Finite Group Databasebeta

← #33107831

Subgroups — #33107831

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×#17988888C211yesmaximal, characteristic
2#22613025C211yesmaximal, characteristic, complemented
3#22612842C211yesmaximal, characteristic
4#22610896C211yesmaximal, characteristic, complemented
5#22610701C211yesmaximal, characteristic, complemented
6#22610664C211yesmaximal, characteristic, complemented
7#22610659C211yesmaximal, characteristic, complemented
8#22610524C211yesmaximal, characteristic, complemented
9#22610517C211yesmaximal, characteristic, complemented
10#22610195C211yesmaximal, characteristic
11#22610181C211yesmaximal, characteristic
12C4×#14556513C211yesmaximal, characteristic
13C4×#14556513C211yesmaximal, characteristic
14C4×#14556513C211yesmaximal, characteristic
15C8×C4⋊C4×C73C211yesmaximal, characteristic, complemented
16C2×#14556583C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20#17988888C418yescomplemented
21C2×#14556525C2211yescharacteristic
22C2×#14556525C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556513C2211yescharacteristic
27C2×#14556513C2211yescharacteristic
28C2×#14556513C2211yescharacteristic
29#17988485C2211yescharacteristic
30#17988485C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988482—21no—
34#17988469—21no—
35#17988468—21no—
36#17988458—21no—
37#17988458—21no—
38#17988453C2211yescharacteristic
39#17988446—21no—
40C4×C4⋊Dic73C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43#17988438—21no—
44#95×D73—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×C4×C8×C73C2211yescharacteristic
55C2×C4×C8×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64#14556583—22no—
65#14556583C2×C414yescomplemented
66C4×D292—22no—
67C4×D292—22no—
68C4×D292—22no—
69C4×D292—41no—
70C4×D292—41no—
71C4×D292C2×C414yes—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87C2×#11924005—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91#14556525—41no—
92#14556525C2×C414yescomplemented
93#14556523—41no—
94#14556523C2×C414yescomplemented
95#14556513—22no—
96#14556513C2×C414yes—
97#14556513C2×C414yes—
98#14556513Q814yes—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C2×C4⋊C4×C73D411yescharacteristic
113C2×C4⋊C4×C73D411yescharacteristic
114C2×C42×C73C2311yescharacteristic
115C2×C42×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C4⋊C8×C73C2×C414yescomplemented
118C4×C8×C73—22no—
119C4×C8×C73—41no—
120C4×C8×C73—41no—
121#95×C73D411yescharacteristic
122#95×C73D411yescharacteristic
123C23×D73—21no—
124C2×D292—21no—
125C2×D292—21no—
126C2×D292—21no—
127C2×D292C22×C412yes—
128C2×D292C2×Q812yes—
129C2×D292#4112yes—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C2×C4×D73—24no—
138C2×C4×D73—42no—
139C2×C4×D73—42no—
140C2×C4×D73—42no—
141C2×C4×D73—42no—
142C2×C4×D73—81no—
143C2×C4×D73—81no—
144C2×C4×D73—81no—
145C2×C4×D73—81no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73C22×C412yes—
157C4⋊Dic73C22×C412yes—
158C4⋊Dic73C22×C412yes—
159C4⋊Dic73C2×Q812yes—
160#11924007—24no—
161#11924007—24no—
162#11924007—42no—
163#11924007—81no—
164#11924007—81no—
165#11924007—81no—
166#11924005—24no—
167#11924005—42no—
168#11924005—42no—
169C4×Dic73—42no—
170C4×Dic73—81no—
171C4×Dic73—81no—
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2411yescharacteristic
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73C22×C412yes—
189C2×C8×C73C22×C412yes—
190C2×C8×C73C2×Q812yes—
191C4⋊C4×C73—22no—
192C4⋊C4×C73—22no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—21no—
197C42×C73C22×C412yes—
198C42×C73#4112yes—
199C23×C73C22×D411yescharacteristic
200C22×D73—22no—
201C22×D73—22no—
202C22×D73—22no—
203C22×D73—41no—
204C22×D73—41no—
205C22×D73—41no—
206C22×D73—41no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73C4×D412yes—
222C2×C4×C73C4×D412yes—
223C2×C4×C73#13512yes—
224C2×C4×C73#13512yes—
225C2×C4×C73C23×C411yescharacteristic
226C2×C4×C73C22×Q811yescharacteristic
227C2×C4×C73C2×#4111yescharacteristic
228C2×C4×C73C2×#4111yescharacteristic
229C2×C4×C73H(5,𝔽2)11yescharacteristic
230C2×C4×C732−1+411yescharacteristic
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—22no—
234C2×Dic73—22no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252D292—22no—
253D292—22no—
254D292—22no—
255D292C4×Q814yes—
256C4×D73—84no—
257C4×D73—84no—
258C4×D73—84no—
259C8×C73—22no—
260C8×C73—22no—
261C8×C73—22no—
262C8×C73—41no—
263C22×C73C2×C4×D411yescharacteristic
264C22×C73#54611yescharacteristic
265C22×C73D4×Q811yescharacteristic
266C22×C73C2×#13511yescharacteristic
267C22×C73C2×#13511yescharacteristic
268C22×C73#57611yescharacteristic
269C22×C73#57711yescharacteristic
270C2×D73—24no—
271C2×D73—42no—
272C2×D73—42no—
273C2×D73—42no—
274C2×D73—42no—
275C2×D73—42no—
276C2×D73—42no—
277C4×C73—22no—
278C4×C73—22no—
279C4×C73—22no—
280C4×C73—22no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73C2×C4×Q811yescharacteristic
284C4×C73#52011yescharacteristic
285C4×C73#55211yescharacteristic
286C4×C73#55411yescharacteristic
287Dic73—42no—
288Dic73—42no—
289Dic73—42no—
290Dic73—42no—
291Dic73—81no—
292Dic73—81no—
293Dic73—81no—
294#22966—731nomaximal, Sylow
295C2×C73#236011yescharacteristic
296C2×C73C4×#13511yescharacteristic
297C2×C73#294611yescharacteristic
298C2×C73#329611yescharacteristic
299C2×C73#329711yescharacteristic
300C2×C73#338011yescharacteristic
301C2×C73#338111yescharacteristic
302D73—44no—
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⋊C4D7311yescharacteristic, complemented
315C4×#102—731no—
316C4×#102—731no—
317C4×#102—731no—
318C73#2296611yescharacteristic, complemented, Sylow
319C2×#125—731no—
320C2×C4×D4—731no—
321C2×C4×D4—731no—
322C2×C4×D4—731no—
323#478—738no—
324C2×#107—731no—
325C2×#107—731no—
326C2×#106—731no—
327C2×#106—731no—
328C2×C4⋊C8C2×D7311yescharacteristic
329C2×#102—731no—
330C2×#102—731no—
331C2×#102—731no—
332C2×#102—731no—
333C2×C4×C8C2×D7311yescharacteristic
334C2×C4×C8C2×D7311yescharacteristic
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#340—731no—
360#340—731no—
361#336C2×D7311yescharacteristic
362#336C2×D7311yescharacteristic
363C22×D4—731no—
364C23×C4—1461no—
365C23×C4—1461no—
366C23×C4—1461no—
367C22×C8C22×D7311yescharacteristic
368C22×C8C22×D7311yescharacteristic
369C22×C8C22×D7311yescharacteristic
370C22×C8C22×D7311yescharacteristic
371#125—734no—
372#125—1462no—
373C4×D4—2921no—
374C4×D4—2921no—
375C4×D4—734no—
376C4×D4—1462no—
377C4×D4—1462no—
378C4×D4—1462no—
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—731no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4—1461no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4D29211yescharacteristic
393C2×C4⋊C4D29211yescharacteristic
394C2×C22⋊C4—1461no—
395C2×C22⋊C4—1461no—
396C2×C22⋊C4—1461no—
397C2×C42—1461no—
398C2×C42—1461no—
399C2×C42—1461no—
400C2×C42C22×D7311yescharacteristic
401C2×C42C22×D7311yescharacteristic
402C2×C42C22×D7311yescharacteristic
403#107—2921no—
404#107—734no—
405#106—2921no—
406#106—734no—
407C4⋊C8C4×D7314yescomplemented
408#102—734no—
409#102—734no—
410#102—734no—
411#102—1462no—
412C4×C8—22no—
413C4×C8—41no—
414C4×C8—41no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424#95D29211yescharacteristic
425#95D29211yescharacteristic
426C24—1461no—
427C2×D4—732no—
428C2×D4—732no—
429C2×D4—732no—
430C2×D4—1461no—
431C2×D4—1461no—
432C2×D4—1461no—
433C22×C4—2922no—
434C22×C4—2922no—
435C22×C4—2922no—
436C22×C4—2922no—
437C22×C4—5841no—
438C22×C4—5841no—
439C22×C4—5841no—
440C22×C4—5841no—
441C22×C4—1461no—
442C22×C4—1461no—
443C22×C4—1461no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1464no—
449C22×C4C2×D29211yescharacteristic
450C22×C4C2×D29211yescharacteristic
451C22×C4C2×D29211yescharacteristic
452C22×C4C2×D29211yescharacteristic
453C22×C4C2×D29211yescharacteristic
454C22×C4C2×D29211yescharacteristic
455C22×C4C23×D7311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8C2×C4×D7312yes—
466C2×C8C2×C4×D7312yes—
467C2×C8Q8×D7312yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—2922no—
471C4⋊C4—5841no—
472C4⋊C4—5841no—
473C4⋊C4—5841no—
474C4⋊C4—732no—
475C4⋊C4—732no—
476C4⋊C4—732no—
477C4⋊C4—732no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
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⋊C4—1461no—
489C4⋊C4—1461no—
490C22⋊C4—2922no—
491C22⋊C4—2922no—
492C22⋊C4—1464no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—2922no—
498C42—5841no—
499C42—5841no—
500C42C2×C4×D7312yes—
501C42#1192402812yes—
502C23—2921no—
503C23—2921no—
504C23—2921no—
505C23—2921no—
506C23—1462no—
507C23—1462no—
508C23—1462no—
509C23C22×D29211yescharacteristic
510D4—734no—
511D4—1462no—
512D4—1462no—
513D4—1462no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
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—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—5844no—
546C2×C4—5844no—
547C2×C4—5844no—
548C2×C4—1462no—
549C2×C4—1462no—
550C2×C4—1462no—
551C2×C4—1462no—
552C2×C4C4×D29212yes—
553C2×C4C4×D29212yes—
554C2×C4#1455661312yes—
555C2×C4#1455661312yes—
556C2×C4C22×C4×D7311yescharacteristic
557C2×C4C2×Q8×D7311yescharacteristic
558C2×C4#41×D7311yescharacteristic
559C2×C4C2×#1192402811yescharacteristic
560C2×C4#1455663611yescharacteristic
561C2×C4#1455663811yescharacteristic
562C8—22no—
563C8—22no—
564C8—22no—
565C8—41no—
566C22—2922no—
567C22—2922no—
568C22—2922no—
569C22—2922no—
570C22—2922no—
571C22—2922no—
572C22—1464no—
573C22C2×C4×D29211yescharacteristic
574C22#1798923211yescharacteristic
575C22Q8×D29211yescharacteristic
576C22C2×#1455661311yescharacteristic
577C22C2×#1455661311yescharacteristic
578C22#1798939211yescharacteristic
579C22#1798939911yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—2922no—
587C4—2922no—
588C4—2922no—
589C4—2922no—
590C4—5841no—
591C4—5841no—
592C4—5841no—
593C4C4×Q8×D7311yescharacteristic
594C4#1798906611yescharacteristic
595C4#1798927611yescharacteristic
596C4#1798929211yescharacteristic
597C2—2924no—
598C2#2261293811yescharacteristic
599C2C4×#1455661311yescharacteristic
600C2#2261876511yescharacteristic
601C2#2262176811yescharacteristic
602C2#2262178311yescharacteristic
603C2#2262279911yescharacteristic
604C2#2262280811yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC8×C4⋊C4×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenterC8×C4⋊C4D73
hypocenterC73#22966
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22613025C2semidirect
4. #22610896C2semidirect
5. #22610701C2semidirect
6. #22610664C2semidirect
7. #22610659C2semidirect
8. #22610524C2semidirect
9. #22610517C2semidirect
15. C8×C4⋊C4×C73C2semidirect
20. #17988888C4semidirect
65. #14556583C2×C4semidirect
92. #14556525C2×C4semidirect
94. #14556523C2×C4semidirect
117. C4⋊C8×C73C2×C4semidirect
314. C8×C4⋊C4D73semidirect
318. C73#22966semidirect
407. C4⋊C8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888882
#226130252
#226128422
#226108962
#226107012
#226106642
#226106592
#226105242
#226105172
#226101952
#226101812
C4×#145565132
C8×C4⋊C4×C732
#2296673