no—
307C2×#529—731no—
308C2×#431—731no—
309C2×C4×M16—731no—
310#1984—732no—
311#1954—732no—
312#1870—732no—
313#1870—732no—
314#1750—731no—
315#1750—731no—
316#1750—731no—
317#1750D7311yescharacteristic, complemented
318C73#2137811yescharacteristic, complemented, Sylow
319C22×M16—732no—
320#529—1464no—
321#529—1464no—
322C2×#126—731no—
323C2×#126—731no—
324C2×#125—731no—
325C2×#124—731no—
326C2×#123—731no—
327C2×#123—731no—
328C2×C22⋊Q8—731no—
329C2×#121—731no—
330C2×C4×Q8—731no—
331C2×C4×D4—731no—
332C2×C4×D4—731no—
333C2×C4×D4—731no—
334C2×#117—731no—
335C2×#117C2×D7311yescharacteristic
336C22×C42—731no—
337#431—1464no—
338C2×C4⋊C8—732no—
339C2×C22⋊C8—732no—
340C2×C22⋊C8—732no—
341C2×C22⋊C8—732no—
342C2×C22⋊C8—732no—
343C2×C22⋊C8C2×D7312yes—
344C4×M16—1464no—
345C2×#97—731no—
346C2×#97C2×D7311yescharacteristic
347C2×C4×C8—732no—
348C2×C4×C8—731no—
349C2×C4×C8C2×D7311yescharacteristic
350#336—732no—
351#336—732no—
352#336—732no—
353#336C2×D7312yes—
354C22×Q8—731no—
355C22×D4—731no—
356C22×D4—731no—
357C22×D4—731no—
358C23×C4—731no—
359C23×C4—731no—
360C23×C4—731no—
361C23×C4—731no—
362C23×C4C22×D7311yescharacteristic
363C2×M16—1464no—
364C2×M16—1464no—
365C2×M16—1464no—
366C22×C8—732no—
367C22×C8—732no—
368C22×C8C22×D7312yes—
369C22×C8C22×D7312yes—
370#126—1464no—
371#126—1464no—
372#125—1464no—
373#124—1464no—
374#123—1464no—
375#123—1464no—
376C22⋊Q8—1464no—
377#121—1464no—
378C4×Q8—1462no—
379C4×Q8—1462no—
380C4×D4—1462no—
381C4×D4—1462no—
382C4×D4—1462no—
383C4×D4—1462no—
384C4×D4—1462no—
385C4×D4—1462no—
386#117—22no—
387#117—22no—
388#117—1462no—
389#117—1462no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4C4×D7311yescharacteristic
399C2×C4⋊C4C4×D7311yescharacteristic
400C2×C22⋊C4—731no—
401C2×C22⋊C4—731no—
402C2×C22⋊C4—731no—
403C2×C22⋊C4—731no—
404C2×C22⋊C4—731no—
405C2×C22⋊C4—731no—
406C2×C22⋊C4—731no—
407C2×C22⋊C4—731no—
408C2×C22⋊C4C4×D7311yescharacteristic
409C2×C22⋊C4C4×D7311yescharacteristic
410C2×C42—731no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—734no—
414C2×C42—734no—
415C2×C42—731no—
416C2×C42C22×D7311yescharacteristic
417C2×C42C22×D7311yescharacteristic
418C4⋊C8—1464no—
419C22⋊C8—24no—
420C22⋊C8—1464no—
421C22⋊C8—1464no—
422C22⋊C8—1464no—
423C22⋊C8—1464no—
424#97—22no—
425#97—1462no—
426C4×C8—22no—
427C4×C8—1464no—
428C4×C8—1462no—
429C24—731no—
430C24—731no—
431C24C2×C4×D7311yescharacteristic
432C2×Q8—1462no—
433C2×Q8—1464no—
434C2×D4—2922no—
435C2×D4—2922no—
436C2×D4—1462no—
437C2×D4—1462no—
438C2×D4—1464no—
439C2×D4—1462no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—2922no—
445C22×C4—734no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—734no—
450C22×C4—732no—
451C22×C4—732no—
452C22×C4—731no—
453C22×C4—731no—
454C22×C4—731no—
455C22×C4—1464no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1462no—
459C22×C4—1462no—
460C22×C4—1462no—
461C22×C4—1462no—
462C22×C4C2×C4×D7311yescharacteristic
463C22×C4C2×C4×D7311yescharacteristic
464C22×C4C2×C4×D7311yescharacteristic
465C22×C4C2×C4×D7311yescharacteristic
466C22×C4C2×C4×D7311yescharacteristic
467C22×C4C23×D7311yescharacteristic
468M16—14616no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—734no—
475C2×C8—1462no—
476C2×C8—1462no—
477C2×C8—1462no—
478C2×C8—1462no—
479C2×C8—1462no—
480C2×C8#1192402814yes—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—1462no—
484C4⋊C4—1462no—
485C4⋊C4—1462no—
486C4⋊C4—1462no—
487C4⋊C4—1462no—
488C4⋊C4—1462no—
489C4⋊C4—1462no—
490C4⋊C4—1462no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—1462no—
494C22⋊C4—1462no—
495C22⋊C4—1462no—
496C22⋊C4—1462no—
497C22⋊C4—1462no—
498C22⋊C4—1462no—
499C22⋊C4—1462no—
500C22⋊C4—1462no—
501C42—21no—
502C42—21no—
503C42—734no—
504C42—734no—
505C42—732no—
506C42—732no—
507C42—1461no—
508C42—1461no—
509C42D4×D7312yes—
510C42D4×D7312yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—732no—
517C23—1461no—
518C23—1461no—
519C23—1461no—
520C23—1462no—
521C23—1462no—
522C23—1461no—
523C23—1461no—
524C23—1462no—
525C23—1462no—
526C23C22×C4×D7311yescharacteristic
527Q8—2924no—
528D4—2924no—
529D4—2924no—
530D4—2924no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—2924no—
543C2×C4—2922no—
544C2×C4—2922no—
545C2×C4—732no—
546C2×C4—734no—
547C2×C4—732no—
548C2×C4—732no—
549C2×C4—1468no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1462no—
556C2×C4—1462no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1462no—
560C2×C4—1464no—
561C2×C4—1464no—
562C2×C4C22⋊C4×D7312yes—
563C2×C4C22⋊C4×D7312yes—
564C2×C4C2×D4×D7311yescharacteristic
565C2×C4C2×D4×D7311yescharacteristic
566C2×C4#41×D7312yes—
567C2×C4C2×#1192402812yes—
568C8—24no—
569C8—42no—
570C8—2922no—
571C8—1464no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—2922no—
578C22—2922no—
579C22—2921no—
580C22—2921no—
581C22—2922no—
582C22—2922no—
583C22—2921no—
584C22—1462no—
585C22—1462no—
586C22—1462no—
587C22C2×C22⋊C4×D7311yescharacteristic
588C22#1798905511yescharacteristic
589C22C4×#1192402811yescharacteristic
590C22#131×D7312yes—
591C22#1798935212yes—
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—2921no—
598C4—2921no—
599C4—2921no—
600C4—734no—
601C4—1462no—
602C4—1462no—
603C4#1798909612yes—
604C4#1798913412yes—
605C2—81no—
606C2—2922no—
607C2—5841no—
608C2#2261241711yescharacteristic
609C2#408×D7311yescharacteristic
610C2#2261837311yescharacteristic
611C2#2261893112yes—
612C2#2261893312yes—
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1750C2
prime radicalC2×#14556564C4
cosocleC22×C4×C73C24
hypercenter#1750D73
hypocenterC73#21378
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C2×#17988638C2semidirect
3. C2×#17988539C2semidirect
4. #22610865C2semidirect
5. #22610723C2semidirect
6. #22610227C2semidirect
7. #22610225C2semidirect
8. #22609700C2semidirect
9. #22609698C2semidirect
10. #22609697C2semidirect
11. C73×#1750C2semidirect
317. #1750D73semidirect
318. C73#21378semidirect

Maximal subgroups

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

subgroupindex
C2×#179891152
C2×#179886382
C2×#179885392
#226108652
#226107232
#226102272
#226102252
#226097002
#226096982
#226096972
C73×#17502
#2137873